Interpolation on uniform meshes by the translates of one function and related attenuation factors

Abstract

Abstract. The exact Fourier coefficients c,(PM/) are proportional to the discrete Fourier coefficients dj-"\f) if Pn is a translation invariant operator which depends only on the values of/on an equidistant mesh of width 2it/n. The proportionality factors which depend only on P" but not on / are called attenuation factors and have been calculated for several operators P" of spline type. Here we analyze first the interpolation problem which is produced by the functions <j(--lirj/n),j•» 0,..., n -1, where a is a suitable 2»r-periodic generating function. It is essential that the associated interpolation matrix is of discrete convolution type. Thus, we can derive conditions guaranteeing the unique solvability of the interpolation problem and representations of the interpolating function. Then the attenuation factors may be expressed in terms of the Fourier coefficients of a. We point especially to the case where o is a reproducing kernel in a suitable Hilbert space. Here we get attenuation factors of a new type which are generated by interpolation with analytic functions.

Approximation by optimal periodic interpolation

“…It is easily seen that 5R(S") is a space of periodic monosplines of degree 2r [2,5]. Then an application of Proposition 4.2 yields the asymptotic error estimate (5.6) ||/ -S B (/)IL = 0(e-bm ) (n = 2m + 1 -o>) .…”

“…It is easily seen that $l(S n ) is translation invariant with respect to t v Thus, the method of interpolation by translation is applicable [2,3,5]. It uses the discrete Fourier transform method.…”

“…Prager [6] continued these investigations and discovered the relationship between optimal approximations of linear functional on periodic Hilbert spaces and in minimum norm interpolation (optimal periodic interpolation). Recently we have shown that optimal periodic interpolation is closely related to interpolation by translation [2,3,5]. In this paper we will study the approximation power of optimal periodic interpolation in the uniform norm.…”

Mean square approximation by optimal periodic interpolation

“…See Locher [7], Cheney [2]. If these conditions are violated for some k, a modified approach is possible.…”

“…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].…”