Abstract
Given α ∈ (0, 1) and c = h + iβ, h, β ∈ R, the function fα,c: R → C defined as follows is considered: (1) fα,c is Hermitian, i.e., \({f_{\alpha ,c}}\left( { - x} \right)\overline {{f_{\alpha ,c}}\left( x \right)} ,x \in \mathbb{R};\), x ∈ R; (2) fα,c(x) = 0 for x > 1; moreover, on each of the closed intervals [0, α] and [α, 1], the function fα,c is linear and satisfies the conditions fα,c(0) = 1, fα,c(α) = c, and fα,c(1) = 0. It is proved that the complex piecewise linear function fα,c is positive definite on R if and only if m(α) ≤ h ≤ 1 − α and |β| ≤ γ(α, h), \(where m\left( \alpha \right) = \left\{ \begin{gathered} 0if1/\alpha \notin \mathbb{N}, \hfill \\ - \alpha if1/\alpha \in \mathbb{N}. \hfill \\ \end{gathered} \right.\) If m(α) ≤ h ≤ 1 − α and α ∈ Q, then γ(α, h) > 0; otherwise, γ(α, 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.
Similar content being viewed by others
References
A. Manov and V. Zastavnyi, “Positive definiteness of piecewise-linear function,” Expo. Math. 35 (3), 357–361 (2017).
E. Lukacs, Characteristic Functions (Hafner Publ., New York, 1970).
R. E. Williamson, “Multiply monotone functions and their Laplace transforms,” Duke Math. J. 23, 189–207 (1956).
V. P. Zastavnyi, “On positive definiteness of some functions,” J. Multivariate Anal. 73 (1), 55–81 (2000).
P. I. Lizorkin, “Bounds for trigonometrical integrals and an inequality of Bernstein for fractional derivatives,” Izv. Akad. Nauk SSSR Ser. Mat. 29 (1), 109–126 (1965).
R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation Functions (Kluwer Acad. Publ., Dordrecht, 2004).
E. A. Gorin, “Bernstein inequalities from the point of view of operator theory,” Vestn. Kharkov. Univ. Ser. Prikl. Mat. Mekh. 205 (45), 77–105 (1980).
V. V. Arestov and P. Yu. Glazyrina, “Bernstein–Szegöinequality for fractional derivatives of trigonometric polynomials,” Trudy Inst. Mat. i Mekh. UrO RAN 20 (1), 17–31 (2014) [Proc. Steklov Inst. Math. (Suppl.) 288, Suppl. 1, 13–28 (2015)].
S. B. Gashkov, “Bernstein’s inequality, Riesz identity, and Euler’s formula for the series of inverse squares,” in Mat. Pros., Ser. 3 (Izd. MTsNMO, Moscow, 2014), Vol. 18, pp. 143–171 [in Russian].
O. L. Vinogradov, “Sharp error estimates for the numerical differentiation formulas on the classes of entire functions of exponential type,” Sibirsk. Mat. Zh. 48 (3), 538–555 (2007) [Sib. Math. J. 48 (3), 430–445 (2007)].
N. I. Akhiezer, Lectures on Integral Transformations (Vyshcha Shkola, Kharkov, 1984) [in Russian].
N. N. Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions in Banach Spaces (Nauka, Moscow, 1985) [in Russian].
Z. Sasvári, Positive Definite and Definitizable Functions (Akademie Verlag, Berlin, 1994).
T. M. Bisgaard and Z. Sasvári, Characteristic Functions and Moment Sequences. Positive Definiteness in Probability (Nova Sci. Publ., Huntington, NY, 2000).
Zh.-P. Kakhan, Absolutely Convergent Fourier Series (Mir, Moscow, 1976) [in Russian].
N. I. Akhiezer, Classical Moment Problem and Some Related Questions of Analysis (Fizmatgiz, Moscow, 1961) [in Russian].
W. Feller, “Completely monotone functions and sequences,” Duke Math. J. 5, 661–674 (1939).
Z. Sasvári, Multivariate Characteristic and Correlation Functions, in De Gruyter Stud. Math. (Walter de Gruyter, Berlin, 2013), Vol.50.
R. L. Schilling, R. Song, and Z. Vondraček, Bernstein Functions, in De Gruyter Stud. Math. (Walter de Gruyter, Berlin, 2010), Vol.37.
D. V. Widder, Laplace Transform (Princeton Univ. Press, Princeton, 1941).
E. Lieb and M. Loss, Analysis (Amer. Math. Soc., Providence, RI, 1997; Nauchn. Kniga, Novosibirsk, 1998), Vol.1.
B. Sz.-Nagy, “über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. I. Periodischer Fall,” Ber. Verh. Sächs. Akad. Leipzig 90, 103–134 (1938).
S. A. Telyakovskii, “Norms of trigonometrical polynomials and approximation of differentiable functions by averages of their Fourier series. I,” in Trudy Mat. Inst. Steklova, Vol. 62: Collection of Articles on Linear Methods of Summation of Fourier Series (Izd. AN SSSR, Moscow, 1961), pp. 61–97 [in Russian].
A. I. Kozko, “The exact constants in the Bernstein–Zygmund–Szegöinequalities with fractional derivatives and the Jackson–Nikolskii inequality for trigonometric polynomials,” East J. Approx. 4 (3), 391–416 (1998).
V. P. Zastavnyi, “Positive definite functions and sharp inequalities for periodic functions,” Ural. Math. J. 3 (2), 82–99 (2017).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V. P. Zastavnyi, A. D. Manov, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 4, pp. 519–535.
Rights and permissions
About this article
Cite this article
Zastavnyi, V.P., Manov, A.D. Positive Definiteness of Complex Piecewise Linear Functions and Some of Its Applications. Math Notes 103, 550–564 (2018). https://doi.org/10.1134/S0001434618030227
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434618030227