Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-24T13:52:57.875Z Has data issue: false hasContentIssue false
  • English
  • Français

On the approximation of analytic functions in a strip

Published online by Cambridge University Press:  24 October 2008

Dieter Klusch
Dieter Klusch
Affiliation:
Tanneck 7, D-2370 Rendsburg, West Germany
Get access Rights & Permissions [Opens in a new window]

Extract

1. Let

and denote by Aδ the class of functions f analytic in the strip Sδ = {z = x + iy| |y| < δ}, real on the real axis, and satisfying |Ref(z)| ≤ 1,z∊Sδ. Then N.I. Achieser ([1], pp. 214–219; [8], pp. 137–8, 149) proved that each f∊Aδ can be uniformly approximated on the whole real axis by an entire function fc of exponential type at most c with an error

where ∥·∥∞ is the sup norm on ℝ. Furthermore ([7], pp. 196–201), if f∊Aδ is 2π-periodic, then the uniform approximation Ẽn (Aδ) of the class Aδ by trigonometric polynomials of degree at most n is given by

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 3 , May 1985 , pp. 381 - 384
Copyright
Copyright © Cambridge Philosophical Society 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Achieser, N. I.. Vorlesungen über Approximationstheorie (Akademie Verlag, 1953).Google Scholar
[2]Berndt, B. C.. Analytic Eisenstein series, theta-functions and series relations in the spirit of Ramanujan. J. reine angew. Math. 303/304 (1978), 332–365.Google Scholar
[3]Klusch, D.. Funktionalgleichungen für die Riemann'sche, die Hurwitz'sche und die Lipschitz-Lerch'sche Zetafunktion. Dissertation, I. Math. Inst., Freie Universität Berlin 1977.Google Scholar
[4]Lerch, M.. Note sur la fonction Acta Math. 11 (1887), 19.24.Google Scholar
[5]Meinardus, G.. Approximation von Funktionen und ihre numerische Behandlung (Springer-Verlag, 1964).CrossRefGoogle Scholar
[6]Ramanujan, S.. Notebooks (2 volumes), vol. I (Tata Institute of Fundamental Research, 1957).Google Scholar
[7]Schönhage, A.. Approximationstheorie (W. de Gruyter, 1971).CrossRefGoogle Scholar
[8]Shapiro, H. S.. Topics in Approximation Theory. Lecture Notes in Mathematics, vol. 187 (Springer-Verlag, 1971).CrossRefGoogle Scholar
[9]Titchmarsh, E. C.. The Theory of Functions (Clarendon Press, 1937).Google Scholar