Borel transform
An integral transform of the type
where f(z) is an entire function of exponential type. The Borel transform is a special case of the Laplace transform. The function \gamma (t) is called the Borel transform of f(z) . If
f(z) = \sum _ { n=0 } ^ \infty \frac{a _ {n} }{n!} z ^ {n} ,
then
\gamma (t) = \sum _ { v=0 } ^ \infty a _ {v} t ^ {-(v+1) } ;
the series converges for | t | > \sigma , where \sigma is the type of f(z) . Let \overline{D}\; be the smallest closed convex set containing all the singularities of the function \gamma (t) ; let
K( \phi ) = \max _ {z \in \overline{D}\; } \ \mathop{\rm Re} (ze ^ {-i \phi } )
be the supporting function of \overline{D}\; ; and let h ( \phi ) be the growth indicator function of f(z) ; then K( \phi ) = h( - \phi ) . If in a Borel transform the integration takes place over a ray \mathop{\rm arg} z = \phi , the corresponding integral will converge in the half-plane x \cos \phi + y \sin \phi > K ( - \phi ) . Let C be a closed contour surrounding \overline{D}\; ; then
f(z) = \frac{1}{2 \pi i } \int\limits _ { C } \gamma (t) e ^ {zt} dt.
If additional conditions are imposed, other representations may be deduced from this formula. Thus, consider the class of entire functions f(z) of exponential type \leq \sigma for which
\int\limits _ {- \infty } ^ \infty | f(x) | ^ {2} dx < \infty .
This class is identical with the class of functions f(z) that can be represented as
f(z) = \ \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \sigma } ^ \sigma e ^ {izt} \phi (t) dt,
where \phi (t) \in {L _ {2} } ( - \sigma , \sigma ) .
References
[1] | E. Borel, "Leçons sur les series divergentes" , Gauthier-Villars (1928) Zbl 54.0223.01 |
[2] | M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian) |
Comments
The statement at the end of the article above is called the Paley–Wiener theorem.
References
[a1] | R.P. Boas, "Entire functions" , Acad. Press (1954) MR0068627 Zbl 0058.30201 |
Borel transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_transform&oldid=46122