Generalized hybrid integral transform of (Legendre 2nd kind)-Fourier-Fourier type on the polar axis (Q2768763)

Using the delta-shaped sequences method the author introduces the integral transform generated on the set \(I_{2}^{+}=\{r: r\in(R_0,R_1)\cup(R_1,R_2)\cup(R_2,\infty); R_0>0\}\) by the hybrid differential operator \(M_{(\mu)}=a_1^2\theta(r-R_0)\theta(R_1-r)\Lambda_{(\mu)}+ a_2^2\theta(r-R_1)\theta(R_2-r)d^2/dr^2+a_3^2\theta(r-R_2)d^2/dr^2\). Here \(a_{j}>0\), \(\theta(x)=\begin{cases} 0,&x<0,\\ 1,&x\geq 0,\end{cases}\) NEWLINENEWLINENEWLINE\(\Lambda_{(\mu)}=d^2/dr^2+\text{cth} r d/dr+1/4+(1/2)(\mu^2_1/(1-\text{ch} r)+\mu^2_2/(1+\text{ch} r))\), \(\mu_1\geq \mu_2\geq 0\). A theorem on the integral representation and a theorem on the fundamental identity are proved.