A classification of continuous wavelet transforms in dimension three

This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups $H<\mathrm{GL}(3,\mathbb{R})$ that give rise to a continuous wavelet transform with associated irreducible quasi-regular representation. For each group in this class, coorbit theory allows to consistently define spaces of sparse signals, and to construct atomic decompositions converging simultaneously in a whole range of these spaces. As an application of the classification, we investigate the existence of compactly supported admissible vectors and atoms for the groups.     

On the Smith normal form of a skew‐symmetric d‐optimal design of order

We show that the Smith normal form of a skew‐symmetric D ‐optimal design of order is determined by its order. Furthermore, we show that the Smith normal form of such a design can be written explicitly in terms of the order , thereby proving a recent conjecture of Armario. We apply our result to show that certain D ‐optimal designs of order are not equivalent to any skew‐symmetric D ‐optimal design. We also provide a correction to a result in the literature on the Smith normal form of D ‐optimal designs.     

Norms and lower bounds of some matrix operators on Fibonacci weighted difference sequence space

Norm of an operator T:X→Y is the best possible value of U satisfying the inequality and lower bound for T is the value of L satisfying the inequality where ‖.‖_X and ‖.‖_Y are the norms on the spaces X and Y, respectively. The main goal of this paper is to compute norms and lower bounds for some matrix operators from the weighted sequence space ?_p(w) into a new space called as Fibonacci weighted difference sequence space. For this purpose, we firstly introduce the Fibonacci difference matrix and the space consisting of sequences whose ‐transforms are in .     

Summability factor relations between absolute weighted and Cesàro means

By , we denote the set of all sequences such that Σ?_na_n is summable V whenever Σa_n is summable U, where U and V are two summability methods. Recently, Sar?göl has characterized the set for k > 1,α > ?1 and arbitrary positive sequences Now, in the present paper, we characterize the sets , k > 1 and , k ≥ 1 for arbitrary positive sequences Hence we extend these results to the range α≥ ? 1. In this way, some open problems in this topic are also completed.     

An optimal perturbation bound

Cai and Zhang establish separate perturbation bounds for distances with spectral and Frobenius norms (Cai T, Zhang A. Rate‐optimal perturbation bounds for singular subspaces with applications to high‐dimensional statistics. The Annals of Statistics. 2018; Vol. 46, No. 1: 60?89). We extend their theorem to each unitarily invariant norm. It turns out that our estimation is optimal as well.