Let M ¢ 2 be a fixed positive integer. A family of closed subspaces Vj , j √ Z , of L 2 , the space of all square integrable functions on the real line, is said to be a multiresolution of L 2 if the following conditions hold: ( i) Vj , Vj/1 , and f √ Vj if and only if f ( Mr) √ Vj/1 for all j √ Z ; ( ii ) < j√Z Vj is dense in L 2 and > j√Z Vj Å M; ( iii ) there exists a function f in V0 such th...

Frames consisting of translates of a single function play an important role in sampling theory as well as in wavelet theory and Gabor analysis. We give a necessary and sufficient condition for a subfamily of regularly spaced translates of a function φ ∈ L(R), (τnbφ)n∈Λ, Λ ⊂ Z, to form a frame (resp. Riesz basis) for its closed linear span. One consequence is that if Λ ⊂ N , then this family is ...

A theorem of Luxemburg and Zaanen on normed Riesz spaces (Theorem 2.4 below) and one of Nakano (Theorem 2.3 below) have been extended by the author in [1] to metrizable locally solid linear topological Riesz spaces. This note gives an example which shows they cannot be further extended to nonmetrizable Hausdorff locally solid linear topological Riesz spaces. 1. Notation and basic concepts. For ...

We study the optimal approximation of the solution of an operator equation A(u) = f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs that are given by an isomorphism A :...

For Gabor sets, (g; a, b), it is known that (g; a, b) is a frame if and only if (g; 1/b, 1/a) is a Riesz basis for its span. In particular, for every g there is a0 such that for every a < a0, there is a bm = bm(a) > 0 so that for every b < bm, (g; a, b) is a frame, and (g; 1/b, 1/a) is a Riesz basis sequence. In this talk we shall consider a similar problem for wavelet sets (Ψ; a, b). The main ...

Image quality assessment is an important topic in the field of digital image processing. In this study, a full-reference image quality assessment method called Riesz transform and Visual contrast sensitivity-based feature SIMilarity index (RVSIM) is proposed. More precisely, a Log-Gabor filter is first used to decompose reference and distorted images, and Riesz transform is performed on the dec...