Watson transform

An integral transform $g$ of a function $f \in {L _ {2} } ( 0, \infty )$, defined as follows:

$$\tag{1 } g( x) = \frac{d}{dx} \int\limits _ { 0 } ^ \infty \widetilde \omega ( xu) f( u) \frac{du}{u} .$$

Here $x$ is a real variable, the kernel ${\widetilde \omega } ( x)$ has the form

$$\tag{2 } \widetilde \omega ( x) = \frac{x}{2 \pi } \mathop{\rm l}.i.m. _ {T \rightarrow \infty } \ \int\limits _ { - } T ^ { T } \frac{\Omega \left ( \frac{1}{2} + it \right ) }{ \frac{1}{2} - it } x ^ {- ( t+ 1/2) } dt$$

(l.i.m. denotes the limit in the mean in $L _ {2}$) and the function $\Omega ( it + 1 / 2)$ satisfies the condition

$$\Omega ( s) \Omega ( 1- s) = 1.$$

The following conditions are sufficient for the existence of the kernel ${\widetilde \omega } ( x)$ and the inclusion ${\widetilde \omega ( x) } / x \in {L _ {2} } ( 0, \infty )$:

$$\Omega \left ( \frac{1}{2} - it \right ) = \Omega \left ( \frac{1}{2} + it \right )$$

and

$$\frac{\Omega \left ( \frac{1}{2} + it \right ) }{ \frac{1}{2} - it } \in \ L _ {2} (- \infty , \infty ).$$

For a function $f \in L _ {2} ( 0, \infty )$, formula (1) defines the function $g \in L _ {2} ( 0, \infty )$ almost-everywhere. The inversion formula for the Watson transform (1) has the form

$$f( x) = \frac{d}{dx} \int\limits _ { 0 } ^ \infty \widetilde \omega ( xu ) g( u) \frac{du}{u} .$$

Named after G.N. Watson [1], who was the first to study this transform.

References

 [1] G.N. Watson, "General transforms" Proc. London Math. Soc. (2) , 35 (1933) pp. 156–199

Quite generally, let $\psi$ be a Lebesgue-measure function in $\mathbf R _ {>} 0$ and let

$$\phi = \int\limits _ { 0 } ^ { x } \psi ( x) dt .$$

The kernel $\psi$( or $\phi$) is called a generalized kernel, or kernel of a generalized transform, if

a) $\psi ( x)$ is real valued on $\mathbf R _ {>} 0$;

b) $x ^ {-} 1 \phi ( x) \in L _ {2} ( \mathbf R _ {>} 0 )$;

c) $\int _ {0} ^ \infty \phi ( xu ) \phi ( yu ) u ^ {-} 2 du = \min ( x, y)$.

The operator $\Phi$ defined on $L _ {2} ( \mathbf R _ {>} 0 )$ by

$$\Phi ( f )( x) = \frac{d}{dx} \int\limits _ { 0 } ^ \infty \frac{\phi ( xt) f( t) }{t} dt$$

is called a generalized transform or Watson transform.

References

 [a1] G.O. Okikiolu, "Aspects of the theory of bounded operators in -spaces" , Acad. Press (1971) pp. §6.7
How to Cite This Entry:
Watson transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Watson_transform&oldid=49174
This article was adapted from an original article by A.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article