scholarly journals About Nodal Systems for Lagrange Interpolation on the Circle

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
E. Berriochoa ◽  
A. Cachafeiro ◽  
J. M. García Amor
Keyword(s):  
Unit Circle ◽  
Modulus Of Continuity ◽  
Lagrange Interpolation ◽  
Continuous Functions ◽  
Trigonometric Interpolation ◽  
Laurent Polynomials ◽  
Class Of Functions

We study the convergence of the Laurent polynomials of Lagrange interpolation on the unit circle for continuous functions satisfying a condition about their modulus of continuity. The novelty of the result is that now the nodal systems are more general than those constituted by thenroots of complex unimodular numbers and the class of functions is different from the usually studied. Moreover, some consequences for the Lagrange interpolation on[-1,1]and the Lagrange trigonometric interpolation are obtained.

scholarly journals The Rate of Convergence of Lupasq-Analogue of the Bernstein Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Heping Wang ◽  
Yanbo Zhang
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Bernstein Operators ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous

We discuss the rate of convergence of the Lupasq-analogues of the Bernstein operatorsRn,q(f;x)which were given by Lupas in 1987. We obtain the estimates for the rate of convergence ofRn,q(f)by the modulus of continuity off, and show that the estimates are sharp in the sense of order for Lipschitz continuous functions.

APPROXIMATIVE PROPERTIES OF ABEL–POISSON-TYPE OPERATORS ON THE GENERALIZED HÖLDER CLASSES

2021 ◽  
Vol 1 ◽  
pp. 76-83
Author(s):  
Yuri I. Kharkevich ◽  
◽  
Alexander G. Khanin ◽  
Keyword(s):  
Modulus Of Continuity ◽  
Applied Mathematics ◽  
Continuous Functions ◽  
Elliptic Type ◽  
Periodic Functions ◽  
French Mathematician ◽  
First Order ◽  
Function Classes ◽  
In The Beginning ◽  
Poisson Type

The paper deals with topical issues of the modern applied mathematics, in particular, an investigation of approximative properties of Abel–Poisson-type operators on the so-called generalized Hölder’s function classes. It is known, that by the generalized Hölder’s function classes we mean the classes of continuous -periodic functions determined by a first-order modulus of continuity. The notion of the modulus of continuity, in turn, was formulated in the papers of famous French mathematician Lebesgue in the beginning of the last century, and since then it belongs to the most important characteristics of smoothness for continuous functions, which can describe all natural processes in mathematical modeling. At the same time, the Abel-Poisson-type operators themselves are the solutions of elliptic-type partial differential equations. That is why the results obtained in this paper are significant for subsequent research in the field of applied mathematics. The theorem proved in this paper characterizes the upper bound of deviation of continuous -periodic functions determined by a first-order modulus of continuity from their Abel–Poisson-type operators. Hence, the classical Kolmogorov–Nikol’skii problem in A.I. Stepanets sense is solved on the approximation of functions from the classes by their Abel–Poisson-type operators. We know, that the Abel–Poisson-type operators, in partial cases, turn to the well-known in applied mathematics Poisson and Jacobi–Weierstrass operators. Therefore, from the obtained theorem follow the asymptotic equalities for the upper bounds of deviation of functions from the Hölder’s classes of order from their Poisson and Jacobi–Weierstrass operators, respectively. The obtained equalities generalize the known in this direction results from the field of applied mathematics.

scholarly journals A note on weakly θ-continuous functions

1989 ◽  
Vol 12 (1) ◽  
pp. 9-13 ◽  
Author(s):  
M. Mrševic ◽  
I. L. Reilly
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Topological Spaces ◽  
New Class ◽  
Weakly Continuous ◽  
Class Of Functions

Recently a new class of functions between topological spaces, called weaklyθ-continuous functions, has been introduced and studied. In this paper we show how an appropriate change of topology on the domain of a weaklyθ-continuous function reduces it to a weakly continuous function. This paper examines some of the consequences of this result.

scholarly journals δ-perfectly continuous functions

2009 ◽  
Vol 42 (1) ◽  
Author(s):  
J. K. Kohli ◽  
D. Singh
Keyword(s):  
Continuous Functions ◽  
New Class ◽  
Class Of Functions

AbstractA new class of functions called ‘

scholarly journals Classical Lagrange Interpolation Based on General Nodal Systems at Perturbed Roots of Unity

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 498
Author(s):  
Elías Berriochoa ◽  
Alicia Cachafeiro ◽  
Alberto Castejón ◽  
José Manuel García-Amor
Keyword(s):  
Rate Of Convergence ◽  
Unit Circle ◽  
Lagrange Interpolation ◽  
Roots Of Unity ◽  
Smooth Functions ◽  
Numerical Examples ◽  
Practical Applications ◽  
Bounded Interval ◽  
Interesting Approach ◽  
Nodal System

The aim of this paper is to study the Lagrange interpolation on the unit circle taking only into account the separation properties of the nodal points. The novelty of this paper is that we do not consider nodal systems connected with orthogonal or paraorthogonal polynomials, which is an interesting approach because in practical applications this connection may not exist. A detailed study of the properties satisfied by the nodal system and the corresponding nodal polynomial is presented. We obtain the relevant results of the convergence related to the process for continuous smooth functions as well as the rate of convergence. Analogous results for interpolation on the bounded interval are deduced and finally some numerical examples are presented.

Frame sets for a class of compactly supported continuous functions

2019 ◽  
Vol 13 (05) ◽  
pp. 2050093 ◽  
Author(s):  
A. Ganiou D. Atindehou ◽  
Yebeni B. Kouagou ◽  
Kasso A. Okoudjou
Keyword(s):  
Continuous Functions ◽  
Gabor Frame ◽  
Frequency Shifts ◽  
Time Frequency ◽  
Compactly Supported ◽  
Frame Set ◽  
Class Of Functions

The frame set of a function [Formula: see text] is the subset of all parameters [Formula: see text] for which the time-frequency shifts of [Formula: see text] along [Formula: see text] form a Gabor frame for [Formula: see text] In this paper, we investigate the frame set of a class of compactly supported continuous functions which includes the [Formula: see text]-splines. In particular, we add some new points to the frame sets of these functions. In the process, we generalize and unify some recent results on the frame sets for this class of functions.

scholarly journals Quasi cl-supercontinuous functions and their function spaces

2012 ◽  
Vol 45 (3) ◽  
Author(s):  
J. K. Kohli ◽  
Jeetendra Aggarwal
Keyword(s):  
Continuous Functions ◽  
Function Spaces ◽  
Strong Form ◽  
General Topology ◽  
Regular Space ◽  
Basic Properties ◽  
New Class ◽  
Mathematical Literature ◽  
Class Of Functions ◽  
Appl Math

AbstractA new class of functions called ‘quasi cl-supercontinuous functions’ is introduced. Basic properties of quasi cl-supercontinuous functions are studied and their place in the hierarchy of variants of continuity that already exist in the mathematical literature is elaborated. The notion of quasi cl-supercontinuity, in general, is independent of continuity but coincides with cl-supercontinuity (≡ clopen continuity) (Applied General Topology 8(2) (2007), 293–300; Indian J. Pure Appl. Math. 14(6) (1983), 767–772), a significantly strong form of continuity, if range is a regular space. The class of quasi cl-supercontinuous functions properly contains each of the classes of (i) quasi perfectly continuous functions and (ii) almost cl-supercontinuous functions; and is strictly contained in the class of quasi

scholarly journals Moduli of continuity, functional spaces,\break and elliptic boundary value problems. The full regularity spaces Cα0,λ(Ω̅)

2018 ◽  
Vol 7 (1) ◽  
pp. 15-34 ◽  
Author(s):  
Hugo Beirão da Veiga
Keyword(s):  
Modulus Of Continuity ◽  
Elliptic Boundary ◽  
Continuous Functions ◽  
Second Order ◽  
Hölder Spaces ◽  
Elliptic Boundary Value ◽  
New Class ◽  
Holder Spaces ◽  
Derivatives Of ◽  
Full Regularity

AbstractLet {\boldsymbol{L}} be a second order uniformly elliptic operator, and consider the equation {\boldsymbol{L}u=f} under the boundary condition {u=0}. We assume data f in generical subspaces of continuous functions {D_{\overline{\omega}}} characterized by a given modulus of continuity{\overline{\omega}(r)}, and show that the second order derivatives of the solution u belong to functional spaces {D_{\widehat{\omega}}}, characterized by a modulus of continuity{\widehat{\omega}(r)} expressed in terms of {\overline{\omega}(r)}. Results are optimal. In some cases, as for Hölder spaces, {D_{\widehat{\omega}}=D_{\overline{\omega}}}. In this case we say that full regularity occurs. In particular, full regularity occurs for the new class of functional spaces {C^{0,\lambda}_{\alpha}(\overline{\Omega})} which includes, as a particular case, the classical Hölder spaces {C^{0,\lambda}(\overline{\Omega})=C^{0,\lambda}_{0}(\overline{\Omega})}. Few words, concerning the possibility of generalizations and applications to non-linear problems, are expended at the end of the introduction and also in the last section.

V.—Les transformations asymptotiquement presque périodiques discontinues et le lemme ergodique. (Première Note.)

1950 ◽  
Vol 63 (1) ◽  
pp. 61-68
Author(s):  
Maurice Fréchet
Keyword(s):  
Ergodic Theorem ◽  
Continuous Functions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Discontinuous Functions ◽  
Extended Class ◽  
Class Of Functions ◽  
Asymptotically Almost Periodic

SynopsisWith the aim of establishing, under wide conditions, the ergodic theorem of G. D. Birkhoff, the author extends the class of asymptotically almost-periodic functions, considering now not only continuous functions, as he had already done in 1943, but discontinuous functions. Definitions and properties of the extended class of functions are set out, some comparisons being made with almost-periodic functions in the sense of Bohr, Stepanoff, Weyl and Besicovitch. Applications to the ergodic theorem are adumbrated.

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