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Bulletin of the American Mathematical Society

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Some extremal functions in Fourier analysis

Author: Jeffrey D. Vaaler
Journal: Bull. Amer. Math. Soc. 12 (1985), 183-216
MSC (1980): Primary 42A10, 42A38; Secondary 10H30, 41A17
MathSciNet review: 776471
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  • [A-K] N. I. Achieser and M. G. Krein, Best approximation of differentiable periodic functions by means of trigonometric sums, Dokl. Akad. Nauk SSSR 15 (1937), 107-112. (Russian)
  • R. C. Baker and G. Harman, Small fractional parts of quadratic forms, Proc. Edinburgh Math. Soc. (2) 25 (1982), no. 3, 269–277. MR 678550, 10.1017/S0013091500016758
  • Paul van Beek, An application of Fourier methods to the problem of sharpening the Berry-Esseen inequality, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), 187–196. MR 0329000
  • [Brn] S. N. Bernstein, Sur une propriété des fonctions entieres, C. R. Acad. Sci. 176 (1923), 1603-1605.
  • Andrew C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates, Trans. Amer. Math. Soc. 49 (1941), 122–136. MR 0003498, 10.1090/S0002-9947-1941-0003498-3
  • [Beu] A. Beurling, Sur les intégrales de Fourier absolument convergentes et leur application à une transformation fonctionelle, Neuvième Congrès Math. Scandinaves, Helsingfors, 1938.
  • Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954. MR 0068627
  • [Che] E. W. Cheney, Introduction to approximation theory, Chelsea, New York, 1982.
  • P. Erdös and P. Turán, On a problem in the theory of uniform distribution. I, Nederl. Akad. Wetensch., Proc. 51 (1948), 1146–1154 = Indagationes Math. 10, 370–378 (1948). MR 0027895
  • Carl-Gustav Esseen, On the Liapounoff limit of error in the theory of probability, Ark. Mat. Astr. Fys. 28A (1942), no. 9, 19. MR 0011909
  • [Fav] J. Favard, Sur les meilleurs procédés d'approximation de certaines classes de fonctions par des polynômes trigonométriques, Bull. Sci. Math. 61 (1937), 209-224, 243-256.
  • [Fej] L. Fejer, Einige Sätze, die sich auf das Vorzeichen einer ganzen rationalen Function bezeihen nebst Anwendungen, Monatsh. Math. Phys. 35 (1928), 305-344.
  • William Feller, An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
  • S. W. Graham and Jeffrey D. Vaaler, Extremal functions for the Fourier transform and the large sieve, Topics in classical number theory, Vol. I, II (Budapest, 1981) Colloq. Math. Soc. János Bolyai, vol. 34, North-Holland, Amsterdam, 1984, pp. 599–615. MR 781154
  • S. W. Graham and Jeffrey D. Vaaler, A class of extremal functions for the Fourier transform, Trans. Amer. Math. Soc. 265 (1981), no. 1, 283–302. MR 607121, 10.1090/S0002-9947-1981-0607121-1
  • L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0419394
  • Michel Loève, Probability theory, Third edition, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748
  • [Log] B. F. Logan, Bandlimited functions bounded below over an interval, Notices Amer. Math. Soc. 24 (1977), A-331.
  • Hugh L. Montgomery, The analytic principle of the large sieve, Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. MR 0466048, 10.1090/S0002-9904-1978-14497-8
  • H. L. Montgomery and R. C. Vaughan, Hilbert’s inequality, J. London Math. Soc. (2) 8 (1974), 73–82. MR 0337775
  • H. Niederreiter and Walter Philipp, Berry-Esseen bounds and a theorem of Erdős and Turán on uniform distribution 𝑚𝑜𝑑1, Duke Math. J. 40 (1973), 633–649. MR 0337873
  • [P-P] M. Plancherel and G. Polya, Fonctions entières et intégrales de Fourier multiples, (Seconde partie) Comment. Math. Helv. 10 (1938), 110-163.
  • Hȧkan Prawitz, Limits for a distribution, if the characteristic function is given in a finite domain, Skand. Aktuarietidskr. (1972), 138–154 (1973). MR 0375429
  • H. Prawitz, Ungleichungen für den absoluten Betrag einer charackteristischen Funktion, Skand. Aktuarietidskr. (1973), 11–16 (German). MR 0350809
  • H. Prawitz, Weitere Ungleichungen für den absoluten Betrag einer charakteristischen Funktion, Scand. Actuar. J. (1975), 21–28 (German). MR 0372947
  • Hȧkan Prawitz, Zur Variationsrechnung für die Verteilungsfunktionen, Skand. Aktuarietidskr. (1972), 202–214 (1973) (German). MR 0346624
  • Hȧkan Prawitz, On the remainder in the central limit theorem. I. One dimensional independent variables with finite absolute moments of third order, Scand. Actuar. J. 3 (1975), 145–156. MR 0397839
  • Atle Selberg, Remarks on sieves, Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972) Univ. Colorado, Boulder, Colo., 1972, pp. 205–216. MR 0389802
  • Harold S. Shapiro, Topics in approximation theory, Springer-Verlag, Berlin-New York, 1971. With appendices by Jan Boman and Torbjörn Hedberg; Lecture Notes in Math., Vol. 187. MR 0437981
  • [SNa] B. Sz. Nagy, Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. II, Ber. Math.-Phys. Kl. Sächs Akad. Wiss. Leipzig 91 (1939).
  • Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
  • A. F. Timan, Theory of approximation of functions of a real variable, Translated from the Russian by J. Berry. English translation edited and editorial preface by J. Cossar. International Series of Monographs in Pure and Applied Mathematics, Vol. 34, A Pergamon Press Book. The Macmillan Co., New York, 1963. MR 0192238
  • V. M. Zolotarev, Absolute estimate of the remainder in the central limit theorem, Teor. Verojatnost. i Primenen. 11 (1966), 108–119 (Russian, with English summary). MR 0198531
  • [Zo2] V. M. Zolotarev, Some inequalities in probability theory and their application in sharpening the Lyapunov theorem, Soviet Math. Dokl. 8 (1967), 1427-1430.
  • V. M. Zolotarev, A sharpening of the inequality of Berry-Esseen, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 332–342. MR 0221570
  • A. Zygmund, Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR 0236587

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