Extremal Functions with Vanishing Condition

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• 作者：Friedrich Littmann ; Mark Spanier
• 关键词：Exponential type ; Bandlimited functions ; Best one ; sided approximation ; De Branges space ; Hermite–Biehler entire function ; Primary 41A30 ; 41A52 ; Secondary 41A05 ; 41A44 ; 42A82
• 刊名：Constructive Approximation
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：42
• 期：2
• 页码：209-229
• 全文大小：542 KB
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• 作者单位：Friedrich Littmann (1)
  Mark Spanier (2)
  
  1. Department of Mathematics, North Dakota State University, Fargo, ND, 58105, USA
  2. College of Arts and Sciences, Dakota State University, Madison, SD, 57042, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Analysis
  
• 出版者：Springer New York
• ISSN：1432-0940

文摘

For a Hermite–Biehler function E of mean type \(\tau \), we determine the optimal (with respect to the de Branges measure of E) majorant \(M_E^+\) and minorant \(M_E^-\) of exponential type \(\tau \) for the truncation of \(x\mapsto (x^2+a^2)^{-1}\). We prove that $$\begin{aligned} \int _\mathbb {R}\left( M_E^+(x) - M_E^-(x)\right) |E(x)|^{-2}\hbox {d}x = \frac{1}{a^2 K(0,0)}, \end{aligned}$$