Michael Ganzburg - Academia.edu (original) (raw)
Papers by Michael Ganzburg
Journal of Mathematical Analysis and Applications, 2023
Asymptotic relations between zeta functions (such as, ζ(s), β(s), and other Dirichlet Lfunctions)... more Asymptotic relations between zeta functions (such as, ζ(s), β(s), and other Dirichlet Lfunctions) and interpolation differences of functions like |y| s and their interpolating entire functions of exponential type 1 are discussed. New criteria for zeros of the zeta functions in the critical strip in terms of integrability of the interpolation differences are obtained as well.
Sbornik Mathematics, Mar 1, 2018
We extend Markov's and Nagy's theorems on best approximation by entire functions of exponential t... more We extend Markov's and Nagy's theorems on best approximation by entire functions of exponential type in the L 1 (R) metric to some complex-valued integrable and locally integrable functions. We use these results for finding sharp constants of best approximation in L 1 (R) and L ∞ (R) on some complex convolution classes. For classes of real-valued convolutions those constants were found by Akhiezer. As an example, we apply these results to the Schwarz-type kernel and to the corresponding convolution classes. The following properties of A σ (f) p for f ∈ L p (R), 1 ≤ p ≤ ∞, are valid: 1. A σ (f) p is a nonincreasing function of σ ∈ (0, ∞), 1 ≤ p ≤ ∞.
Russian Mathematical Surveys, Feb 28, 1979
Journal of Approximation Theory, Feb 1, 2023
Let V be a symmetric convex body in R m. We prove sharp Bernstein-type inequalities for entire fu... more Let V be a symmetric convex body in R m. We prove sharp Bernstein-type inequalities for entire functions of exponential type with the spectrum in V and discuss certain properties of the extremal functions. Markov-type inequalities with sharp constants for algebraic polynomials on V and certain non-symmetric convex bodies are proved as well.
arXiv (Cornell University), Jul 16, 2020
Let V ⊂ R m be a convex body, symmetric about all coordinate hyperplanes, and let PaV , a ≥ 0, be... more Let V ⊂ R m be a convex body, symmetric about all coordinate hyperplanes, and let PaV , a ≥ 0, be a set of all algebraic polynomials whose Newton polyhedra are subsets of aV. We prove a limit equality as a → ∞ between the sharp constant in the multivariate Markov-Bernstein-Nikolskii type inequalities for polynomials from PaV and the corresponding constant for entire functions of exponential type with the spectrum in V .
Математический сборник, 2018
Точные значения величины наилучшего приближения для комплекснозначных периодических функций Теоре... more Точные значения величины наилучшего приближения для комплекснозначных периодических функций Теорема Секефальви-Надя о наилучшем приближении тригонометрическими полиномами в метрике L1 обобщается на случай приближения некоторых комплекснозначных периодических функций. Полученный результат применяется для нахождения точных констант наилучшего приближения в метриках L1 и L∞ на некоторых комплексных классах сверток. Для классов вещественнозначных сверток эти величины были найдены С. М. Никольским. В качестве примера данные результаты применяются для ядра Шварца и соответствующих классов сверток. Библиография: 20 названий.
Springer proceedings in mathematics, Oct 20, 2011
Page 1. Lagrange Interpolation and New Asymptotic Formulae for the Riemann Zeta Function Michael ... more Page 1. Lagrange Interpolation and New Asymptotic Formulae for the Riemann Zeta Function Michael I. Ganzburg Abstract An asymptotic representation for the Riemann zeta function ζ(s) in terms of the Lagrange interpolation ...
Journal of Mathematical Analysis and Applications, 2010
Let a > 0 be a fixed number. A function f : R → R is said to be a-shift-generating (a-SG) if for ... more Let a > 0 be a fixed number. A function f : R → R is said to be a-shift-generating (a-SG) if for every x ∈ R, { f (an − x)} ∞ n=−∞ is a totally positive sequence and it does not coincide with a sequence of the form {Aλ n } ∞ n=−∞ , where A 0 and λ > 0. In this paper, we describe all a-SG functions and obtain a new characterization of totally positive functions in the terms of a-SG functions. In addition, using characteristic properties of a-SG functions, we generalize the famous Jacobian identity in theory of elliptic functions.
Journal of Approximation Theory, Feb 1, 2008
We compute the best constants of approximation by entire functions of spherical type and by trigo... more We compute the best constants of approximation by entire functions of spherical type and by trigonometric polynomials of spherical degree on classes of functions f satisfying the condition k f L p 1, where p = 1 or 2 and is the Laplace operator.
Journal of Approximation Theory, Jul 1, 2008
We show that if {s k } ∞ k=1 is the sequence of all zeros of the L-function L(s,) := ∞ k=0 (−1) k... more We show that if {s k } ∞ k=1 is the sequence of all zeros of the L-function L(s,) := ∞ k=0 (−1) k (2k + 1) −s satisfying Re s k ∈ (0, 1), k = 1, 2,. .. , then any function from span {|x| s k } ∞ k=1 satisfies the pointwise rapid convergence property, i.e. there exists a sequence of polynomials Q n (f, x) of degree at most n such that f − Q n C[−1,1] C(f)E n (f), n=1, 2,. .. , and for every x ∈ [−1, 1], lim n→∞ (|f (x)−Q n (f, x)|)/E n (f)= 0, where E n (f) is the error of best polynomial approximation of f in C[−1, 1]. The proof is based on Lagrange polynomial interpolation to |x| s , Re s > 0, at the Chebyshev nodes. We also establish a new representation for |L(x,
Journal of Approximation Theory, Jun 1, 2003
In this paper we prove three conjectures of Revers on Lagrange interpolation for f l ðtÞ ¼ jtj l ... more In this paper we prove three conjectures of Revers on Lagrange interpolation for f l ðtÞ ¼ jtj l ; l40; at equidistant nodes. In particular, we describe the rate of divergence of the Lagrange interpolants L N ð f l ; tÞ for 0ojtjo1; and discuss their convergence at t ¼ 0: We also establish an asymptotic relation for max jtjp1 j jtj l À L N ð f l ; tÞj: The proofs are based on strong asymptotics for jtj l À L N ð f l ; tÞ;
arXiv (Cornell University), Jun 13, 2014
We prove a general M. Riesz-Schur-type inequality for entire functions of exponential type. If f ... more We prove a general M. Riesz-Schur-type inequality for entire functions of exponential type. If f and Q are two functions of exponential type σ > 0 and τ ≥ 0, respectively, and if Q is real-valued and the real and distinct zeros of Q are bounded away from each other, then |f (x)| ≤ (σ + τ)(A σ+τ (Q)) −1/2 Qf C(R) , x ∈ R , where A s (Q) def = inf x∈R [Q ′ (x)] 2 + s 2 [Q(x)] 2. We apply this inequality to the weights Q(x) def = sin (τ x) and Q(x) def = x and describe all extremal functions in the corresponding inequalities.
Математический сборник, 2015
Journal of Mathematical Analysis and Applications, Oct 1, 2008
Let E be a subset of R. We say that f : R → R is an E-Pólya frequency function if the kernel f (u... more Let E be a subset of R. We say that f : R → R is an E-Pólya frequency function if the kernel f (u − x) is totally positive on E × R and f is integrable on R. In this paper, we show that under some conditions on E, an E-Pólya frequency function is a Pólya frequency function. This strengthens Schoenberg's result on Pólya frequency functions.
Analysis Mathematica, Dec 20, 2022
In this paper we first introduce the unified definition of the sharp constant that includes const... more In this paper we first introduce the unified definition of the sharp constant that includes constants in three major problems of approximation theory, such as, inequalities for approximating elements, approximation of individual elements, and approximation on classes of elements. Second, we find sufficient conditions that imply limit relations between various sharp constants of approximation theory in general settings. Third, a number of examples from various areas of approximation theory illustrates the general approach.
Journal of Mathematical Analysis and Applications, Jul 1, 2021
Let V ⊂ R m be a convex body, symmetric about all coordinate hyperplanes, and let PaV , a ≥ 0, be... more Let V ⊂ R m be a convex body, symmetric about all coordinate hyperplanes, and let PaV , a ≥ 0, be a set of all algebraic polynomials whose Newton polyhedra are subsets of aV. We prove a limit equality as a → ∞ between the sharp constant in the multivariate Markov-Bernstein-Nikolskii type inequalities for polynomials from PaV and the corresponding constant for entire functions of exponential type with the spectrum in V .
Ukrainian Mathematical Journal, 1982
Ukrainian Mathematical Journal, 1983
... Proof of Theorem i. Using Lemma 3 for k =i, we obtain sup [] Vf(X)]Iv~ = sup sup ](y, V/(x))]... more ... Proof of Theorem i. Using Lemma 3 for k =i, we obtain sup [] Vf(X)]Iv~ = sup sup ](y, V/(x))]-~ o max [[ 9 ][v~ [[ [ ]ciRm; ... Mat. Zh., 22, No. 2, 74-83 (1981). 8. DR Wilhelmsen, "A Markov inequality in several dimensions," J. Approx. Theory, ii, No. 3, 216-220 (1974). ...
Bulletin of The London Mathematical Society, May 1, 2003
A relatively long-standing problem concerning the best constants of approximation by entire funct... more A relatively long-standing problem concerning the best constants of approximation by entire functions of exponential type on the class W r H ω is solved. The central tool is a limit relation between the best constants of approximation by trigonometric polynomials and entire functions of exponential type on W r H ω and its periodic analogue.
Journal of Approximation Theory, Sep 1, 2000
We study Remez-type inequalities for univariate and multivariate polynomials on bounded sets in w... more We study Remez-type inequalities for univariate and multivariate polynomials on bounded sets in weighted spaces and discuss their applications to Nikolskii-type inequalities and local approximation.
Journal of Mathematical Analysis and Applications, 2023
Asymptotic relations between zeta functions (such as, ζ(s), β(s), and other Dirichlet Lfunctions)... more Asymptotic relations between zeta functions (such as, ζ(s), β(s), and other Dirichlet Lfunctions) and interpolation differences of functions like |y| s and their interpolating entire functions of exponential type 1 are discussed. New criteria for zeros of the zeta functions in the critical strip in terms of integrability of the interpolation differences are obtained as well.
Sbornik Mathematics, Mar 1, 2018
We extend Markov's and Nagy's theorems on best approximation by entire functions of exponential t... more We extend Markov's and Nagy's theorems on best approximation by entire functions of exponential type in the L 1 (R) metric to some complex-valued integrable and locally integrable functions. We use these results for finding sharp constants of best approximation in L 1 (R) and L ∞ (R) on some complex convolution classes. For classes of real-valued convolutions those constants were found by Akhiezer. As an example, we apply these results to the Schwarz-type kernel and to the corresponding convolution classes. The following properties of A σ (f) p for f ∈ L p (R), 1 ≤ p ≤ ∞, are valid: 1. A σ (f) p is a nonincreasing function of σ ∈ (0, ∞), 1 ≤ p ≤ ∞.
Russian Mathematical Surveys, Feb 28, 1979
Journal of Approximation Theory, Feb 1, 2023
Let V be a symmetric convex body in R m. We prove sharp Bernstein-type inequalities for entire fu... more Let V be a symmetric convex body in R m. We prove sharp Bernstein-type inequalities for entire functions of exponential type with the spectrum in V and discuss certain properties of the extremal functions. Markov-type inequalities with sharp constants for algebraic polynomials on V and certain non-symmetric convex bodies are proved as well.
arXiv (Cornell University), Jul 16, 2020
Let V ⊂ R m be a convex body, symmetric about all coordinate hyperplanes, and let PaV , a ≥ 0, be... more Let V ⊂ R m be a convex body, symmetric about all coordinate hyperplanes, and let PaV , a ≥ 0, be a set of all algebraic polynomials whose Newton polyhedra are subsets of aV. We prove a limit equality as a → ∞ between the sharp constant in the multivariate Markov-Bernstein-Nikolskii type inequalities for polynomials from PaV and the corresponding constant for entire functions of exponential type with the spectrum in V .
Математический сборник, 2018
Точные значения величины наилучшего приближения для комплекснозначных периодических функций Теоре... more Точные значения величины наилучшего приближения для комплекснозначных периодических функций Теорема Секефальви-Надя о наилучшем приближении тригонометрическими полиномами в метрике L1 обобщается на случай приближения некоторых комплекснозначных периодических функций. Полученный результат применяется для нахождения точных констант наилучшего приближения в метриках L1 и L∞ на некоторых комплексных классах сверток. Для классов вещественнозначных сверток эти величины были найдены С. М. Никольским. В качестве примера данные результаты применяются для ядра Шварца и соответствующих классов сверток. Библиография: 20 названий.
Springer proceedings in mathematics, Oct 20, 2011
Page 1. Lagrange Interpolation and New Asymptotic Formulae for the Riemann Zeta Function Michael ... more Page 1. Lagrange Interpolation and New Asymptotic Formulae for the Riemann Zeta Function Michael I. Ganzburg Abstract An asymptotic representation for the Riemann zeta function ζ(s) in terms of the Lagrange interpolation ...
Journal of Mathematical Analysis and Applications, 2010
Let a > 0 be a fixed number. A function f : R → R is said to be a-shift-generating (a-SG) if for ... more Let a > 0 be a fixed number. A function f : R → R is said to be a-shift-generating (a-SG) if for every x ∈ R, { f (an − x)} ∞ n=−∞ is a totally positive sequence and it does not coincide with a sequence of the form {Aλ n } ∞ n=−∞ , where A 0 and λ > 0. In this paper, we describe all a-SG functions and obtain a new characterization of totally positive functions in the terms of a-SG functions. In addition, using characteristic properties of a-SG functions, we generalize the famous Jacobian identity in theory of elliptic functions.
Journal of Approximation Theory, Feb 1, 2008
We compute the best constants of approximation by entire functions of spherical type and by trigo... more We compute the best constants of approximation by entire functions of spherical type and by trigonometric polynomials of spherical degree on classes of functions f satisfying the condition k f L p 1, where p = 1 or 2 and is the Laplace operator.
Journal of Approximation Theory, Jul 1, 2008
We show that if {s k } ∞ k=1 is the sequence of all zeros of the L-function L(s,) := ∞ k=0 (−1) k... more We show that if {s k } ∞ k=1 is the sequence of all zeros of the L-function L(s,) := ∞ k=0 (−1) k (2k + 1) −s satisfying Re s k ∈ (0, 1), k = 1, 2,. .. , then any function from span {|x| s k } ∞ k=1 satisfies the pointwise rapid convergence property, i.e. there exists a sequence of polynomials Q n (f, x) of degree at most n such that f − Q n C[−1,1] C(f)E n (f), n=1, 2,. .. , and for every x ∈ [−1, 1], lim n→∞ (|f (x)−Q n (f, x)|)/E n (f)= 0, where E n (f) is the error of best polynomial approximation of f in C[−1, 1]. The proof is based on Lagrange polynomial interpolation to |x| s , Re s > 0, at the Chebyshev nodes. We also establish a new representation for |L(x,
Journal of Approximation Theory, Jun 1, 2003
In this paper we prove three conjectures of Revers on Lagrange interpolation for f l ðtÞ ¼ jtj l ... more In this paper we prove three conjectures of Revers on Lagrange interpolation for f l ðtÞ ¼ jtj l ; l40; at equidistant nodes. In particular, we describe the rate of divergence of the Lagrange interpolants L N ð f l ; tÞ for 0ojtjo1; and discuss their convergence at t ¼ 0: We also establish an asymptotic relation for max jtjp1 j jtj l À L N ð f l ; tÞj: The proofs are based on strong asymptotics for jtj l À L N ð f l ; tÞ;
arXiv (Cornell University), Jun 13, 2014
We prove a general M. Riesz-Schur-type inequality for entire functions of exponential type. If f ... more We prove a general M. Riesz-Schur-type inequality for entire functions of exponential type. If f and Q are two functions of exponential type σ > 0 and τ ≥ 0, respectively, and if Q is real-valued and the real and distinct zeros of Q are bounded away from each other, then |f (x)| ≤ (σ + τ)(A σ+τ (Q)) −1/2 Qf C(R) , x ∈ R , where A s (Q) def = inf x∈R [Q ′ (x)] 2 + s 2 [Q(x)] 2. We apply this inequality to the weights Q(x) def = sin (τ x) and Q(x) def = x and describe all extremal functions in the corresponding inequalities.
Математический сборник, 2015
Journal of Mathematical Analysis and Applications, Oct 1, 2008
Let E be a subset of R. We say that f : R → R is an E-Pólya frequency function if the kernel f (u... more Let E be a subset of R. We say that f : R → R is an E-Pólya frequency function if the kernel f (u − x) is totally positive on E × R and f is integrable on R. In this paper, we show that under some conditions on E, an E-Pólya frequency function is a Pólya frequency function. This strengthens Schoenberg's result on Pólya frequency functions.
Analysis Mathematica, Dec 20, 2022
In this paper we first introduce the unified definition of the sharp constant that includes const... more In this paper we first introduce the unified definition of the sharp constant that includes constants in three major problems of approximation theory, such as, inequalities for approximating elements, approximation of individual elements, and approximation on classes of elements. Second, we find sufficient conditions that imply limit relations between various sharp constants of approximation theory in general settings. Third, a number of examples from various areas of approximation theory illustrates the general approach.
Journal of Mathematical Analysis and Applications, Jul 1, 2021
Let V ⊂ R m be a convex body, symmetric about all coordinate hyperplanes, and let PaV , a ≥ 0, be... more Let V ⊂ R m be a convex body, symmetric about all coordinate hyperplanes, and let PaV , a ≥ 0, be a set of all algebraic polynomials whose Newton polyhedra are subsets of aV. We prove a limit equality as a → ∞ between the sharp constant in the multivariate Markov-Bernstein-Nikolskii type inequalities for polynomials from PaV and the corresponding constant for entire functions of exponential type with the spectrum in V .
Ukrainian Mathematical Journal, 1982
Ukrainian Mathematical Journal, 1983
... Proof of Theorem i. Using Lemma 3 for k =i, we obtain sup [] Vf(X)]Iv~ = sup sup ](y, V/(x))]... more ... Proof of Theorem i. Using Lemma 3 for k =i, we obtain sup [] Vf(X)]Iv~ = sup sup ](y, V/(x))]-~ o max [[ 9 ][v~ [[ [ ]ciRm; ... Mat. Zh., 22, No. 2, 74-83 (1981). 8. DR Wilhelmsen, "A Markov inequality in several dimensions," J. Approx. Theory, ii, No. 3, 216-220 (1974). ...
Bulletin of The London Mathematical Society, May 1, 2003
A relatively long-standing problem concerning the best constants of approximation by entire funct... more A relatively long-standing problem concerning the best constants of approximation by entire functions of exponential type on the class W r H ω is solved. The central tool is a limit relation between the best constants of approximation by trigonometric polynomials and entire functions of exponential type on W r H ω and its periodic analogue.
Journal of Approximation Theory, Sep 1, 2000
We study Remez-type inequalities for univariate and multivariate polynomials on bounded sets in w... more We study Remez-type inequalities for univariate and multivariate polynomials on bounded sets in weighted spaces and discuss their applications to Nikolskii-type inequalities and local approximation.