Related papers: A sharp bound for the Chebyshev functional
Some new sufficient conditions for the weighted Chebyshev's inequality for real numbers to hold are provided.
Chebyshev's inequality provides an upper bound on the tail probability of a random variable based on its mean and variance. While tight, the inequality has been criticized for only being attained by pathological distributions that abuse the…
Let X be a separable Banach space which admits a separating polynomial; in particular X a separable Hilbert space. Let $f:X \rightarrow R$ be bounded, Lipschitz, and $C^1$ with uniformly continuous derivative. Then for each {\epsilon}>0,…
We prove the following estimate \[ \|{e^{it\partial_x^2}f}\|_{L_{(t,x)\in \mathbb{T}^2}^6}\leq C (\log N)^{{1/6}} \|f\|_{L^2_x(\mathbb{T})}, \] assuming $\mbox{supp} (\hat f)\subset [-N,N]$ for $N>1$. The bound $(\log N)^{{1/6}}$ is sharp…
In this paper, we give a sharp lower bound for the minimum deviation of the Chebyshev polynomial on a compact subset of the real line in terms of the corresponding logarithmic capacity. Especially if the set is the union of several real…
Firstly we establish a sharp pointwise estimate for the arbitrary derivative of the function $f\in F_{\alpha}^{p},$ where $F_{\alpha}^{p}$ denotes the Fock space for $1\leq p<\infty.$ Then, in a particular Hilbert case when $p=2$ we…
Chebyshev coefficients of a coordinate representation can be used to form the corresponding velocity representation. One way is to directly apply them to the derivatives of Chebyshev polynomials, another is to compute from them the…
In this paper our aim is to deduce some sufficient (and necessary) conditions for the close-to-convexity of some special functions and their derivatives, like Bessel functions, Struve functions, and a particular case of Lommel functions of…
We extend the classical Lyapunov inequality on the measurable space with infinite measure and on the so-called Grand Lebesgue spaces (GLS). We find also the exact value for correspondent constant. Possible applications: Functional Analysis…
We present a criterion that provides an easy sufficient condition in order that a collection of Abelian integrals has the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be…
We consider a family of pseudo differential operators $\{\Delta+ a^\alpha \Delta^{\alpha/2}; a\in [0, 1]\}$ on $\R^d$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $d\geq 1$ and $\alpha \in (0, 2)$. It gives…
Ordinary differential equations and boundary value problems arise in many aspects of mathematical physics. Chebyshev differential equation is one special case of the Sturm-Liouville boundary value problem. Generating function, recursive…
We establish a sharp norm estimate of the Schwarzian derivative for a function in the classes of convex functions introduced by Ma and Minda [Proceedings of the Conference on Complex Analysis, International Press Inc., 1992, 157-169]. As…
We show that, for $0<s<1$, $0<p<\infty$, $0<q<\infty$, Haj\l asz-Besov and Haj\l asz-Triebel-Lizorkin functions can be approximated in the norm by discrete median convolutions. This allows us to show that, for these functions, the limit of…
In this paper we present a result about analytic functions f defined on the open unit disc and with a finite number of exceptional values containedin the real interval (0, 1). We find an upper bound for the modulus of f' in 0. This bound is…
Two Morrey-Sobolev inequalities (with support-bound and $L^1-$bound, respectively) are investigated on complete Riemannian manifolds with their sharp constants in $\mathbb R^n$. We prove the following results in both cases: $\bullet$ If…
In this note we study the multiplier norm estimates for the multiplication operators between weighted Bergman spaces, whose symbols are the higher-order Schwarzian derivatives of univalent functions. We establish sharp multiplier estimates…
A brief presentation of basics of the theory of Tchebycheff and Markov systems of functions and its applications to extremal problems for integrals of such functions is given. The results, as well as all the necessary definitions, are…
We obtain an analog of Shvedov theorem for convex multivariate approximation by algebras of continuous functions.
A Chebyshev expansion is a series in the basis of Chebyshev polynomials of the first kind. When such a series solves a linear differential equation, its coefficients satisfy a linear recurrence equation. We interpret this equation as the…