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''Skal'skaya–Lebedev transform''
 
''Skal'skaya–Lebedev transform''
  
 
The [[Integral transform|integral transform]]
 
The [[Integral transform|integral transform]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l1200501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} F ( \tau ) = \int _ { 0 } ^ { \infty } \operatorname { Re } K _ { 1 / 2  + i \tau} ( x ) f ( x ) d x, \end{equation}
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l1200502.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l1200503.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l1200505.png" />-transform.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l1200507.png" />-transform was initiated by them as well:
+
The $\operatorname{Im}$-transform was initiated by them as well:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l1200508.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} F ( \tau ) = \int _ { 0 } ^ { \infty } \operatorname { Im } K _ { 1 / 2  + i \tau} ( x ) f ( x ) d x, \end{equation}
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l1200509.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005010.png" /> is an integrable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005011.png" /> with respect to the weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005012.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005013.png" />, 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]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005014.png" />. Then the Lebedev–Skal'skaya transforms (a1), (a2) converge in the mean-square sense to functions belonging to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005015.png" /> and isomorphically map these two spaces onto each other. Moreover, the [[Parseval equality|Parseval equality]] holds (see [[#References|[a5]]])
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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]]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005016.png" /></td> </tr></table>
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\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
 
as well as the inversion formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005017.png" /></td> </tr></table>
+
\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*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005018.png" /></td> </tr></table>
+
\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.
 
for the two transforms, respectively, where the integrals are understood in the mean-square sense.
  
If two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005020.png" /> are from the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005022.png" /> defines a convolution (cf. also [[Convolution of functions|Convolution of functions]]), for instance for the Lebedev–Skalskaya transform (a1),
+
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),
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005023.png" /></td> </tr></table>
+
\begin{equation*} ( f ^ { * } g ) ( x ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005024.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005025.png" /> belongs to the same space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005026.png" /> and satisfies the norm estimate
+
The convolution $f * g$ belongs to the same space $L _ { 1 } ( \mathbf{R} _ { + } ; e ^ { - x } / \sqrt { x } )$ and satisfies the norm estimate
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005027.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005029.png" /> is the Lebedev–Skal'skaya transform of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005030.png" />.
+
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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005032.png" />, then the following integral representation of the convolution holds:
+
If, moreover, $f , g \in L _ { 1 } ( \mathbf{R} _ { + } ; e ^ { - \beta x } / \sqrt { x } )$, $0 < \beta < 1$, then the following integral representation of the convolution holds:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005033.png" /></td> </tr></table>
+
\begin{equation*} ( f ^ { * } g ) ( x ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120050/l12005034.png" /></td> </tr></table>
+
\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====
 
====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>
+
<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>

Latest revision as of 14:42, 27 January 2024

Skal'skaya–Lebedev transform

The 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. 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).

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 holds (see [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), 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, $f , g \in L _ { 1 } ( \mathbf{R} _ { + } ; e ^ { - \beta x } / \sqrt { x } )$, $0 < \beta < 1$, 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

[a1] 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)
[a2] Yu.M. Rappoport, "Integral equations and Parseval equalities for the modified Kontorovich–Lebedev transforms" Diff. Uravn. , 17 (1981) pp. 1697–1699 (In Russian)
[a3] 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
[a4] S.B. Yakubovich, Yu.F. Luchko, "The hypergeometric approach to integral transforms and convolutions" , Kluwer Acad. Publ. (1994)
[a5] S.B. Yakubovich, "Index transforms" , World Sci. (1996)
How to Cite This Entry:
Lebedev-Skal'skaya transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebedev-Skal%27skaya_transform&oldid=22715
This article was adapted from an original article by S.B. Yakubovich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article