Babes -Bolyai Universty of Cluj-Napoca [616066]

"Babes »-Bolyai" Universty of Cluj-Napoca
Faculty of Mathematics and Computer Science
PhD Thesis
Contributions to the Approximation Theory
of Functions of Real and Complex Variable
PhD student: [anonimizat] »Supervisor :
Professor dr. Sorin Gal
Cluj-Napoca
2017

2

Contents
1 General Introduction 5
2 Approximation by nonlinear integral operators 1
2.1 Approximation by Durrmeyer-Choquet operators . . . . . . 1
2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 3
2.1.3 Main Results . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Approximation by possibilistic integral operators . . . . . . . 8
2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Feller's scheme in terms of possibilistic integral . . . 10
2.2.3 Approximation by convolution possibilistic operators 17
3 Arbitrary order by Sz¶ asz and Baskakov operators 23
3.1 Generalized real Baskakov operators . . . . . . . . . . . . . . 24
3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.3 The case of q-Baskakov operators, 0 < q < 1 . . . . . 29
3.2 Generalized real Sz¶ asz-Stancu operators . . . . . . . . . . . 33
3.3 Generalized real Baskakov-Stancu operators . . . . . . . . . 33
3

4 CONTENTS
4 Complex Sz¶ asz and Baskakov operators 35
4.1 Arbitrary order in compact disks . . . . . . . . . . . . . . . 35
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.2 Sz¶ asz and Sz¶ asz-Kantorovich complex operators . . . 37
4.1.3 Generalized complex Baskakov operators . . . . . . . 39
4.2 Arbitrary order by Baskakov-Faber operators . . . . . . . . . 42
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 43
4.2.3 Main results . . . . . . . . . . . . . . . . . . . . . . . 44
References 46

Ch. 1
General Introduction
This thesis contains the results I have obtained in the topic of approximation
of functions of real and complex variable.
Approximation Theory is a part of Mathematical Analysis, having its
roots in the 19th century. It deals, in essence, with the approximation of
some complicated elements of a space (most of the time functions), with
simpler elements (most of the time algebraic polynomials, trigonometric
polynomials, spline functions, so on). Moreover, quantitative characteriza-
tions of this approximation are obtained, most of the time in terms of the
so-called moduli of continuity (smoothness).
From historical point of view, in the case of approximation of functions
of real variable, probably that the ¯rst result was obtained by the German
mathematician K. Weierstrass in 1895, who proved the following result.
Theorem A. Iff: [a; b]!Ris continuous on [a; b], then there exists a
sequence of algebraic polynomials with real coe±cients, Pmn(x) =a0xmn+
:+amn¡1x+amn, such that limn!1Pmn(x) =f(x), uniformly with respect
tox2[a; b].
A constructive proof of the above result was obtained by the Russian
5

6 CH. 1. GENERAL INTRODUCTION
mathematician S.N. Bernstein in 1912, who proved that the sequence of alge-
braic polynomials (called in our days "Bernstein polynomials"), Bn(f)(x) =
Pn
k=0¡n
k¢
xk(1¡x)n¡kf)k=n), converges uniformly to the continuous func-
tionf.
The ¯rst quantitative result in the Weierstrass' and Bernstein's result
was obtained by the Romanian mathematician Tiberiu Popoviciu in 1942,
who proved
jBn(f)(x)¡f(x)j ·3
2!1(f; 1=pn);8×2[0;1]; n2N;
where !1(f;±) = sup fjf(x)¡f(y)j;x; y2[0;1];jx¡yj ·±gdenotes the
modulus of continuity of f.
In the case of approximation of continuous and 2 ¼periodic functions, the
¯rst constructive result was obtained by the Hungarian mathematician L.
Fej¶ er in 1900, who proved that if f:R!Ris a continuous and 2 ¼periodic
function onR, denoting Sn(f)(x) =Pn
k=0akcos(kx) +bksin(kx), were ak
andbkare the Fourier coe±cients of f, then Tn(f)(x) =S0(f)(x)+:::+Sn(f)(x)
n+1
represents a sequence of trigonometric polynomials converging uniformly to
fonR.
The ¯rst quantitative and constructive result in approximation by trigono-
metric polynomials was obtained by the American mathematician D. Jack-
son in the doctoral thesis in 1911, who proved that if f:R!Ris contin-
uous and 2 ¼periodic, then a sequence of trigonometric polynomials can be
constructed, Jn(f)(x),n2N, with the property that
jJn(f)(x)¡f(x)j ·C!2(f; 1=n);8x2R; n2N;
where !2(f;±) = sup fjf(x+h)¡2f(x) +f(x¡h)j; 0·h·±; x2Rg
represents the second order modulus of smoothness of f.

7
An important direction in approximation of functions is represented by
the theory of approximation by positive and linear operators, having its
roots between 1950 and 1970, by the classical results of Tiberiu Popoviciu,
Bohman, Korovkin, Shisha-Mond and others. In essence, these results state
(see Korovkin's results) that in order that a sequence of positive and linear
operators, ( Ln(f))n2N, be uniformly convergent on [ a; b] to a continuous
function f, is that Ln(ek)!ek, for k= 0;1 and 2, where e0(x) = 1,
e1(x) =x» sie2(x) =x2.
In the case of complex approximation, the roots of this theory can be
found, for the approximation of continuous functions by polynomials or en-
tire functions in the MÄ untz-Sz¶ asz Carleman's papers, while for approxima-
tion of analytic functions of a complex variable by polynomials or rational
functions, can be mentioned the results obtained by Runge, Walsh, Faber,
Mergelyan, Arakelyan and Dzyadyk.
This thesis is structured in four chapters.
In the present Chapter 1, we make a general introduction in Approxi-
mation Theory and we shortly describe the thesis.
In Chapter 2 titled "Approximation by nonlinear integral operators",
the basic idea is the replacement of the classical integral in the expressions
of some integral linear operators, by more general integrals (which are not
linear) and to study the approximation properties of the new obtained op-
erators.
The chapter has two sections.
Thus, in the ¯rst section, titled "Approximation by Durrmeyer-Choquet
operators", in the expression of the classical Bernstein-Durrmeyer opera-
tors, the Lebesgue integral is replaced by the nonlinear Choquet integral
with respect to a monotone and submodular set function. We show that

8 CH. 1. GENERAL INTRODUCTION
the new obtained nonlinear operators remain uniformly convergent to the
approximated function.
In the second section, in the classical Feller's scheme of generation of lin-
ear and positive operators with good approximation properties, we replace
the classical (linear) integral with respect to the Lebesgue measure, with
the nonlinear possibilistic integral. In this way, we generate new (nonlinear)
operators with good approximation properties, including the so-called max-
product operators studied in a long series of papers by B. Bede, L. Coroianu
and S.G. Gal (which culminates with the research monograph [10] published
at Springer).
In the same section, we study the quantitative approximation properties
of the convolution possibilistic operators obtained by the Feller's scheme.
In Chapter 3 titled "Arbitrary order by Sz¶ asz and Baskakov operators",
starting from a sequence ¸n>0,n2N, converging to zero as fast we want
(that is, arbitrary fast), we construct sequences of Baskakov, q-Baskakov,
Sz¶ asz-Stancu and Baskakov-Stancu operators, converging to the approxi-
mated function f: [0;1)!Rwith the order of convergence !1(f;p¸n)
(in fact, arbitrary good, because ¸ncan be chosen to converge to zero,
arbitrarily rapid).
For this reason, the results in this chapter can be considered of de¯nitive
type (that is, the best possible). In the same time, the results obtained have
also a strong unifying character, in the sense that one can recapture from
them all the results previously obtained by other authors, for various choices
of the nodes ¸n.
In Chapter 4 titled "Complex Sz¶ asz and Baskakov operators", we apply
the ideas in Chapter 3 to the case of aproximation of analytic functions of
complex variable, by complex Sz¶ asz, Sz¶ asz-Kantorovich and Baskakov.

9
In the ¯rst section of the chapter, starting again from a sequence ¸n>0,
n2N, converging to zero as fat we want (arbitrarily rapid), we construct
sequences of Sz¶ asz, Sz¶ asz-Kantorovich and Baskakov operators attached to
an analytic function of exponential growth in a compact disk centered at
origin, which approximate fwith the order O(¸n) and for which quantitative
Voronovskaja type results with the order O(¸2
n) are obtained.
In the second section of the chapter, we consider the same problem as
in the previous section, for the so-called complex Baskakov-Faber opera-
tors, attached through the Faber polynomials to an analytic function of
exponential growth in a compact set of C(not necessarily a disk).
The results presented in this thesis were obtained by the author in collab-
oration with professor dr. Sorin Gal, Nazim Mahmodov, Lucian Coroianu,
Sorin Trifa, or as a single author, in 6 papers, published by the following
journals :
1) Gal, Sorin G.; Opri» s, Bogdan D., Approximation with an ar-
bitrary order by modi¯ed Baskakov type operators. Appl. Math.
Comput., 265 (2015), 329-332 (Impact Factor ISI (IF)on 2015 :
1.345, Relative Score of In°uence (RSI) on 2015 : 0.694)
2) Gal, Sorin G.; Opri» s, Bogdan D., Uniform and pointwise
convergence of Bernstein-Durrmeyer operators with respect to
monotone and submodular set functions. J. Math. Anal. Appl.
424 (2015), no. 2, 1374-1379 (IF on 2015 : 1.014, RSI on 2015 :
1.121)
3) Gal, Sorin G.; Opri» s, Bogdan D., Approximation of analytic
functions with an arbitrary order by generalized Baskakov-Faber
operators in compact sets. Complex Anal. Oper. Theory 10
(2016), no. 2, 369-377 (IF on 2015 : 0.663, RSI on 2016 : 0.724)

10 CH. 1. GENERAL INTRODUCTION
4) Gal, Sorin G.; Mahmudov, Nazim I.; Opri» s, Bogdan D., Ap-
proximation with an arbitrary order of Sz¶ asz, Sz¶ asz-Kantorovich
and Baskakov complex operators in compact disks. Azerb. J.
Math. 6 (2016), no. 2, 3-12 (indexed in Mathematical Reviews
and Zentralblatt fÄ ur Mathematik)
5) Coroianu, Lucian ; Gal, Sorin G. ; Opri» s, Bogdan D.; Trifa,
Sorin, Feller's scheme in approximation by nonlinear possibilistic
integral operators, submitted for publication.
6) Opri» s, Bogdan, D., Approximation with an arbitrary or-
der by generalized Sz¶ asz-Stancu and Baskakov-Stancu operators,
submitted for publication.
The original results obtained in the thesis are the following :
Chapter 2. Section 2.1 : Lemma 2.1.2, Theorem 2.1.3, Theorem 2.14 ;
The results were published in the paper [46];
Section 2.2 : Theorem 2.2.2, Lemma 2.2.3, Theorem 2.2.4, Theorem
2.2.5, Corollary 2.2.6, Theorem 2.2.7, Corollary 2.2.8, Theorem 2.2.9, Corol-
lary 2.2.9 ; The results were published in the paper [21] ;
Chapter 3. Section 3.1 : Lemma 3.1.1, Corollary 3.1.2, Theorem 3.1.3,
Corollary 3.1.4, Lemma 3.1.5, Theorem 3.1.6, Corollary 3.1.7, Corollary
3.1.8 ; The results were published in the paper [45];
Section 3.2 : [61]
Section 3.3 :
Chapter 4. Section 4.1 : Theorem 4.1.1, Theorem 4.1.2, Theorem 4.1.3
;The results were published in the paper [48];
Section 4.2 : De¯nition 4.2.1, Lemma 4.2.2, Lemma 4.2.3, Theorem
4.2.4. The results were published by the paper [47].
Key words : monotone and submodular set function, Choquet inte-

11
gral, Bernstein-Durrmeyer operator, uniform convergence, pointwise con-
vergence ; theory of possibility, Feller's scheme, Chebyshev type inequality,
nonlinear possibilistic integral, possibilistic Picard operators, possibilistic
Gauss-Weierstrass operators, possibilistic Poisson-Cauchy operators, max-
product (possibilistic) Bernstein kind operators ; generalized Baskakov op-
erator of real variable, linear and positive operators, modulus of continuity,
order of approximation, q-calculus ; generalized Sz¶ asz, Sz¶ asz-Kantorovich
and Baskakov complex operators, Voronovskaja-type results ; compact sets,
Faber polynomials, generalized Baskakov-Faber operator.
I want to express my deep gratitude to professor dr. Sorin Gal for his
constant support in the elaboration of this thesis.

12 CH. 1. GENERAL INTRODUCTION

Ch. 2
Approximation by nonlinear
integral operators
In this chapter we deal with the study of the approximation properties of the
integral operators, in the case when the classical linear integral is replaced
with the nonlinear Choquet integral and the nonlinear possibilistic integral.
The chapter consists in two sections : in the ¯rst section we deal with the
Durrmeyer-Choquet operators and in the second section we deal with the
possibilistic operators.
2.1 Approximation by Durrmeyer-Choquet
operators
In this section we study the multivariate Bernstein-Durmayer operator Mn;¹,
in terms of the Choquet integral with respect to a monotone and submod-
ular set function ¹, on the standard d-dimensional simplex. This operator
is nonlinear and generalizes the linear Bernstein-Durrmeyer operator with
1

2CH. 2. APPROXIMATION BY NONLINEAR INTEGRAL OPERATORS
respect to a nonnegative, bounded Borel measure (inlcuding the Lebesgue
measure). We prove the uniform and pointwise convergence of Mn;¹(f)(x)
tof(x) for n! 1 , thus generalizing the results obtained be the recent
papers [11] and [12].
2.1.1 Introduction
Starting from the paper [13], in other three recent papers [11], [12] and
[54], uniform, pointwise and Lpconvergence (respectively) of Mn;¹(f)(x)
tof(x) (as n! 1 ) were obtained, where Mn;¹(f)(x) denotes the multi-
variate Bernstein-Durrmeyer linear operator with respect to a nonnegative,
bounded Borel measure ¹, de¯ned on the standard simplex
Sd=f(x1; :::; x d); 0·x1; :::; x d·1;0·x1+:::+xd·1g;
by
Mn;¹(f)(x)
=X
j®j=nR
Sdf(t)B®(t)d¹(t)R
SdB®(t)d¹(t)¢B®(x) :=X
j®j=nc(®; ¹)¢B®(x); x2Sd; n2N;
(2.1)
where fis supposed to be ¹-integrable on Sd. Also, in formula (2.1), we
have denoted ®= (®0; ®1; :::; ® n), with ®j¸0 for all j= 0; :::; n ,j®j=
®0+®1+:::+®n=nand
B®(x) =n!
®0!¢®1!¢:::¢®n!(1¡x1¡x2¡:::¡xd)®0¢x®1
1¢:::¢x®d
d
:=n!
®0!¢®1!¢:::¢®n!¢P®(x):
In what follows we show that the results in [11] and [12] on pointwise
and uniform convergence, remain valid in the more general setting when ¹is

2.1. APPROXIMATION BY DURRMEYER-CHOQUET OPERATORS 3
a monotone, bounded and submodular set function on Sdand the integrals
appearing in the expression of the coe±cients c(®; ¹) in formula (2.1), are
Choquet integrals with respect to ¹.
2.1.2 Preliminaries
In this subsection we present concepts and results used in the next subsec-
tions.
De¯nition 2.1.1. Let (­ ;C) be a measurable space, i.e. ­ is a nonempty
set and Cis a¾-algebra of subsets in ­.
(i) (see, e.g., [65], p. 63) The set function ¹:C ! [0;+1] is called a
monotone set function (or capacity) if ¹(;) = 0 and A; B2 C, with A½B,
implies ¹(A)·¹(B). Also, ¹is called submodular if
¹(A[
B) +¹(A\
B)·¹(A) +¹(B);for all A; B2 C:
If¹(­) = 1, then ¹is called normalized.
(ii) (see [16], or [65], p. 233) Let ¹be a normalized, monotone set
function de¯ned on C. Recall that f: ­!Ris called C-measurable if for
anyB, Borel subset in R, we have f¡1(B)2 C.
Iff: ­!RisC-measurable, then for any A2 C, the Choquet integral
is de¯ned by
(C)Z
Afd¹=Z+1
0¹(F¯(f)\
A)d¯+Z0
¡1[¹(F¯(f)\
A)¡¹(A)]d¯;
where F¯(f) =f!2­;f(!)¸¯g. If ( C)R
Afd¹exists inR, then fis
called Choquet integrable on A. Note that if f¸0 on A, then the term
integralR0
¡1in the above formula becomes equal to zero.
When ¹is the Lebesgue measure (i.e. countably additive), then the
Choquet integral ( C)R
Afd¹reduces to the Lebesgue integral.

4CH. 2. APPROXIMATION BY NONLINEAR INTEGRAL OPERATORS
In what follows, we list some known properties we need.
Remarks. Let us suppose that ¹:C ! [0;+1] is a monotone set
function. Then, the following properties hold :
(i) (C)R
Ais positively homogeneous, i.e. for a¸0 we have ( C)R
Aafd¹ =
a¢(C)R
Afd¹(forf¸0 see, e.g., [65], Theorem 11.2, (5), p. 228 and for f
of arbitrary sign, see, e.g., [23], p. 64, Proposition 5.1, (ii)).
(ii) In general, ( C)R
A(f+g)d¹6= (C)R
Afd¹+(C)R
Agd¹. However, we
have
(C)Z
A(f+c)d¹= (C)Z
Afd¹+c¢¹(A);
for all c2Randfof arbitrary sign (see, e.g., [65], pp. 232-233, or [23], p.
65).
If¹is submodular too, then the Choquet integral is sublinear, that is
(C)Z
A(f+g)d¹·(C)Z
Afd¹+ (C)Z
Agd¹;
for all f; gof arbitrary sign and lower bounded (see, e.g., [23], p. 75, Theo-
rem 6.3).
(iii) If f·gonAthen ( C)R
Afd¹·(C)R
Agd¹(see, e.g., [65], p. 228,
Theorem 11.2, (3) for f; g¸0 and p. 232 for f; gof arbitrary sign).
(iv) Let f¸0. By the de¯nition of the Choquet integral, it is immediate
that if A½Bthen
(C)Z
Afd¹·(C)Z
Bfd¹
and if, in addition, ¹is ¯nitely subadditive, then
(C)Z
ASBfd¹·(C)Z
Afd¹+ (C)Z
Bfd¹:
(v) By the de¯nition of the Choquet integral, it is immediate that
(C)Z
A1¢d¹(t) =¹(A):

2.1. APPROXIMATION BY DURRMEYER-CHOQUET OPERATORS 5
(vi) Simple concrete examples of monotone and submodular set functions
¹, can be obtained from a probability measure Mon a ¾-algebra on ­
(i.e. M(;) = 0, M(­) = 1 and Mis countably additive), by the formula
¹(A) =°(M(A)), where °: [0;1]![0;1] is an increasing and concave
function, with °(0) = 0, °(1) = 1 (see, e.g., [23], pp. 16-17, Example 2.1).
Note that in fact if Mis only ¯nitely additive, then ¹(A) =°(M(A)) still
is submodular.
Also, any possibility measure ¹is monotone and submodular. Indeed,
the monotonicity and the submodularity are immediate from the axioms
(respectively)
¹(A[
B) = max f¹(A); ¹(B)g; ¹(A\
B)·minf¹(A); ¹(B)g:
Recall here that a set function ¹:P(­)![0;1] (P(­) denotes the
family of all subset of ­) is called a possibility measure on the non-empty
set ­, if it satis¯es the axioms ¹(;) = 0, ¹(­) = 1 and ¹(S
i2IAi) =
supf¹(Ai);i2Igfor all Ai2­, and any I, family of indices.
It is known that any given possibility distribution (on ­), that is a
function ¸: ­![0;1], such that sup f¸(s);s2­g= 1, induces a possibility
measure ¹¸:P(­)![0;1], given by the formula ¹¸(A) = sup f¸(s);s2Ag,
for all A½­,A6=;,¹¸(;) = 0 (see, e.g., [27], Chapter 1).
2.1.3 Main Results
LetBSdbe the sigma algebra of all Borel measurable subsets in P(Sd) and
¹:BSd![0;+1) be a normalized, monotone and submodular set function
onBSd.
We say that ¹is strictly positive if ¹(A\Sd)>0, for every open set
A½Rnwith A\Sd6=;.

6CH. 2. APPROXIMATION BY NONLINEAR INTEGRAL OPERATORS
Also, by de¯nition, the support of ¹, denoted by supp(¹), is the set of
allx2Sdwith the property that for every open neighborhood Nx2 B Sdof
x, we have ¹(Nx)>0.
Denote by C+(Sd) the space of all positive-valued continuous functions
onSdand by L1
¹(Sd) the space of all real-valued BSd-measurable functions
f, such that there exists a set E½Sd(depending on f) with ¹(E) = 0 and
fis bounded on SdnE.
Denote
Mn;¹(f)(x) =X
j®j=nc(®; ¹)¢B®(x); x2Sd; n2N;
where applying Remark 2.2, (i), we easily get
c(®; ¹) =(C)R
Sdf(t)B®(t)d¹(t)
(C)R
SdB®(t)d¹(t)=(C)R
Sdf(t)P®(t)d¹(t)
(C)R
SdP®(t)d¹(t):
It is worth noting here that we did not loose any generality by the
normalization condition on the set valued function ¹and that the condition
supp(¹)n@Sd6=;, guarantees that ( C)R
SdB®(t)d¹(t)>0, for all B®.
For the proof of the main results, we need the following auxiliary result.
Lemma 2.1.2. Let us suppose that ¹is a normalized, monotone and
submodular set function. If we de¯ne Tn:C+(Sd)!R+by
Tn(f) = (C)Z
Sdf(t)P®(t)d¹(t); f2C+(Sd); n2N;j®j=n;
then for all f; g2C+(Sd), we have
jTn(f)¡Tn(g)j ·Tn(jf¡gj) = (C)Z
Sdjf(t)¡g(t)j ¢P®(t)d¹(t):
The ¯rst main result is an analogous result to Theorem 1 in [11] and
refers to uniform approximation.

2.1. APPROXIMATION BY DURRMEYER-CHOQUET OPERATORS 7
Theorem 2.1.3. Let¹be a normalized, monotone, submodular and
strictly positive set function on BSd, such that supp(¹)n@Sd6=;. For every
f2C+(Sd)we have
lim
n!1kMn;¹(f)¡fkC(Sd)= 0;
where kFkC(Sd)= max fjF(x)j;x2Sdg.
The second main result is an analogue result to Theorem 1 in [12] and
refers to pointwise convergence. In this sense, analysing the reasonings
in the proof of Theorem 1 in [12] and using the same properties of the
Choquet integral as in the proof of the above Theorem 2.1.3, we easily get
the following.
Theorem 2.1.4. Let¹be a normalized, monotone, submodular set
function on BSd, such that supp(¹)n@Sd6=;. Iff2L1
¹(Sd)andf(x)¸0,
for all x2Sd, then at any point x2supp(¹)where fis continuous, we
have
lim
n!1jMn;¹(f)(x)¡f(x)j= 0:
Remarks. 1) According to the previous Remark, (vi), an example of
submodular set function ¹satisfying all the requirements in the statements
of Theorems 2.1.3 and 2.1.4, can simply be de¯ned by ¹(A) =p
ș(A),
where șis a Borel probability measure as in [11] and [12]. Also, it is worth
noting that due to the nonlinearity of the Choquet integral (see Remark
(ii)), unlike the case in [11], [12], the Bernstein-Durrmeyer operator in the
present paper is nonlinear.
2) The positivity of function fin Theorems 2.1.3 and 2.1.4 is necessary
because of the positive homogeneity of the Choquet integral applied in the
proof. However, if fis of arbitrary sign on Sd, then it is immediate that

8CH. 2. APPROXIMATION BY NONLINEAR INTEGRAL OPERATORS
the statements of Theorems 2.1.3 and 2.1.4 can be restated for the slightly
modi¯ed Bernstein-Durrmeyer operator de¯ned by
M¤
n;¹(f)(x) =Mn;¹(f¡m)(x) +m;
where m2Ris a lower bound for f, that is f(x)¸m, for all x2Sd.
2.2 Approximation by possibilistic integral
operators
By analogy with the Feller's general probabilistic scheme used in the con-
struction of many classical convergent sequences of linear operators, in this
paper we consider a Feller-kind scheme based on the possibilistic integral,
for the construction of convergent sequences of nonlinear operators. As par-
ticular cases, in the discrete case, all the so-called max-product Bernstein
type operators and their qualitative convergence properties are recovered.
In addition, discrete non-discrete nonlinear possibilistic convergent opera-
tors of Picard type, Gauss-Weierstrass type and Poisson-Cauchy type are
considered.
2.2.1 Introduction
In the very recent paper [33], the so-called max-product operators of Bern-
stein, of Favard-Sz¶ asz-Mirakjan kind, of Baskakov kind, of Bleimann-Butzer-
Hahn kind and of Meyer-KÄ onig-Zeller kind (whose quantitative approxima-
tion properties were intensively studied in many previously published pa-
pers, see, e.g., [8], [9], [17]-[20] and the References in [33]), were naturally
interpreted as possibilistic expectations of particular discrete fuzzy variables

2.2. APPROXIMATION BY POSSIBILISTIC INTEGRAL OPERATORS 9
having various possibilistic distributions. By using the Bernstein's idea in
[14], (see also the more accessible paper [53]), but based on a Chebyshev-
type inequality in possibility theory, these interpretations allowed to obtain
for them qualitative convergence results.
It is worth mentioning here that possibility theory is a well-established
mathematical theory dealing with certain types of uncertainties and is con-
sidered as an alternative to probability theory (see, e.g., [27], [22]) .
The main aim of this section is to present the well-known Feller's prob-
abilistic scheme in approximation, in the setting of possibility theory. In
particular, this scheme will allow not just another natural approach of the
max-product operators, but also to introduce and study many other possi-
bilistic approximation operators too.
Firstly, let us recall that a classical scheme in constructing linear and
positive approximation operators, is the Feller's probabilistic scheme (see
[29], Chapter 7, or more detailed, [3], Section 5.2, pp. 283-319). Described
shortly, it consists in attaching to a continuous and bounded function f:
R!R, approximation operators of the form
Ln(f)(x) =Z
­f±Z(n; x)dP=Z
RfdP Z(n;x);
where Pis a probability on the measurable space (­ ;C),Z:N£I! M 2(­),
with Ia subinterval of R,M2(­) represents the space of all random vari-
ables whose square is integrable on ­ with respect to the probability Pand
PZ(n;x)denotes the distribution of the random variable Z(n; x) with respect
toPde¯ned by PZ(n;x)(B) =P(Z¡1(n; x)(B)), for al B-Borel measurable
subset ofR. Then, denoting by E(Z(n; x)) and V ar(Z(n; x)) the expectance
and the variance of the random variable Z(n; x), respectively, and suppos-
ing that lim n!1E(Z(n; x)) = x, lim n!1V ar(Z(n; x)) = 0, uniformly on

10CH. 2. APPROXIMATION BY NONLINEAR INTEGRAL OPERATORS
I, it is proved that for all fas above, Ln(f) converges to funiformly on
each compact subinterval of I.
In addition, if for the random variable Z(n; x), its probability density
function ¸n;xis known, then for any fwe can write
Z
RfdP Z(n;x)=Z
Rf(t)¢¸n;x(t)dP(t);
formula which is useful in the concrete construction of the approximation
operators Ln(f)(x).
In the very recent paper [34], the Feller's scheme was generalized to
the case when the above classical integral is replaced with the nonlinear
Choquet integral with respect to a monotone and subadditive set function.
By analogy with the above considerations, in the next subsection we
consider a Feller kind scheme based on the possibilistic integral, for the
construction of convergent sequences of nonlinear operators. In particular,
in the discrete case, all the so-called max-product Bernstein type operators
and their qualitative convergence are reobtained through this scheme. In
Section 3, new discrete nonperiodic nonlinear possibilistic convergent op-
erators of Picard type, Gauss-Weierstrass type and Poisson-Cauchy type
suggested by Feller's scheme are considered. At the end, for future studies
we consider discrete periodic(trigonometric) nonlinear possibilistic opera-
tors of de la Vall¶ ee-Poussin type, of Fej¶ er type and of Jackson type.
2.2.2 Feller's scheme in terms of possibilistic integral
Firstly we summarize some known concepts for the discrete or non-discrete
fuzzy variables in possibility theory, which will be useful in the next section.
As it is easily seen, in fact they are the corresponding concepts for those
in probability theory, like random variable, probability distribution, mean

2.2. APPROXIMATION BY POSSIBILISTIC INTEGRAL OPERATORS 11
value, probability, so on. For details, see, e.g., [27] or [22].
De¯nition 2.2.1. Let ­ be a non-empty, discrete (i.e. at most count-
able) or non-discrete set.
(i) A fuzzy variable Xis an application X: ­!R. If ­ is a discrete
set, then Xis called discrete fuzzy variable. If ­ is ¯nite then Xis called
a ¯nite fuzzy variable. If ­ is not discrete, then Xis called non-discrete
fuzzy variable.
(ii) A possibility distribution (on ­), is a function ¸: ­![0;1], such
that sup f¸(s);s2­g= 1.
(iii) The possibility expectation of a fuzzy variable X(on ­), with the
possibility distribution ¸is de¯ned by Msup(X) = sups2­X(s)¸(s). The
possibility variance of XisVsup(X) = sup f(X(s)¡Msup(X))2¸(s);s2­g.
(iv) If ­ is a non-empty set, then a possibility measure is a mapping P:
P(­)![0;1], satisfying the axioms P(;) = 0, P(­) = 1 and P(S
i2IAi) =
supfP(Ai);i2Igfor all Ai2­, and any I, an at most countable family
of indices (if ­ is ¯nite then obviously Imust be ¯nite too). Note that if
A; B½­, satisfy A½B, then by the last property it easily follows that
P(A)·P(B) and that P(ASB)·P(A) +P(B).
It is well-known (see, e.g., [27]) that any possibility distribution ¸on
­, induces a possibility measure P¸:P(­)![0;1], given by the formula
P¸(A) = sup f¸(s);s2Ag, for all A½­.
For each fuzzy (possibilistic) variable X: ­!R, we can de¯ne its
distribution measure with respect to a possibility measure Pinduced by a
possibility distribution ¸, by the formula
PX:B !R+; PX(B) =P(X¡1(B)) =P(f!2­;X(!)2Bg); B2 B;
whereR+= [0;+1) andBis the class of all Borel measurable subsets in R.
It is clear that PXis a possibility measure on B, induced by the possibility

12CH. 2. APPROXIMATION BY NONLINEAR INTEGRAL OPERATORS
distribution de¯ned by
¸¤
X:R![0;1]; ¸¤
X(t) = sup f¸(!);!2X¡1(t)g;ifX¡1(t)6=;;
¸¤
X(t) = 0 ;ifX¡1(t) =;:
(v) (see, e.g., [22]) The possibilistic integral of f: ­!R+onA½­,
with respect to the possibilistic measure P¸induced by the possibilistic
distribution ¸, is de¯ned by
(Pos)Z
Af(t)dP¸(t) = sup ff(t)¢¸(t);t2Ag:
It is clear that this de¯nition is a particular case of the possibilistic integral
with respect to a semi-norm t, introduced in [22], by taking there t(x; y) =
x¢y. Also, denoting ¤ 1: ­![0;1], ¤ 1(x) = 1, for all x2­, it is immediate
that we can write
(Pos)Z
Af(t)dP¤1(t) = sup ff(t);t2Ag;
(Pos)Z
Af(t)dP¸(t) = (Pos)Z
Af(t)¢¸(t)dP¤1
anddP¸(t) =¸(t)¢dP¤1(t).
It is also worth noting that the above de¯nition of the concept of possi-
bilistic integral has a good sense only for positive-valued functions, because,
for example, if we denote R¡= (¡1;0], then for any f: ­!R¡with
f(!0) = 0 for a certain !02A½­, we get ( Pos)R
Af(t)dP¸(t) = 0.
In what follows, we also need in the frame of the possibility theory, a
simple analogue of the Chebyshev's inequality in probability theory.
Theorem 2.2.2. (see [33]) Let­be a discrete or non-discrete non-
empty set, ¸: ­![0;1]and consider X: ­!Rbe with the possibility
distribution ¸. Then, for any r >0, we have
P¸(fs2­;jX(s)¡Msup(X)j ¸rg)·Vsup(X)
r2;

2.2. APPROXIMATION BY POSSIBILISTIC INTEGRAL OPERATORS 13
where P¸is the possibilistic measure induced by ¸.
This result was proved by Theorem 2.2 in [33] for ­ discrete set, but
analysing its proof it is obvious that it remains valid in the non-discrete
case too.
In the particular case when X: ­!R+, in terms of the possibility
integral, the above Chebyshev inequality can be written as
P¸(fs2­;jX(s)¡(Pos)Z
­X(t)dP¸(t)j ¸rg)
·(Pos)R
­(X¡(Pos)R
­X(t)dP¸(t))2dP¸
r2:
In what follows, by analogy with the Feller's random scheme in prob-
ability theory which produces nice linear and positive approximation op-
erators, we will consider a similar approximation scheme, but which will
produce nonlinear approximation operators constructed with the aid of the
possibilistic integral.
For that purpose, let us denote by V arb(­) the class of all bounded
X: ­!Rand by V arb
+(­) the class of all bounded X: ­!R+. Also, for
I½Ra real interval (bounded or unbounded), let us consider the mapping
Zde¯ned onN£I!Ywhere Y=V arb(­) or Y=V arb
+(­), depending
on the context.
Notice that if for any ( n; x)2N£Iwe have Z(n; x)2V arb
+(­), then for
the concepts of possibility expectation and possibility variance of Z(n; x)
(de¯ned at the above De¯nition 2.1, (iii)) we can write the integral formulas
Msup(Z(n; x)) = ( Pos)Z
­Z(n; x)(t)dP¸(t) :=®n;x; (2.2)
Vsup(Z(n; x)) = ( Pos)Z
­(Z(n; x)(t)¡®n;x)2dP¸(t) :=¾2
n;x: (2.3)

14CH. 2. APPROXIMATION BY NONLINEAR INTEGRAL OPERATORS
Now, according to the Feller's scheme, to f:R!R+let us attach a
sequence of operators by the formula
Ln(f)(x) := ( Pos)Z
Rf(t)dPZ(n;x)(t); n2N; x2I; (2.4)
where PZ(n;x)is de¯ned as in De¯nition 2.1, (iv), i.e. with respect to the
possibility measure P¸induced by the possibility distribution ¸.
Firstly, for the operators given by (2.4) the following representation
holds.
Lemma 2.2.3. With the above notations, if Z:N£I!V arb(­)and,
in addition, f:R!R+is bounded on R, then the formula
Ln(f)(x) = (Pos)Z
Rf(t)dPZ(n;x)(t) = (Pos)Z
­f±Z(n; x)dP¸; x2I(2.5)
holds and both integrals are ¯nite.
Iff:I!R+is bounded on I, where I½Ris a subinterval and
P¸(f!2­;Z(n; x)(!)2Ig) = 1 , then we have
Ln(f)(x) = (Pos)Z
If(t)dPZ(n;x)(t) = (Pos)Z
­f±Z(n; x)dP¸:
Remark. Explicitly, formula (2.5) can be written as
Ln(f)(x) = sup ff(t)¢¸¤
Z(n;x)(t);t2Rg= supff[Z(n; x)(t)]¢¸(t);t2­g;
where ¸¤
Z(n;x)(t)is de¯ned with respect to ¸as in De¯nition 2.2.1, (iv).
Since the next main result will involve the quantity ®n;xgiven by formula
(2.2), it will be necessarily to suppose that Z(n; x)2V arb
+(­).
The following Feller-type result holds.
Theorem 2.2.4. LetI½Rbe a subinterval, Z(n; x)2V arb
+(­)for all
(n; x)2N£Iand let us suppose that f:R!R+is uniformly continuous

2.2. APPROXIMATION BY POSSIBILISTIC INTEGRAL OPERATORS 15
and bounded on R. With the notations in the formulas (2.2), (2.3) and in
the statement of Lemma 2.3, if limn!+1®n;x=xandlimn!+1¾2
n;x= 0,
uniformly with respect to x2I, then limn!1Ln(f)(x) =f(x), uniformly
with respect to x2I.
Remarks. 1) Analysing the proof of Theorem 2.2.4, it easily follows
that without any change in its proof, the construction of the operators
Ln(f)(x) can be slightly generalized by considering that not just Zdepends
onnandx, but also that ¸(and consequently P¸too) may depend on n
andx. More exactly, we can consider Ln(f)(x) of the more general form
Ln(f)(x) := ( Pos)Z
Rf(t)dPZ(n;x)(t) = (Pos)Z
­f±Z(n; x)dP¸n;x; x2I;
where P¸n;x:P(­)![0;1], (n; x)2N£I, is a family of possibility measures
induced by the families of distributions ¸n;x, (n; x)2N£I. This remark is
useful in producing several concrete examples of such operators.
Also, let us note here that if we suppose that P¸(f!2­;Z(n; x)(!)2
Ig= 1, then the operators Lncan be attached to continuous, bounded
functions de¯ned on a subinterval I½R,f:I!R+, by extending fto
a function continuous and bounded, f¤:R!R+and taking into account
the obvious relationship
(Pos)Z
Rf¤dPZ(n;x)= (Pos)Z
IfdP Z(n;x):
2) If f:I!Ris not necessarily positive, but bounded, then evidently
that there exists a constant c >0 such that f(x) +c¸0, for all x2Iand
in this case, for n2N, we can attach to fthe approximation operators
Ln(f)(x)
= (Pos)Z
I(f(t) +c)dPZ(n;x)(t)¡c= (Pos)Z
­(f+c)±Z(n; x)dP¸n;x¡c:

16CH. 2. APPROXIMATION BY NONLINEAR INTEGRAL OPERATORS
3) As particular cases of operators for which qualitative approximation
properties can be derived by the Feller's scheme in Theorem 2.2.4, are all
the so-called max-product Bernstein-type operators. Thus, for example,
if we take ­ = f0;1; :::; ng,I= [0;1],Z(n; x)(k) =k
n,f: [0;1]!R+,
¸n;x(k) =pn;k(x)Wn
j=0pn;j(x), with pn;k(x) =¡n
k¢
xk(1¡x)n¡kandWn
j=0pn;j(x) =
max j=f0;:::;ngfpn;j(x)g, then by the formula in Lemma 2.2.3 and by the def-
inition of the possibility integral, we get
Ln(f)(x) = (Pos)Z
­f±Z(n; x)dP¸n;x=nW
k=0pn;k(x)f¡k
n¢
nW
k=0pn;k(x);
which are exactly the max-product Bernstein operators B(M)
n(f)(x). The
qualitative approximation properties of B(M)
n(f)(x) can follow now from
Theorem 2.2.4.
Analogously, if, for example, we take the countable ­ = f0;1; :::; k; :::; g
andP¸n;xthe possibility measure induced by the possibility distribution
¸n;x(k) =sn;k(x)W1
k=0sn;k(x); x2[0;+1); k2N[
f0g;
with sn;k(x) =(nx)k
k!andW1
k=0sn;k(x) = max k=f0;1;:::;k;:::; gfsn;k(x)g, then the
formula in Lemma 2.3 gives the max-product Favard-Sz¶ asz-Mirakjan oper-
ators.
In a similar way, from Theorem 2.2.4 can be obtained qualitative ap-
proximation properties for the other max-product operators, like those of
Baskakov kind, of Bleimann-Butzer-Hahn kind and of Meyer-KÄ onig-Zeller
kind.
It is worth nothing that by using other (direct) methods, quantitative
estimates in approximation by max-product type operators were obtained

2.2. APPROXIMATION BY POSSIBILISTIC INTEGRAL OPERATORS 17
by the ¯rst two authors in a long series of papers (see, e.g., [8], [9], [17]-[20]
and their References).
2.2.3 Approximation by convolution possibilistic op-
erators
In this subsection, by using the above possibilistic Feller's scheme, we in-
troduce and study possibilistic variants of the classical linear convolution
operators of Picard, Gauss-Weierstrass and Poisson-Cauchy, formally given
by the formulas
Pn(f)(x) =n
2Z
Rf(t)e¡njx¡tjdt; W n(f)(x) =pnp¼Z
Rf(t)e¡njt¡xj2dt;
Qn(f)(x) =n
¼Z
Rf(t)
n2(t¡x)2+ 1;
respectively, where n2Nandx2R.
Denoting ­ = f0;1; :::; k; :::; gandZ(n; x) as in the previous Remark 3
and de¯ning ¸n;x(k) =e¡njx¡k=njW1
k=¡1e¡njx¡k=nj, by the formula in Lemma 2.3
Ln(f)(x) = (Pos)Z
­f±Z(n; x)dP¸n;x;
we obtain the following discrete possibilistic (max-product !) Picard oper-
ators
P(M)
n(f)(x) =W+1
k=¡1f(k=n)¢e¡njx¡k=nj
W+1
k=¡1e¡njx¡k=nj:
Similarly, for ¸n;x(k) =e¡n(x¡k=n)2
W1
k=¡1e¡n(x¡k=n)2and¸n;x(k) =1=(n2(x¡k=n)2+1)W1
k=01=(n2(x¡k=n)2+1)
we obtain the following discrete possibilistic (max-product !) operators,
W(M)
n(f)(x) =W+1
k=¡1f(k=n)¢e¡n(x¡k=n)2
W+1
k=¡1e¡n(x¡k=n)2;- of Gauss-Weierstrass kind ;

18CH. 2. APPROXIMATION BY NONLINEAR INTEGRAL OPERATORS
Q(M)
n(f)(x) =W+1
k=¡1f(k=n)¢1
n2(x¡k=n)2+1W+1
k=¡11
n2(x¡k=n)2+1;- of Poisson-Cauchy kind :
Let us denote by BUC +(R), the space of all uniformly continuous, bounded
and with positive values functions. The convergence of these operators can
be proved by using Theorem 2.2.4. However, we can obtain quantitative
estimates too, by direct proofs, as follows.
Theorem 2.2.5. For all f2BUC +(R)we have
jP(M)
n(f)(x)¡f(x)j ·2¢!1(f; 1=n)R:
We also can consider truncations of the operator P(M)
n. In this sense, we
can state the following.
Corollary 2.2.6. Let(m(n))n2Nbe a sequence of natural numbers with
the property that limn!1m(n)
n= +1and for f2BUC +(R)let us de¯ne
T(M)
n(f)(x) =W+m(n)
k=¡m(n)f(k=n)¢e¡njx¡k=nj
W+m(n)
k=¡m(n)e¡njx¡k=nj:
Then, T(M)
n(f)converges uniformly (as n! 1 ) to f, on any compact
subinterval of the form [¡A; A],A >0.
In what follows, similar results we present for the other possibilistic oper-
ators, W(M)
n(f)(x),Q(M)
n(f)(x) and their corresponding truncated operators
given by
S(M)
n(f)(x) =W+m(n)
k=¡m(n)f(k=n)¢e¡n(x¡k=n)2
W+m(n)
k=¡m(n)e¡n(x¡k=n)2
and
U(M)
n(f)(x) =W+m(n)
k=¡m(n)f(k=n)¢1
n2(x¡k=n)2+1W+m(n)
k=¡m(n)1
n2(x¡k=n)2+1:
Theorem 2.2.7. For all f2BUC +(R)we have
jW(M)
n(f)(x)¡f(x)j ·2¢!1(f; 1=pn)R:

2.2. APPROXIMATION BY POSSIBILISTIC INTEGRAL OPERATORS 19
Corollary 2.2.8. Let(m(n))n2Nbe a sequence of natural numbers
with the property that limn!1m(n)
n= +1. Then, for any f2BUC +(R),
S(M)
n(f)converges uniformly (as n! 1 ) tof, on any compact subinterval
of the form [¡A; A],A >0(S(M)
n(f)is de¯ned just above the statement of
Theorem 2.2.7).
Theorem 2.2.9. For all f2BUC +(R)we have
jQ(M)
n(f)(x)¡f(x)j ·2¢!1(f; 1=(2n))R:
Corollary 2.2.10. Let(m(n))n2Nbe a sequence of natural numbers with
the property that limn!1m(n)
n= +1. Then, for any f2BUC +(R),
U(M)
n(f)converges uniformly (as n! 1 ) tof, on any compact subinterval
of the form [¡A; A],A >0(U(M)
n(f)is de¯ned just above the statement of
Theorem 2.2.7).
Remarks. 1) We note that in [28] Favard introduced the discrete version
of the above Gauss-Weierstrass singular integral by the formula
Fn(f)(x) =1p¼n¢+1X
k=¡1f(k=n)¢e¡n(x¡k=n)2; n2N; x2R
and proved that if f:R!Ris continuous on R, of the exponential growth
jf(t)j ·MeAt2for all t2R(here M; A > 0), then Fn(f)(x) converges to
f(x) pointwise for each x2Rand uniformly on any compact subinterval
ofR. Other approximation properties of Fn(f)(x), especially in various
weighted spaces, were studied in many papers, see, e.g., [1] and the Refer-
ences therein.
Exactly as it was proved for other max-product operators studied in
previous papers (see, e.g., [17]-[20]), with respect to its linear counterpart
Fn(f)(x), for the max-product operators W(M)
n(f)(x) can be proved that in

20CH. 2. APPROXIMATION BY NONLINEAR INTEGRAL OPERATORS
some subclasses of functions f, have better global approximation properties
and that present much stronger localization results. More precisely, they
represent locally much better (probably best possible) the approximated
function, in the sense that if fandgcoincides on a strict subinterval I½R,
then for any subinterval I0strictly included in I,W(M)
n(f) and W(M)
n(g)
coincide in I0for su±ciently large n.
2) By using the above possibilistic Feller's scheme, we can introduce for
study possibilistic variants of the classical linear convolution trigonometric
operators of de la Vall¶ ee-Poussin, Fej¶ er and Jackson, formally de¯ned by
the formulas
Vn(f)(x) =1
2¼Z¼
¡¼f(t)kn(x¡t)dt; F n(f)(x) =1
2¼Z¼
¡¼f(t)bn(x¡t)dt;
Jn(f)(x) =1
¼Z¼
¡¼f(t)cn(x¡t)dt;
respectively, where fis 2¼-periodic,
kn(t) =(n!)2
(2n)!(2 cos( t=2))2n; bn(t) =1
nµsin(nt=2)
sin(t=2)¶2
andcn(t) =3
2n(2n2+1)³
sin(nt=2)
sin(t=2)´4
.
More precisely, denoting ­ = f¡n; :::;¡1;0;1; :::; ngandZn;x(k) =k¼
n,
forf: [¡¼; ¼]!Rand¸n;x(k) =kn(x¡k¼=n )Wn
k=¡nkn(x¡k¼=n ), by the formula in Lemma
2.2.3 and by the de¯nition of the possibility integral, we get the possibilistic
de la Vall¶ ee-Poussin operators
V(M)
n(f)(x) = (Pos)Z
­f±Z(n; x)dP¸n;x=Wn
k=¡nf(k¼=n )kn(x¡k¼=n )Wn
k=¡nkn(x¡k¼=n ):
Similarly, we can obtain the possibilistic operators of Fej¶ er type
F(M)
n(f)(x) =Wn
k=¡nf(k¼=n )bn(x¡k¼=n )Wn
k=¡nbn(x¡k¼=n )

2.2. APPROXIMATION BY POSSIBILISTIC INTEGRAL OPERATORS 21
and of Jackson type
J(M)
n(f)(x) =Wn
k=¡nf(k¼=n )cn(x¡k¼=n )Wn
k=¡ncn(x¡k¼=n ):
The study of the approximation properties of these operators will be made
elsewhere.

22CH. 2. APPROXIMATION BY NONLINEAR INTEGRAL OPERATORS

Ch. 3
Approximation with an
arbitrary order by Sz¶ asz and
Baskakov operators of real
variable
Given an arbitrary sequence ¸n>0,n2N, with the property that
limn!1¸n= 0 as fast we want, in this note we introduce modi¯ed Baskakov
operators in such a way that on each compact subinterval in [0 ;+1) the
order of uniform approximation is !1(f;p¸n).
The idea of construction of these generalized operators is simple : in
their classical formulas, we replace everywhere nwith1
¸n.
For example, starting from the classical formula for the Sz¶ asz operators
Sn(f)(x) =e¡nx1X
k=0(nx)k
k!f(k=n);
23

24CH. 3. ARBITRARY ORDER BY SZ ¶ASZ AND BASKAKOV OPERATORS
by replacing nby1
¸nwe get the generalized Sz¶ asz operator
Sn(f;¸n)(x) =e¡x=¸n1X
k=01
k!µx
¸n¶k
¢f(k¸n);
while starting from the classical formula for the Baskakov operator
Vn(f)(x) = (1 + x)¡n1X
j=0µn+j¡1
j¶µx
1 +x¶j
¢fµj
n¶
= (1 + x)¡n1X
j=0(n+j¡1)!
(n¡1)!j!µx
1 +x¶j
¢fµj
n¶
= (1 + x)¡n1X
j=01
j!¢n(n+ 1)¢:::¢(n+j¡1)µx
1 +x¶j
¢fµj
n¶
;
by replacing nby1
¸nwe get the generalized Baskakov operator
Vn(f;¸n)(x)
= (1 + x)¡1=¸n1X
j=01
j!¢1
¸nµ
1 +1
¸n¶
¢:::¢µ
j¡1 +1
¸n¶
¢µx
1 +x¶j
f(j¸n):
3.1 Generalized real Baskakov operators
Given an arbitrary sequence ¸n>0,n2N, with the property that
limn!1¸n= 0 as fast we want, in this section we introduce modi¯ed
Baskakov operators in such a way that on each compact subinterval in
[0;+1) the order of uniform approximation is !1(f;p¸n). These modi-
¯ed operators can uniformly approximate a Lipschitz 1 function, on each
compact subinterval of [0 ;1) with the arbitrary good order of approxima-
tionp¸n. Also, similar considerations are made for modi¯ed qn-Baskakov
operators, with 0 < qn<1, lim n!1qn= 1.

3.1. GENERALIZED REAL BASKAKOV OPERATORS 25
3.1.1 Introduction
Let ( ¸n)nbe a sequence of real positive numbers with the properties that
limn!1¸n= 0.
In [15] Cetin and Ispir introduced a remarkable generalization of the
Sz¶ asz-Mirakjan operators attached to analytic functions fof exponential
growth in a compact disk,
Sn(f;¸n)(z) =e¡z=¸n1X
k=01
k!µz
¸n¶k
¢f(k¸n);
which approximate fin any compact disk jzj ·r,r < R , with the approx-
imation order ¸n.
Involving in their construction the Faber polynomials too, these oper-
ators and their order of approximation were extended in Gal [32] in order
to approximate analytic functions in compact subsets (continuums) of the
complex plane. Then, in Gal [32], Sn(f;¸n)(x) was strongly generalized on
the real axis through the She®er polynomials, proving for them the approx-
imation order !1(f;p¸n). The great advantage of all these constructions
is that the sequence ¸n,n2N, can evidently be chosen to converge to zero
with an arbitrary small order. Note that in fact, all the above mentioned
results were obtained for ¸nwritten in the unnecessary more complicated
form, ¸n=¯n
®n.
The ¯rst main aim of this section is to introduce and study the linear
and positive modi¯ed Baskakov-type operators, de¯ned by
Ln(f;¸n)(x) =1X
j=0(¡1)j'(j)(¸n;x)xj
j!f(j¸n); (3.1)
for functions f: [0; b)!R(here bcan be + 1too) such that the above
series converges (e.g. if fis bounded or uniformly continuous on [0 ; b)),

26CH. 3. ARBITRARY ORDER BY SZ ¶ASZ AND BASKAKOV OPERATORS
where the sequence of analytic functions 'n: [0; b)!R,n2N, satisfy
the hypothesis : (i) '(¸n; 0) = 1; (ii) ( ¡1)j'(j)(¸n;x)¸0, for all n; j2N,
x2[0; b].
It is worth noting that for the particular case ¸n=1
nand under the
additional hypothesis
(iii) there exists a sequence m(n),n2Nwith lim n!1n
m(n)= 1 such that
'(k)
n(¸n;x) =¡n'(k¡1)
n(¸n;x), for all x2[0; b),n2N,k2N, the operators
in (3.1) were introduced and investigated in Baskakov [7].
Choosing '(¸n;x) = (1 + x)¡1=¸nin (3.1), because of the formula
'(j)(¸n;x) = (¡1)j1
¸nµ
1 +1
¸n¶
¢:::¢µ
j¡1 +1
¸n¶
¢(1 +x)¡j¡1=¸n;(3.2)
we immediately get the modi¯ed Baskakov-type operators de¯ned by
Vn(f;¸n)(x)
= (1 + x)¡1=¸n1X
j=01
j!¢1
¸nµ
1 +1
¸n¶
¢:::¢µ
j¡1 +1
¸n¶
¢µx
1 +x¶j
f(j¸n);
(3.3)
x¸0, where by convention1
¸n³
1 +1
¸n´
¢:::¢³
j¡1 +1
¸n´
= 1 for j= 0.
For these operators Vn(f;¸n)(x) in (3.3), in the next Subsection we prove
that on each compact subinterval in [0 ;+1), the order of uniform approx-
imation obtained is !1(f;p¸n), and consequently uniformly approximate
a Lipschitz 1 function, on each compact subinterval of [0 ;1) with an ar-
bitrary good order of approximationp¸n. In other words, from the point
of view of approximation theory, between all kinds of Baskakov-type op-
erators in literature, these modi¯ed Baskakov operators represent the best
possible construction. In the same time, the results obtained have also a
strong unifying character, in the sense that one can recapture from them all
the results previously obtained by other authors, for various choices of the

3.1. GENERALIZED REAL BASKAKOV OPERATORS 27
nodes ¸n. It is also remarked that by modifying a Baskakov-type operator
introduced in Lopez-Moreno [55], similar considerations can be made for
the operator de¯ned by
Ln;r(f;¸n)(x) =1X
j=0(¡1)rf(j¸n)¢'(j+r)(¸n;x)¢(¡x)j
j!¢(¸n)r; r; n2N:
(3.4)
Then, in the next Subsection we make similar considerations for modi¯ed
q-Baskakov-type operators, 0 < q < 1.
3.1.2 Main results
Firstly, we need the following two auxiliary results.
Lemma 3.1.1. Let¸n>0,n2N, be with limn!1¸n= 0.
(i) If Ln(f;¸n)(x)given by (3.1) is well-de¯ned, then we can write
Ln(f;¸n)(x) =1X
j=0(¸n)j¢(¡1)j¢'(j)(¸n; 0)¢[0; ¸n; :::; j¸ n;f]¢xj; x2[0; b];
where [0; ¸n; :::; j¸ n;f]is the divided di®erence of fon the knots 0; ¸n; :::; j¸ n.
(ii) Denoting ek(x) =xk, we have Ln(e0;¸n)(x) = 1 ,Ln(e1;¸n)(x) =
¡x¸n'0(¸n; 0),
Ln(e2;¸n)(x) = (¸n)2¢[x2'00(¸n; 0)¡x'0(¸n; 0)]:
Remark. In the case when ¸n=1
n, the formula in Lemma 3.1.1, (i)
was obtained by Lupas [56].
Corollary 3.1.2. (i) If by convention, (1 +¸n):::(1 + ( j¡1)¸n) = 1
forj= 0, then for Vn(f;¸n)(x)given by (3.3), we have
Vn(f;¸n)(x) =1X
j=0(1 +¸n):::(1 + ( j¡1)¸n)¢[0; ¸n; :::; j¸ n;f]xj; x¸0:

28CH. 3. ARBITRARY ORDER BY SZ ¶ASZ AND BASKAKOV OPERATORS
(ii)Vn(e0;¸n)(x) = 1 ,Vn(e1;¸n)(x) =x,Vn(e2;¸n)(x) =x2+¸n¢x(1+x)
;
Vn((¢ ¡x)2;¸n)(x) =¸nx(1 +x):
Since Vn(f;¸n),n2N, are positive and linear operators, we can state the
following result.
Theorem 3.1.3. Letf: [0;1)!Rbe uniformly continuous on [0;1).
Denote !1(f;±) = sup fjf(x)¡f(y)j;jx¡yj ·±; x; y 2[0;1)g. For all
x2[0;1),n2Nwe have
jVn(f;¸n)(x)¡f(x)j ·2¢!1³
f;p
¸n¢p
x(1 +x)´
:
As an immediate consequence of Theorem 3.1.3 we get the following.
Corollary 3.1.4. If there exists L >0such that jf(x)¡f(y)j ·Ljx¡yj,
for all x; y2[0;1), then
jVn(f;¸n)(x)¡f(x)j ·2Lp
x(1 +x)¢p
¸n; n2N; x¸0:
Remarks. 1) If xbelong to a compact subinterval of [0 ;+1), then evi-
dently that we get uniform convergence in that subinterval.
2) The optimality of the estimates in Theorem 3.1.3 and Corollary 3.1.4
consists in the fact that given an arbitrary sequence of strictly positive
numbers ( °n)n, with lim n!1°n= 0 and a compact subinterval [0 ; b], we
can ¯nd a sequence ¸n, satisfying 2 !1(f;p¸n¢p
x(1 +x))·°nfor all
n2N,x2[0; b] in the case of Theorem 3.1.3 and 2 Lp¸n¢p
x(1 +x)·°n
for all n2N,x2[0; b] in the case of Corollary 3.1.4.
3) If fis uniformly continuous on [0 ;+1) then it is well known that
its growth on [0 ;+1) is linear, i.e. there exist ®; ¯ > 0 such that jf(x)j ·
®x+¯, for all x2[0;+1) (see e.g. [25], p. 48, Problµ eme 4, or [26]). This
implies that in this case Vn(f;¸n)(x) is well-de¯ned for all x2[0;1).

3.1. GENERALIZED REAL BASKAKOV OPERATORS 29
4) If in Lemma 3.1.1, Corollary 3.1.2, Theorem 3.1.3 and Corollary 3.1.4
we consider '(¸n;x) =e¡x=¸n, then for x¸0,Ln(f;¸n)(x),x¸0 becomes
a particular case of the modi¯ed Sz¶ asz-Mirakjan operators studied in [32],
while for z2Cbecomes the operator Sn(f;¸n)(z) studied in [15].
5) In the paper [55], the Baskakov-type approximation operators of the
form
Ln;r(f)(x) =1X
j=0(¡1)rfµj
n¶
¢'(j+r)
n(x)¢(¡x)j
j!¢µ1
n¶r
; r; n2N
were studied, obtaining for example if 'n(x) = (1 + x)¡n, quantitative es-
timates of the order ­( f;n¡1=2) +C
n, where ­( f;±) is a suitable weighted
modulus of continuity. By following the lines of proofs in [55], choosing
'(¸n;x) = (1+ x)¡1=¸nin the modi¯ed Baskakov-type operator Ln;r(f;¸n)(x)
given by formula (3.4), the order of approximation ­¡
f;p¸n¢
+C¸nis ob-
tained, where ¸ncan be chosen to converge to 0 as fast we want.
3.1.3 The case of q-Baskakov operators, 0< q < 1
Firstly, we need the following concepts in quantum calculus (see e.g. [52],
pp. 7-13).
For 0 < q,q6= 1, and a2R, deqanalogue of ais de¯ned by [ a]q=1¡qa
1¡q.
Forn2N[ f0g, we get [ n]q= 1 + q+:::+qn¡1,n2N, [0] q= 1. The q-
factorial is de¯ned by [ n]q! = [1] q¢[2]q¢:::¢[n]qand the q-binomial coe±cient
is given by¡n
k¢
q=[n]q!
[k]q!¢[n¡k]q!,k= 0;1; :::n.
Note that for q= 1 we get [ n]q=nand as a consequence, [ n]q! =n! and
¡n
k¢
q=¡n
k¢
.
Theq-derivative of a function f:R!Ris de¯ned by Dq(f)(x) =
f(x)¡f(qx)
x(1¡q); x6= 0, Dq(f)(0) = lim x!0Dq(f)(x), and the q-derivatives of

30CH. 3. ARBITRARY ORDER BY SZ ¶ASZ AND BASKAKOV OPERATORS
higher order are given recursively by D0
q(f) =f,Dn
q(f) =Dq(Dn¡1
q(f)),
n2N.
Everywhere in what follows, we consider 0 < q < 1.
Various kinds of q-Baskakov operators were studied in the e.g. the papers
[2], [62], [4]{[6], [51], [30].
Following the previous ideas and suggested by the q-Baskakov operators
introduced and studied in [62] and [2], we introduce here a modi¯ed q-
Baskakov operator, as follows.
Let¸n>0,n2Nbe with lim n!11
¸n= +1. It is clear that with-
out any lost of generality, we may suppose that1
¸n¸1,n2N. For
'(¸n;¢) : [0 ;1)!R,n2N, a sequence of analytic functions satisfying
the hypothesis (i) '(¸n; 0) = 1; (ii) ( ¡1)j'(j)(¸n;x)¸0, for all n; j2N,
x2[0;1), let us introduce the q-Baskakov operator given by
Tn;q(f;¸n)(x) =1X
j=0(¡x)j
[j]q!¢q(k(k¡1)=2Dk
q'(¸n;x)fµ[j]q
qk¡1¢1
[1=¸n]q¶
;(3.5)
attached to functions for which Tn;q(f;¸n)(x) is well-de¯ned.
Note that for 1 =¸n=nwe recapture the q-Baskakov operators in [62],
[2].
Following exactly the lines in the proof of Lemma 1 in [62] and also using
relationships (21) and(22) in [2], we immediately get the following.
Lemma 3.1.5. Let¸n>0,1
¸n¸1,n2Nbe with limn!11
¸n= +1.
For all n2N; x¸0and0< q < 1, we have :
(i)Tn;q(e0;¸n)(x) = 1 ;Tn;q(e1;¸n)(x) =¡x¢Dq('(¸n;¢))(0)¢1
[1=¸n]q;
(ii)Tn;q(e2;¸n)(x) =x2¢D2
q('(¸n;¢))(0)¢1
q¢[1=¸n]2q¡x¢Dq('(¸n;¢))(0)¢
1
[1=¸n]2q;
(iii)Tn;q((¢ ¡x)2;¸n)(x) =An;qx2+Bn;qx, where
An;q= 1 + D2
q('(¸n;¢))(0)¢1
q¢[1=¸n]2
q+ 2¢Dq('(¸n;¢))(0)¢1
[1=¸n]q

3.1. GENERALIZED REAL BASKAKOV OPERATORS 31
and
Bn;q=¡Dq'(¸n; 0)
[1=¸n]2
q:
Denoting by CB(R+) the space of all bounded continuous real-valued func-
tions on [0 ;1) and following exactly the lines in the proof of Theorem 2 in
[2], we can state the following.
Theorem 3.1.6. Let¸n>0,1
¸n¸1,n2Nbe with limn!11
¸n=
+1and let (qn)n2Nbe a sequence such that 0< q n<1for all n2N
andlimn!1qn= 1. Then, for f2CB(R+)uniformly continuous, the
qn-operators given by (3.5) satisfy
jTn;qn(f;¸n)(x)¡f(x)j ·(1 +p
maxfx; x2g)¢!1(f;p
Cn;qn); n2N; x¸0;
where Cn;qn=jAn;qnj+Bn;qn,(An;qn)n;(Bn;qn)nare given in Lemma 3.1.5,
(iii) and !1(f;±) = sup fjf(x)¡f(y)j;x; y2[0;1);jx¡yj ·±g.
As consequences of Theorem 3.1.6, we get the following two corollaries.
Corollary 3.1.7. Let¸n>0,1
¸n¸1,n2Nbe with limn!11
¸n= +1
and(qn)n2Nbe a sequence such that 0< q n<1for all n2Nand
limn!1qn= 1. Then, for f2CB(R+)uniformly continuous, the qn-
operators given by
Tn;qn(f;¸n)(x) =1
(1 +x)1=¸n¢1X
j=0[1=¸n]qn¢[1=¸n+ 1] qn¢:::¢[1=¸n+j¡1]qn
[j]qn!
¢qj(j¡1)=2¢xj
(1 +x)j¢fµ[j]qn
qj¡1
n¢1
[1=¸n]qn¶
; (3.6)
for all n2N; x¸0, satisfy the estimate
jTn;qn(f;¸n)(x)¡f(x)j ·(1+p
maxfx; x2g)¢!1Ã
f;r1 +qn
qn¢1p
[1=¸n]qn!
:
Corollary 3.1.8. Let¸n>0,1
¸n¸1,n2Nbe with limn!11
¸n= +1and
(qn)n2Nbe a sequence such that 0< qn<1for all n2Nandlimn!1qn= 1.

32CH. 3. ARBITRARY ORDER BY SZ ¶ASZ AND BASKAKOV OPERATORS
Then, for f2CB(R+)uniformly continuous, the qn-operators given by
Sn;qn(f;¸n)(x)
=1X
j=0([1=¸n]qnx)j
[j]qn!¢qj(j¡1)
n¢Eqn(¡[1=¸n]qnqj
nx)¢fµ[j]qn
qj¡1
n¢1
[1=¸n]qn¶
;(3.7)
for all n2N; x¸0, satisfy the estimate
jSn;qn(f;¸n)(x)¡f(x)j ·(1 +p
maxfx; x2g)¢!1Ã
f;1p
[1=¸n]qn!
:
Remarks. 1) Corollary 3.1.8 is a generalization of the result in [32], to
the case of q-Sz¶ asz-Mirakjan operators, with 0 < q·1.
2) The order of approximation for the qn-Baskakov-type operators in
Corollary 3.1.7 and for the qn-Sz¶ asz-Mirakjan operators in Corollary 3.4 is
O(1=p
[1=¸n]qn). On the other hand, for qn= 1, for all n2N, the order of
approximation is O(1=p
1=¸n) =O(p¸n) (see Theorem 3.1.3 in the case of
Baskakov-type operators and [32] in the case of Sz¶ asz-Mirakjan operators).
However, for 0 < q n<1 for all n2N, it is easy to see thatp¸n·
p
2p
[1=¸n]qn, because [1 =¸n]qn·2=¸n.
Indeed, denoting with [ a]¤the integer part of a, we have 1 =¸n·[1=¸n]¤+
1, which by 0 < q n<1 implies q[1=¸n]¤+1
n ·q1=¸nn, leading to [1 =¸n]qn·
[[1=¸n]¤+ 1] qn·[1=¸n]¤+ 1·2=¸n.
On the other hand, by [24], Lemma 3.4, n·C0[n]qn, for all n2N(with
C0>0 independent of n), if and only if there exists a constant c >0 and
n02N(independent of n) such that qn
n¸c, for all n¸n0. Therefore, in
this case, we obtain
1=¸n·[1=¸n]¤+ 1·C0[[1=¸n]¤+ 1] qn
·C0[2[1=¸n]¤]qn·2C0[[1=¸n]¤]qn·2C0[1=¸n]qn:

3.2. GENERALIZED REAL SZ ¶ASZ-STANCU OPERATORS 33
In conclusion, if in Corollaries 3.1.7 and 3.1.8 qnis chosen to satisfy qn
n¸c,
for all n¸n0, 0< q n<1,n2N, and lim n!1qn= 1, then the approx-
imation orders for the corresponding qn-Baskakov and qn-Sz¶ asz-Mirakjan
operators are !1¡
f;p¸n¢
, which can be chosen to converge to 0 as fast we
want.
3.2 Generalized real Sz¶ asz-Stancu operators
3.3 Generalized real Baskakov-Stancu oper-
ators

34CH. 3. ARBITRARY ORDER BY SZ ¶ASZ AND BASKAKOV OPERATORS

Ch. 4
Approximation with an
arbitrary order by Sz¶ asz and
Baskakov kinds operators of
complex variable
In this chapter we consider the ideas in the previous chapter, but applied
now to the case of approximation of analytic functions by complex Sz¶ asz
and Baskakov operators, in compact sets in C. Two cases are studied : (i)
approximation in compact disks with center at origin ; (ii) approximation
in arbitrary compacts by using the Faber polynomials attached to these
compact sets.
4.1 Arbitrary order in compact disks
By using a sequence ¸n>0,n2Nwith the property that ¸n!0 as fast
we want, in this section we obtain the approximation order O(¸n) for some
35

36 CH. 4. COMPLEX SZ ¶ASZ AND BASKAKOV OPERATORS
generalized Sz¶ asz, Sz¶ asz-Kantorovich, and Baskakov complex operators at-
tached to entire functions or to analytic functions of exponential growth in
compact disks and without to involve the values on [0 ;+1).
4.1.1 Introduction
In [15], with the notations there for two sequences anandbn,n2N, and
denoting here ¸n=bn
an, the authors introduced the generalized complex
Sz¶ asz operator by
Sn(f;¸n)(z) =e¡z=¸n1X
j=0(z=¸ n)j
j!¢f(j¸n); (4.1)
where ¸n>0,¸n!0.
For this operator, attached to functions f:DRS[R;+1)!Cof expo-
nential growth in DRS[R;+1), analytic in the disk DR=fz2C;jzj< Rg,
R >1 and continuous on [0 ;+1), the exact order of approximation O(¸n)
is obtained in [15]. Also, in the same paper a Voronovskaja-type result with
an upper estimate of order O(¸2
n) is proved.
The ¯rst goal of the present section is to extend the results in [15] to
the case of entire functions and then, to a kind od Sz¶ asz operator which
does not involve the values of fon [0 ;+1). Also, a complex operator of
Sz¶ asz-Kantorovich type is introduced, for which similar results are proved,
essentially improving the order of approximation O(1=n) obtained in [60].
The second goal is to introduce generalized complex Baskakov operators,
for which similar results with those obtained for the Sz¶ asz operators are
proved.

4.1. ARBITRARY ORDER IN COMPACT DISKS 37
4.1.2 Sz¶ asz and Sz¶ asz-Kantorovich complex operators
In the case of complex Sz¶ asz operator, we can prove the following result.
Theorem 4.1.1. Let¸n>0,n2Nbe with ¸n!0as fast we want. Let
f:DR!C,1< R·+1, i.e. f(z) =P1
k=0ckzk, for all z2DR. Suppose
that there exist M > 0andA2(1=R;1), with the property jckj ·MAk
k!, for
allk= 0;1; :::;(which implies jf(z)j ·MeAjzjfor all z2DR). Consider
1·r <1
A.
(i) If R= +1, (1=R= 0), i.e. fis an entire function, then Sn(f;¸n)(z)
is entire function, we have Sn(f;¸n)(z) =P1
k=0ckSn(ek;¸n)(z)for all z2
C,n2Nand for all jzj ·rthe following estimates hold :
jSn(f;¸n)(z)¡f(z)j ·Cr;M;A¢¸n;
jS(p)
n(f;¸n)(z)¡f(p)(z)j ·p!r1¢Cr1;M;A
(r1¡r¢¸n;
¯¯¯¯Sn(f;¸n)(z)¡f(z)¡¸n
2zf00(z)¯¯¯¯·Mr(f)(z)¢¸2
n·Cr(f)¢¸2
n;
kS(p)
n(f;¸n)¡f(p)kr»¸n;
the last equivalence holding if fis not a polynomial of degree ·p2Nand
the constants in the equivalence depend on f,r,p.
Above, Cr;M;A =M
2rP1
k=2(k+ 1)( rA)k<1,p2N,1·r < r 1<1
A,
Mr(f)(z) =3MAjzj
r2¢P1
k=2(k+ 1)( rA)k¡1<1,Cr(f) =3MA
r¢P1
k=2(k+
1)(rA)k¡1andkfkr= max fjf(z)j;jzj ·rg.
(ii) If R <+1, then the complex approximation operator
S¤
n(f;¸n)(z) =1X
k=0ck¢Sn(ek;¸n)(z); z2Dr;
is well-de¯ned and S¤
n(f;¸n)(z)satis¯es all the estimates from the point (i),
for all 1·r <1
A< R.

38 CH. 4. COMPLEX SZ ¶ASZ AND BASKAKOV OPERATORS
In what follows, we can de¯ne the generalized complex Sz¶ asz-Kantorovich
operator by the formula
Kn(f;¸n)(z) =e¡z=¸n1X
j=0(z=¸ n)j
j!¢1
¸n¢Z(j+1)¸n
j¸nf(v)dv
=e¡z=¸n1X
j=0(z=¸ n)j
j!¢Z1
0f((t+j)¸n)dt:
Denoting F(z) =Rz
0f(t)dt, simple calculation leads to the formula (under
the hypothesis that the series Sn(F;¸n)(z) is uniformly convergent)
Kn(f;¸n)(z) =S0
n(F;¸n)(z): (4.2)
We can prove the following results.
Theorem 4.1.2. Let¸n>0,n2Nbe with ¸n!0as fast we want. Let
f:DR!C,1< R·+1, i.e. f(z) =P1
k=0ckzk, for all z2DR. Suppose
that there exist M > 0andA2(1=R;1), with the property jckj ·MAk
k!,
for all k= 0;1; :::;(which implies jf(z)j ·MeAjzjfor all z2DR). Also,
consider 1·r <1=A.
(i) If R= +1, (1=R= 0), i.e. fis an entire function, then, Kn(f;¸n)(z)
is entire function, we have Kn(f;¸n)(z) =P1
k=0ckKn(ek;¸n)(z)for all
z2C,n2Nand for all jzj ·rthe following estimates hold :
¯¯¯¯Kn(f;¸n)(z)¡f(z)¡¸n
2[f0(z) +zf00(z)]¯¯¯¯·C0
r(f)¢¸2
n;
kK(p)
n(f;¸n)¡f(p)kr»¸n;
the last equivalence holding if fis not a polynomial of degree ·pand the
constants in the equivalence depend on f,r,p.
Above p2NSf0g,C0
r(f)<1is a constant independent of nandz
andkfkr= max fjf(z)j;jzj ·rg.

4.1. ARBITRARY ORDER IN COMPACT DISKS 39
(ii) If R <+1, then the complex approximation operator
K¤
n(f;¸n)(z) =1X
k=0ck¢Kn(ek;¸n)(z); z2Dr;
is well-de¯ned and K¤
n(f;¸n)(z)satis¯es all the estimates from the point
(i), for all 1·r <1
A< R.
Remarks. 1) It is worth noting that in the case of real variable, the
generalized Sz¶ asz operators de¯ned by (4.1) were considered in [39], where,
denotingbn
an:=¸n, the approximation order !1(f;p¸n) was obtained, with
!1denoting the modulus of continuity of fon [0;+1). In conclusion, the
results in the real case in [39] and those in the complex case in Theorems
4.1.1 and 4.1.2, seem to be of de¯nitive type, in the sense that they exhibit
operators which can approximate the functions with an arbitrary chosen
order.
2) The ¯rst estimate in the statement of Theorem 4.1.1, (i), was ex-
tended (with a di®erent constant, of course) in [37] to the approximation
by generalized Sz¶ asz-Faber polynomials in compact sets in C.
4.1.3 Generalized complex Baskakov operators
Forxreal and ¸0, the original formula of the classical now Baskakov
operator is given by (see [7])
Zn(f)(x) = (1 + x)¡n1X
k=0µn+k¡1
k¶µx
1 +x¶k
f(k=n)
and many approximation results of this operators were published.
According to [56], Theorem 2, under the same hypothesis on fthat
Zn(f)(x) is well de¯ned and denoting by [0 ;1=n; :::; j=n ;f] the divided dif-
ference of fon the knots 0, …, j=n, for x¸0 we can write Zn(f)(x) =

40 CH. 4. COMPLEX SZ ¶ASZ AND BASKAKOV OPERATORS
Wn(f)(x),x¸0, where
Wn(f)(x) :=1X
j=0µ
1 +1
n¶
¢:::¢µ
1 +j¡1
n¶
¢[0;1=n; :::; j=n ;f]xj; x¸0;
(4.3)
(here for j= 0 and j= 1 we take (1 + 1 =n)¢:::¢(1 + ( j¡1)=n) = 1).
For¸n&0, arbitrary, by formula (1) in the paper [45] (particularizing
there 'n(¸n;x) = (1 + x)¡1=¸n),Zn(f)(x) can be generalized to
Zn(f;¸n)(x)
= (1+ x)¡1=¸n¢1X
j=01
j!¢1
¸nµ
1 +1
¸n¶
¢:::¢µ
j¡1 +1
¸n¶
¢µx
1 +x¶j
f(j¸n);
x¸0, where by convention1
¸n³
1 +1
¸n´
¢:::¢³
j¡1 +1
¸n´
= 1 for j= 0.
For this generalization, in [45] the order of approximation !1(f;p¸n¢
p
x(1 +x)) was obtained.
Accordingly, Wn(f)(x) given by (4.3), can be generalized to
Wn(f;¸n)(x) =1X
j=0(1 +¸n):::(1 + ( j¡1)¸n)¢[0; ¸n; :::; j¸ n;f]xj; x¸0;
where by convention, (1 + ¸n):::(1 + ( j¡1)¸n) = 1 for j= 0.
It is clear that Zn(f;¸n)(x) =Wn(f;¸n)(x) for all x¸0, but as it was
remarked in [35], p. 124, in the particular case ¸n=1
n, ifjxj<1 is not
positive then Wn(f;¸n)(x) and Zn(f;¸n)(x) do not necessarily coincide and
because of this reason in Section 1.8 of the book [35], pp. 124-138, they were
studied separately, under di®erent hypothesis on fandz2C.
In what follows we study the approximation properties of the complex
generalized Baskakov operators Wn(f;¸n)(z) attached to analytic functions
satisfying some exponential-type growth condition.
In this sense, we can state the following.

4.1. ARBITRARY ORDER IN COMPACT DISKS 41
Theorem 4.1.3. Let0< ¸ n·1
2,n2Nbe with ¸n!0as fast
we want. Let f:DR!C,1< R·+1, i.e. f(z) =P1
k=0ckzk, for
allz2DR. Suppose that there exist M > 0andA2(1=R;1), with the
property jckj ·MAk
k!, for all k= 0;1; :::;(which implies jf(z)j ·MeAjzjfor
allz2DR). Consider 1·r <1
A.
(i) If R= +1, (1=R= 0), i.e. fis an entire function, then for jzj ·r
Wn(f;¸n)(z)is analytic, we have Wn(f;¸n)(z) =P1
k=0ckWn(ek;¸n)(z)and
the following estimates hold :
jWn(f;¸n)(z)¡f(z)j ·Cr;M;A¢¸n;
jW(p)
n(f;¸n)(z)¡f(p)(z)j ·p!r1¢Cr1;M;A
(r1¡r¢¸n;
¯¯¯¯Wn(f;¸n)(z)¡f(z)¡¸n
2zf00(z)¯¯¯¯·Mr(f)¢¸2
n;
kW(p)
n(f;¸n)¡f(p)kr»¸n;
the last equivalence holding if fis not a polynomial of degree ·p2Nand
the constants in the equivalence depend on f,r,p.
Above, Cr;M;A = 6MP1
k=2(k+ 1)( k¡1)(rA)k<1,p2N,1·r <
r1<1
A,Mr(f) = 16 M¢P1
k=3(k¡1)(k¡2)(rA)k<1andkfkr=
maxfjf(z)j;jzj ·rg.
(ii) If R <+1, then the complex approximation operator
W¤
n(f;¸n)(z) =1X
k=0ck¢Wn(ek;¸n)(z); z2Dr;
is well-de¯ned and W¤
n(f;¸n)(z)satis¯es all the estimates from the point
(i), for all 1·r <1
A< R.
Remark. Due to the results in the real case in [45] and to those in
the complex case in Theorem 4.1.3, we can say that they seem to be of
de¯nitive type, in the sense that exhibit Baskakov type operators which can
approximate the functions with an arbitrary chosen order.

42 CH. 4. COMPLEX SZ ¶ASZ AND BASKAKOV OPERATORS
4.2 Arbitrary order by Baskakov-Faber op-
erators
By using a sequence ¸n>0,n2Nwith the property that ¸n!0 as fast we
want, in this paper we obtain the approximation order O(¸n) for a general-
ized Baskakov-Faber operator attached to analytic functions of exponential
growth in a continuum G½C. Several concrete examples of continuums G
are given for which this operator can explicitly be constructed.
In this way, the results obtained in the previous section for compact
disks, are generalized to the case when the disk is replaced by a compact
set inC.
4.2.1 Introduction
According to the considerations in Subsection 4.1.1, denoting
Wn(f)(z) =1X
j=0µ
1 +1
n¶
¢:::¢µ
1 +j¡1
n¶
¢[0;1=n; :::; j=n ;f]zj;
for analytic functions satisfying some exponential-type growth condition,
quantitative estimates of order O¡1
n¢
in approximation by Wn(f)(z) in
compact disks with center at origin were obtained in [35], Section 1.9, pp.
124-138. For f(z) =P1
k=0akzk, all the quantitative results are based on
the formula Wn(f)(z) =P1
k=0ak¢Wn(ek)(z), with ek(z) =zk, i.e. by using
(4.3) too,
Wn(f)(z) =1X
k=0ak¢kX
j=0µ
1 +1
n¶
¢:::¢µ
1 +j¡1
n¶
¢[0;1=n; :::; j=n ;ek]zj:
(4.4)
Also, it is worth noting that similar quantitative estimates in approximation
by other complex operators can be found in, e.g., the books [35], [36], [50]

4.2. ARBITRARY ORDER BY BASKAKOV-FABER OPERATORS 43
and in the papers [15], [38], [40]-[49], [57]-[59].
By using a sequence of real positive numbers, ( ¸n)n2N, with the proper-
ties that ¸n!0 as fast we want, suggested by the formula (4.4) too, the aim
of this note is to generalize the approximation by the operators Wn(f)(z),
to the approximation by the so-called by us generalized Baskakov-Faber
operators attached to analytic functions of some exponential growth in a
continuum in C, obtaining the approximation order O(¸n).
Since ¸n!0, obviously that without to loose the generality, everywhere
in the paper we may suppose that 0 < ¸ n·1
2, for all n2N.
4.2.2 Preliminaries
Firstly, we brie°y recall some basic concepts on Faber polynomials and
Faber expansions.
ForG½Ca compact set such that ~CnGis connected, let A(G) be the
Banach space of all functions that are continuous on Gand analytic in the
interior of G, endowed with the norm kfkG= supfjf(z)j;z2Gg. Denoting
Dr=fz2C;jzj< rg, according to the Riemann Mapping Theorem,
there exists a unique conformal mapping ă of ~CnD1onto ~CnGsuch that
ă(1) =1and ă0(1)>0. Then, to Gone may attach the polynomial
of exact degree n,Fn(z), called Faber polynomial , de¯ned byă0(w)
ă(w)¡z=
P1
n=0Fn(z)
wn+1; z2G;jwj>1.
Iff2A(G) then an(f) =1
2¼iR
juj=1f(ă(u))
un+1du=1
2¼R¼
¡¼f(ă(eit))e¡intdt; n2
N[f0gare called the Faber coe±cients of fandP1
n=0an(f)Fn(z) is called
the Faber expansion (series) attached to fonG. It is worth noting that the
Faber series represent a natural generalization of the Taylor series, when
the unit disk is replaced by an arbitrary simply connected domain bounded
by a "nice" curve.

44 CH. 4. COMPLEX SZ ¶ASZ AND BASKAKOV OPERATORS
Detailed properties of Faber polynomials and Faber expansions can be
found in e.g. [31], [64].
LetGbe a connected compact subset in C(that is a continuum) and
suppose that fis analytic on G, that is there exists R > 1 such that
fis analytic in GR, given by f(z) =P1
k=0ak(f)Fk(z),z2GR. Recall
here that GRdenotes the interior of the closed level curve ¡ Rgiven by
¡R=fă(w);jwj=Rg(and that G½Grfor all 1 < r < R ).
Suggested by the formula (4.4), we can introduce the following.
De¯nition 4.2.1. The generalized Baskakov-Faber operators attached
toGandfis de¯ned by Wn(f;¸n; G;z) =P1
k=0ak(f)¢Wn(ek;¸n; G;z),
i.e.,
Wn(f;¸n; G;z)
=1X
k=0ak(f)¢kX
j=0(1 +¸n)¢:::¢(1 + ( j¡1)¸n)¢[0; ¸n; :::; j¸ n;ek]¢Fj(z);(4.5)
where for j= 0 and j= 1, by convention (1 + ¸n)¢:::¢(1 + ( j¡1)¸n) = 1.
Remark. For¸n= 1=n,n2NandG=D1, since Fj(z) =zj, the
above generalized Baskakov-Faber operators reduce to the classical complex
Baskakov operators, introduced and studied in [35], Section 1.9.
4.2.3 Main results
For the proof of the main result, we need two lemmas, as follows.
Lemma 4.2.2. Let0< ¸ n·1
2<1,n2N, be with ¸n!0. For all
k; n2Nwithk·[1=¸n](here [a]denotes the integer part of a) we have the
inequality
Ek;n:=k¡1X
j=0(1 +¸n)¢:::¢(1 + ( j¡1)¸n)¢[0; ¸n; :::; j¸ n;ek]·¸n¢(k+ 3)!:

4.2. ARBITRARY ORDER BY BASKAKOV-FABER OPERATORS 45
Here, by convention, for j= 0andj= 1we take (1+¸n)¢:::¢(1+(j¡1)¸n) =
1:
Also, we can prove the following.
Lemma 4.2.3. Let0< ¸ n·1
2,n2N, be with ¸n!0. For all k¸0
andn2N, we have
Gk;n:=kX
j=0(1 +¸n)¢:::¢(1 + ( j¡1)¸n)¢[0; ¸n; :::; j¸ n;ek]·(k+ 1)!:
The main result is the following.
Theorem 4.2.4. Letfbe analytic on the continuum G, that is there ex-
istsR >1such that fis analytic in GR, given by f(z) =P1
k=0ak(f)Fk(z),
z2GR. Also, suppose that there exist M > 0andA2¡1
R;1¢
, with
jak(f)j ·MAk
k!, for all k= 0;1; :::;(which implies jf(z)j ·C(r)MeArfor
allz2Gr,1< r < R ).
Let1< r <1
Abe arbitrary ¯xed. Then, there exist an index n02Nand
a constant C(r; f)>0depending on randfonly, such that for all z2Gr
andn¸n0we have
jWn(f;¸n; G;z)¡f(z)j ·C(r; f)¢¸n:
Remarks. 1) Theorem 4.2.4 generalizes Theorem 1.9.1, p. 126 in [35],
in two senses : ¯rstly, it is extended from compact disks with center at
origin to compact sets and secondly, the order of approximation O¡1
n¢
is
essentially improved to the order O(¸n), with ¸n!0 as fast we want.
2) It is clear that Theorem 4.2.4 holds under the more general hypothesis
jak(f)j ·Pm(k)¢Ak
k!, for all k¸0, where Pmis an algebraic polynomial of
degree mwith Pm(k)>0 for all k¸0.
3) There are many concrete examples for Gwhen the conformal mapping
ă and the Faber polynomials associated to G, and consequently when the

46 CH. 4. COMPLEX SZ ¶ASZ AND BASKAKOV OPERATORS
Baskakov-Faber operators too, can explicitly be written (see, e.g., [36], pp.
81-83, or [37]), as follows : G= [¡1;1],Gis the continuum bounded by
them-cusped hypocycloid, Gis the regular m-star ( m= 2;3; :::;),Gis the
m-leafed symmetric lemniscate, m= 2;3; :::;,Gis a semidisk, or Gis a
circular lune.

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