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Equiconvergence of expansions in eigenfunctions of integral operators with kernels that can have discontinuities on the diagonals. (English. Russian original) Zbl 1032.47027

Sb. Math. 192, No. 10, 1451-1469 (2001); translation from Mat. Sb. 192, No. 10, 33-50 (2001).
Let \[ Af=\int_{0}^{1-x} A(1-x,t)f(t) dt+ \int_{0}^{x} A(x,t)f(t) dt,\quad t\in [0,1], \] where \(\alpha^2 \neq 1\) and the kernel \(A(x,t)\) has \(n\) continuous derivatives with respect to \(x\) and a continuous derivative with respect to \(t\) and, moreover, \(\frac{\partial^j}{\partial x^j}A(x,t)|_{t=x}=\delta_{n-1,j}, j=0,1,\dots ,n, \) where \(\delta_{i,j}\) is the Kronecker delta. The authors state sufficient conditions for the equiconvergence of Fourier expansions of \(f(x)\) in \(L[0,1]\) in eigenfunctions and associated functions of the operator \(A\). The expansions in trigonometric Fourier series for \(f(x)\) and \(f(1-x)\) are also considered.

MSC:

47G10 Integral operators
45P05 Integral operators
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