FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 3, PAGES 205-226

**Recursive expansions with respect to a chain of subspaces**

A. V. Slovesnov

Abstract

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In this work, recursive expansions in Hilbert space $H\; =\; L$_{2}[0,1] are
considered.
We discuss a related notion of frames in finite-dimensional
spaces.
We also suggest a constructive approach to extend an arbitrary
basis to obtain a tight frame.
The algorithm of extending is applied to bases of a special form,
whose Gram matrix is circulant.
A construction of a chain of nested subspaces $\{Vn\}$_{n=1}^{¥} is given, and
in its foundation lies an example of a function that can be
expressed as a linear combination of its contractions and
translations.
The main result of the paper is the theorem that provides the uniform
convergence of recursive Fourier series with respect to the chain
$\{Vn\}$_{n=1}^{¥} for continuous
functions.

Location: http://mech.math.msu.su/~fpm/eng/k10/k103/k10311h.htm

Last modified: March 24, 2011