Sharp conditions for the convergence of greedy expansions with prescribed coefficients - Valiullin Artur R. (De Gruyter, 2021)
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 cnn=1∞{\left\{{c}_{n}\right\}}_{n=1}^{∞ } 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∞cn=∞{\sum }_{n=1}^{∞ }{c}_{n}=∞ and ∑n=1∞cn2<∞{\sum }_{n=1}^{∞ }{c}_{n}^{2}\lt ∞ . In this paper, we study a possibility of relaxing the latter condition. More specifically, we show that the convergence is guaranteed for cn=o1nc_{n}=o\left(\frac{1}{\sqrt{n}}\right), but can be violated if cn≍1nc_{n}\hspace{0.33em}\asymp \hspace{0.33em}\frac{1}{\sqrt{n}}.