Abstract
We show that the classical results of L.Fejér and M. Riesz concerning norm convergence of trigonometric Fourier series can be carried over to eigenfunction expansions arising from regular indefinite boundary eigenvalue problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benzinger, H. E.: TheL p behavior of eigenfunction expansions;Trans. Amer. Math. Soc. 174 (1972), 333–344.
Beals, R.: Indefinite Sturm-Liouville problems and half-range completeness;J. Diff. Equ. 56 (1985), 391–407.
Braß, H.: Approximation mittels Linearkombinationen von Projektionsoperatoren;Abh. Braunschw. Wiss. Ges. 18 (1966), 50–69.
Braß, H.: Approximation durch Linearkombinationen von Eigenfunktionen;Abh. Braunschw. Wiss. Ges. 21 (1969), 429–479.
Butzer, P. L.; Nessel, R. J.: Fourier Analysis and Approximation, Vol 1; Birkhäuser, Basel 1971.
Carleson, L.: On the convergence and growth of partial sums of Fourier series;Acta Math. 116 (1966), 135–157.
Eberhard, W.; Freiling, G.; Schneider, A.: On the distribution of the eigenvalues of a class of indefinite eigenvalue problems.to appear in J. Diff. Int. Equ.
Eberhard, W.; Freiling, G.; Schneider, A.: Expansion theorems for a class of regular indefinite eigenvalue problems.to appear in J. Diff. Int. Equ.
Edwards, R. E.: Fourier Series — A Modern Introduction, Vol. 2; Springer, New York 1979.
Fichtenholz, G. M.: Differential-und Integralrechnung Bd. III; VEB Deutscher Verlag der Wissenschaften, Berlin 1979.
Haar, A.: Zur Theorie der orthogonalen Funktionensysteme I;Math. Ann. 69 (1910), 331–371.
Hunt, R. A.: On the convergence of Fourier series. In: Orthogonal expansions and their continuous analogues. (Proc. Conf. Southern. Illinois Univ., Edwardsville, III., 1967; Ed. D. Tepper Haimo) Southern Illinois Univ. Press; Feffer and Simons 1968; 235–255.
Kaper H. G., Kwong M. K., Lekkerkerker, C. G., Zettl A.: Full-and partial-range eigenfunction expansions for Sturm-Liouville problems with indefinite weights.Proc. Roy. Soc. Edinburgh Sect. A 98 (1984), 69–88.
Kaufmann, F.-J.: Abgeleitete Birkhoff-Reihen bei Randeigenwertproblemen zuN(y)=λP(y) mit λ-abhängigen Randbedingungen; Thesis, Aachen 1988. Abriged version: Derived Birkhoff-series associated withN(y)=λP(y); To appear inResults in Mathematics.
Mingarelli, A. B.: A survey of the regular weighted Sturm-Liouville Problem—The non-definite case. In: Proc. Workshop on Appl. Diff. Equ., Beijing, Aiao Shutie and Pu Fuquan (eds.), World Scientific Publ. Co., Philadelphia 1986.
Naimark, M. A.: Linear differential operators, Part I; F. Ungar, New York 1967.
Stone, M. H.: A comparison of the series of Fourier and Birkhoff;Trans. Amer. Math. Soc. 28 (1926), 695–761.
Tamarkin, J.: Sur quelques points de la théorie des équations différentielles linéaires ordinaires et sur la généralisation de la série de Fourier;Rend. Circ. Mat. Palermo 34 (1912), 345–382.
Tamarkin, J.: Some general problems of the theory of ordinary linear differential equations and expansions of an arbitrary function in series of fundamental functions;Math. Z. 27 (1928), 1–54.
Wermuth, E. M. E.: Konvergenzuntersuchungen bei Eigenfunktionsentwicklungen zu Randeigenwertproblemenn-ter Ordnung mit parameterabhängigen Randbedingungen; Thesis Aachen, 1984.
Zygmund, A.: Trigonometric Series I, II; University Press, Cambridge 1977.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Freiling, G., Kaufmann, F.J. On uniform and Lp-convergence of eigenfunction expansions for indefinite eigenvalue problems. Integr equ oper theory 13, 193–215 (1990). https://doi.org/10.1007/BF01193756
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01193756