Friedrich Littmann

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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 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)
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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