Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The $ L\sp{p}$ behavior of eigenfunction expansions


Author: Harold E. Benzinger
Journal: Trans. Amer. Math. Soc. 174 (1972), 333-344
MSC: Primary 34B25
DOI: https://doi.org/10.1090/S0002-9947-1972-0328189-0
MathSciNet review: 0328189
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the extent to which the eigenfunction expansions arising from a large class of two-point boundary value problems behave like Fourier series expansions in the norm of $ {L^p}(0,1),1 < p < \infty $. We obtain our results by relating Green's function to the Hilbert transform.


References [Enhancements On Off] (What's this?)

  • [1] R. E. Edwards, Fourier series: A modern introduction. Vol. II, Holt, Rinehart and Winston, New York, 1967. MR 36 #5588. MR 0222538 (36:5588)
  • [2] N. Dunford and J. Schwartz, Linear operators. III, Interscience, New York, 1971.
  • [3] M. A. Naĭmark, Linear differential operators, GITTL, Moscow, 1954; English transl., Ungar, New York, 1967; German transl., Akademie-Verlag, Berlin, 1960. MR 16, 702; MR 41 #7485.
  • [4] H. E. Benzinger, The $ {L^2}$ behavior of eigenfunction expansions, J. Applicable Anal. (to appear). MR 0414981 (54:3073)
  • [5] D. R. Smart, Eigenfunction expansions in $ {L^p}$ and C, Illinois J. Math. 3 (1959), 82-97. MR 20 #6568. MR 0100134 (20:6568)
  • [6] M. H. Stone, A comparison of the series of Fourier and Birkhoff, Trans. Amer. Math. Soc. 28 (1926), 695-761. MR 1501372
  • [7] H. E. Benzinger, Green's function for ordinary differential operators, J. Differential Equations 7 (1970), 478-496. MR 41 #7189. MR 0262583 (41:7189)
  • [8] -, Equiconvergence for singular differential operators, J. Math. Anal. Appl. 32 (1970), 338-351. MR 42 #895. MR 0265986 (42:895)
  • [9] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, 2nd ed., Clarendon Press, Oxford, 1948.
  • [10] Y. Katznelson, An introduction to harmonic analysis, Wiley, New York. 1968. MR 40 #1734. MR 0248482 (40:1734)
  • [11] D. Rutovitz, On the $ {L_p}$-convergence of eigenfunction expansions, Quart. J. Math. Oxford Ser. (2) 7 (1956), 24-38. MR 18, 309. MR 0080830 (18:309d)
  • [12] R. E. L. Turner, Eigenfunction expansions in Banach spaces, Quart. J. Math. Oxford Ser. (2) 19 (1968), 193-211. MR 39 #2024. MR 0240678 (39:2024)
  • [13] H. E. Benzinger, Completeness of eigenvectors in Banach spaces, Proc. Amer. Math. Soc. (to appear). MR 0318941 (47:7487)
  • [14] N. Wiener and R. E. A. C. Paley, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publ., vol. 19, Amer. Math. Soc., Providence, R. I., 1934. MR 1451142 (98a:01023)
  • [15] John E. Gilbert, Maximal theorems for some orthogonal series. I, Trans. Amer. Math. Soc 145 (1969), 495-515. MR 40 #6156. MR 0252941 (40:6156)
  • [16] -, Maximal theorems for some orthogonal series. II, J. Math. Anal. Appl. 31 (1970), 349-368. MR 0419922 (54:7939)
  • [17] J. L. Kazdan, Peturbation of complete orthonormal sets and eigenfunction expansions, Proc. Amer. Math. Soc. 27 (1971), 506-510. MR 42 #6648. MR 0271767 (42:6648)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34B25

Retrieve articles in all journals with MSC: 34B25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0328189-0
Keywords: Two-point boundary value problem, eigenfunction expansions, Green's function, asymptotic estimates, Hilbert transform
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society