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''Skal'skaya–Lebedev transform''

The [[Integral transform|integral transform]]

\begin{equation} \tag{a1} F ( \tau ) = \int _ { 0 } ^ { \infty } \operatorname { Re } K _ { 1 / 2  + i \tau} ( x ) f ( x ) d x, \end{equation}

where

\begin{equation*} \operatorname { Re } K _ { 1 / 2 + i \tau } ( x ) = \frac { K _ { 1 / 2 + i \tau } ( x ) + K _ { 1 / 2 - i \tau } ( x ) } { 2 } \end{equation*}

and $K _ { \nu } ( x )$ is the [[Macdonald function|Macdonald function]]. This transformation was introduced by N.N. Lebedev and I.P. Skal'skaya and investigated in connection with possible applications to certain problems in mathematical physics. It is also called the $\operatorname{Re}$-transform.

The $\operatorname{Im}$-transform was initiated by them as well:

\begin{equation} \tag{a2} F ( \tau ) = \int _ { 0 } ^ { \infty } \operatorname { Im } K _ { 1 / 2  + i \tau} ( x ) f ( x ) d x, \end{equation}

where

\begin{equation*} \operatorname{Im}K _ { 1 / 2 + i \tau } ( x ) = \frac { K _ { 1 / 2 + i \tau } ( x ) - K _ { 1 / 2 - i \tau } ( x ) } { 2 i }. \end{equation*}

If $f$ is an integrable function on $\mathbf{R} _ { + }$ with respect to the weight $e ^ { - x } / \sqrt { x }$, i.e. $f \in L _ { 1 } ( {\bf R} _ { + } ; e ^ { - x } / \sqrt { x } )$, then the Lebedev–Skal'skaya transforms (a1), (a2) exist and represent bounded continuous functions on the positive half-axis which tend to zero at infinity (an analogue of the Riemann–Lebesgue lemma, cf. also [[Fourier series|Fourier series]]).

Let $f \in L _ { 2 } ( \mathbf{R} _ { + } )$. Then the Lebedev–Skal'skaya transforms (a1), (a2) converge in the mean-square sense to functions belonging to the space $L _ { 2 } ( \mathbf{R}_ { + } ; \operatorname { cosh } ( \pi \tau ) )$ and isomorphically map these two spaces onto each other. Moreover, the [[Parseval equality|Parseval equality]] holds (see [[#References|[a5]]])

\begin{equation*} \frac { 4 } { \pi ^ { 2 } } \int _ { 0 } ^ { \infty } \operatorname { cosh } ( \pi \tau ) | F ( \tau ) | ^ { 2 } d \tau = \int _ { 0 } ^ { \infty } | f ( x ) | ^ { 2 } d x, \end{equation*}

as well as the inversion formulas

\begin{equation*} f ( x ) = \operatorname { lim } _ { N \rightarrow \infty } \frac { 4 } { \pi ^ { 2 } } \int _ { 0 } ^ { N } \operatorname { cosh } ( \pi \tau ) \operatorname { Re } K _ { 1 / 2 + i \tau } ( x ) F ( \tau ) d \tau, \end{equation*}

\begin{equation*} f ( x ) = \operatorname { lim } _ { N \rightarrow \infty } \frac { 4 } { \pi ^ { 2 } } \int _ { 0 } ^ { N } \operatorname { cosh } ( \pi \tau ) \operatorname { Im } K _ { 1 / 2 + i \tau } ( x ) F ( \tau ) d \tau , \end{equation*}

for the two transforms, respectively, where the integrals are understood in the mean-square sense.

If two functions $f$, $g$ are from the space $L _ { 1 } ( \mathbf{R} _ { + } ; e ^ { - x } / \sqrt { x } )$, then $( f ^ { * } g ) ( x )$ defines a convolution (cf. also [[Convolution of functions|Convolution of functions]]), for instance for the Lebedev–Skalskaya transform (a1),

\begin{equation*} ( f ^ { * } g ) ( x ) = \end{equation*}

\begin{equation*} =\frac { 1 } { 2 \sqrt { 2 \pi } } \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \operatorname { exp } \left( - \frac { 1 } { 2 } \left( \frac { x u } { v } + \frac { x v } { u } + \frac { u v } { x } \right) \right) \times \times \left( \frac { 1 } { x } + \frac { 1 } { u } + \frac { 1 } { v } \right) f ( u ) g ( v ) d u d v. \end{equation*}

The convolution $f * g$ belongs to the same space $L _ { 1 } ( \mathbf{R} _ { + } ; e ^ { - x } / \sqrt { x } )$ and satisfies the norm estimate

\begin{equation*} \| f ^ { * } g \| \leq \| f \|  \| g \| \end{equation*}

in this space. The result of the action of the Lebedev–Skal'skaya transform (a1) on this convolution gives the product $\sqrt { 2 / \pi } F ( \tau ) G ( \tau )$, where $G ( \tau )$ is the Lebedev–Skal'skaya transform of the function $g$.

If, moreover, $f , g \in L _ { 1 } ( \mathbf{R} _ { + } ; e ^ { - \beta x } / \sqrt { x } )$, $0 &lt; \beta &lt; 1$, then the following integral representation of the convolution holds:

\begin{equation*} ( f ^ { * } g ) ( x ) = \end{equation*}

\begin{equation*} = \left( \frac { 2 } { \pi } \right) ^ { 5 / 2 } \int _ { 0 } ^ { \infty } \operatorname { cosh } ( \pi \tau ) \operatorname { Re } K _ { 1 / 2  + i \tau} ( x ) F ( \tau ) G ( \tau ) d \tau . \end{equation*}

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  N.N. Lebedev,   N.P. Skalskaya,   "Some integral transforms related to the Kontorovich–Lebedev transform"  ''Probl. Math. Phys. St. Petersburg''  (1976)  pp. 68–79  (In Russian)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  Yu.M. Rappoport,   "Integral equations and Parseval equalities for the modified Kontorovich–Lebedev transforms"  ''Diff. Uravn.'' , '''17'''  (1981)  pp. 1697–1699  (In Russian)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  S.B. Yakubovich,   "On the new properties of the Kontorovich–Lebedev like integral transforms"  ''Rev. Tec. Ing. Univ. Zulia'' , '''18''' :  3  (1995)  pp. 291–299</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  S.B. Yakubovich,   Yu.F. Luchko,   "The hypergeometric approach to integral transforms and convolutions" , Kluwer Acad. Publ.  (1994)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  S.B. Yakubovich,   "Index transforms" , World Sci.  (1996)</td></tr></table>