Positive Definiteness of Complex Piecewise Linear Functions and Some of Its Applications

V. P. Zastavnyi ; A. Manov

Matematičeskie zametki, Tome 103 (2018) no. 4, pp. 519-535.

Résumé

Given $\alpha\in(0,1)$ and $c=h+i\beta$, $h,\beta\in\mathbb R$, the function $f_{\alpha,c}\colon\mathbb R\to\mathbb C$ defined as follows is considered: (1) $f_{\alpha,c}$ is Hermitian, i.e., $f_{\alpha,c}(-x)=\overline{f_{\alpha,c}(x)}$, $x\in\mathbb R$; (2) $f_{\alpha,c}(x)=0$ for $x>1$; moreover, on each of the closed intervals $[0,\alpha]$ and $[\alpha,1]$, the function $f_{\alpha,c}$ is linear and satisfies the conditions $f_{\alpha,c}(0)=1$, $f_{\alpha,c}(\alpha)=c$, and $f_{\alpha,c}(1)=0$. It is proved that the complex piecewise linear function $f_{\alpha,c}$ is positive definite on $\mathbb R$ if and only if $$ m(\alpha)\le h\le 1-\alpha\quad \text{and}\quad |\beta|\le\gamma(\alpha,h), $$ where $$ m(\alpha)= \begin{cases} 0{} \text{if } 1/\alpha\notin\mathbb N, \\ -\alpha{} \text{if }1/\alpha\in\mathbb N. \end{cases} $$ If $m(\alpha)$ and $\alpha\in\mathbb Q$, then $\gamma(\alpha,h)>0$; otherwise, $\gamma(\alpha,h)=0$. This result is used to obtain a criterion for the complete monotonicity of functions of a special form and prove a sharp inequality for trigonometric polynomials.

Détail

Keywords: positive definite function, piecewise linear function, completely monotone function, Bochner–Khinchine theorem, Bernstein's inequality.

Bibliographie

[1] A. Manov, V. Zastavnyi, “Positive definiteness of piecewise-linear function”, Expo. Math., 35:3 (2017), 357–361 | DOI | MR | Zbl

[2] E. Lukacs, Characteristic Functions, Hafner Publ., New York, 1970 | MR | Zbl

[3] R. E. Williamson, “Multiply monotone functions and their Laplace transforms”, Duke Math. J., 23 (1956), 189–207 | DOI | MR | Zbl

[4] V. P. Zastavnyi, “On positive definiteness of some functions”, J. Multivariate Anal., 73:1 (2000), 55–81 | DOI | MR | Zbl

[5] P. I. Lizorkin, “Otsenki trigonometricheskikh integralov i neravenstvo Bernshteina dlya drobnykh proizvodnykh”, Izv. AN SSSR. Ser. matem., 29:1 (1965), 109–126 | MR | Zbl

[6] R. M. Trigub, E. S. Belinsky, Fourier Analysis and Approximation of Functions, Kluwer Acad. Publ., Dordrecht, 2004 | MR | Zbl

[7] E. A. Gorin, “Neravenstva Bernshteina s tochki zreniya teorii operatorov”, Vestn. Kharkovsk. un-ta. Ser. prikl. matem. i mekh., 205:45 (1980), 77–105 | MR | Zbl

[8] V. V. Arestov, P. Yu. Glazyrina, “Neravenstvo Bernshteina–Sege dlya drobnykh proizvodnykh trigonometricheskikh polinomov”, Tr. IMM UrO RAN, 20, no. 1, 2014, 17–31 | MR

[9] S. B. Gashkov, “Neravenstvo Bernshteina, tozhdestvo Rissa i formula Eilera dlya ryada obratnykh kvadratov”, Matem. prosv., ser. 3, 18, Izd-vo MTsNMO, M., 2014, 143–171

[10] O. L. Vinogradov, “Tochnye otsenki pogreshnostei formul tipa chislennogo differentsirovaniya na klassakh tselykh funktsii konechnoi stepeni”, Sib. matem. zhurn., 48:3 (2007), 538–555 | MR | Zbl

[11] N. I. Akhiezer, Lektsii ob integralnykh preobrazovaniyakh, Vischa shkola, Kharkov, 1984 | MR | Zbl

[12] N. N. Vakhaniya, V. I. Tarieladze, S. A. Chobanyan, Veroyatnostnye raspredeleniya v banakhovykh prostranstvakh, Nauka, M., 1985 | MR | Zbl

[13] Z. Sasvári, Positive Definite and Definitizable Functions, Akademie Verlag, Berlin, 1994 | MR | Zbl

[14] T. M. Bisgaard, Z. Sasvári, Characteristic Functions and Moment Sequences. Positive Definiteness in Probability, Nova Sci. Publ., Huntington, NY, 2000 | MR | Zbl

[15] Zh.-P. Kakhan, Absolyutno skhodyaschiesya ryady Fure, Mir, M., 1976 | MR | Zbl

[16] N. I. Akhiezer, Klassicheskaya problema momentov i nekotorye voprosy analiza, svyazannye s neyu, Fizmatgiz, M., 1961 | MR | Zbl

[17] W. Feller, “Completely monotone functions and sequences”, Duke Math. J., 5 (1939), 661–674 | MR | Zbl

[18] Z. Sasvári, Multivariate Characteristic and Correlation Functions, De Gruyter Stud. Math., 50, Walter de Gruyter, Berlin, 2013 | MR | Zbl

[19] R. L. Schilling, R. Song, Z. Vondraček, Bernstein Functions, De Gruyter Stud. Math., 37, Walter de Gruyter, Berlin, 2010 | MR | Zbl

[20] D. V. Widder, The Laplace Transform, Princeton Univ. Press, Princeton, 1941 | MR | Zbl

[21] E. Lib, M. Loss, Analiz, Universitetskaya seriya, 1, Nauchnaya kniga, Novosibirsk, 1998 | MR | Zbl

[22] B. Sz.-Nagy, “Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. I. Periodischer Fall”, Ber. Verh. Sächs. Akad. Leipzig, 90 (1938), 103–134 | Zbl

[23] S. A. Telyakovskii, “O normakh trigonometricheskikh polinomov i priblizhenii differentsiruemykh funktsii lineinymi srednimi ikh ryadov Fure. I”, Sbornik rabot po lineinym metodam summirovaniya ryadov Fure, Tr. MIAN SSSR, 62, Izd-vo AN SSSR, M., 1961, 61–97 | MR | Zbl

[24] A. I. Kozko, “The exact constants in the Bernstein–Zygmund–Szegö inequalities with fractional derivatives and the Jackson–Nikolskii inequality for trigonometric polynomials”, East J. Approx., 4:3 (1998), 391–416 | MR | Zbl

[25] V. P. Zastavnyi, “Positive definite functions and sharp inequalities for periodic functions”, Ural. Math. J., 3:2 (2017), 82–99