Frank Zeilfelder: [616070]
Frank Zeilfelder:
Interpolation and Best Approximation for Periodic Spline Func-
tions
[Mai 1996, Supervisor: G. Walz, Universitaet Mannheim, Germany]
e-mail: zeilfeld @fourier.math.uni-mannheim.de
Abstract: We investigate interpolation and best approximation by spaces
of periodic splines of degree m. It is well known, that periodic spline spaces
are weak chebychev if and only if their dimension Nis odd. Only in this case
interpolation can be characterized by a condition of Schoenberg-Whitney
type. In this thesis periodic spline spaces of even dimension are considered.
We present a general method of constructing a large class of interpo-
lation sets for arbitrary knot sets by using a special basis and results on
non-interpolation sets which satisfy the Schoenberg-Whitney condition. For
equidistant knot sets the following result is known: a set T¸such that every
point of T¸lies between two knots, with the uniform shift ¸, is an interpo-
lation set if and only if ( mis odd and ¸6=1
2) or ( mis even and ¸6= 1).
An application of our method shows that for even degree man analogous
result holds for arbitrary knot sets. On the other hand, we show that this
characterization is not true for odd degree. Based on eigenvalue arguements,
we give a large class of knot sets for which the characterization holds.
Then we analyze Hermite-interpolation for spaces of periodic splines
with defect and even dimension. First we give some results for arbitra-
ry knot sets. Then for equidistant knots, we derive a general criterion to
characterize the shifting factors ¸such that T¸admits unique Hermite-
interpolation. This is done by using eigenvalue arguements and the theo-
ry of oscillatory matrices. As an application of this method we determine
the non-interpolation shifting factors ¸of periodic splines of smoothness
q;q= 0;1;2. Finally for splines with defect up to degree seven, we com-
pute all non-interpolation shifting factors.
In the last part of the dissertation, we consider best approximation by
periodic splines in the uniform norm. We give a complete characterization
of strongly unique best approximation. For the case when the space is weak
chebyshev strong unicity can be characterized by alternation properties of
the error function alone. In the other case, strong unicity depends on these
alternation properties and on additional interpolation conditions.
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