Inequalities for exponential sums

T. Erdélyi

Sbornik. Mathematics, Tome 208 (2017) no. 3, pp. 433-464

Résumé

We study the classes \begin{gather*} {\mathscr E}_n:= \biggl\{f\colon f(t)=\sum_{j=1}^n{a_j e^{\lambda_jt}}, \ a_j, \lambda_j\in {\mathbb C} \biggr\}, \\ {\mathscr E}_n^+:= \biggl\{f\colon f(t)=\sum_{j=1}^n{a_j e^{\lambda_jt}}, \ a_j, \lambda_j\in {\mathbb C}, \ \operatorname{Re}(\lambda_j) \geqslant 0 \biggr\}, \\ {\mathscr E}_n^-:= \biggl\{f\colon f(t)=\sum_{j=1}^n{a_j e^{\lambda_jt}}, \ a_j, \lambda_j\in {\mathbb C}, \ \operatorname{Re}(\lambda_j)\leqslant 0 \biggr\}, \end{gather*} and $$ {\mathscr T}_n:= \biggl\{f\colon f(t)=\sum_{j=1}^n{a_j e^{i\lambda_jt}}, \ a_j\in {\mathbb C}, \ \lambda_1<\lambda_2<\dots<\lambda_n \biggr\}. $$ A highlight of this paper is the asymptotically sharp inequality $$ |f(0)|\leqslant (1+\varepsilon_n)3n\|f(t)e^{-9nt/2}\|_{L_2[0,1]}, \qquad f\in {\mathscr T}_n , $$ where $\varepsilon_n$ converges to $0$ rapidly as $n$ tends to $\infty$. The inequality $$ \sup_{0 \not \equiv f\in {\mathscr T}_n}{ \frac{|f(0)|}{\|f\|_{L_2{[0,1]}}}} \geqslant n $$ is also established. Our results improve an old result due to Halász and a recent result due to Kós. We prove several other related order-sharp results in this paper. Bibliography: 33 titles.

Détail

Keywords: exponential sums, Nikol'skii-, Bernstein- and Markov-type inequalities, infinite-finite range inequalities.

Bibliographie

[1] D. Benko, T. Erdélyi, J. Szabados, “The full Markov–Newman inequality for Müntz polynomials on positive intervals”, Proc. Amer. Math. Soc., 131:8 (2003), 2385–2391 | DOI | MR | Zbl

[2] S. N. Bernstein, Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d'une variable réelle, Collection de monographies sur la théorie des fonctions, Gauthier-Villars, Paris, 1926, x+208 pp. | Zbl

[3] R. P. Boas, Jr., “The derivative of a trigonometric integral”, J. London Math. Soc., S1-12:3 (1937), 164–165 | DOI | MR | Zbl

[4] P. Borwein, T. Erdélyi, Polynomials and polynomial inequalities, Grad. Texts in Math., 161, Springer-Verlag, New York, 1995, x+480 pp. | DOI | MR | Zbl

[5] P. Borwein, T. Erdélyi, “A sharp Bernstein-type inequality for exponential sums”, J. Reine Angew. Math., 476 (1996), 127–141 | MR | Zbl

[6] P. Borwein, T. Erdélyi, “Generalizations of Müntz's theorem via a Remez-type inequality for Müntz spaces”, J. Amer. Math. Soc., 10:2 (1997), 327–349 | DOI | MR | Zbl

[7] P. Borwein, T. Erdélyi, “Pointwise Remez- and Nikolskii-type inequalities for exponential sums”, Math. Ann., 316:1 (2000), 39–60 | DOI | MR | Zbl

[8] P. Borwein, T. Erdélyi, “Nikolskii-type inequalities for shift invariant function spaces”, Proc. Amer. Math. Soc., 134:11 (2006), 3243–3246 | DOI | MR | Zbl

[9] P. Borwein, T. Erdélyi, J. Zhang, “Müntz systems and orthogonal Müntz–Legendre polynomials”, Trans. Amer. Math. Soc., 342:2 (1994), 523–542 | DOI | MR | Zbl

[10] P. Borwein, T. Erdélyi, G. Kós, “Littlewood-type problems on $[0,1]$”, Proc. London Math. Soc. (3), 79:1 (1999), 22–46 | DOI | MR | Zbl

[11] P. Borwein, T. Erdélyi, G. Kós, “The multiplicity of the zero at $1$ of polynomials with constrained coefficients”, Acta Arith., 159:4 (2013), 387–395 | DOI | MR | Zbl

[12] S. Denisov, “On the size of the polynomials orthonormal on the unit circle with respect to a measure which is a sum of the Lebesgue measure and $p$ point masses”, Proc. Amer. Math. Soc., 144:3 (2016), 1029–1039 | DOI | MR | Zbl

[13] R. A. DeVore, G. G. Lorentz, Constructive approximation, Grundlehren Math. Wiss., 303, Springer-Verlag, Berlin, 1993, x+449 pp. | MR | Zbl

[14] R. D. Duffin, A. C. Schaeffer, “Some inequalities concerning functions of exponential type”, Bull. Amer. Math. Soc., 43:8 (1937), 554–556 | DOI | MR | Zbl

[15] T. Erdélyi, “Extremal properties of the derivatives of the Newman polynomials”, Proc. Amer. Math. Soc., 131:10 (2003), 3129–3134 | DOI | MR | Zbl

[16] T. Erdélyi, “Markov–Nikolskii type inequalities for exponential sums on finite intervals”, Adv. Math., 208:1 (2007), 135–146 | DOI | MR | Zbl

[17] T. Erdélyi, “Inequalities for exponential sums via interpolation and Turán-type reverse Markov inequalities”, Frontiers in interpolation and approximation, Pure Appl. Math. (Boca Raton), 282, Chapman Hall/CRC, Boca Raton, FL, 2007, 119–144 | MR | Zbl

[18] T. Erdélyi, “Coppersmith–Rivlin type inequalities and the order of vanishing of polynomials at $1$”, Acta Arith., 172:3 (2016), 271–284 | DOI | MR | Zbl

[19] T. Erdélyi, P. Nevai, “Generalized Jacobi weights, Christoffel functions and zeros of orthogonal polynomials”, J. Approx. Theory, 69:2 (1992), 111–132 | DOI | MR | Zbl

[20] G. Halász, “On the first and second main theorem in Turán's theory of power sums”, Studies in pure mathematics, Birkhäuser, Basel, 1983, 259–269 | DOI | MR | Zbl

[21] G. Kós, “Two Turán type inequalities”, Acta Math. Hungar., 119:3 (2008), 219–226 | DOI | MR | Zbl

[22] M. Lachance, E. B. Saff, R. S. Varga, “Inequalities for polynomials with a prescribed zero”, Math. Z., 168:2 (1979), 105–116 | DOI | MR | Zbl

[23] D. S. Lubinsky, “Dirichlet orthogonal polynomials with Laguerre weight”, J. Approx. Theory, 194 (2015), 146–156 | DOI | MR | Zbl

[24] F. L. Nazarov, “Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type”, St. Petersburg Math. J., 5:4 (1994), 663–717 | MR | Zbl

[25] S. M. Nikolskii, “Neravenstva dlya tselykh funktsii konechnoi stepeni i ikh primenenie v teorii differentsiruemykh funktsii mnogikh peremennykh”, Sbornik statei. Posvyaschaetsya akademiku Ivanu Matveevichu Vinogradovu k ego 60-letiyu, Tr. MIAN SSSR, 38, Izd-vo AN SSSR, M., 1951, 244–278 | MR | Zbl

[26] Q. I. Rahman, G. Schmeisser, Analytic theory of polynomials, London Math. Soc. Monogr. (N. S.), 26, The Clarendon Press, Oxford Univ. Press, Oxford, 2002, xiv+742 pp. | MR | Zbl

[27] P. W. Smith, “An improvement theorem for Descartes systems”, Proc. Amer. Math. Soc., 70:1 (1978), 26–30 | DOI | MR | Zbl

[28] G. Szegő, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., 23, 4th ed., Amer. Math. Soc., Providence, RI, 1975, xiii+432 pp. | MR | Zbl | Zbl

[29] G. Szegö, A. Zygmund, “On certain mean values of polynomials”, J. Analyse Math., 3 (1954), 225–244 | DOI | MR | Zbl

[30] V. P. Sklyarov, “The sharp constant in Markov's inequality for the Laguerre weight”, Sb. Math., 200:6 (2009), 887–897 | DOI | DOI | MR | Zbl

[31] P. Turán, “Remark on a theorem of Erhard Schmidt”, Mathematica (Cluj), 2(25) (1960), 373–378 | MR | Zbl

[32] P. Turán, On a new method of analysis and its applications, Pure Appl. Math. (N. Y.), John Wiley Sons, Inc., New York, 1984, xvi+584 pp. | MR | Zbl

[33] A. Zygmund, Trigonometric series, v. I, 2nd ed., Cambridge Univ. Press, New York, 1959, xii+383 pp. | MR | MR | Zbl | Zbl