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Some extremal functions in Fourier analysis. (English) Zbl 0575.42003

This is a survey paper concerning a collection of topics in the areas of entire functions of finite type, almost periodic functions and extremal properties. One of the key theorems concerns the class \(E^ p\) of entire functions F of exponential type \(\leq 2\pi\) such that \(\int_{{\mathbb{R}}}| F(x)|^ p dx<\infty\), (where \(0<p<\infty)\). The following interpolation formula is established: \[ F(z)=((\sin \pi z)/\pi)^ 2\{\sum^{\infty}_{m=-\infty}F(m)(z-m)^{- 2}+\sum^{\infty}_{n=-\infty}F'(n)(z-n)^{-1}\}, \] with uniform convergence on compact sets; further, if \(p=2\) then \[ \int^{1}_{- 1}F(x)e^{-itx} dx=(1-| t|)(\sum^{\infty}_{m=- \infty}F(m)e^{-2\pi imt})+ \]
\[ (1/(2\pi i))(sgn t)(\sum^{\infty}_{n=-\infty}F'(n)e^{-2\pi int}). \] In the interpolation formula set \(F(n)=1\) for \(n\geq 0\), \(F(n)=0\) for \(n<0\), \(F'(0)=2\), \(F'(n)=0\) for \(n\neq 0\), and denote the resulting sum by B(z). The author discusses this function, which was introduced by A. Beurling. Among entire functions F of type \(2\pi\) satisfying F(x)\(\geq sgn x\), (x\(\in {\mathbb{R}})\), B achieves the (unique) minimum of the functional \(\int_{{\mathbb{R}}}(F(x)-sgn x)dx\), namely, 1. These ideas are also applied to almost periodic (trigonometric) polynomials; for example, it is shown that \(\| f\|_{\infty}\leq (1/(4\delta))\| f'\|_{\infty},\) where \(f(x)=\sum^{n}_{j=1}a_ j\exp (2\pi i\lambda_ jx)\) and \(| \lambda_ j| \geq \delta >0\) for each j, (and \(\| \cdot \|_{\infty}\) denotes the supremum over \({\mathbb{R}})\).
Reviewer: Ch.Dunkl

MSC:

42A10 Trigonometric approximation
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
30D10 Representations of entire functions of one complex variable by series and integrals
11N35 Sieves
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