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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)

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)

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)

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)

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