A stability estimate for convolution equation

https://doi.org/10.1016/S0096-3003(01)00140-0Get rights and content

Abstract

In this study we obtain an estimate for the solution of a convolution equation. These types of equations arise in the integral equations theory [Linear Integral Equation, Applied Mathematical Sciences, Springer, Berlin, 1989].

Introduction

Many problems of mathematical physics and technology lead to the consideration of an equation of the formAx=y,where A is a (not necessarily linear) continuous operator acting from a subset X of a Banach space into a subset Y of another Banach space and xX is to be found for a given yY [1], [2]. Such problems, to determine x from indirect measurements of y, are called inverse problems [3]. Let us consider Eq. (1.1) (with x replaced by a function f defined on Ω and y replaced by a function F defined on Ω1) whenAf(x)=∫ΩK(x,y)f(y)dywith the kernel K continuous on Ω1×Ω2. Another important well-known equation is convolution equation [4]. We consider the one-dimensional case and let Ω=Ω1=R, X=Y=L2(Ω) and K(x,y)=k(xy). Eq. (1.1) takes the formRnk(x−y)f(y)dy=F(x),x∈R.To study and solve such equations, one can solve the Fourier (or Laplace) transform f→f, which transforms Eq. (1.2) into its multiplicative formk(ξ)f(ξ)=F(ξ).Now, we will obtain the ill-posedness condition and a stability estimate for Eq. (1.2).

Section snippets

Statement and proof of main result

Lemma 2.1

The equationAf=∫Rkx−yfydy=F(x),x∈R,is ill-posed if and only if for any natural number l, the function k−1(ξ)(1+|ξ|)−l is (essentially) unbounded on R.

Proof

If we assume that k−1(ξ)(1+|ξ|)−l is bounded on the R, then for any M>0, ∃ a set EM with μ(EM)>0 such thatk−1(ξ)(1+|ξ|)−l⩾M∀ξ∈EM,l=1,2,…,lm.From Eq. (1.2) and from the property of Fourier transform, we havek(ξ)f(ξ)=F(ξ).Therefore,Rf2dξ=∫RF(ξ)kξ2dξ=∫R1kξ1+ξl2Fξ1+ξl2dξ.Now, let us define FM such thatFMξ=GM/1+ξlm,ξ∈EM,0,ξ∈R⧹EM.Here, we require lM→∞

References (5)

  • V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, vol. 127, Springer, New...
  • B. Yıyldız et al.

    On a regularization problem

    Appl. Math. Comput.

    (2000)
There are more references available in the full text version of this article.

Cited by (0)

View full text