Lebedev transform

The integral transform

$$F(\tau)=\int\limits_0^\infty[I_{i\tau}(x)+I_{-i\tau}(x)]K_{i\tau}(x)f(x)dx,\quad0\leq\tau<\infty,$$

where $I_\nu(x)$ and $K_\nu(x)$ are the modified cylinder functions. It was introduced by N.N. Lebedev [1]. If

$$x^{-1/2}f(x)\in L(0,1),\quad x^{1/2}f(x)\in L(1,\infty),$$

then for almost-all $x$ one has the inversion formula

$$f(x)=-\frac{4}{\pi^2}\int\limits_0^\infty F(\tau)\tau\sinh\pi\tau K_{i\tau}^2(x)d\tau.$$

References

[1]	N.N. Lebedev, "On an integral representation of an arbitrary function in terms of squares of MacDonald functions with imaginary index" Sibirsk. Mat. Zh. , 3 : 2 (1962) pp. 213–222 (In Russian)

Comments

The following transform pair is also called a Lebedev transform (or Kontorovich–Lebedev transform)

$$G(\tau)=\int\limits_0^\infty g(x)x^{-1/2}K_{i\tau}(x)dx,$$

$$g(x)=\frac{2}{\pi^2}\frac{1}{\sqrt x}\int\limits_0^\infty\tau\sinh\pi\tau K_{i\tau}(x)G(\tau)d\tau.$$

References

[a1]	N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)
[a2]	I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972) pp. Chapt. 6