Finite hybrid Fourier-Bessel-Fourier-\dots-Bessel-Fourier integral transforms (Q2703330)

The paper deals with the construction of a bounded nontrivial solution on \(I_{2n}=\{r: r\in\bigcup\limits _{k=1}^ {2n+1} (R _{k-1},R_{k}); R_{0}\geq0\}\) of the separate system of Fourier and Bessel equations NEWLINE\[NEWLINE\left({d^{2}\over dr^{2}}+ q_{2k+1}^{2}\right) v_{2k+1}(r)=0,\;r\in (R_{2k}, R_{2k+1}),\;k=0,\dots,n,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\left({d^{2}\over dr^{2}}+ {2\alpha_{k}+1\over r} {d\over dr}-{\nu_{k} ^{2}-\alpha_ {k}^{2}\over r^{2}}+q_{2k}^{2} \right)v_{2k}(r)=0,\quad r\in (R_{2k-1},R_{2k}),\quad k=1, \dots,n,NEWLINE\]NEWLINE with boundary value conditions NEWLINE\[NEWLINE\left( \alpha _{11} ^{0}{d\over dr}+ \left. \beta_{11}^{0}\right)v_{1}(r) \right | _{r=R_{0}}=0, \quad \left.\left(\alpha_{22}^{2n+1}{d\over dr}+ \beta_{22} ^{2n+1}\right)v _{2n+1}(r)\right| _{r=R _ {2n+1}}=0NEWLINE\]NEWLINE and conjugation conditions NEWLINE\[NEWLINE\left.\left[\left(\alpha _{j1} ^{k} {d\over dr}+\beta_{j1}^{k}\right) v_{k}-\left( \alpha_ {j2}^{k}{d\over dr}+\beta_{j2}^{k}\right) v_{k+1} \right]\right| _{r=R_{k}}=0,\quad j=1,2;\quad k=1,\dots,2n.NEWLINE\]NEWLINE One of the results is the following. Let \(f(r)\in C^{(2)}(I_{2n})\) satisfy the above-mentioned boundary value and conjugation conditions, then NEWLINE\[NEWLINE f(r)=\sum \limits_{N=1} ^ {\infty}\int \limits_{R_{0}} ^{R_{2n+1}} f(\rho)V_{(\nu, \alpha)} ^{(N)}(\rho,\lambda _ {N})d\rho {V_{(\nu,\alpha)} ^ {(N)}(r,\lambda_{N})\over \| V_{(\nu,\alpha)}^ {(N)} (r,\lambda_{N})\|^{2}},NEWLINE\]NEWLINE where the Fourier series is absolutely and uniformly convergent, \(V_ {(\nu,\alpha)} ^{(N)} (r,\lambda_{N})\) are the corresponding spectral functions.