Orthonormal sets of periodic functions of the type {𝑓(𝑛𝑥)}

Bourgin, D. G.; Mendel, C. W.

doi:10.1090/S0002-9947-1945-0012158-8

Orthonormal sets of periodic functions of the type $\{f(nx)\}$

Authors: D. G. Bourgin and C. W. Mendel
Journal: Trans. Amer. Math. Soc. 57 (1945), 332-363
MSC: Primary 42.4X
DOI: https://doi.org/10.1090/S0002-9947-1945-0012158-8
MathSciNet review: 0012158
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S. Banach, Théorie des opérations linéaires, Warsaw, 1932.

Marshall Harvey Stone, Linear transformations in Hilbert space, American Mathematical Society Colloquium Publications, vol. 15, American Mathematical Society, Providence, RI, 1990. Reprint of the 1932 original. MR 1451877

E. Hopf, Ergoden Theorie, Ergebnisse der Mathematik, 1937.

E. C. Titchmarsh, Han-shu lun, Science Press, Peking, 1964 (Chinese). Translated from the English by Wu Chin. MR 0197687
David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923

A. Zygmund, Trigonometrical series, Warsaw, 1933.

E. E. Kummer, Beitrag zur Theorie der Funktion $\Gamma (x)$, Zeitschrift für reine und angewandte Mathematik vol. 35 (1847) pp. 1-4.

G. H. Hardy and J. E. Littlewood, On the absolute convergence of Fourier series, J. London Math. Soc. vol. 3 (1928) pp. 250-253.

M. H. Stone, A Characteristic Property of Certain Sets of Trigonometric Functions, Amer. J. Math. 49 (1927), no. 4, 535–542. MR 1506643, DOI https://doi.org/10.2307/2370837

E. Landau, Über den Wertevorrat von $\zeta (s)$ in der Halbebene $\sigma > 1$, Nachr. Ges. Wiss. Göttingen (1933) pp. 81-91.