×

zbMATH — the first resource for mathematics

Some extremal functions in Fourier analysis. (English) Zbl 0575.42003
This is a survey paper concerning a collection of topics in the areas of entire functions of finite type, almost periodic functions and extremal properties. One of the key theorems concerns the class \(E^ p\) of entire functions F of exponential type \(\leq 2\pi\) such that \(\int_{{\mathbb{R}}}| F(x)|^ p dx<\infty\), (where \(0<p<\infty)\). The following interpolation formula is established: \[ F(z)=((\sin \pi z)/\pi)^ 2\{\sum^{\infty}_{m=-\infty}F(m)(z-m)^{- 2}+\sum^{\infty}_{n=-\infty}F'(n)(z-n)^{-1}\}, \] with uniform convergence on compact sets; further, if \(p=2\) then \[ \int^{1}_{- 1}F(x)e^{-itx} dx=(1-| t|)(\sum^{\infty}_{m=- \infty}F(m)e^{-2\pi imt})+ \]
\[ (1/(2\pi i))(sgn t)(\sum^{\infty}_{n=-\infty}F'(n)e^{-2\pi int}). \] In the interpolation formula set \(F(n)=1\) for \(n\geq 0\), \(F(n)=0\) for \(n<0\), \(F'(0)=2\), \(F'(n)=0\) for \(n\neq 0\), and denote the resulting sum by B(z). The author discusses this function, which was introduced by A. Beurling. Among entire functions F of type \(2\pi\) satisfying F(x)\(\geq sgn x\), (x\(\in {\mathbb{R}})\), B achieves the (unique) minimum of the functional \(\int_{{\mathbb{R}}}(F(x)-sgn x)dx\), namely, 1. These ideas are also applied to almost periodic (trigonometric) polynomials; for example, it is shown that \(\| f\|_{\infty}\leq (1/(4\delta))\| f'\|_{\infty},\) where \(f(x)=\sum^{n}_{j=1}a_ j\exp (2\pi i\lambda_ jx)\) and \(| \lambda_ j| \geq \delta >0\) for each j, (and \(\| \cdot \|_{\infty}\) denotes the supremum over \({\mathbb{R}})\).
Reviewer: Ch.Dunkl

MSC:
42A10 Trigonometric approximation
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
30D10 Representations of entire functions of one complex variable by series and integrals
11N35 Sieves
PDF BibTeX XML Cite
Full Text: DOI
References:
[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. (2) 25 (1982), no. 3, 269 – 277. · Zbl 0499.10037
[3] 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. · Zbl 0238.60020
[4] S. N. Bernstein, Sur une propriété des fonctions entieres, C. R. Acad. Sci. 176 (1923), 1603-1605. · JFM 49.0215.02
[5] Andrew C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates, Trans. Amer. Math. Soc. 49 (1941), 122 – 136. · Zbl 0025.34603
[6] A. Beurling, Sur les intégrales de Fourier absolument convergentes et leur application à une transformation fonctionelle, Neuvième Congrès Math. Scandinaves, Helsingfors, 1938. · JFM 65.0483.02
[7] Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954. · Zbl 0058.30201
[8] E. W. Cheney, Introduction to approximation theory, Chelsea, New York, 1982. · Zbl 0535.41001
[9] 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). · Zbl 0031.25402
[10] Carl-Gustav Esseen, On the Liapounoff limit of error in the theory of probability, Ark. Mat. Astr. Fys. 28A (1942), no. 9, 19. · Zbl 0027.33902
[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. · Zbl 0017.25101
[12] L. Fejer, Einige Sätze, die sich auf das Vorzeichen einer ganzen rationalen Function bezeihen nebst Anwendungen, Monatsh. Math. Phys. 35 (1928), 305-344. · JFM 54.0314.03
[13] William Feller, An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. · Zbl 0077.12201
[14] 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. · Zbl 0552.10028
[15] 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. · Zbl 0483.42007
[16] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. · Zbl 0281.10001
[17] Michel Loève, Probability theory, Third edition, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. · Zbl 0095.12201
[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), no. 4, 547 – 567. · Zbl 0408.10033
[20] H. L. Montgomery and R. C. Vaughan, Hilbert’s inequality, J. London Math. Soc. (2) 8 (1974), 73 – 82. · Zbl 0281.10021
[21] H. Niederreiter and Walter Philipp, Berry-Esseen bounds and a theorem of Erdős and Turán on uniform distribution \?\?\?1, Duke Math. J. 40 (1973), 633 – 649. · Zbl 0273.10043
[22] M. Plancherel and G. Polya, Fonctions entières et intégrales de Fourier multiples, (Seconde partie) Comment. Math. Helv. 10 (1938), 110-163. · Zbl 0018.15204
[23] HÈ&sect;kan Prawitz, Limits for a distribution, if the characteristic function is given in a finite domain, Skand. Aktuarietidskr. (1972), 138 – 154 (1973).
[24] H. Prawitz, Ungleichungen für den absoluten Betrag einer charackteristischen Funktion, Skand. Aktuarietidskr. (1973), 11 – 16 (German). · Zbl 0275.60026
[25] H. Prawitz, Weitere Ungleichungen für den absoluten Betrag einer charakteristischen Funktion, Scand. Actuar. J. , posted on (1975), 21 – 28 (German). · Zbl 0317.60008
[26] HÈ&sect;kan Prawitz, Zur Variationsrechnung für die Verteilungsfunktionen, Skand. Aktuarietidskr. (1972), 202 – 214 (1973) (German). · Zbl 0273.49036
[27] HÈ&sect;kan 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. · Zbl 0317.60009
[28] Atle Selberg, Remarks on sieves, Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972) Univ. Colorado, Boulder, Colo., 1972, pp. 205 – 216. · Zbl 0341.10041
[29] 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. · Zbl 0213.08501
[30] B. Sz. Nagy, Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. II, Ber. Math.-Phys. Kl. Sächs Akad. Wiss. Leipzig 91 (1939). · JFM 65.0272.01
[31] 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. · Zbl 0232.42007
[32] 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. · Zbl 0117.29001
[33] 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).
[34] V. M. Zolotarev, Some inequalities in probability theory and their application in sharpening the Lyapunov theorem, Soviet Math. Dokl. 8 (1967), 1427-1430. · Zbl 0185.46802
[35] V. M. Zolotarev, A sharpening of the inequality of Berry-Esseen, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 332 – 342. · Zbl 0157.25501
[36] A. Zygmund, Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.