Sharp conditions for the convergence of greedy expansions with prescribed coefficients

Abstract

Abstract Greedy expansions with prescribed coefficients were introduced by V. N. Temlyakov in a general case of Banach spaces. In contrast to Fourier series expansions, in greedy expansions with prescribed coefficients, a sequence of coefficients { c n } n = 1 ∞ {\left\{{c}_{n}\right\}}_{n=1}^{\infty } is fixed in advance and does not depend on an expanded element. During the expansion, only expanding elements are constructed (or, more precisely, selected from a predefined set – a dictionary). For symmetric dictionaries, V. N. Temlyakov obtained conditions on a sequence of coefficients sufficient for a convergence of a greedy expansion with these coefficients to an expanded element. In case of a Hilbert space these conditions take the form ∑ n = 1 ∞ c n = ∞ {\sum }_{n=1}^{\infty }{c}_{n}=\infty and ∑ n = 1 ∞ c n 2 < ∞ {\sum }_{n=1}^{\infty }{c}_{n}^{2}\lt \infty . In this paper, we study a possibility of relaxing the latter condition. More specifically, we show that the convergence is guaranteed for c n = o 1 n {c}_{n}=o\left(\frac{1}{\sqrt{n}}\right) , but can be violated if c n ≍ 1 n {c}_{n}\hspace{0.33em}\asymp \hspace{0.33em}\frac{1}{\sqrt{n}} .