Fourier integral transforms with spectral parameter on piecewise-homogeneous Cartesian axis (Q2753485)

Hybrid Fourier integral transforms with one and two conjugation points for differential operators \({\mathcal L}_1 = [a_1^2\theta(-x) + a_2^2\theta(x)]d^2/dx^2\), \({\mathcal L}_2 = [a_1^2\theta(l_1-x) + a_2^2\theta(x-l_1)\theta(l_2-x) + a_3^2\theta(x-l_2)]d^2/dx^2\) are considered. Here \(\theta\) is the Heaviside step function, \(a_j>0\). It is assumed that the spectral parameter enters the conjugation conditions at points \(a_j\). For derivation of integral transforms the delta-like Cauchy sequences are used. These sequences are given by the Cauchy kernels, i.e., fundamental matrices of solutions of the Cauchy problem for equations of heat conduction of parabolic type corresponding to the operators \({\mathcal L}_1,{\mathcal L}_2\).