Turan's Inequalities for Trigonometric Polynomials

We present here a technique for establishing inequalities ofthe form in the set of alltrigonometric polynomials of order n which have only real zeros.The function is assumed to be convex and increasing on [0,). As a corollary of the main result we get Turan's inequalities with the exact constantc(n, k, q) for each 1 q , n and k.     

A Sharp Remez Inequality for Trigonometric Polynomials

We obtain a sharp Remez inequality for the trigonometric polynomial T _n of degree n on [0,2π): $$\|T_n \|_{L_\infty([0,2\pi))} \le \biggl(1+2\tan^2 \frac{n\beta}{4m} \biggr) { \|T_n \|_{L_\infty ([0,2\pi) \setminus B )}}, $$ where $\frac{2\pi}{m}$ is the minimal period of T _n and $|B|=\beta<\frac {2\pi m}{n}$ is a measurable subset of $\mathbb {T}$ . In particular, this gives the asymptotics of the sharp constant in the Remez inequality: for a fixed n, $$\mathcal{C}(n, \beta)=1+ \frac{(n\beta)^2}{8} +O \bigl(\beta^4\bigr), \quad\beta \to0, $$ where $$\mathcal{C}(n,\beta):= \sup_{|B|=\beta}\sup_{T_n} \frac{ \|T_n \|_{L_\infty([0,2\pi ))}}{ \|T_n \|_{L_\infty ([0,2\pi) \setminus B )}}. $$ We also obtain sharp Nikol’skii’s inequalities for the Lorentz spaces and net spaces. Multidimensional variants of Remez and Nikol’skii’s inequalities are investigated.     

Sharp Inequalities for the Product of Polynomials

Let f₁(z),..., f_m(z) be polynomials with complex coefficients,and let their product be of degree n. For any polynomial, let||f|| be the maximum of |f(z)| on the unit circle. We determineconstants C_m < 2 for which for any n. The inequalities are asymptotically sharp as n .This improves earlier results of Gel'fond and Mahler, who gavethe constants e and 2 respectively. If f₁,..., f_m have realcoefficients, we show that for all m 2 and that this is asymptotically sharp. That is,in the real case, the best constant does not depend upon m form 2.     

Some Inequalities for Derivatives of Trigonometric and Algebraic Polynomials

We investigate inequalities for derivatives of trigonometric and algebraic polynomials in weighted L ^P spaces with weights satisfying the Muckenhoupt A _p condition. The proofs are based on an identity of Balázs and Kilgore [1] for derivatives of trigonometric polynomials. Also an inequality of Brudnyi in terms of rth order moduli of continuity ω_r will be given. We are able to give values to the constants in the inequalities.     

Application of Conformal Mappings to Inequalities for Trigonometric Polynomials

In this paper, we obtain inequalities for trigonometric and algebraic polynomials supplementing and strengthening the classical results going back to papers of S. N. Bernstein and I. I. Privalov. The method of proof is based on the construction of the conformal and univalent mapping from a given trigonometric polynomial and on the application of results of the geometric theory of functions of a complex variable to this mapping.     

Inequalities for Moduli of Continuity in Abstract Banach Spaces

Suppose that X is a Banach space, K denotes the set of real numbers R or the set of nonnegative real numbers R _{+}, is a family of linear operators from X into X such that T _0=I is the identity operator in X, for all , and there exists M such that for all . The expression is called the rth order modulus of continuity of an element x with step h in the space X with respect to the family A(K). The properties of are studied. Bibliography: 3 titles.     

One-Sided Integral Approximations of the Generalized Poisson Kernel by Trigonometric Polynomials

We consider the generalized Poisson kernel Π _q,α = cos(απ/2)P + sin(απ/2)Q with q ∈ (?1, 1) and α ∈ ?, which is a linear combination of the Poisson kernel \(P(t) = 1/2 + \sum\nolimits_{k = 1}^\infty {{q^k}} \cos kt\)and the conjugate Poisson kernel \(Q(t) = \sum\nolimits_{k = 1}^\infty {{q^k}} \sin kt\). The values of the best integral approximation to the kernel Π _q,α from below and from above by trigonometric polynomials of degree not exceeding a given number are found. The corresponding polynomials of the best one-sided approximation are obtained.     

Sharp Inequalities for the Zeros of Polynomials and Power Series

Let f be a monic polynomial of degree n with zeros z₁,…, z _n. We establish sharp estimates for $$\mathop \sum \limits_{\nu=1}^k \mid z_\nu \mid \quad \quad {\rm and}\quad \quad \mathop \sum \limits_{\nu=1}^k \mid \Im z_\nu \mid \quad \quad (k=1,2,\cdots,n).$$ Results for the zeros of power series are obtained as consequences.     

Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines

We obtain new inequalities of different metrics for differentiable periodic functions. In particular, for p, q (0, ], q > p, and s [p, q], we prove that functions satisfy the unimprovable inequality

Sharp Estimates for Errors of Numerical Differentiation Type Formulas on Trigonometric Polynomials

Sharp estimates for errors of formulas of numerical differentiation type on trigonometric polynomials are established. These estimates generalize the well-known Bernshtein-type inequalities. Bibliography: 18 titles.     

On the Problem of Describing Sequences of Best Trigonometric Rational Approximations

For a strictly decreasing sequence a_{n n=0} of nonnegative real numbers converging to zero, we construct a continuous 2-periodic function f such that R^T _n(f) = a_n, n=0,1,2,..., where R^T _n(f) are best approximations of the function f in uniform norm by trigonometric rational functions of degree at most n.     

Sharp Inequalities between the Best Root-Mean-Square Approximations of Analytic Functions in the Disk and Some Smoothness Characteristics in the Bergman Space

Mathematical Notes - In Jackson–Stechkin type inequalities for the smoothness characteristic $$\Lambda_m(f)$$ , $$m\in\mathbb N$$ , we find exact constants determined by averaging the norms...     

On the Exact Asymptotics of the Best Relative Approximations of Classes of Periodic Functions by Splines

We obtain the exact asymptotics (as n ) of the best L ₁-approximations of classes of periodic functions by splines s S _{2n, r – 1} and s S _{2n, r + k – 1} (S _{2n, r} is the set of 2-periodic polynomial splines of order r and defect 1 with nodes at the points k/n, k Z) under certain restrictions on their derivatives.     

On the Continuity of Best Approximations by Constants on Balls in Metric Measure Spaces

Mathematical Notes - Conditions for the constants of best approximation in the metric of the spaces Lp(B) to be continuous or semicontinuous as functions of the center of a ball B of fixed radius...     

Sharp Estimates for the Norms of Differences in Terms of the Norms of the Derivatives and the Best Majorants of Moduli of Continuity of Periodic Functions

Let C be the space of 2-periodic continuous functions with uniform norm, let _r(f,h) be a modulus of continuity of order r of a function f, and let

Lacunary Interpolation by Antiperiodic Trigonometric Polynomials

The problem of lacunary trigonometric interpolation is investigated. Does a trigonometric polynomial T exist which satisfies T(x _k) = a _k, D ^m T(x _k) = b _k, 0 k n – 1, where x _k = k/n is a nodal set, a _k and b _k are prescribed complex numbers, and m N. Results obtained by several authors for the periodic case are extended to the antiperiodic case. In particular solvability is established when n as well as m are even. In this case a periodic solution does not exist.