On the representation of functions by an integral in the eigenfunctions of the Bessel operator (Q2704022)

The authors have presented the expansion in an integral of the form NEWLINE\[NEWLINEf(x)=\int^\infty_0 Y_\nu(x\xi)\sqrt{x\xi} F(\xi)d\xi,\quad \nu\in (-1,1),NEWLINE\]NEWLINE where \(Y_\nu(z)\) is the Neumann function of order \(\nu\), and an estimation of the norm of the function \(f\) in the space \(L_2(0,\infty)\) via the norm of the function \(F\). It is shown that the function \(F(\xi)\) can be found by the formula NEWLINE\[NEWLINEF(\xi)=\int^\infty_0\sqrt{t\xi} \mathbb{H}_\nu(\xi t)f(t)dt, \text{ for }\nu\in (-1,0),NEWLINE\]NEWLINE where \(\mathbb{H}_\nu(z)\) is the Struve function, and by the formula NEWLINE\[NEWLINEF(\xi)=\int^\infty_0\sqrt{t\xi} \overline{\mathbb{H}}_\nu(\xi t)f(t)dt, \text{ for }\nu\in(0,1),NEWLINE\]NEWLINE where \(\overline\mathbb{H}_\nu(z)=\mathbb{H}_\nu(z)-2^{1-\nu}z^{\nu-1}/[\sqrt\pi \Gamma(\nu+1/2)]\). Moreover, the estimate NEWLINE\[NEWLINEm\|f\|_{L_2(0,\infty)}\leq \|F\|_{L_2(0,\infty)}\leq M\|f\|_{L_2(0,\infty)}NEWLINE\]NEWLINE is valid.