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We determine extremal entire functions for the problem of ma-jorizing, minorizing, and approximating the Gaussian function e −πλx 2 by entire functions of exponential type. The combination of the Gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously… (More)

In this note we study the connection between best approximation and interpolation by entire functions on the real line. A general representation for entire interpolants is outlined. As an illustration, best upper and lower approximations from the class of functions of fixed exponential type to the Gaussian are constructed. The Fourier transform of ϕ ∈ L 1… (More)

- Peter Borwein, Tam´as Erdélyi, Friedrich Littmann
- 2006

In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of C is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem. This will allow us to give an easy proof of a recent… (More)

The zero sets of (D + a) n g(t) with D = d/dt in the (t, a)-plane are investigated for g(t) = te αt (e t − 1) −1 and g(t) = e θt (e t + 1) −1. The results are used to determine entire interpolations to functions x n + e −λx , which give representations for the best approximation and best one-sided approximation from the class of functions of exponential… (More)

- Felipe Gonç, Alves And, Friedrich Littmann
- 2015

We consider the problem of reconstruction of entire functions of exponential type τ that are elements of certain weighted L p (µ)-spaces from their values and the values of their derivatives up to order ν. In this paper we extend the interpolation results of [24] in which the case ν = 1 was solved. Using the theory of de Branges spaces we find a discrete… (More)

Let f : R → R have an nth derivative of finite variation V f (n) and a locally absolutely continuous (n − 1)st derivative. Denote by E ± (f, δ)p the error of onesided approximation of f (from above and below, respectively) by entire functions of exponential type δ > 0 in L p (R)–norm. For 1 ≤ p ≤ ∞ we show the estimate E ± (f, δ)p ≤ C 1−1/p n π 1/p V f (n)… (More)

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