Borel transform

An integral transform of the type

$$\gamma (t) = \int\limits _ { 0 } ^ \infty f(z)e ^ {-zt} dz,$$

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 $\phi (t) \in {L _ {2} } ( - \sigma , \sigma )$.

Comments

The statement at the end of the article above is called the Paley–Wiener theorem.

References

[1]	E. Borel, "Leçons sur les séries 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)
[a1]	R.P. Boas, "Entire functions" , Acad. Press (1954) MR0068627 Zbl 0058.30201