Some extremal functions in fourier analysis

Bulletin of the American Mathematical Society

Volumn 12, Issue 2, 1985, Pages 183-216

  Vaaler, Jeffrey D ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84968464817     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0273-0979-1985-15349-2     Document Type: Article

References (36)

1. 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)
2. R. C. Baker and G. Harman, Small fractional parts of quadratic forms, Proc. Edinburgh Math. Soc. 25 (1982), 269–277.
3. P. van Beek, An application of Fourier methods to the problem of sharpening the Berry-Esseen inequality, Z. Wahrsch. verw. Gebiete. 23 (1972), 187–196.
4. S. N. Bernstein, Sur une propriété des fonctions entieres, C. R. Acad. Sci. 176 (1923), 1603–1605.
5. A. C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates, Trans. Amer. Math. Soc. 49 (1941), 122–136.
6. A. Beurling, Sur les intégrales de Fourier absolument convergentes et leur application à une transformation fonctionelle, Neuvième Congrès Math. Scandinaves, Helsingfors, 1938.
7. R. P. Boas, Entire functions, Academic Press, New York, 1954.
8. E. W. Cheney, Introduction to approximation theory, Chelsea, New York, 1982.
9. P. Erdös and P. Turân, On a problem in the theory of uniform distribution. I, Indag. Math. 10 (1948), 370–378.
10. C. G. Esseen, On the Liapunov limit of error in the theory of probability, Ark. Mat.-Astr. Fys. 28A (1942), 1–19.
11. 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.
12. L. Fejer, Einige Sätze, die sich auf das Vorzeichen einer ganzen rationalen Function bezeihen nebst Anwendungen, Monatsh. Math. Phys. 35 (1928), 305–344.
13. W. Feller, An introduction to probability theory and its applications, Vol. II, Wiley, New York, 1971.
14. S. W. Graham and J. D. Vaaler, Extremal functions for the Fourier transform and the large sieve, Topics in Classical Number Theory (G. Halász, ed.), North-Holland, New York, 1984.
15. ——, A class of extremal functions for the Fourier transform, Trans. Amer. Math. Soc. 265 (1981), 283–302.
16. L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley, New York, 1974.
17. M. Loève, Probability theory, 3rd ed. van Nostrand, Princeton, N. J., 1963.
18. B. F. Logan, Bandlimited functions bounded below over an interval, Notices Amer. Math. Soc. 24 (1977), A-331.
19. Hugh L. Montgomery, The analytic principle of the large sieve, Bull. Amer. Math. Soc. 84 (1978), 547–567.
20. H. L. Montgomery and R. C. Vaughan, Hilbert’s inequality, J. London Math. Soc. (2) 8 (1974), 73–81.
21. H. Niederreiter and W. Philipp, Berry-Esseen bounds and a theorem of Erdös and Turón on uniform distribution mod 1, Duke Math. J. 40 (1973), 633–649.
22. M. Plancherel and G. Polya, Fonctions entières et intégrales de Fourier multiples, (Seconde partie) Comment. Math. Helv. 10 (1938), 110–163.
23. H. Prawitz, Limits for a distribution, if the characteristic function is given in a finite domain, Skand. Aktur Tidskr, (1972), 138–154.
24. ——, Ungleichungen für den absoluten Betrag einer charakteristischen Funktion, Skand. Akturatidskr (1973), 11–16.
25. ——, Weiter Ungleichungen für den absoluten Betrag einer charakteristischen Funktion, Scand. Actuarial J. (1975), 21–28.
26. ——, Zur Variationsrechnung für die Verteilungsfuntionen, Skand. Aktuar Tidskr. (1972), 202–208.
27. ——, On the remainder in the central limit theorem, Scand. Actuarial J. (1975), 145–156.
28. A. Selberg, Remarks on sieves, Proc. 1972 Number Theory Conf. (Univ. of Colorado, Boulder), pp. 205–216.
29. H. S. Shapiro, Topics in approximation theory, Lecture Notes in Math., vol. 187, SpringerVerlag, New York, 1971.
30. B. Sz. Nagy, Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. II, Ber. Math.-Phys. Kl. Sachs Akad. Wiss. Leipzig 91 (1939).
31. E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N. J., 1971.
32. A. F. Timan, Theory of approximation of functions of a real variable, Pergamon Press-Macmillan, New York, (1963).
33. V. M. Zolotarev, An absolute estimate of the remainder term in the central limit theorem, Theor. Probab. Appl. 11 (1966), 95–105.
34. ——, Some inequalities in probability theory and their application in sharpening the Lyapunov theorem, Soviet Math. Dokl. 8 (1967), 1427–1430.
35. ——, A sharpening of the inequality of Berry-Esseen, Z. Wahrsch. verw. Gebiete. 8 (1967), 332–342.
36. Zygmund, Trigonometric series, Vols. I, II, Cambridge Univ. Press, 1968.