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A singular perturbation problem. (English) Zbl 0587.45002

For a fixed \(f\geq 0\), unimodal and integrable with compact support in [- 1,1], the authors consider the following modified convolution operator \(L_{\epsilon}\), defined by \[ (L_{\epsilon}g)(x)=(2\epsilon)^{- 1}\int^{x+\epsilon}_{x-\epsilon}f(y)g(y)dy\quad (\epsilon >0). \] This operator maps \(L^ 2(-1,1)\) into itself. Under some additional assumptions of f, it is shown that the eigenfunction \(\phi_{\epsilon}\) which corresponds to the largest eigenvalue of \(L_{\epsilon}\) converges weakly as \(\epsilon \to 0^+\) to a normal distribution.
Reviewer: J.Burbea

MSC:

45C05 Eigenvalue problems for integral equations
45P05 Integral operators
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