Asymptotic properties of Gabor frame operators as sampling density tends to infinity

Abstract

We study the asymptotic properties of Gabor frame operators defined by the Riemannian sums of inverse windowed Fourier transforms. When the analysis and the synthesis window functions are the same, we give necessary and sufficient conditions for the Riemannian sums to be convergent as the sampling density tends to infinity. Moreover, we show that Gabor frame operators converge to the identity operator in operator norm whenever they are generated with locally Riemann integrable window functions in the Wiener space.

Behavior of Gabor frame operators on Wiener amalgam spaces

“…Since g · γ is locally Riemann integrable, we see from Also as p = q we have W (L p , ℓ p ) = L p (R d ), so this result extends Sun's ( [16]) result.…”

Convergence of Riemannian sums of inverse wavelet transforms

“…In fact, although the theory of wavelet frames has been developed very fast in the past twenty years, e.g., see [1-4, 8, 11, 16-20] for an overview, it is still a hard work to find the dual frame for a given wavelet frame [5,6,14]. Motivated by [9,10,12,13,22,23], where the convergence of Riemannian sums of the inverse windowed Fourier transform was studied, the authors [15] studied the approximation of f with the Riemannian sums defined by (5). It was shown that under certain conditions, S a,b;ψ1,ψ2 f converges to f as (a, b) tends to (1,0).…”

Frame sequences and dual frames for operators

“…Owing to the redundancy and flexibility, frames have applications such as in wireless communication 1 , Σ∆ quantization 2 , sampling theory 3 , and image processing 4 . For details and background on frames see Refs.…”