A stability estimate for convolution equation
Introduction
Many problems of mathematical physics and technology lead to the consideration of an equation of the formwhere 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 x∈X is to be found for a given y∈Y [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 and y replaced by a function F defined on ) whenwith the kernel K continuous on . Another important well-known equation is convolution equation [4]. We consider the one-dimensional case and let , and K(x,y)=k(x−y). Eq. (1.1) takes the formTo study and solve such equations, one can solve the Fourier (or Laplace) transform , which transforms Eq. (1.2) into its multiplicative formNow, 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 equationis ill-posed if and only if for any natural number l, the function is (essentially) unbounded on . Proof If we assume that is bounded on the , then for any M>0, ∃ a set EM with μ(EM)>0 such thatFrom Eq. (1.2) and from the property of Fourier transform, we haveTherefore,Now, let us define FM such thatHere, we require lM→∞
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