Hilbert空间中特殊字典类的规定系数贪婪展开
Greedy expansions with prescribed coefficients in Hilbert spaces for special classes of dictionaries
AI总结：
本文研究Hilbert空间中规定系数贪婪展开的收敛条件能否在特殊空间或字典类中减弱，证明有限维空间及无穷维空间中与正交规范序列相关的某些字典类可以满足更弱的条件。
摘要：
规定系数的贪婪展开由V. N. Temlyakov在Banach空间框架中引入。其思想是选取一列固定的（实）系数$\{c_n\}_{n=1}^\infty$以及Banach空间的一个固定元素集（字典）；然后，在系数和字典满足适当条件时，可以将Banach空间中的所有元素展开为仅包含固定系数和字典元素的级数。在Hilbert空间中，规定系数的贪婪算法的收敛性已被刻画，即存在关于系数的充分必要条件，使得该算法对所有字典都收敛。本文关注的问题是：对于特定类别的空间或字典，这些条件是否可以减弱；我们证明对于有限维空间，以及无穷维空间中与正交规范序列相关的某些字典类，情况确实如此。
英文摘要：
Greedy expansions with prescribed coefficients have been introduced by V. N. Temlyakov in the frame of Banach spaces. The idea is to choose a sequence of fixed (real) coefficients $\{c_n\}_{n=1}^\infty$ and a fixed set of elements (dictionary) of the Banach space; then, under suitable conditions on the coefficients and the dictionary, it is possible to expand all the elements of the Banach space in series that contain only the fixed coefficients and the elements of the dictionary. In Hilbert spaces the convergence of greedy algorithm with prescribed coefficients is characterized, in the sense that there are necessary and sufficient conditions on the coefficients in order that the algorithm is convergent for all the dictionaries. This paper is concerned with the question if such conditions can be weakened for particular classes of spaces or dictionaries; we prove that this is the case for finite dimensional spaces, and for some classes of dictionaries related to orthonormal sequences in infinite dimensional spaces.
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