Greedy expansions with prescribed coefficients in Hilbert spaces (Q1728990)

Summary: Greedy expansions with prescribed coefficients, which have been studied by \textit{V. N. Temlyakov} [Adv. Comput. Math. 26, No. 4, 431--449 (2007; Zbl 1119.41034); J. Fourier Anal. Appl. 13, No. 1, 71--86 (2007; Zbl 1123.41019)] in Banach spaces, are considered here in a narrower case of Hilbert spaces. We show that in this case the positive result on the convergence does not require monotonicity of coefficient sequence \(\mathcal{C}\). Furthermore, we show that the condition sufficient for the convergence, namely, the inclusion \(\mathcal{C} \in \ell^2 \backslash \ell^1\), can not be relaxed at least in the power scale. At the same time, in finite-dimensional spaces, the condition \(\mathcal{C} \in \ell^2\) can be replaced by convergence of \(\mathcal{C}\) to zero.