scholarly journals Jackson’s rational singular integral on the cut

2019 ◽  
Vol 63 (4) ◽  
pp. 398-407
Author(s):  
Yevgeniy A. Rovba ◽  
Pavel G. Potsejko
Keyword(s):  
Uniform Convergence ◽  
Singular Integral ◽  
Singular Integrals ◽  
Uniform Estimate ◽  
Approximation Error ◽  
Special Choice ◽  
Order Of Approximation ◽  
Computational Mathematics ◽  
Optimal Value ◽  
Rational Chebyshev

The introduction presents the main results of previously known papers on Jackson’s singular integral in polynomial and rational cases. Next, we introduce Jackson’s singular integral on the interval [–1, 1] with the kernel obtained by one system of rational Chebyshev–Markov fractions and establish its basic approximative properties: a theorem on uniform convergence of a sequence of Jackson’s singular integrals for an even function is obtained, and conditions are specified that the parameter must satisfy in order for uniform convergence to take place; the approximative properties of sequences of Jackson’s singular integrals on classes of functions satisfying on the interval [–1, 1] the condition of Lipschitz class with constant M. are investigated. The obtained estimates are asymptotically exact as n → ∞; an estimate of deviation of Jackson’s rational singular integral from the function |x|s, 0 < s < 2 depending on the position of the point on the segment, a uniform estimate of the deviation on the segment [–1, 1] and its asymptotics are found. The optimal value of the parameter is obtained, for which the deviation error of the studied approximation apparatus from the function |x|s, 0 < s < 2 on the interval [–1, 1] has the highest rate of zero; the order of approximation of the function |x| on the interval [–1, 1] byJackson’s considered singular integral is found. It is shown that with a special choice of the parameter, the velocity of the approximation error tending to zero is higher in comparison with the polynomial case. All results of this article are new. The work is both theoretical and applied. It is possible to apply the results to solve specific problems of computational mathematics and to read special courses at mathematical faculties.

scholarly journals On approximations of the function |x|s by the Vallee Poussin means of the Fourier series by the system of the Chebyshev – Markov rational fractions

2019 ◽  
Vol 55 (3) ◽  
pp. 263-282
Author(s):  
P. G. Patseika ◽  
Y. A. Rovba
Keyword(s):  
Fourier Series ◽  
Integral Representation ◽  
Asymptotic Expression ◽  
Uniform Estimate ◽  
Main Section ◽  
Computational Mathematics ◽  
Power Singularity ◽  
Asymptotic Estimation ◽  
Optimal Value ◽  
Pointwise Estimation

The approximative properties of the Valle Poussin means of the Fourier series by the system of the Chebyshev – Markov rational fractions in the approximation of the function |x|s, 0 < s < 2 are investigated. The introduction presents the main results of the previously known works on the Vallee Poussin means in the polynomial and rational cases, as well as on the known literature data on the approximations of functions with power singularity. The Valle Poussin means on the interval [–1,1] as a method of summing the Fourier series by one system of the Chebyshev – Markov rational fractions are introduced. In the main section of the article, a integral representation for the error of approximations by the rational Valle Poussin means of the function |x|s, 0 < s < 2, on the segment [–1,1], an estimate of deviations of the Valle Poussin means from the function |x|s, 0 < s < 2, depending on the position of the point on the segment, a uniform estimate of deviations on the segment [–1,1] and its asymptotic expression are found. The optimal value of the parameter is obtained, at which the deviation error of the Valle Poussin means from the function |x|s, 0 < s <2, on the interval [–1,1] has the highest velocity of zero. As a consequence of the obtained results, the problem of approximation of the function |x|s, s > 0, by the Valle Poussin means of the Fourier series by the system of the Chebyshev first-kind polynomials is studied in detail. The pointwise estimation of approximation and asymptotic estimation are established.The work is both theoretical and applied. Its results can be used to read special courses at mathematical faculties and to solve specific problems of computational mathematics.

Multi-parameter Singular Integrals II: General Theory

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
General Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Main Issue

This chapter turns to a general theory which generalizes and unifies all of the examples in the preceding chapters. A main issue is that the first definition from the trichotomy does not generalize to the multi-parameter situation. To deal with this, strengthened cancellation conditions are introduced. This is done in two different ways, resulting in four total definitions for singular integral operators (the first two use the strengthened cancellation conditions, while the later two are generalizations of the later two parts of the trichotomy). Thus, we obtain four classes of singular integral operators, denoted by A1, A2, A3, and A4. The main theorem of the chapter is A1 = A2 = A3 = A4; i.e., all four of these definitions are equivalent. This leads to many nice properties of these singular integral operators.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

scholarly journals A note on singular integrals with angular integrability

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Function Spaces ◽  
Riesz Transforms ◽  
Hilbert Transforms ◽  
Rotation Method ◽  
Vector Valued Function ◽  
Vector Valued ◽  
L Log L

Abstract In this note we study the rough singular integral $$ T_{\varOmega }f(x)=\mathrm{p.v.} \int _{\mathbb{R}^{n}}f(x-y)\frac{\varOmega (y/ \vert y \vert )}{ \vert y \vert ^{n}}\,dy, $$ T Ω f ( x ) = p . v . ∫ R n f ( x − y ) Ω ( y / | y | ) | y | n d y , where $n\geq 2$ n ≥ 2 and Ω is a function in $L\log L(\mathrm{S} ^{n-1})$ L log L ( S n − 1 ) with vanishing integral. We prove that $T_{\varOmega }$ T Ω is bounded on the mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}( \mathbb{R}^{n})$ L | x | p L θ p ˜ ( R n ) , on the vector-valued mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}(\mathbb{R}^{n},\ell ^{\tilde{p}})$ L | x | p L θ p ˜ ( R n , ℓ p ˜ ) and on the vector-valued function spaces $L^{p}(\mathbb{R}^{n}, \ell ^{\tilde{p}})$ L p ( R n , ℓ p ˜ ) if $1<\tilde{p}\leq p<\tilde{p}n/(n-1)$ 1 < p ˜ ≤ p < p ˜ n / ( n − 1 ) or $\tilde{p}n/(\tilde{p}+n-1)< p\leq \tilde{p}<\infty $ p ˜ n / ( p ˜ + n − 1 ) < p ≤ p ˜ < ∞ . The same conclusions hold for the well-known Riesz transforms and directional Hilbert transforms. It should be pointed out that our proof is based on the Calderón–Zygmund’s rotation method.

scholarly journals A note on maximal singular integrals with rough kernels

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiao Zhang ◽  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Maximal Operators ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Homogeneous Mapping ◽  
Recent Developments ◽  
Maximal Singular Integrals ◽  
Maximal Singular Integral

Abstract In this note we study the maximal singular integral operators associated with a homogeneous mapping with rough kernels as well as the corresponding maximal operators. The boundedness and continuity on the Lebesgue spaces, Triebel–Lizorkin spaces, and Besov spaces are established for the above operators with rough kernels in $H^{1}({\mathrm{S}}^{n-1})$ H 1 ( S n − 1 ) , which complement some recent developments related to rough maximal singular integrals.

A criterion on oscillation and variation for the commutators of singular integral operators

2015 ◽  
Vol 27 (1) ◽  
Author(s):  
Feng Liu ◽  
Huoxiong Wu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Weighted Norm ◽  
Norm Estimates ◽  
Variation Operators

AbstractThis paper gives a criterion on the weighted norm estimates of the oscillatory and variation operators for the commutators of Calderón–Zygmund singular integrals in dimension 1. As applications, the weighted

scholarly journals Precompact Sets, Boundedness, and Compactness of Commutators for Singular Integrals in Variable Morrey Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Wei Wang ◽  
Jingshi Xu
Keyword(s):  
Integral Operator ◽  
Singular Integral Operator ◽  
Singular Integral ◽  
Singular Integrals ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Bmo Function

We give sufficient conditions for subsets to be precompact sets in variable Morrey spaces. Then we obtain the boundedness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces. Finally, we discuss the compactness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces.

On the Scenario-Tree Optimal-Value Error for Stochastic Programming Problems

2020 ◽  
Vol 45 (4) ◽  
pp. 1572-1595
Author(s):  
Julien Keutchayan ◽  
David Munger ◽  
Michel Gendreau
Keyword(s):  
Stochastic Programming ◽  
Large Scale ◽  
Probability Distributions ◽  
Approximation Error ◽  
Scenario Tree ◽  
Branching Structures ◽  
Tree Generation ◽  
Optimal Value ◽  
Scenario Trees ◽  
Scenario Tree Generation

Stochastic programming problems generally lead to large-scale programs if the number of random outcomes is large or if the problem has many stages. A way to tackle them is provided by scenario-tree generation methods, which construct approximate problems from a reduced subset of outcomes. However, it is well known that the number of scenarios required to keep the approximation error within a given tolerance grows rapidly with the number of random parameters and stages. For this reason, to limit the fast growth of complexity, scenario-tree generation methods tailored to problems must be developed. These will use more information about the problem than just the underlying probability distributions; namely, they will also take into account the objective function and the constraints. In this paper, we develop a general framework to build problem-driven scenario trees. We do so by studying how the optimal-value error arises as a sum of lower-level errors made at each node of the tree. We show how these small but numerous node errors depend on the specific features of the problem and how they can be controlled by designing scenario trees with appropriate branching structures and discretization points and weights. We illustrate our approach on two examples.

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