Approximation and spectral properties of periodic spline operators

Abstract

We consider discrete convolution operators tij*' whose range is the fc-dimensional space ^ spanned by the translates of a single function. Examples of Sf k include the space of trigonometric polynomials, periodic polynomial splines and trigonometric splines. The eigenfunctions of these operators corresponding to the nonzero eigenvalues are independent of a, and they form an orthogonal basis for £f k . The limiting behaviour of t k a) as a,k-*co, is also considered. The corresponding limiting semigroups are computed explicitly.

L 2-approximation by the translates of a function and related attenuation factors

“…Furthermore ( [7,8]), k-1 {Pj}j=O forms an orthogonal basis for ~ with respect to the inner product where 2m + 1 = k (see [1,12]), then 5~k is the space of trigonometric polynomials of degree m, and [10,13]) and trigonometric splines (see [14,15]) which will be considered in Sect. 6 and Sect.…”

“…Some of these results are contained in [7] and [8], but short proofs are given here for completeness. Throughout this paper we shall assume that pj4=O,j=O, 1 ..... k-1, so that 5{k is of dimension k. For f, geC2,, let…”

Mean square approximation by optimal periodic interpolation

“…Babuska [1] introduced the concept of the periodic Hilbert space for studying optimal quadrature formulas, Prager [8] continued these investigations and related these problems to the minimum norm interpolation (optimal periodic interpolation) in periodic Hilbert spaces. These ideas have been further developed in a number of papers [2,3,4,5,6,7]. In this paper we will study the approximation power of optimal periodic interpolation in the mean square norm an thereby extended results of [4].…”