Ofcontinuous Functions Research Articles

On the degree of approximation of continuous functions by a linear transformation of their Fourier series

In this paper, we have proved four theorems on the degree of approximation of continuous functions by matrix means of their Fourier series which is expressed in terms of the modulus of continuity and a non-negative mediate function.

SOME FIXED POINT THEOREMS AND CYCLIC CONTRACTIONS IN DISLOCATED QUASI-METRIC SPACES

In this paper, we established some common xed point theoremsfor types of cyclic contractions in the setting of dislocated quasi-metric spaces.Using type of contraction introduced by C-class function in[26] and a class ofcontinuous functions G3 in [10] we extend, generalize and unify some resultsin the existing literature.

On the continuity of global attractors

1. GlobalattractorsThe global attractor of a dynamical system is the unique compact invariant setthat attracts the trajectories starting in any bounded set at a uniform rate. Intro-duced by Billotti & LaSalle [3], they have been the subject of much research sincethe mid-1980s, and form the central topic of a number of monographs, includingBabin & Vishik [1], Hale [9], Ladyzhenskaya [13], Robinson [16], and Temam [18].The standard theory incorporates existence results [3], upper semicontinuity [10],and bounds on the attractordimension [7]. Global attractorsexist for many infinite-dimensional models [18], with familiar low-dimensional ODE models such as theLorenz equations providing a testing ground for the general theory [8].While upper semicontinuity with respect to perturbations is easy to prove, lowersemicontinuity (and hence full continuity) is more delicate, requiring structuralassumptions on the attractor or the assumption of a uniform attraction rate. How-ever, Babin & Pilyugin [2] proved that the global attractor of a parametrised setof semigroups is continuous at a residual set of parameters, by taking advantage ofthe known upper semicontinuity and then using the fact that upper semicontinuousfunctions are continuous on a residual set.Here we reprove results on equi-attraction and residual continuity in a moredirect way, which also serves to demonstrate more clearly why these results aretrue. Given equi-attraction the attractor is the uniform limit of a sequence ofcontinuous functions, and hence continuous (the converse requires a generalisedversion of Dini’s Theorem); more generally, it is the pointwise limit of a sequence ofcontinuous functions, i.e. a ‘Baire one’ function, and therefore the set of continuitypoints forms a residual set.

From coinductive proofs to exact real arithmetic: theory and applications

Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. We discuss several examples including the extraction of programs for polynomials up to degree two and the definite integral of continuous maps.

On the Recovery of Continuous Functions from Noisy Fourier Coefficients

AbstractWe consider the classical ill-posed problem of the recovery of continuous functions from noisy Fourier coefficients. For the classes of functions given in terms of generalized smoothness, we present a priori and a posteriori regularization parameter choice realizing an order-optimal error bound.

Fuzzy Function Aproximation in Simulation Model

The paper deals with utilization of fuzzy logic for simulationmodel parameter, which is changed during simulation experiment.Additive fuzzy system is used for approximation ofcontinuous functions, whereas the relevant membership functionsare based on ellipsoidal rules. Application of that methodologyis demonstrated on the parameter adaptation relatedto simulation model of an asynchronous machine. The mentionedadaptation depends on the changing magnetic properties.

Subdifferentiability of Real Functions

In this paper, we show that nowhere monotone functions are the key ingredients to construction of continuous functions, absolutely continuous functions, and Lipschitz functions with large subdifferentials on the real line. Let $\partial_{c}f, \partial_{a}f$ denote the Clarke subdifferential and approximate subdifferential respectively. We construct absolutely continuous functions on $\R$ such that $\partial_{a}f=\partial_{c}f\equiv \R$. In the Banach space of continuous functions defined on $[0,1]$, denoted by $C[0,1]$, with the uniform norm, we show that there exists a residual and prevalent set $D\subset C[0,1]$ such that $\partial_{a}f=\partial_{c}f\equiv \R$ on $[0,1]$ for every $f\in D$. In the space of automorphisms we prove that most functions $f$ satisfy $\partial_{a}f=\partial_{c}f\equiv [0,+\infty)$ on $[0,1]$. The subdifferentiability of the Weierstrass function and the Cantor function are completely analyzed. Similar results for Lipschitz functions are also given.

Generation of Analytic Semigroups by Elliptic Operators with Dirichlet Boundary Conditions in a Cylindrical Domain

We consider an elliptic operator of second order with Dirichlet boundary conditions in a cylindrical domain. We show that a suitable interpretation of this operator in a certain space of continuous functions vanishing on the boundary is the infinitesimal generator of an analytic semigroup in the space of continuous functions. We prove several inclusions of the domain of the infinitesimal generator and of the real interpolation spaces between this domain and the basic space of continuous functions.

Correspondances singulières par parallélisme des plans tangents des deux surfaces

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