Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T08:47:47.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Jacobi series which converge to zero, with applications to a class of singular partial differential equations

Published online by Cambridge University Press:  24 October 2008

Jet Wimp  and
David Colton
Jet Wimp
Affiliation:
Midwest Research Institute, Kansas City; The University, Edinburgh
David Colton
Affiliation:
Midwest Research Institute, Kansas City; The University, Edinburgh
Get access Rights & Permissions [Opens in a new window]

Extract

Expansions in series of functions are one of the most important tools of the applied mathematician, particularly expansions in series of the classical orthogonal polynomials, e.g. Laguerre, Jacobi and Hermite polynomials. In applied problems, the uniqueness of the particular expansion is usually intrinsic to the analysis, and often implicitly assumed. Indeed, in those cases where the functions in the series are orthogonal, uniqueness can often be proved by an argument that runs as follows. Let {φn(x)} (n = 0, 1, 2, …) be a sequence of functions orthogonal with respect to the weight function ρ(x) over the interval [0, 1], and suppose that

the series being boundedly convergent for 0 ≤ x ≤ 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 65 , Issue 1 , January 1969 , pp. 101 - 106
Copyright
Copyright © Cambridge Philosophical Society 1969

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)Titchmarsh, E. C.The Theory of Functions, 2nd ed. p. 339, (10.51), (Oxford University Press, 1939).Google Scholar
(2)Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.Higher Transcendental Functions, vol. 3 (McGraw-Hill, 1953).Google Scholar
(3) Reference (1), p. 218, (7.31).Google Scholar
(4) Reference (1), p. 229, (7.61).Google Scholar
(5) Reference (2), vol. 3, p. 264, eq. (9).Google Scholar
(6) Reference (2), vol. 2, p. 198, eq. (10).Google Scholar
(7) Reference (1): the example of (1.12), p. 5 also applies to cosine series, and reference 2, vol. 2, p. 169, eq. (3).Google Scholar
(8)Fields, Jerry L.Asymptotic expansions for a class of hypergeometric polynomials with respect to the order, III. J. Math. Anal. Appl. 12, (1965), 593–601.CrossRefGoogle Scholar
(9) Reference (2), vol. 2, p. 173, eq. (35).Google Scholar
(10)Rainville, Earl D.Generating Functions for Bessel and Related Polynomials. Canad. J. Math. 5 (1953), 104–106.CrossRefGoogle Scholar
(11)Kogbetliantz, E.Recherches sur l'Unicité des Séries Ultrasphériques. J. Math. Pures Appl. (9), 5 (1926), 125–196.Google Scholar
(12)Kogbetliantz, E.Recherches sur la Sommabilité des Séries Ultrasphériques par le Méthode des Moyennes Arithmétiques. J. Math. Pures. Appl. 9 (1924), 107–187.Google Scholar
(13)Courant, R. and Hilbert, D.Methods of mathematical physics, vol. 2, Interscience (1962).Google Scholar
(14)Parter, S. V.On the existence and uniqueness of symmetric axially symmetric potentials. Arch. Rational Mech. Anal. 20 (1965), 279–286.CrossRefGoogle Scholar
(15) Reference (2), vol. 2, p. 206, eq. (8), (9).Google Scholar