Elias m. Stein - Localization and Summability of Multiple Fourier Series
Summability Methods m
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Summability methods m From Wikipedia, the free encyclopedia In mathematics, Mittag-Leffler summation is any of several variations of the Borel summation method for summing possibly divergent formal power series, introduced by Mittag-Leffler (1908) 1 Definition Let y(z)= ∞ ∑ k=0 y k z k be a formal power series in z. Define the transform B α y of y by B α y(t) ≡ ∞ ∑ k=0 y k Γ(1 + αk) t k Then the Mittag-Leffler sum of y is given by lim α→0 B α y(z) if each sum converges and the limit exists. A closely related summation method, also called Mittag-Leffler summation, is given as follows (Sansone & Gerretsen 1960). Suppose that the Borel transform converges to an analytic function near 0 that can be analytically continued along the positive real axis to a function growing sufficiently slowly that the following integral is well defined (as an improper integral). Then the Mittag-Leffler sum of y is given by ∫ ∞ 0 e -t B α y(t α z) dt When α = 1 this is the same as Borel summation. 2 See also • Mittag-Leffler function 3 References • Hazewinkel, Michiel, ed. (2001), “Mittag-Leffler summation method”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 1
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Transcript of Summability Methods m
Summability methods mFrom Wikipedia, the free encyclopediaIn mathematics, Mittag-Leer summation is any of several variations of the Borel summation method for summingpossibly divergent formal power series, introduced by Mittag-Leer (1908)1 DenitionLety(z) =k=0ykzkbe a formal power series in z.Dene the transform By ofy byBy(t) k=0yk(1 + k)tkThen the Mittag-Leer sum of y is given bylim0By(z)if each sum converges and the limit exists.A closely related summation method, also called Mittag-Leer summation, is given as follows (Sansone & Gerretsen1960). Suppose that the Borel transform converges to an analytic function near 0 that can be analytically continuedalong the positive real axis to a function growing suciently slowly that the following integral is well dened (as animproper integral). Then the Mittag-Leer sum of y is given by0etBy(tz) dtWhen = 1 this is the same as Borel summation.2 See alsoMittag-Leer function3 ReferencesHazewinkel, Michiel, ed. (2001), Mittag-Leer summation method, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-412 3 REFERENCESMittag-Leer, G. (1908), Sur la reprsentation arithmtique des fonctions analytiques d'une variable com-plexe, Atti del IV Congresso Internazionale dei Matematici (Roma, 611 Aprile 1908) I, pp. 6786Sansone, Giovanni; Gerretsen, Johan (1960), Lectures on the theory of functions of a complex variable. I.Holomorphic functions, P. Noordho, Groningen, MR 011398834 Text and image sources, contributors, and licenses4.1 Text Mittag-Leer summation Source: https://en.wikipedia.org/wiki/Mittag-Leffler_summation?oldid=671037802 Contributors: MichaelHardy, Rgdboer, R.e.b. and Deltahedron4.2 Images4.3 Content license Creative Commons Attribution-Share Alike 3.0
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