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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
##### Keywords:
extremal properties; interpolation formula
Full Text:
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