Kolmogorov-type Inequalities Research Articles

Some Landau–Kolmogorov Type Inequalities for Differential Operators Generated by Polynomials

In this paper, we establish some Landau–Kolmogorov type inequalities for differential operators generated by polynomials in the following formfor all , where 0 < g ≤ p ≤ ∞, and the differential operator P (D) is obtained from the polynomial P (x) by substituting . Moreover, the explicit form of and are given.

On Kolmogorov type inequalities for the norms of operators of intermediate derivatives in the space of smooth vector functions on a finite segment

On Kolmogorov type inequalities for the norms of operators of intermediate derivatives in the space of smooth vector functions on a finite segment

Kolmogorov-Type Inequalities for the Norms of Fractional Derivatives of Functions Defined on the Positive Half Line

Kolmogorov-Type Inequalities for the Norms of Fractional Derivatives of Functions Defined on the Positive Half Line

Inequalities of Kolmogorov type for fractional derivatives of multivariable functions

We prove new sharp inequality of Kolmogorov type that estimates the norm of mixed fractional Marchaud derivative of n-variable function by C-norm of this function and its norms in Lipschitz spaces.

On inequalities of Kolmogorov type for fractional derivatives of functions defined on the real domain

We obtain new inequalities that generalize known result of Geisberg, which was obtained for fractional Marchaud derivatives, to the case of higher derivatives, at that the fractional derivative is a Riesz one. The inequality with second higher derivative is sharp.

On inequalities of Kolmogorov type for fractional Hadamard derivatives of functions defined on the half-line

New exact inequalities for Hadamard fraction derivatives of functions, defined on the half-line, are obtained.

On inequalities of Kolmogorov type for classes of functions that are multiply monotonic on a finite interval

We solve the problem about exact constants in additive Kolmogorov-type inequalities on the class of multiply monotonic functions, defined on a finite interval.

Exact inequalities of Kolmogorov type for Hadamard fractional derivatives of functions defined on half-line

New exact inequalities for Hadamard fractional derivatives of functions, defined on the half-line, are obtained.

Kolmogorov type inequalities for hypersingular integrals with sign-alternating characteristic

New sharp Kolmogorov type inequalities for hypersingular integrals with homogeneous characteristic of the form $\Omega(t) = \mathrm{sgn} \prod\limits_{k=1}^m t_k$ for multivariate functions from Hölder spaces are obtained.

Kolmogorov inequalities for norms of Marchaud-type fractional derivatives of multivariate functions

We obtain new sharp Kolmogorov type inequalities, estimating the norm of mixed Marchaud type derivative of multivariate function through the C-norm of function itself and its norms in Hölder spaces.

Strengthening of Kolmogorov Type Inequalities for Derivatives and Differences

Strengthening of Kolmogorov Type Inequalities for Derivatives and Differences

Equivalence of the global and local Markov inequalities in the complex plane

We prove that a compact subset of the complex plane satisfies the Global Markov Inequality if and only if it admits an extension property and a Kolmogorov type inequality in Jackson norms, generalizing a result established by Bos and Milman in the real case. We also show that the Global Markov Inequality is equivalent to the Local Markov Property for all compact subsets of the complex plane admitting the Jackson Property. The latter is a generalization of Jackson's approximation inequality, closely connected with the boundary behavior of the Green's function. Finally, we construct an example showing that there is no equivalence in general.

Kolmogorov inequalities for the norms of the Riesz derivatives of functions of many variables

New sharp Kolmogorov-type inequalities for the norms of the Riesz derivatives ∥D α f∥∞ of functions $$ f\in {L}_{\infty, E}^{\nabla}\left({\mathrm{\mathbb{R}}}^m\right) $$ are obtained. Some applications of obtained inequalities are investigated.

Sharp Inequalities for Seminorms Defined on Spaces of Periodic Functions

We present a polynomial approximation method for realizing modified Landau–Kolmogorov type inequalities with small order derivatives. Bibliography: 6 titles.

Some results on dependent random variables and a connection with the multivariate s-increasing convex order

ABSTRACTIn this paper, some new concepts of dependence are introduced that generalize the concepts of positive and negative association. The new concepts of dependence are linked to the multivariate s-increasing convex order. Furthermore, a Kolmogorov-type inequality and a Hàjek–Rènyi inequality are proven that lead to an asymptotic result for these new random objects.

Equivalence theorem for additive inequalities of Kolmogorov type

We prove the equivalence theorem for additive inequalities on a finite interval. Besides, we describe a pair of constants so that the additive inequalities with that constants are valid on the whole class of functions $L_s$.

Kolmogorov-type inequalities for derived functions of two variables with application for approximation by an “Angle”

In the present paper we obtain sharp Kolmogorov-type inequalities for partial and mixed intermediate derivatives of functions of two variables. We also consider certain applications of the results for approximation of functions of two variables by an “angle” and obtain a number of relations, which are exact in certain sense.

Sharp inequalities of Kolmogorov type for non-periodic functions on the real domain

We obtained sharp inequalities of Kolmogorov type for non-periodic functions on the real domain. The obtained results were applied to solve some extremum problems for non-periodic functions and splines on the real domain.

Inequalities of Kolmogorov type for norms of hypersingular integrals with homogeneous characteristic of constant sign

We obtained new sharp Kolmogorov-type inequalities for norms of hypersingular integrals with homogeneous characteristic of constant sign ${\| D^{\alpha}_{\Omega} \|}_{\infty}$. We provided some applications of the obtained inequalities.

Best Recovery of the Laplace Operator of a Function and Sharp Inequalities

The paper is concerned with the problem of optimal recovery of fractional powers of the Laplace operator in the uniform norm on the multivariate generalized Sobolev class of functions from incomplete data on the Fourier transform of these functions on a ball of radius r centered at the origin. An optimal recovery method is constructed, and a number \( \widehat{r} \)> 0 is specified such that for r ≤\( \widehat{r} \) the method makes use of all the information on the Fourier transform, smoothing thereof; and if r >\( \widehat{r} \), then the information on the Fourier transform proves superfluous and hence is not used by the optimal method. For fractional powers of the Laplace operator, a sharp inequality is proved. This inequality turns out to be closely related to the recovery problem and is an analogue of Kolmogorov-type inequalities for derivatives.