Example Of Continuous Function Research Articles

Computability in Harmonic Analysis

We study the question of constructive approximation of the harmonic measure \(\omega _x^\varOmega \) of a bounded domain \(\varOmega \) with respect to a point \(x\in \varOmega \). In particular, using a new notion of computable harmonic approximation, we show that for an arbitrary such \(\varOmega \), computability of the harmonic measure \(\omega ^\varOmega _x\) for a single point \(x\in \varOmega \) implies computability of \(\omega _y^\varOmega \) for any \(y\in \varOmega \). This may require a different algorithm for different points y, which leads us to the construction of surprising natural examples of continuous functions that arise as solutions to a Dirichlet problem, whose values can be computed at any point, but cannot be computed with the use of the same algorithm on all of their domains. We further study the conditions under which the harmonic measure is computable uniformly, that is by a single algorithm, and characterize them for regular domains with computable boundaries.

From curves to currents

Abstract Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.

On the uniform convergence of negative order Cesaro means of Fourier series

The paper investigates the uniform convergence of negative order Cesaro means of Fourier series of continuous functions after correction of these functions on a set of small measure. We also give an example of continuous function, the (C,α) means of Fourier series of which diverge over any subsequence on the set of positive measure for any α∈(−1,−12).

The Mañé–Conze–Guivarc’h lemma for intermittent maps of the circle

AbstractWe study the existence of solutions g to the functional inequality f≤g ∘T−g+β, where f is a prescribed continuous function, T is a weakly expanding transformation of the circle having an indifferent fixed point, and β is the maximum ergodic average of f. Using a method due to T. Bousch, we show that continuous solutions g always exist when the Hölder exponent of f is close to 1. In the converse direction, we construct explicit examples of continuous functions f with low Hölder exponent for which no continuous solution g exists. We give sharp estimates on the best possible Hölder regularity of a solution g given the Hölder regularity of f.

Simultaneous Confidence Intervals for Functions of Variance Components in Random Models

Abstract Interval estimation of certain functions of variance components is of interest to research workers in all fields of applications in which the variance component model is used. Confidence intervals for linear functions and ratios of variance components have been proposed by several authors. For the most part, these intervals are approximate with unknown exact probabilities associated with their coverage. In this article a technique is given for the construction of simultaneous confidence intervals for the values of all continuous functions of the variance components in a balanced, general random linear model. These confidence intervals are conservative; that is, the actual confidence level cannot be less than any preset value. The proposed technique is easy to apply as it only requires the optimization of a given continuous function of the variance components over a bounded region. Several examples of continuous functions are considered, including linear functions and ratios of variance components. Furthermore, the technique can be applied to unbalanced, random two-way classification models.

Simultaneous Confidence Intervals for Functions of Variance Components in Random Models

Abstract Interval estimation of certain functions of variance components is of interest to research workers in all fields of applications in which the variance component model is used. Confidence intervals for linear functions and ratios of variance components have been proposed by several authors. For the most part, these intervals are approximate with unknown exact probabilities associated with their coverage. In this article a technique is given for the construction of simultaneous confidence intervals for the values of all continuous functions of the variance components in a balanced, general random linear model. These confidence intervals are conservative; that is, the actual confidence level cannot be less than any preset value. The proposed technique is easy to apply as it only requires the optimization of a given continuous function of the variance components over a bounded region. Several examples of continuous functions are considered, including linear functions and ratios of variance components...