## 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 $$\varepsilon\to 0$$.

### MSC:

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

### Keywords:

integral operator; convolution type; symbol; asymptotic