Characterising Extended Lipschitz Type Conditions with Moduli of Continuity

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

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On Estimating Integral Moduli of Continuity of Functions of Several Variables in Terms of Fourier Coefficients

Georgian Mathematical Journal ◽

10.1515/gmj.1999.307 ◽

1999 ◽

Vol 6 (4) ◽

pp. 307-322

Author(s):

L. Gogoladze

Keyword(s):

Fourier Coefficients ◽

Moduli Of Continuity ◽

Several Variables

Abstract Inequalities are derived which enable one to estimate integral moduli of continuity of functions of several variables in terms of Fourier coefficients.

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Preserving the Shape of Functions by Applying Multidimensional Schoenberg-Type Operators

Symmetry ◽

10.3390/sym13061016 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1016

Author(s):

Camelia Liliana Moldovan ◽

Radu Păltănea

Keyword(s):

Applied Sciences ◽

Degree Of Approximation ◽

Higher Order ◽

Second Order ◽

Dimensional Case ◽

Moduli Of Continuity ◽

One Dimensional ◽

Multidimensional Generalization ◽

The One

The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.

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Pointwise estimates for higher order convexity preserving polynomial approximation

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000010365 ◽

1994 ◽

Vol 36 (2) ◽

pp. 213-233 ◽

Cited By ~ 6

Author(s):

Jia-Ding Cao ◽

Heinz H. Gonska

Keyword(s):

Polynomial Approximation ◽

Convex Functions ◽

Higher Order ◽

Second Order ◽

Convolution Type ◽

Shape Preservation ◽

Moduli Of Continuity ◽

Pointwise Estimates ◽

Higher Order Convexity ◽

Boolean Sum

AbstractDeVore-Gopengauz-type operators have attracted some interest over the recent years. Here we investigate their relationship to shape preservation. We construct certain positive convolution-type operators Hn, s, j which leave the cones of j-convex functions invariant and give Timan-type inequalities for these. We also consider Boolean sum modifications of the operators Hn, s, j show that they basically have the same shape preservation behavior while interpolating at the endpoints of [−1, 1], and also satisfy Telyakovskiῐ- and DeVore-Gopengauz-type inequalities involving the first and second order moduli of continuity, respectively. Our results thus generalize related results by Lorentz and Zeller, Shvedov, Beatson, DeVore, Yu and Leviatan.

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On the Third Moduli of Continuity

Ukrainian Mathematical Journal ◽

10.1007/s11253-015-1034-z ◽

2015 ◽

Vol 66 (10) ◽

pp. 1589-1594 ◽

Cited By ~ 1

Author(s):

S. I. Bezkryla ◽

O. N. Nesterenko ◽

A. V. Chaikovs’kyi

Keyword(s):

Moduli Of Continuity ◽

The Third

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Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

Symmetry ◽

10.3390/sym13071251 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1251

Author(s):

Wensheng Wang

Keyword(s):

Heat Equation ◽

White Noise ◽

Fourth Order ◽

Gaussian Processes ◽

Three Dimensional ◽

Space Time ◽

Symmetry Analysis ◽

Stochastic Heat Equation ◽

Moduli Of Continuity

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.

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