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24.—Mean-square Convergence of Non-harmonic Trigonometrical Series

Published online by Cambridge University Press:  14 February 2012

J. Cossar
J. Cossar
Affiliation:
Department of Mathematics, University of Edinburgh
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Synopsis

The series considered are of the form , where Σ | cn |2 is convergent and the real numbers λn (the exponents) are distinct. It is known that if the exponents are integers, the series is the Fourier series of a periodic function of locally integrable square (the Riesz-Fischer theorem); and more generally that if the exponents are not necessarily integers but are such that the difference between any pair exceeds a fixed positive number, the series is the Fourier series of a function of the Stepanov class, S2, of almost periodic functions.

We consider in this paper cases where the exponents are subject to less stringent conditions (depending on the coefficients cn). Some of the theorems included here are known but had been proved by other methods. A fuller account of the contents of the paper is given in Sections 1-5.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 75 , Issue 4 , 1976 , pp. 297 - 323
Copyright
Copyright © Royal Society of Edinburgh 1976

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References

1Amerio, L.. Sulla convergenza in media della serie . Ann. Scuola Norm. Sup. Pisa 10 (1941), 191–198. In the review of this paper in Math. Reviews 8 (1947), inline 2 should be omitted.Google Scholar
2Berry, A. C.. The Fourier transform identity theorem. Ann. of Math. 32 (1931), 227–232.CrossRefGoogle Scholar
3Besicovitch, A. S.. On generalised almost periodic functions. Proc. London Math. Soc. 25 (1926), 495–512.CrossRefGoogle Scholar
4Besicovitch, A. S.. Almost periodic functions (Cambridge: University Press, 1932).Google Scholar
5Bochner, S.. Properties of Fourier series of almost periodic functions. Proc. London Math. Soc. 26 (1927), 433–452.CrossRefGoogle Scholar
6Bochner, S.. Konvergenzsätze für Fourierreihen grenzperiodischer Funktionen. Math. Z. 27 (1928), 187–211.CrossRefGoogle Scholar
7Bochner, S. and Chandrasekharan, K.. Fourier transforms (Princeton: University Press, 1949).Google Scholar
8Bohr, H. and Følner, E.. On some types of functional spaces. Acta Math. 76 (1944), 31–155.CrossRefGoogle Scholar
9Bredihina, E. A.. The divergence of Fourier series of almost periodic functions. Soviet Math.Dokl. 7 (1966), 1530–1533.Google Scholar
10Cossar, J.. Hilbert transforms and almost periodic functions. Proc. Cambridge Philos. Soc. 56 (1960), 354–366.CrossRefGoogle Scholar
11Favard, J.. Sur les fonctions harmoniques presque périodiques. J. Math. Pures Appl. 6 (1927), 229–336.Google Scholar
12Favard, J.. Sur la fonction conjuguée d'une fonction presque-périodique. Mat. Tidsskr. B (1934), 57–60.Google Scholar
13Hardy, G. H.. Divergent series (Oxford: Clarendon Press, 1949).Google Scholar
14Hartman, P.. The divergence of non–harmonic gap series. Duke Math. J. 9 (1942), 404–405.CrossRefGoogle Scholar
15Hartman, S.. Les intégrales de fonctions presque-périodiques et les sections de séries de Fourier. Studia Math. 22 (1962), 147–160.CrossRefGoogle Scholar
16Jessen, B.. Some aspects of the theory of almost periodic functions. Proc. Int. Congr. Math. I (1954), 305–314 (Amsterdam: North-Holland, 1957).Google Scholar
17Levitan, B. M.. Počti-periodičeskie funkcii (Moscow: Gosudarstv. Izdat. Tehn.-Teor. Lit., 1953).Google Scholar
18Paley, R. E. A. C. and Wiener, N.. Fourier transforms in the complex domain. Colloquium Publications 19 (New York: Amer. Math. Soc., 1934).Google Scholar
19Pitt, H. R.. On the Fourier coefficients of almost periodic functions. J. London Math. Soc. 14 (1939), 143–150.CrossRefGoogle Scholar
20Stepanoff, W.. Über einige Verallgemeinerungen der fastperiodischen Funktionen. Math. Ann. 95 (1926), 473–498.CrossRefGoogle Scholar
21Stepanov, V. V.. O metrike v prostranstve počti periodičeskih funkciĭ S2. Dokl. Akad. Nauk SSSR 64 (1949), 171–174.Google Scholar
22Stepanov, V. V.. Ob odnom klasse počti periodičeskih funkciĭ. Dokl. Akad. Nauk SSSR 64 (1949), 297–300.Google Scholar
23Titchmarsh, E. C.. A class of trigonometrical series. J. London Math. Soc. 3 (1928), 300–304.CrossRefGoogle Scholar
24Titchmarsh, E. C.. Introduction to the theory of Fourier integrals (Oxford: Clarendon Press, 1937).Google Scholar
25Titchmarsh, E. C.. On the mean convergence of trigonometrical series. J. London Math. Soc. 40 (1965), 594–596, and 43 (1968), 367. The result had been stated without proof in Proc. London Math. Soc. 24 (1925), xxii–xxiii.CrossRefGoogle Scholar
26Tornehave, H.. On the Fourier series of Stepanov almost periodic functions. Math. Scand. 2 (1954), 237–242.CrossRefGoogle Scholar
27Wiener, N.. On the representation of functions by trigonometrical integrals. Math. Z. 24 (1926), 575–616.CrossRefGoogle Scholar
28Zygmund, A.. Trigonometric series Vol. I (Cambridge: University Press, 1959).Google Scholar