Inversion of an integral transform related to a general form of heat equation
Let \(L_ x\) be a selfadjoint elliptic operator, which has the properties: (i) its spectrum is discrete and bounded below, (ii) the eigenvalues \(\lambda_ n\) are such that \(\lim_{n\to \infty}n/\lambda_ n<\infty\), (iii) the normalized eigenfunctions \(\psi_ n\) are such that \(\psi_ n(x)=O(n\) m) for \(n\to \infty\) and some nonnegative integer m, (iv) \(L_ x\) is the generator of the analytic semi-group \(\{\exp (-tL_ x)\}_{Re t>0}\) in L 2(0,\(\infty)\), and for \(f\in L\) 2(0,\(\infty)\) one has \(\lim_{t\to 0\quad +}e^{-tL_ x}f(x)=f(x)\). The associated source solution of the parabolic equation \(- L_ xu(x,t)=(\partial /\partial t)u(x,t)\) is \(g(x,y;t)=\sum^{\infty}_{n=0}\psi_ n(x)\psi_ n(y)e^{-\lambda_ nt}\), Re t\(>0\). The authors introduce the P-transform of a function \(\phi\) (x), \(x\in [0,\infty)\) by \({\tilde \phi}(x,t)=\int^{\infty}_{0}\phi (y)g(x,y;t)dy,\) t real and the W- transform of \(\phi\) (x) by \(\tilde{\tilde \phi}(x)={\tilde \phi}(x,1)\). If there exists a continuous bounded function \(\eta\) (x,y;t) such that \[ \int^{\infty}_{0}g(ax,y;t)\eta (x,y;t)\psi_ n(ay)dy=\psi_ n(x)e^{\lambda_ nt} \] for some complex number a, then for a function \(\phi\in L\) 2(0,\(\infty)\) and her W-transform the inversion formula holds \[ \phi (y)=\lim_{t\to 1\quad -}\int^{\infty}_{0}\tilde{\tilde \phi}(ax)g(ay,x;t)\eta (x,y;t)dx \] in L 2(0,\(\infty)\) uniformly in the region \(0<\delta \leq Re t\leq \delta '<1\), \(| \arg (t-1)| \geq \theta >\pi /2\). The examples \(L_ x=-d^ 2/dx^ 2+x^ 2\) and \(L_ x=-d^ 2/dx^ 2-d/dx+x/4+\alpha^ 2/4x\) illustrate the inversion theorem.
scientific article; zbMATH DE number 4046358
scientific article