Inequalities For Trigonometric Polynomials Research Articles

On Turan-type inequalities for trigonometric polynomials of half-integer order

Some exact inequalities of the Turan type are obtained in the paper for trigonometric polynomials $h(x)$ of half-interger order $n+\frac {1}{2}$, $n=1, 2, ...$, such that all $2n+1$ their zeros are real and located on a segment $[0;2\pi )$. Namely, the inequality that relates the norms in the space $C$ of the trigonometric polynomials $h(x)$ of half-integer order $n+\frac {1}{2}$, $n=1, 2, ...$, and its second derivative $h''(x)$, $\|h''\|_c\ge \frac {2n+1}{4}\|h\|_c$, that is the inequalities that connect the norms in the space $L_2$ of the trigonometric polynomials $h(x)$ of half-interger order $n+\frac {1}{2}$, $n=1, 2, ...$, and its first derivative $h'(x)$, that is $\|h'\|_{L_2}\ge \sqrt {\frac {2n+1}{8}}\|h\|_{L_2}$. The resulting inequalities cannot be improved. In proving the theorems, we use the method that was developed by V.F. Babenko and S.A. Pichugov for trigonometric polynomials, all of whose roots are real.

On identities for derivative operators

Let $X$ be a commutative algebra with unity $e$ and let $D$ be a derivative on $X$ that means the Leibniz rule is satisfied: $D(f\cdot g)=D(f)\cdot g+f\cdot D(g)$. If $D^{(k)}$ is $k$-th iteration of $D$ then we prove that the following identity holds for any positive integer $k$ $$\frac{1}{k!}\sum\limits_{j=0}^k(-1)^j\binom{k}{j}f^jD^{(m)}(gf^{k-j})=\Phi_{k,m}(g,f)=\begin{cases}0,\ 0\leq m <k,\\ gD(f)^k,\ k=m.\end{cases}$$ As an application we prove a sharp version of Bernstein's inequality for trigonometric polynomials.

Bernstein–Nikol’skii-Type Inequalities for Trigonometric Polynomials with an Arbitrary Choice of Harmonics

We obtain the Bernstein–Nikol’skii-type inequalities for trigonometric polynomials with an arbitrary choice of harmonics.

Some Inequalities for Trigonometric Polynomials and Fourier Coefficients

The Bernstein inequalities for trigonometric polynomials are generalized. For sums of Fourier coefficients, upper bounds with certain constants in terms of quantities characterizing structural properties of functions are obtained. Bibliography: 9 titles.

Weighted analogues of Bernstein-type inequalities on several intervals

We give weighted analogues of Bernstein-type inequalities for trigonometric polynomials and rational functions on several intervals. MSC: 41A17; 42A05; 41A20

Inequalities for trigonometric polynomials in spaces with integral metric

In the spaces L ψ (T ) of periodic functions with metric $$ \rho {\left( {f,0} \right)_\psi } = \int\limits_T {\psi \left| {f(x)} \right|dx,} $$ where ψ is a function of the modulus-of-continuity type, we investigate analogs of the classic Bern-stein inequalities for the norms of derivatives and increments of trigonometric polynomials.

An Inequality for Trigonometric Polynomials

An Inequality for Trigonometric Polynomials

Sharp inequalities for trigonometric polynomials with respect to integral functionals

The problem on sharp inequalities for linear operators on the set of trigonometric polynomials with respect to integral functionals \(\int_0^{2\pi } {\phi \left( {\left| {f\left( x \right)} \right|} \right)dx}\) is discussed. A solution of the problem on trigonometric polynomials with given leading harmonic of least deviation from zero with respect to such functionals over the set of all functions φ defined, nonnegative, and nondecreasing on the semiaxis [0,+∞) is given.

SOME ESTIMATES FOR POLYNOMIALS

We give an elementary alternative proof of the Bernstein inequalities and the Szegö inequalities for trigonometric polynomials or polynomials.

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 Lp.

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.

An extremal problem for polynomials with a prescribed multiple zero

We obtain a sharp inequality for trigonometric polynomials which have a double zero at a specified point. A similar extremal problem has been studied by R.P. Boas Jr.

Inequalities for Trigonometric Polynomials and Some Integral Means

Some inequalities associated with the Laplacian for trigonometric polynomials are given, which will be applied to investigate the behavior in approximation by trigonometric polynomials in higher dimensions and the best lower and upper estimates for some linear operators. In particular, we obtain a complete characterization for the approximation behavior of the classical Jackson operator. Thus the approximation error of a function by Jackson polynomials is equivalent to a K-functional defined by the Laplacian.

On some extensions of Bernstein's inequality for trigonometric polynomials

On some extensions of Bernstein's inequality for trigonometric polynomials

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 \(x \in L_\infty ^{{\text{ }}r}\) satisfy the unimprovable inequality $$|| (x-c_{s+1} (x))_{\pm} ||_q \leqslant \frac{|| (\phi_r)_{\pm} ||_q}{|| \phi_r ||_p^{\frac{r+1/q}{r+1/p}}} || x-c_{s+1}(x) ||_p^{\frac{r+1/q}{r+1/P}} || x^(r) ||_\infty^{\frac{1/p-1/q}{r+1/p}},$$ where ϕ r is the perfect Euler spline of order r and c s + 1(x) is the constant of the best approximation of the function x in the space L s + 1. By using the inequality indicated, we obtain a new Bernstein-type inequality for trigonometric polynomials τ whose degree does not exceed n, namely, $$|| (\tau^(k))_{\pm} ||_q \leqslant n^{k+1/p-1/q} \frac{|| (\cos(\cdot))_{\pm} ||_q}{|| \cos(\cdot) ||_p} || \tau ||_p,$$ where k ∈ N, p ∈ (0, 1], and q ∈ [1, ∞]. We also consider other applications of the inequality indicated.

Fractional derivatives and inequalities for trigonometric polynomials in spaces with asymmetric norms

We consider the Bernstein-Jackson-Nikol'skii inequalities for fractional derivatives in the case when the norm is asymmetric. Assume that , , and . Then where and the functions and are given by an explicit formula. The asymptotic behaviour is with respect to for fixed , , , and .

Inequalities for trigonometric polynomials

Let tn(x) be any real trigonometric polynomial of degreen n such that ∥tm∥∞⩽1. Here we are concerned with obtaining the best possible upper estimate of $$\int_0^{2k} {|t_m^{(h)} (x)|^q } dx/\int_0^{2k} {|t_n^{(h)} (x)|^{q - 2} } dx,$$ where q>2. In addition, we shall obtain the estimate of\(||t_m^{(k)} ||_q \) in terms of ∥tm∥q and ∥t n (r) .

Analogs of the jackson-nikol'skii inequalities for trigonometric polynomials in spaces with nonsymmetric norm

The paper is concerned with the evaluation of Open image in new window where ▮·▮p1, p2 is a nonsymmetric norm. The order of this number is obtained. Lower bounds involve new polynomials whose properties are studied in detail. In the casep1=p2,q1=q2, the estimate obtained is reduced to the well-known Jackson-Nikol'skii inequality.

Turan's Inequalities for Trigonometric Polynomials

Turan's Inequalities for Trigonometric Polynomials

Lagrange interpolation on extended generalized Jacobi nodes

We consider theL p -convergence of interpolatory processes for nonsmooth functions. Therefore we use generalizations of the well-known Marcinkiewicz-Zygmund inequality for trigonometric polynomials to the case of algebraic polynomials, extending a result of Y. Xu. Particularly, we obtain the order of convergence for certain Lagrange and quasi-Lagrange interpolatory processes on generalized Jacobi nodes. Our approach enables us also to discuss the influence of additional nodes near the endpoints ±1.