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On the divergence in rth mean of a class of eigenfunction expansions

Published online by Cambridge University Press:  14 November 2011

C. G. C. Pitts
School of Mathematics and Physics, University of East Anglia, Norwich NR4 7TJ


We consider the expansion of a function in Lr (the class of measurable functions whose rth powers are Lebesgue integrable over some interval) in terms of the eigenfunctions arising from a singular Sturm-Liouville problem defined over an infinite or semi-infinite interval. We show that if l ≦ r ≦ inline1 or if r ≧ 4 there exists f in Lr whose eigenfunction expansion is divergent in the rth mean sense, and that the terms of the series form an unbounded sequence in Lr The result extends some work of Askey and Wainger concerning the Hermite series expansions of functions in Lr(–∞, ∞).

Research Article
Copyright © Royal Society of Edinburgh 1979

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1Askey, R., Norm inequalities for some orthogonal series. Bull. Amer. Math. Soc. 73 (1966), 802823.Google Scholar
2Askey, R.. Mean convergence of orthogonal series and Lagrange interpolation. Acta Math. Acad. Sci. Hungar. 23 (1972), 7185.CrossRefGoogle Scholar
3Askey, R.. and Wainger, S.. Mean convergence of expansions in Laguerre and Hermite series. Amer. J. Math. 87 (1965), 695708.CrossRefGoogle Scholar
4Benzinger, H. E.. The Lp behaviour of eigenfunction expansions. Trans. Amer. Math. Soc. 174 (1972), 333344.Google Scholar
5Hardy, G. H.. Orders of infinity, 2nd edn (Cambridge Univ. Press, 1924).Google Scholar
6Milne, W. E.. On the degree of convergence of expansions in an infinite interval. Trans. Amer. Math. Soc. 31 (1929), 907918.CrossRefGoogle Scholar
7Muckenhoupt, B.. Mean convergence of Hermite and Laguerre series I, II. Trans. Amer. Math. Soc. 147 (1970), 419431, 433–460.CrossRefGoogle Scholar
8Newman, J. and Rudin, W.. Mean convergence of orthogonal series. Proc. Amer. Math. Soc. 3 (1952), 219222.CrossRefGoogle Scholar
9Pitts, C. G. C.. Asymptotic approximations to solutions of a second-order differential equation. Quart. J. Math. Oxford Ser. (2) 17 (1966), 307320.CrossRefGoogle Scholar
10Pitts, C. G. C.. On eigenfunction expansions for a positive potential function increasing slowly to infinity. J. Differential Equations 13 (1973), 358373.CrossRefGoogle Scholar
11Pitts, C. G. C.. An equiconvergence theorem for a class of eigenfunction expansions. Trans. Amer. Math. Soc. 189 (1974), p337350.CrossRefGoogle Scholar
12Riesz, M.. Sur les fonctions conjugées. Math. Z. 27 (1927), 218244.CrossRefGoogle Scholar
13Titchmarsh, E. C.. Eigenfunction expansions associated with second-order differential equations, Pt 1, 2nd edn (Oxiord Univ. Press, 1962).CrossRefGoogle Scholar
14Watson, G. N.. A treatise on the theory of Bessel functions, 2nd edn (Cambridge Univ. Press, 1944).Google Scholar
15Zygmund, A.. Trigonometric Series, Vol. 1, 2nd edn (Cambridge Univ. Press, 1959).Google Scholar