Computes Methods to Integral Equations and Their Integral Transforms

References

89:69-128

101-117

161-169

248 p

42

[1] Dôku I. Star-product functional and unbiased estimator of solutions to nonlinear integral equations. Far East Journal of Mathematical Sciences. 2014;

Recent Advances in Integral Equations

[11] Le Gall J-F. Random trees and applications. Probability Surveys. 2005;

[12] Drmota M. Random Trees. Wien:

[13] Dynkin EB. Markov Processes. Vol. 1. Berlin: Springer-Verlag; 1965. 380 p

Springer-Verlag; 2009. 458 p

2:245-311

[2] Dôku I. On a limit theorem for environment-dependent models. Institute of Statistical Mathematics Research Reports. 2016;352:103-111

[3] Dôku I. A recursive inequality of empirical measures associated with EDM. Journal of Saitama University. Faculty of Education (Mathematics for Natural Science). 2016;65(2):253-259

[4] Dôku I. A support problem for superprocesses in terms of random measure. RIMS Kôkyûroku (Kyoto University). 2017;2030:108-115

[5] Dôku I. Exponential moments of solutions for nonlinear equations with catalytic noise and large deviation. Acta Applicandae Mathematicae. 2000;63:

[6] Dôku I. Removability of exceptional sets on the boundary for solutions to some nonlinear equations. Scientiae Mathematicae Japonicae. 2001;54:

[7] Le Jan Y, Sznitman AS. Stochastic cascades and 3-dimensional Navier-Stokes equations. Probability Theory and Related Fields. 1997;109:343-366

[8] Kallenberg O. Foundations of Modern Probability. 2nd ed. New York:

[9] Harris TE. The Theory of Branching Processes. Berlin: Springer-Verlag; 1963.

[10] Aldous D. Tree-based models for random distribution of mass. Journal of Statistical Physics. 1993;73:625-641

Springer; 2002. 638 p

Chapter 4

Weidong Chen

Abstract

Computation of Two-Dimensional

The computation of the two-dimensional Fourier transform by the sampling points creates an ill-posed problem. In this chapter, we will cover this problem for the band-limited signals in the noisy case. We will present a regularized algorithm based on the two-dimensional Shannon Sampling Theorem, the two-dimensional Fourier series, and the regularization method. First, we prove the convergence property of the regularized solution according to the maximum norm. Then an error

regularized Fourier series is given in theory, and some examples are given to compare the numerical results of the regularized Fourier series with the numerical

Keywords: Fourier transform, band-limited signal, ill-posedness, regularization

The two-dimensional Fourier transform is widely applied in many fields [1–9]. In this chapter, the ill-posedness of the problem for computing two-dimensional Fourier transform is analyzed on a pair of spaces by the theory and examples in detail. A two-dimensional regularized Fourier series is presented with the proof of

Definition. For two positive Ω1, Ω<sup>2</sup> ∈ R, a function f ∈L<sup>2</sup> R<sup>2</sup> � � is said to be band-

f tð Þ <sup>1</sup>; t<sup>2</sup> e

\½ �� � �Ω1; Ω<sup>1</sup> ½ � Ω2; Ω<sup>2</sup> :

it1ω1þit2ω2dt1dt2,ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>

: (1)

AMS subject classifications: 65T40, 65R20, 65R30, 65R32

the convergence property and some experimental results.

^fð Þ¼ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> <sup>0</sup>, <sup>∀</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>

ð<sup>∞</sup> �∞ ð<sup>∞</sup> �∞

We will consider the problem of computing ^fð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> from f tð Þ <sup>1</sup>; <sup>t</sup><sup>2</sup> .

First, we describe the band-limited signals.

Here ^f is the Fourier transform of:

F fð Þð Þ¼ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> ^fð Þ¼ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup>


Fourier Transforms for Noisy

Band-Limited Signals

estimation is given according to the L<sup>2</sup>

results of the Fourier series.

1. Introduction

limited if

45
