Trigonometric wavelets for Hermite interpolation

Abstract

Abstract. The aim of this paper is to investigate a multiresolution analysis of nested subspaces of trigonometric polynomials. The pair of scaling functions which span the sample spaces are fundamental functions for Hermite interpolation on a dyadic partition of nodes on the interval [0, 2π). Two wavelet functions that generate the corresponding orthogonal complementary subspaces are constructed so as to possess the same fundamental interpolatory properties as the scaling functions. Together with the corresponding dual functions, these interpolatory properties of the scaling functions and wavelets are used to formulate the specific decomposition and reconstruction sequences. Consequently, this trigonometric multiresolution analysis allows a completely explicit algorithmic treatment.

Numerical Solution of Green’s Function for Solving Inhomogeneous Boundary Value Problems with Trigonometric Functions by New Technique

“…In this section, we will give a brief introduction of Quak's work on the construction of Hermite interpolatory trigonometric wavelets and their basic properties (see [12]). For all n ∈ » , the Dirichlet kernel …”

“…and their derivations are given by ( ) [12].) For 0 j ∈ » , The following interpolatory properties hold for each…”

“…Nowadays, the trigonometric interpolant wavelet has arisen in the approximation of operators [9]- [11]. Quack [12] has constructed a multiresolution analysis (MRA) of nested subspace of trigonometric Hermite polynomials. The trigonometric Hermite interpolation enables a completely explicit description of the corresponding decomposition and reconstruction coefficients by means of some circular matrices.…”

Numerical solution ofnth-order integro-differential equations using trigonometric wavelets

“…In this section, we give a brief introduction of Quak's work on the construction of Hermite interpolatory trigonometric wavelets and their basic properties [24]. Throughout this study, we denote by L 2 2 the set of 2 -periodic square-integrable functions f as…”

Algorithms for Trigonometric Wavelet Packets

“…The first approach to develop a trigonometric multiresolution analysis of nested spaces of trigonometric polynomials was introduced by Chui & Mhaskar [2], with their investigations being based on quasi-interpolants. Various aspects of MRA's such as decomposition and reconstruction algorithms, dual and orthogonal bases, localization properties, etc., derived from Privalov's Lagrange interpolants, were studied by Prestin, Privalov, Quak, and Selig in [19][20][21][22][23][24]26].…”