Quasiclassical integral equations. (English. Russian original) Zbl 0769.45009

Sov. Math., Dokl. 44, No. 1, 127-131 (1992); translation from Dokl. Akad. Nauk SSSR 319, No. 3, 527-530 (1991).
The authors study the integral operator \(A[\varepsilon]\) defined by \((A[\varepsilon]f)(x)=a(-i\varepsilon\partial_ x)\theta(x)f(x)=\varepsilon^{-1}\int^ 1_{-1} {\mathcal A}(\varepsilon^{-1}(x-y))f(y)dy\), where \(x\in (-1,1)\), \(\theta\) is the characteristic function of the interval \((-1,1)\), and \(\varepsilon>0\). It is assumed that the symbol of the operator \(a(\varepsilon\xi)=\int_ R\exp(-i\xi x)\cdot\varepsilon^{-1}{\mathcal A}(\varepsilon^{-1}x)dx\) may have jumps (or jumps of the derivatives) and/or roots. The aim is the asymptotic investigation of the operator \((A[\varepsilon])^{-1}\) as \(\varepsilon\to 0\).


45P05 Integral operators
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45M05 Asymptotics of solutions to integral equations