Greedy Expansions with Prescribed Coefficients in Hilbert Spaces

Artur R Valiullin,Vladimir V Galatenko,Albert R Valiullin

doi:10.1155/2018/4867091

Artur R Valiullin, Vladimir V Galatenko + Show 1 more

Open Access

https://doi.org/10.1155/2018/4867091

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Abstract

Greedy expansions with prescribed coefficients, which have been studied by V. N. Temlyakov 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 C. Furthermore, we show that the condition sufficient for the convergence, namely, the inclusion C∈l2∖l1, can not be relaxed at least in the power scale. At the same time, in finite-dimensional spaces, the condition C∈l2 can be replaced by convergence of C to zero.

Highlights

Expansion in Fourier series [1] is a classical and comprehensively studied tool of theoretical and applied mathematics which takes an expanded function as an input and constructs a sequence of its expansion coefficients
Greedy expansions [2, 3], which are equivalent in the simplest case to Fourier series reordered by decreasing norms of terms and known in statistics and signal processing as Projection Pursuit Regression [4, 5] and Matching Pursuit [6], respectively, perform parallel computation of expansion coefficients and selection of expansion elements from a predefined dictionary
We start with a positive result which states that in Hilbert spaces the monotonicity is not required for the standard convergence

Summary

Introduction

Expansion in Fourier series [1] is a classical and comprehensively studied tool of theoretical and applied mathematics which takes an expanded function as an input and constructs a sequence of its expansion coefficients. Set The rsner=iersn−∑1∞ n−=1ccnneenn.(f) is called a greedy expansion of f in the dictionary D with the prescribed coefficients C and the weakness parameter t. It immediately follows from the definition of a greedy expansion that rN = f − ∑Nn=1 cnen(f) (N ∈ N), and the convergence of the expansion to an expanded element is equivalent to the convergence of remainders rn to zero as n 󳨀→ ∞. As a selection of an expanding element en is potentially not unique, there may exist different realizations of a greedy expansion for a given dictionary D, weakness parameter t and sequence of coefficients C. It remained unknown whether the condition C ∈ l2 and the monotonicity condition could be essentially relaxed without violating the guaranteed convergence to an expanded element

Main Results

Proof of Theorem 2

Proof of Theorem 3

The Case of Finite-Dimensional Spaces

Conclusion