We obtain strong converse inequalities for the Bernstein polynomials with explicit asymptotic constants. We give different estimation procedures in the central and non-central regions of [0,1]. The main ingredients in our approach are the following: representation of the derivatives of the Bernstein polynomials in terms of the Krawtchouk polynomials, estimates of different inverse moments of various random variables, sharp estimates of both absolute central moments of Bernstein polynomials and the total variation distance between binomial and Poisson distributions, and iterates of the Bernstein polynomials, together with their probabilistic representations.
It is demonstrated that multiplier methods naturally yield better constants in strong converse inequalities for the Bernstein-Durrmeyer operator. The absolute constants obtained in some of the inequalities are independent of the weight and the dimension. The estimates are stated in terms of the $K$-functional that is naturally associated to the operator.
We prove a weak converse estimate for the simultaneous approximation by several forms of the Bernstein polynomials with integer coefficients. It is stated in terms of moduli of smoothness. In particular, it yields a big $O$-characterization of the rate of that approximation. We also show that the approximation process generated by these Bernstein polynomials with integer coefficients is saturated. We identify its saturation rate and the trivial class.
Bernstein inequalities and inverse theorems are a recent development in the theory of radial basis function(RBF) approximation. The purpose of this paper is to extend what is known by deriving $L^p$ Bernstein inequalities for RBF networks on $\mathbb{R}^d$. These inequalities involve bounding a Bessel-potential norm of an RBF network by its corresponding $L^p$ norm in terms of the separation radius associated with the network. The Bernstein inequalities will then be used to prove the corresponding inverse theorems.
Higher order Bernstein- and Markov-type inequalities are established for trigonometric polynomials on compact subsets of the real line and algebraic polynomials on compact subsets of the unit circle. In the case of Markov-type inequalities we assume that the compact set satisfies an interval condition.
We prove limit relations between the sharp constants in the multivariate Bernstein-Nikolskii type inequalities for trigonometric polynomials and entire functions of exponential type with the spectrum in a centrally symmetric convex body.
This survey discusses the classical Bernstein and Markov inequalities for the derivatives of polynomials, as well as some of their extensions to general sets.
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.
We prove that the sharp constant in the univariate Bernstein--Nikolskii inequality for entire functions of exponential type is the limit of the sharp constant in the V. A. Markov type inequality with an exponential weight for coefficients of an algebraic polynomials of degree n as $n\to\iy$.
Borwein and Erdélyi proved a Bernstein type inequality for rational functions on the unit circle and on the real line. Here we establish asymptotically sharp extensions of their inequalities for rational functions on analytic Jordan arcs and curves. In the proofs key roles are played by Borwein-Erdélyi inequality on the unit circle, Gonchar-Grigorjan type estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials.
The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrödinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates f…
We use geometric methods to calculate a formula for the complex Monge-Ampère measure $(dd^cV_K)^n$, for $K \Subset \RR^n \subset \CC^n$ a convex body and $V_K$ its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies $K$. We apply this to show that two methods for deriving Bernstein-Markov-type inequalities, i.e., pointwise estimates of gradients of polynomials, yield the same results for all convex bodies. A key role is played by the geometric result that the extremal inscribed ellipses appearing in approximation theory are the maximal area elli…
This is a continuation of our papers \cite{AP2} and \cite{AP3}. In those papers we obtained estimates for finite differences $(\D_Kf)(A)=f(A+K)-f(A)$ of the order 1 and $(\D_K^mf)(A)\df\sum\limits_{j=0}^m(-1)^{m-j}(m\j)f\big(A+jK\big)$ of the order $m$ for certain classes of functions $f$, where $A$ and $K$ are bounded self-adjoint operator. In this paper we extend results of \cite{AP2} and \cite{AP3} to the case of unbounded self-adjoint operators $A$. Moreover, we obtain operator Bernstein type inequalities for entire functions of exponential type. This allows us to obtain alternative proofs…
Under certain conditions on an integrable function f having a real-valued Fourier transform Tf=F, we obtain a certain estimate for the oscillation of F in the interval [-C||f'||/||f||,C||f'||/||f||] with C>0 an absolute constant. Given q>0 and an integrable positive definite function f, satisfying some natural conditions, the above estimate allows us to construct a finite linear combination P of translates f(x+kq)(with k running the integers) such that ||P'||>c||P||/q, where c>0 is another absolute constant. In particular, our construction proves sharpness of an inequality of H. N. Mhaskar for…
Starting out from a question posed by T. Erdélyi and J. Szabados, we consider Schur-type inequalities for the classes of complex algebraic polynomials having no zeroes within the unit disk D. The class of polynomials with no zeroes in D - also known as Bernstein- or Lorentz-class - was studied in detail earlier. For real polynomials utilizing the Bernstein-Lorentz representation as convex combinations of fundamental polynomials (1-x)^k(1+x)^{n-k}, G. Lorentz, T. Erdélyi and J. Szabados proved a number of improved versions of Schur- (and also Bernstein- and Markov-) type inequalities. Here we i…
We present a new, very short proof of a conjecture by I. Raşa, which is an inequality involving basic Bernstein polynomials and convex functions. It was affirmed positively very recently by J. Mrowiec, T. Rajba and S. Wąsowicz (2017) by the use of stochastic convex orders, as well as by Abel (2017) who simplified their proof. We give a useful sufficient condition for the verification of some stochastic convex order relations, which in the case of binomial distributions are equivalent to the I. Raşa inequality. We give also the corresponding inequalities for other distributions. Our methods all…
We obtain the following dimension independent Bernstein-Markov inequality in Gauss space: for each $1\leq p<\infty$ there exists a constant $C_p>0$ such that for any $k\geq 1$ and all polynomials $P$ on $\mathbb{R}^{k}$ we have $$ \| \nabla P\|_{L^{p}(\mathbb{R}^{k}, \mathrm{d}γ_k)} \leq C_p (\mathrm{deg}\, P)^{\frac{1}{2}+\frac{1}π\arctan\left(\frac{|p-2|}{2\sqrt{p-1}}\right)}\|P\|_{L^{p}(\mathbb{R}^{k}, \mathrm{d}γ_k)}, $$ where $\mathrm{d}γ_k$ is the standard Gaussian measure on $\mathbb{R}^{k}$. We also show that under some mild growth assumptions on any function $B \in C^{2}((0,\infty))\c…
We present a short proof of a conjecture proposed by I. Raşa (2017), which is an inequality involving basic Bernstein polynomials and convex functions. This proof was given in the letter to I. Raşa (2017). The methods of our proof allow us to obtain some extended versions of this inequality as well as other inequalities given by I. Raşa. As a tool we use stochastic convex ordering relations. We propose also some generalizations of the binomial convex concentration inequality. We use it to insert some additional expressions between left and right sides of the Raşa inequalities.
The classical A. Markov inequality establishes a relation between the maximum modulus or the $L^{\infty}\left([-1,1]\right)$ norm of a polynomial $Q_{n}$ and of its derivative: $\|Q'_{n}\|\leqslant M_{n} n^{2}\|Q_{n}\|$, where the constant $M_{n}=1$ is sharp. The limiting behavior of the sharp constants $M_{n}$ for this inequality, considered in the space $L^{2}\left([-1,1], w^{(α,β)}\right)$ with respect to the classical Jacobi weight $w^{(α,β)}(x):=(1-x)^α(x+1)^β$, is studied. We prove that, under the condition $|α- β| < 4 $, the limit is $\lim_{n \to \infty} M_{n} = 1/(2 j_ν)$ where $j_ν$ i…
This paper studies the following weighted, fractional Bernstein inequality for spherical polynomials on $\sph$: \begin{equation}\label{4-1-TD-ab} \|(-Δ_0)^{r/2} f\|_{p,w}\leq C_w n^{r} \|f\|_{p,w}, \ \ \forall f\in Π_n^d, \end{equation} where $Π_n^d$ denotes the space of all spherical polynomials of degree at most $n$ on $\sph$, and $(-Δ_0)^{r/2}$ is the fractional Laplacian-Beltrami operator on $\sph$. A new class of doubling weights with conditions weaker than the $A_p$ is introduced, and used to fully characterize those doubling weights $w$ on $\sph$ for which the weighted Bernstein inequal…
We establish weighted Bernstein inequalities in $L^p$ space for the doubling weight on the conic surface $\mathbb{V}_0^{d+1} = \{(x,t): \|x\| = t, x \in \mathbb{R}^d, t\in [0,1]\}$ as well as on the solid cone bounded by the conic surface and the hyperplane $t =1$, which becomes a triangle on the plane when $d=1$. While the inequalities for the derivatives in the $t$ variable behave as expected, there are inequalities for the derivatives in the $x$ variables that are stronger than what one may have expected. As an example, on the triangle $\{(x_1,x_2): x_1 \ge 0, \, x_2 \ge 0,\, x_1+x_2 \le 1\…
Let ${\mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${\mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let $$\|f\|_A := \sup_{x \in A}{|f(x)|}$$ for real-valued functions $f$ defined on a set $A \subset {\Bbb R}$. Let $$V_a^b(f) := \int_a^b{|f^{\prime}(x)| \, dx}$$ denote the total variation of a continuously differentiable function $f$ on an interval $[a,b]$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 \frac nk\leq \min_{P \…
Let $V\subset\R^m$ be a centrally symmetric convex body and let $V^*\subset\R^m$ be its polar. We prove limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities for algebraic polynomials on $V^*$ and the corresponding constants for entire functions of exponential type with the spectrum in $V$.
Let $(X,\|\cdot\|_X)$ be a Banach space. The purpose of this article is to systematically investigate dimension independent properties of vector valued functions $f:\{-1,1\}^n\to X$ on the Hamming cube whose spectrum is bounded above or below. Our proofs exploit contractivity properties of the heat flow, induced by the geometry of the target space $(X,\|\cdot\|_X)$, combined with duality arguments and suitable tools from approximation theory and complex analysis. We obtain a series of improvements of various well-studied estimates for functions with bounded spectrum, including moment compariso…
We prove two inequalities for the Mittag-Leffler function, namely that the function $\log E_α(x^α)$ is sub-additive for $0<α<1,$ and super-additive for $α>1.$ These assertions follow from two new binomial inequalities, one of which is a converse to the neo-classical inequality. The proofs use a generalization of the binomial theorem due to Hara and Hino (Bull. London Math. Soc. 2010). For $0<α<2,$ we also show that $E_α(x^α)$ is log-concave resp. log-convex, using analytic as well as probabilistic arguments.
Approximation on the spherical cap is different from that on the sphere which requires us to construct new operators. This paper discusses the approximation on the spherical cap. That is, so called Jackson-type operator $\{J_{k,s}^m\}_{k=1}^{\infty}$ is constructed to approximate the function defined on the spherical cap $D(x_0,γ)$. We thus establish the direct and inverse inequalities and obtain saturation theorems for $\{J_{k,s}^m\}_{k=1}^{\infty}$ on the cap $D(x_0,γ)$. Using methods of $K$-functional and multiplier, we obtain the inequality \begin{eqnarray*} C_1\:\| J_{k,s}^m(f)-f\|_{D,p}\…
We prove invariance theorems for general inequalities of different metrics and apply them to limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities with the polyharmonic operator for algebraic polynomials on the unit sphere and the unit ball in $\R^m$ and the corresponding constants for entire functions of spherical type on $\R^m$. Certain relations in the univariate weighted spaces are discussed as well.
We summarize researches - in great deal jointly with my host Y. Sarantopoulos and his PhD. students V. Anagnostopoulos and A. Pappas - started by a Marie Curie fellowship in 2001 and is still continuing. The project was to study multivariate polynomial inequalities. In the course of work we realized the role of the ``generalized Minkowski functional'', to which we devoted a throughout survey. Resulting from this, infinite dimensional extensions of Chebyshev's extremal problems were tackled successfully. Investigating Bernstein-Markov constants for homogeneous polynomials of real normed spaces …
A Bernstein type inequality is obtained for the Jacobi polynomials $P_n^{α,β}(x)$, which is uniform for all degrees $n\ge0$, all real $α,β\ge0$, and all values $x\in [-1,1]$. It provides uniform bounds on a complete set of matrix coefficients for the irreducible representations of $\mathrm{SU}(2)$ with a decay of $d^{-1/4}$ in the dimension $d$ of the representation. Moreover it complements previous results of Krasikov on a conjecture of Erdélyi, Magnus and Nevai.
For a convex domain $K$ in the complex plane, the well-known general Bernstein-Markov inequality holds asserting that a polynomial $p$ of degree $n$ must have $||p'|| < c(K) n^2 ||p||$. On the other hand for polynomials in general, $||p'||$ can be arbitrarily small as compared to $||p||$. The situation changes when we assume that the polynomials in question have all their zeroes in the convex body $K$. This was first investigated by Turán, who showed the lower bounds $||p'|| \ge (n/2) ||p||$ for the unit disk $D$ and $||p'|| > c \sqrt{n} ||p||$ for the unit interval $I:=[-1,1]$. Although parti…
In 1939 P. Turán started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm 1 on $I:=[-1,1]$ (interval) and $D:=\{z\in\mathbb{C}~:~|z|\le 1\}$ (disk), under the normalization condition that the zeroes of the polynomial in question all lie in $I$ or $D$, respectively. For the maximum norm he found that with $n:=\mathop{\mathrm{deg}} p$ tending to infinity, the precise growth order of the minimal possible derivative norm is $\sqrt{n}$ for $I$ and $n$ for $D$. J. Erőd continued the work of Turán considering other domains. Finally, a decade ago the growth o…
A recent conjecture by I. Raşa asserts that the sum of the squared Bernstein basis polynomials is a convex function in $[0,1]$. This conjecture turns out to be equivalent to a certain upper pointwise estimate of the ratio $P_n^{\prime}(x)/P_n(x)$ for $x\geq 1$, where $P_n$ is the $n$-th Legendre polynomial. Here, we prove both upper and lower pointwise estimates for the ratios $\big(P_n^{(λ)}(x)\big)^{\prime}/P_n^{(λ)}(x)$, $~x\geq 1$, where $P_n^{(λ)}$ is the $n$-th ultraspherical polynomial. In particular, we validate Raşa's conjecture.
Let W: R to (0,1] be continuous. Bernstein's approximation problem, posed in 1924, deals with approximation by polynomials in the weighted uniform norm ||fW|| Linfinity(R) . The qualitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950's. Quantitative forms of the problem were actively investigated starting from the 1960's. We survey old and recent aspects of this topic, including the Bernstein problem, weighted Jackson and Bernstein Theorems, Markov-Bernstein and Nikolskii inequalities, orthogonal expansions and Lagrange interpolation. We present the main id…
