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 ) $.

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