FPM 2010, vol. 16, no. 3, pp. 205-226

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 ${V$ ⁿ}_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 ${V$ ⁿ}_n=1^¥ for continuous functions.

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