Entire Approximations to the Truncated Powers

Abstract

Abstract. For a variation diminishing function g which is analytic on a set containing the real line and any real polynomial P , we prove that g + P has at most deg(P ) + 2 real zeros.Based on this estimate, we present a way to construct entire approximations Gn to the truncated powers x

Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials

“…The following facts come from Section 6 of [10]. Let φ = / , where is the Euler Gamma function, and α ∈ [0, 1].…”

“…Here we start by recalling the corresponding extremal problem in the real line, solved by F. Littmann in [10]. In this paper he finds the best L 1 (R)-approximation to the function f (x) = x n + ( f (x) = x n for x ≥ 0 and f (x) = 0 for x < 0) by an entire function of exponential type π δ.…”

“…Recently, F. Littmann extended the ideas of Beurling and Selberg to solve the extremal problem (1.1) for the functions f (x) = sgn(x)x n , n ∈ N 0 = N ∪ {0}. He found not only the unique extremal majorants and minorants (see [9]), but also the best entire approximation in the L 1 (R)-norm (see [10]) to these functions. The purpose of this paper is to transfer these two works of Littmann to periodic versions, namely to solve the extremal problems (1.2) and (1.3) for the Bernoulli periodic functions B n (x), thus generalizing the periodic machinery developed by Vaaler in [14].…”

One-sided band-limited approximations of some radial functions

“…Such problems are considered in [5,6,11,12,13,14,34,35,36,51]. Some work has also been done in the several variables setting.…”

Some Extremal Functions in Fourier Analysis, III

“…An account of these functions, the history of their discovery, and many applications can be found in [5], [6], [11], [16], [17], and [18]. Further examples have been given by F. Littmann [9], [10], and extensions of the problem to several variables are considered in [1], [4], and [8].…”