Inequality of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in <Emphasis Type="Italic">L</Emphasis><Subscript>0</Subscript>

We study inequalities of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in the quasinorm of L ₀ and derivatives of any order. We present expressions for constants in these inequalities and obtain double-sided estimates for them.     

Inequalities of Bernstein and Jackson-Nikol’skii type and estimates of the norms of derivatives of Dirichlet kernels

We obtain Bernstein and Jackson-Nikol’skii inequalities for trigonometric polynomials with spectrum generated by the level surfaces of a function Λ(t), and study their sharpness under a specific choice of Λ(t). Estimates of the norms of derivatives of Dirichlet kernels with harmonics generated by the level surfaces of the function Λ(t) are established in L _p .     

The Turán-type inequalities for complex polynomials in <Emphasis Type="Italic">L</Emphasis><Subscript>0</Subscript>-metrics

In this paper we obtain inequalities for measures of trigonometric polynomials of power (P _n (e ^iφ )) and general (T _n (t)) types with the help of measures and their mth derivatives.     

Functions with derivatives given by polynomials in the function itself or a related function

We construct the set of holomorphic functions S ₁ = {f: Ω_f ? ? → ?} whose members have n-th order derivatives which are given by a polynomial of degree n+1 in the function itself. We also construct the set of holomorphic functions S ₂ = {g: Ω_g ? ? → ?} whose members have n-th order derivatives which are given by the product of the function itself with a polynomial of degree n in an element of S ₁. The union S ₁ ∪ S ₂ contains all the hyperbolic and trigonometric functions. We study the properties of the polynomials involved and derive explicit expressions for them. As particular results, we obtain explicit and elegant formulas for the n-th order derivatives of the hyperbolic functions tanh, sech, coth and csch and the trigonometric functions tan, sec, cot and csc.     

On the exact Markov inequality for k-monotone polynomials in uniform and L 1-norms

We consider the classical extremal problem of estimating norms of higher order derivatives of algebraic polynomials when their norms are given. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A. Markov, while Bernstein found the exact constant in the Markov inequality for monotone polynomials. In this note we give Markov-type inequalities for higher order derivatives in the general class of k-monotone polynomials. In particular, in case of first derivative we find the exact solution of this extremal problem in both uniform and L ₁-norms. This exact solution is given in terms of the largest zeros of certain Jacobi polynomials.     

Basic Polynomial Inequalities on Intervals and Circular Arcs

We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for complex algebraic polynomials of degree n having no zeros in the open unit disk. This is done by establishing the right Bernstein–Szeg?–Videnskii type inequality for real trigonometric polynomials of degree at most n on intervals shorter than the period. The paper is closely related to recent work by B. Nagy and V. Totik. In fact, their asymptotically sharp Bernstein-type inequality for complex algebraic polynomials of degree at most n on subarcs of the unit circle is recaptured by using more elementary methods. Our discussion offers a somewhat new way to see V.S. Videnskii’s Bernstein and Markov type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, two classical polynomial inequalities first published in 1960. A new Riesz–Schur type inequality for trigonometric polynomials is also established. Combining this with Videnskii’s Bernstein-type inequality gives Videnskii’s Markov-type inequality immediately.     

Multiplicative inequalities for the L 1 norm: Applications in analysis and number theory

The paper is devoted to multiplicative lower estimates for the L ₁ norm and their applications in analysis and number theory. Multiplicative inequalities of the following three types are considered: martingale (for the Haar system), complex trigonometric (for exponential sums), and real trigonometric. A new method for obtaining sharp bounds for the integral norm of trigonometric and power series is proposed; this method uses the number-theoretic and combinatorial characteristics of the spectrum. Applications of the method (both in H ¹ and L ₁) to an important class of power density spectra, including [n ^α] with 1 ≤ α < ∞, are developed. A new combinatorial theorem is proved that makes it possible to estimate the arithmetic characteristics of spectra under fairly general assumptions.     

Best polynomial approximations in L 2 of classes of 2π-periodic functions and exact values of their widths

We find sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity ofmth order in the space L ₂ as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of n-widths in L ₂.     

On Bernstein–Markov-type inequalities for multivariate polynomials in -norm

Bernstein–Markov-type inequalities provide estimates for the norms of derivatives of algebraic and trigonometric polynomials. They play an important role in Approximation Theory since they are widely used for verifying inverse theorems of approximation. In the past decades these inequalities were extended to the multivariate setting, but the main emphasis so far was on the uniform norm. It is considerably harder to derive Bernstein–Markov-type inequalities in the L_q-norm, and it requires introduction of new methods. In this paper we verify certain Bernstein–Markov-type inequalities in L_q-norm on convex and star-like domains. Special attention is given to the question of how the geometry of the domain affects the corresponding estimates.     

Kernels and best approximations related to the system of ultraspherical polynomials

We study the uniformly bounded orthonormal system of functions

where is the normalized system of ultraspherical polynomials. We investigate some approximation properties of the system and we show that these properties are similar to one's of the trigonometric system. First, we obtain estimates of L^p-norms of the kernels of the system . These estimates enable us to prove Nikol'skiı˘-type inequalities for -polynomials. Next, we prove directly that is a basis in each , where w is an arbitrary A_p-weight function. Finally, we apply these results to get sharp inequalities for the best -approximations in L^q in terms of the best -approximations in . For the trigonometric system such inequalities have been already known.     

On Polynomial Inequalities on Exponential Curves in ℂ n

We discuss Bernstein–Walsh type inequalities for holomorphic polynomials restricted to curves of the form $$\bigl(z,e^{P_{1}(z)},e^{P_{2}(z)},\ldots,e^{P_{d}(z)}\bigr)\in\mathbb{C}^{d+1},$$ where P ₁,P ₂,…,P _d are fixed polynomials on ? (such that the functions z and $e^{P_{k}(z)}$ are algebraically independent). The existence of such inequalities automatically implies the existence of associated Bernstein–Markov inequalities on the derivatives of polynomials restricted to the curve. The d=1 case has been much discussed in the recent literature. However, the d>1 case requires different techniques, and that is the subject of this work.     

Best approximations of periodic functions of several variables from the classes B p, θ Ω

We obtain order-sharp estimates of best approximations for the classes B _{p, θ} ^Ω of periodic functions of several variables by trigonometric polynomials whose spectra are generated by the level surfaces of the function Ω(t).     

Decomposition and approximation of multivariate functions on the cube

In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jφjψj , where each φj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each φjψj is a product of separated variable type and its smoothness is same as f . Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.     

A weak-type inequality for uniformly bounded trigonometric polynomials

This paper is devoted to refining the Bernstein inequality. Let D be the differentiation operator. The action of the operator Λ = D/n on the set of trigonometric polynomials T _n is studied: the best constant is sought in the inequality between the measures of the sets {x ∈ T: |Λt(x)| > 1} and {x ∈ T: |t(x)| > 1}. We obtain an upper estimate that is order sharp on the set of uniformly bounded trigonometric polynomials T _n ^C = {t ∈ T _n : ‖t‖ ≤ C}.     

On the best constants in inequalities of the Markov and Wirtinger types for polynomials on the half-line

The main topic of the paper is best constants in Markov-type inequalities between the norms of higher derivatives of polynomials and the norms of the polynomials themselves. The norm is the L² norm with Laguerre weight. The leading term of the asymptotics of the constants is determined and tight bounds for the principal coefficient in this term, which is the operator norm of a Volterra operator, are given. For best constants in inequalities of the Wirtinger type, the limit is computed and an asymptotic formula for the error term is presented.     

Comparison of approximation properties of generalized polynomials and splines

We establish that, for p ∈ [2, ∞), q = 1 or p = ∞, q ∈ [ 1, 2], the classes W _p^rof functions of many variables defined by restrictions on the L _p-norms of mixed derivatives of order r = (r ₁, r ₂, ..., r _m) are better approximated in the L _q-metric by periodic generalized splines than by generalized trigonometric polynomials. In these cases, the best approximations of the Sobolev classes of functions of one variable by trigonometric polynomials and by periodic splines coincide. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1011–1020, August, 1998.     

Inequalities for upper bounds of functionals on the classes w r H ω and their applications

We show that the well-known results on estimates of upper bounds of functionals on the classes W ^r H ^ω of periodic functions can be regarded as a special case of Kolmogorov-type inequalities for support functions of convex sets. This enables us to prove numerous new statements concerning the approximation of the classes W ^r H ^ω , establish the equivalence of these statements, and obtain new exact inequalities of the Bernstein-Nikol’skii type that estimate the value of the support function of the class H ^ω on the derivatives of trigonometric polynomials or polynomial splines in terms of the L ^ϱ -norms of these polynomials and splines.     

Widths in L2 of classes of differentiable functions,defined by higher-order moduli of continuity

Sharp inequalities are obtained in the space L₂, connecting the best approximations of differentiable 2-periodic functions by trigonometric polynomials with integrals containing the higher-order moduli of continuity of the derivatives of these functions. The Kolmogorov widths of certain classes of functions, defined by these moduli of continuity, are computed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 125–129, January, 1991.     

Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials

In the spaceL _q, 1<q<∞ we establish estimates for the orders of the best approximations of the classes of functions of many variablesB _1,θ ^r andB _p,α ^r by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials. It is shown that, in many cases, the estimates obtained in the present work are better in order than in the case of approximation by polynomials with harmonics from the hyperbolic cross.     

On Trigonometric Blending Interpolation and Cubature Formulae

We discuss error representations for Hermite-Lagrange trigonometric interpolation introduced in Dryanov and Petrov (Interpolation and L ₁-approximation by trigonometric polynomials and blending functions, J. Approx. Theory 164, 1049–1064 (2012)) and obtain one-sided trigonometric quadratures for approximate integration of one-dimensional integrals. Next, we study error representations of multivariate Hermite-Lagrange transfinite trigonometric interpolation and derive one-sided trigonometric blending interpolants to multivariate functions, under some restrictions. Then, we construct one-sided transfinite cubature formulae for approximate integration of multivariate integrals. We construct also cubature formulae with positive coefficients, based on line integrals and exact in a vector space of trigonometric blending functions with prescribed order.