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}} .

中文翻译：

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具有规定系数的贪婪展开收敛的尖锐条件

在一般的Banach空间中，VN Temlyakov引入了具有规定系数的贪婪展开式。与傅立叶级数展开相反，在具有规定系数的贪婪展开中，系数序列{cn} n = 1∞{\ left \ {{c} _ {n} \ right \}} _ {n = 1} ^ { \ infty}是预先固定的，并且不依赖于扩展元素。在扩展过程中，仅会构建扩展元素（或更准确地说，是从预定义的集合中选择-字典）。对于对称字典，VN Temlyakov在一系列系数上获得了条件，这些条件足以使贪婪展开与这些系数收敛为展开元素。在希尔伯特空间的情况下，这些条件的形式为∑ n = 1∞cn =∞{\ sum} _ {n = 1} ^ {\ infty} {c} _ {n} = \ infty和∑ n = 1∞ cn 2 < ∞{\ sum} _ {n = 1} ^ {\ infty} {c} _ {n} ^ {2} \ lt \ infty。在本文中，我们研究了放松后一种情况的可能性。更具体地说，我们证明了对于cn = o 1 n {c} _ {n} = o \ left（\ frac {1} {\ sqrt {n}} \ right）可以保证收敛，但是如果cn ≍1 n {c} _ {n} \ hspace {0.33em} \ asymp \ hspace {0.33em} \ frac {1} {\ sqrt {n}}。