# Kontorovich-Lebedev transform

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Lebedev–Kontorovich transform where is the Macdonald function.

This transform was introduced in [a1] and later investigated in [a2]. If is an integrable function with the weight , i.e. , then is a bounded continuous function, which tends to zero at infinity (an analogue of the Riemann–Lebesgue lemma, cf. Fourier series, for the Fourier integral). If is a function of bounded variation in a neighbourhood of a point and if  then the following inversion formula holds:  If the Mellin transform of , denoted by , belongs to the space , then can be represented by an integral (see [a6]):  where is the Euler gamma-function.

Let . Then the integral converges in mean square and isomorphically maps the space onto the space . The inverse operator has the form [a10]  The Kontorovich–Lebedev transform of distributions was considered in [a7], [a8]. A transform table for the Kontorovich–Lebedev transform can be found in [a4]. Special properties in -spaces are given in [a10].

For two functions , , define the operator of convolution for the Kontorovich–Lebedev transform as ([a9], [a10])  The following norm estimate is true: and the space forms a normed ring with the convolution as operation of multiplication.

If , are the Kontorovich–Lebedev transforms of two functions , , then the factorization property is true: If in the ring , then at least one of the functions , is equal to zero almost-everywhere on (an analogue of the Titchmarsh theorem).

The Kontorovich–Lebedev transform is the simplest and most basic in the class of integral transforms of non-convolution type, which forms a special class of so-called index transforms (cf. also Index transform), depending upon parameters, subscripts (indices) of the hypergeometric functions (cf. Hypergeometric function) as kernels.