Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-05T02:36:06.351Z Has data issue: false hasContentIssue false
  • English
  • Français

A quasi-equivalence between Borel summability and convergence for Fourier-Laguerre series at the end-point

Published online by Cambridge University Press:  20 January 2009

Lee Lorch  and
Jean Tzimbalario
Lee Lorch
Affiliation:
York University, Downsview, Ontario, Canada.
Jean Tzimbalario
Affiliation:
University of Alberta, Edmonton, Alberta, Canada.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A suitable function f(x) 0 ≤ x < ∞, can be expanded into a Fourier series of Laguerre polynomials , whose interval of orthogonality is 0 ≤ x < ∞. The usual problems as to convergence and, lacking convergence, summability, and also the asymptotic behaviour of Lebesgue constants, arise for such developments. A summary of work on these convergence and summability problems, together with extensive references to the literature, can be found in the standard treatise by G. Szegö (5, especially Chapter IX) to whom many of these results are due.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 22 , Issue 3 , October 1979 , pp. 217 - 226
Copyright
Copyright © Edinburgh Mathematical Society 1979

References

REFERENCES

1Gatteschi, L., Formulae asintotiche “ritoccate” per le funzioni di Bessel, Atti Accad. Sci. Torino, Cl. Sci. Hz. Mat. Nat. 93 (1958/1959), 506–514, MR 21-6077.Google Scholar
(2) Hardy, G. H., Divergent series, (Oxford, 1949).Google Scholar
(3) Lebedev, N. N., Special functions and their applications, (Prentice Hall, 1965) [English Translation].CrossRefGoogle Scholar
(4) Lorch, L., Asymptotic expressions for some integrals which include certain Lebesgue and Fejér constants, Duke Math. J. 20 (1953), 89–104. MR 14–635.CrossRefGoogle Scholar
(5) Szegö, G., Orthogonal polynomials, (American Mathematical Society Colloquium Publications 23, 4th ed., 1975).Google Scholar
(6) Szegö, G., Beiträge zur Theorie der Laguerreschen Polynome. I. Entwicklungssätze, Mathematische Zeitschrift, 25 (1926), 87–115.CrossRefGoogle Scholar
(7) Sz.-Nagy, B., Introduction to real functions and orthogonal expansions (Oxford University Press, New York, 1965).Google Scholar
(8) Watson, G. N., A treatise on the theory of Bessel functions, 2nd ed. (Cambridge, 1944).Google Scholar