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Signal reconstruction from given phase of the Fourier transform using Fejér monotone methods

100%

Schott D.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2000

tom 20

nr 1

27-40

The aim is to reconstruct a signal function x ∈ L₂ if the phase of the Fourier transform [x̂] and some additional a-priori information of convex type are known. The problem can be described as a convex feasibility problem. We solve this problem by different Fejér monotone iterative methods comparing the results and discussing the choice of relaxation parameters. Since the a-priori information is partly related to the spectral space the Fourier transform and its inverse have to be applied in each iterative step numerically realized by FFT techniques. The computation uses MATLAB routines.

Riesz means of Fourier transforms and Fourier series on Hardy spaces

86%

Weisz F.

Studia Mathematica

1998

tom 131

nr 3

253-270

Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from $H_p(ℝ)$ to $L_p(ℝ)$ (1/(α+1) < p < ∞) and is of weak type (1,1), where $H_p(ℝ)$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function $⨍ ∈ L_1(ℝ)$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on $H_p(ℝ)$ whenever 1/(α+1) < p < ∞. Thus, in case $⨍ ∈ H_p(ℝ)$, the Riesz means converge to ⨍ in $H_p(ℝ)$ norm (1/(α+1) < p < ∞). The same results are proved for the conjugate Riesz means and for Fourier series of distributions.

Ograniczanie wyników

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Studia Mathematica

1 Schott D.

1 Weisz F.

1 2000

1 1998

Wyniki wyszukiwania

Signal reconstruction from given phase of the Fourier transform using Fejér monotone methods

Riesz means of Fourier transforms and Fourier series on Hardy spaces