Skii Inequality Research Articles

On reverse Markov–Nikol’skii inequalities for polynomials with restricted zeros

Let Πn be the class of algebraic polynomials P of degree n, all of whose zeros lie on the segment [−1,1]. In 1995, S. P. Zhou has proved the following Turán type reverse Markov–Nikol’skii inequality: ‖P′‖Lp[−1,1]>c(n)1−1/p+1/q‖P‖Lq[−1,1], P∈Πn, where 0<p≤q≤∞, 1−1/p+1/q≥0 (c>0 is a constant independent of P and n). We show that Zhou’s estimate remains true in the case p=∞, q>1. Some of related Turán type inequalities are also discussed.

On sampling discretization in L2

We prove a sampling discretization theorem for the square norm of functions from a finite dimensional subspace satisfying Nikol'skii's inequality with an upper bound on the number of sampling points of the order of the dimension of the subspace.

Hardy–Littlewood and Ulyanov inequalities

We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness ω α ( f , t ) q \omega _\alpha (f,t)_q and ω β ( f , t ) p \omega _\beta (f,t)_p for 0 &gt; p &gt; q ≤ ∞ 0&gt;p&gt;q\le \infty . A similar problem for the generalized K K -functionals and their realizations between the couples ( L p , W p ψ ) (L_p, W_p^\psi ) and ( L q , W q φ ) (L_q, W_q^\varphi ) is also solved. The main tool is the new Hardy–Littlewood–Nikol’skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity sup T n ‖ D ( ψ ) ( T n ) ‖ q ‖ D ( φ ) ( T n ) ‖ p , 0 &gt; p &gt; q ≤ ∞ , \begin{equation*} \sup _{T_n} \frac {\Vert \mathcal {D}(\psi )(T_n)\Vert _q}{\Vert \mathcal {D}({\varphi })(T_n)\Vert _p},\qquad 0&gt;p&gt;q\le \infty , \end{equation*} where the supremum is taken over all nontrivial trigonometric polynomials T n T_n of degree at most n n and D ( ψ ) , D ( φ ) \mathcal {D}(\psi ), \mathcal {D}({\varphi }) are the Weyl-type differentiation operators. We also prove the Ulyanov and Kolyada-type inequalities in the Hardy spaces. Finally, we apply the obtained estimates to derive new embedding theorems for the Lipschitz and Besov spaces.

Nikol’skii–Type Inequalities for Trigonometric Polynomials for Lorentz–Zygmund Spaces

Nikol’skii–type inequalities, that is inequalities between different metrics of trigonometric polynomials on the torus Td for the Lorentz–Zygmund spaces, are obtained. The results of previous paper “Nikol’skii inequalities for Lorentz–Zygmund spaces” are extended. Applications to approximation spaces in Lorentz–Zygmund spaces and to Besov spaces are given.

A Bohr-Nikol’skii Inequality for Weighted Lebesgue Spaces

In this paper, we give a new inequality for weighted Lebesgue spaces called Bohr-Nikol’skii inequality, which combines the inequality of Bohr-Favard and the Nikol’skii idea of inequality for functions in different metrics.

Nikol’skii inequalities for Lorentz–Zygmund spaces

Analogs of the Nikol’skii inequality in Lorentz–Zygmund spaces are obtained for functions of the form $$\sum _{k=1}^{n}c_{k}\varphi _{k}$$ , where $$\left\{ \varphi _{k}\right\} $$ is a finite orthonormal system in $$L_2$$ bounded in $$L_\infty $$ . No assumptions about smoothness of considered functions are required. Used technique relies only on the properties of Lorentz–Zygmund spaces and Fourier series map. In addition, real interpolation method is applied.

Nikol’skii Inequality Between the Uniform Norm and Integral Norm with Bessel Weight for Entire Functions of Exponential Type on the Half-Line

We study a Nikol’skii type inequality for even entire functions of given exponential type between the uniform norm on the half-line [0,∞) and the norm (∫ 0 ∞ |f(x)| q x2α+1dx)1/q of the space L q ((0,∞), x2α+1) with the Bessel weight for 1 ≤ q −1/2. An extremal function is characterized. In particular, we prove that the uniform norm of an extremal function is attained only at the end point x = 0 of the half-line. To prove these results, we use the Bessel generalized translation.

On Nikol’skii type inequality between the uniform norm and the integral [formula omitted]-norm with Laguerre weight of algebraic polynomials on the half-line

We study the Nikol’skii type inequality for algebraic polynomials on the half-line [0,∞) between the “uniform” norm sup{|f(x)|e−x∕2:x∈[0,∞)} and the norm ∫0∞|f(x)e−x∕2|qxαdx1∕q of the space Lαq with the Laguerre weight for 1≤q<∞ and α≥0. It is proved that the polynomial with a fixed leading coefficient that deviates least from zero in the space Lα+1q is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the Laguerre translation. The properties of the norm of the Laguerre translation in the spaces Lαq are studied.

A CHARACTERIZATION OF EXTREMAL ELEMENTS IN SOME LINEAR PROBLEMS

We give a characterization of elements of a subspace of a complex Banach space with the property that the norm of a bounded linear functional on the subspace is attained at those elements. In particular, we discuss properties of polynomials that are extremal in sharp pointwise Nikol'skii inequalities for algebraic polynomials in a weighted \(L_q\)-space on a finite or infinite interval.

Nikol’skii inequality between the uniform norm and L q -norm with jacobi weight of algebraic polynomials on an interval

We study the Nikol’skii inequality for algebraic polynomials on the interval [−1, 1] between the uniform norm and the norm of the space Lq(α,β), 1 ≤ q −1. We prove that, in the case α > β ≥ −1/2, the polynomial with unit leading coefficient that deviates least from zero in the space Lq(α+1,,β) with the Jacobi weight ϕ(α+1,β)(x) = (1−x)α+1(1+x)β is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation operator associated with the Jacobi weight. We describe the set of all functions at which the norm of this operator in the space Lq(α,β) for 1 ≤ q β ≥ −1/2 is attained.

On Nikol’skii Inequalities for Domains in $${\mathbb {R}}^d$$ R d

Nikol'skii inequalities for various sets of functions, domains and weights will be discussed. Much of the work is dedicated to the class of algebraic polynomials of total degree $n$ on a bounded convex domain $D$. That is, we study $\sigma:= \sigma(D,d)$ for which \[ \|P\|_{L_q(D)}\le c n^{\sigma(\frac1p-\frac1q)}\|P\|_{L_p(D)},\quad 0<p\le q\le\infty, \] where $P$ is a polynomial of total degree $n$. We use geometric properties of the boundary of $D$ to determine $\sigma(D,n)$ with the aid of comparison between domains. Computing the asymptotics of the Christoffel function of various domains is crucial in our investigation. The methods will be illustrated by the numerous examples in which the optimal $\sigma(D,n)$ will be computed explicitly.

A Bohr–Nikol'skii inequality

ABSTRACTIn this paper, we give a new inequality called Bohr–Nikol'skii inequality which combines the inequality of Bohr–Favard and the Nikol'skii idea of inequality for functions in different metrics.

Nikol’skii Inequality Between the Uniform Norm and $$\varvec{L_q}$$ L q -Norm with Ultraspherical Weight of Algebraic Polynomials on an Interval

We study the Nikol’skii inequality for algebraic polynomials on the interval \([-1,1]\) between the uniform norm and the norm of the space \(L^{\phi }_q,\)\({1\le q<\infty },\) with the ultraspherical weight \(\phi (x)=\phi ^{(\alpha )}(x)=(1-x^2)^\alpha ,\)\({\alpha \ge -1/2.}\) We prove that the polynomial with unit leading coefficient that deviates least from zero in the space \(L_q^\psi \) with the Jacobi weight \(\psi (x)=\phi ^{(\alpha +1,\alpha )}(x)=(1-x)^{\alpha +1}(1+x)^{\alpha }\) is an extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation generated by the ultraspherical weight.

Sharp Markov–Nikol’skii inequality with respect to the uniform norm and the integral norm with Chebyshev weight

We study the inequality between the uniform norm of the derivative of an algebraic polynomial of degree n and the integral norm of this polynomial with Chebyshev weight on a closed interval. We found exact constants and extremal polynomials for derivatives of any order.

Nikol’skii inequality for algebraic polynomials on a multidimensional Euclidean sphere

We study the sharp Nikol’skii inequality between the uniform norm and the L q norm of algebraic polynomials of a given (total) degree n ≥ 1 on the unit sphere \(\mathbb{S}^{m - 1} \) of the Euclidean space ℝ m for 1 ≤ q < ∞. We prove that the polynomial ϱ n in one variable with unit leading coefficient that deviates least from zero in the space L q ψ (−1, 1) of functions f such that |f| q is summable over (−1, 1) with the Jacobi weight ψ(t) = (1 - t)α(1 + t)β, α = (m - 1)/2, β = (m - 3)/2 as a zonal polynomial in one variable t = ξ m , where x = (ξ 1, ξ 2, …, ξ m ) ∈ \(\mathbb{S}^{m - 1} \), is (in a certain sense, unique) extremal polynomial in the Nikol’skii inequality on the sphere \(\mathbb{S}^{m - 1} \). The corresponding one-dimensional inequalities for algebraic polynomials on a closed interval are discussed.

On the Urysohn number of a topological space II

Some variations of Arhangel'skii inequality ∣X∣ = 2χ(X)L(X) for every Hausdorff space X [3], given in [2] and [6] are improved.

Approximation of periodic functions in the classes H{sub q}{sup {Omega}} by linear methods

The following result is proved: if approximations in the norm of L{sub {infinity}} (of H{sub 1}) of functions in the classes H{sub {infinity}}{sup {Omega}} (in H{sub 1}{sup {Omega}}, respectively) by some linear operators have the same order of magnitude as the best approximations, then the set of norms of these operators is unbounded. Also Bernstein's and the Jackson-Nikol'skii inequalities are proved for trigonometric polynomials with spectra in the sets Q(N) (in {Gamma}(N,{Omega})). Bibliography: 15 titles.

On the sharp Jackson-Nikol’skii inequality for algebraic polynomials on a multidimensional Euclidean sphere

The best constant Cn,m in the Jackson-Nikol’skii inequality between the uniform and integral norms of algebraic polynomials of a given total degree n ≥ 0 on the unit sphere \( \mathbb{S}^{m - 1} \) of the Euclidean space ℝm is studied. Two-sided estimates for the constant Cn,m are obtained, which, in particular, give the order nm−1 of its behavior with respect to n as n → +∞ for a fixed m.

The sharp Markov-Nikol’skii inequality for algebraic polynomials in the spaces L q and L 0 on a closed interval

In this paper, an inequality between the Lq-mean of the kth derivative of an algebraic polynomial of degree n ≥ 1 and the L0-mean of the polynomial on a closed interval is obtained. Earlier, the author obtained the best constant in this inequality for k = 0, q ∈ [0,∞] and 1 ≤ k ≤ n, q ∈ {0} ∪ [1,∞]. Here a newmethod for finding the best constant for all 0 ≤ k ≤ n, q ∈ [0,∞], and, in particular, for the case 1 ≤ k ≤ n, q ∈ (0, 1), which has not been studied before is proposed. We find the order of growth of the best constant with respect to n as n → ∞ for fixed k and q.

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.