Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm

Xu Xu, Jinyu Guo, Peixin Ye and Wenhui Zhang ()
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Xu Xu: School of Science, China University of Geosciences, Beijing 100083, China
Jinyu Guo: School of Mathematics and LPMC, Nankai University, Tianjin 300071, China
Peixin Ye: School of Mathematics and LPMC, Nankai University, Tianjin 300071, China
Wenhui Zhang: School of Mathematics and LPMC, Nankai University, Tianjin 300071, China

Mathematics, 2023, vol. 11, issue 9, 1-23

Abstract: We first study the error performances of the Vector Weak Rescaled Pure Greedy Algorithm for simultaneous approximation with respect to a dictionary D in a Hilbert space. We show that the convergence rate of the Vector Weak Rescaled Pure Greedy Algorithm on A 1 ( D ) and the closure of the convex hull of the dictionary D is optimal. The Vector Weak Rescaled Pure Greedy Algorithm has some advantages. It has a weaker convergence condition and a better convergence rate than the Vector Weak Pure Greedy Algorithm and is simpler than the Vector Weak Orthogonal Greedy Algorithm. Then, we design a Vector Weak Rescaled Pure Greedy Algorithm in a uniformly smooth Banach space setting. We obtain the convergence properties and error bound of the Vector Weak Rescaled Pure Greedy Algorithm in this case. The results show that the convergence rate of the VWRPGA on A 1 ( D ) is sharp. Similarly, the Vector Weak Rescaled Pure Greedy Algorithm is simpler than the Vector Weak Chebyshev Greedy Algorithm and the Vector Weak Relaxed Greedy Algorithm.

Keywords: greedy algorithm; vector approximation; Hilbert spaces; modulus of smoothness; uniformly smooth Banach spaces; convergence; error bound (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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