THE EFFECT OF THE WEYL FRACTIONAL INTEGRAL ON FUNCTIONS

This paper mainly discusses the influence of the Weyl fractional integrals on continuous functions and proves that the Weyl fractional integrals can retain good properties of many functions. For example, a bounded variation function is still a bounded variation function after the Weyl fractional integral. Continuous functions that satisfy the Holder condition after the Weyl fractional integral still satisfy the Holder condition, furthermore, there is a linear relationship between the order of the Holder conditions of the two functions. At the end of this paper, the classical Weierstrass function is used as an example to prove the above conclusion.

Extending the Applicability and Convergence Domain of a Fifth-Order Iterative Scheme under Hölder Continuous Derivative in Banach Spaces

The most significant contribution made by this study is that the applicability and convergence domain of a fifth-order convergent nonlinear equation solver is extended. We use Hölder condition on the first Fréchet derivative to study the local analysis, and this expands the applicability of the formula for such problems where the earlier study based on Lipschitz constants cannot be used. This study generalizes the local analysis based on Lipschitz constants. Also, we avoid the use of the extra assumption on boundedness of the first derivative of the involved operator. Finally, numerical experiments ensure that our analysis expands the utility of the considered scheme. In addition, the proposed technique produces a larger convergence domain, in comparison to the earlier study, without using any extra conditions.

Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles

<p style='text-indent:20px;'>In this paper, we first prove the strong Birkhoff Ergodic Theorem for subharmonic functions with the irrational shift on the Torus. Then, we apply it to the analytic quasi-periodic Jacobi cocycles and show that for suitable frequency and coupling number, if the Lyapunov exponent of these cocycles is positive at one point, then it is positive on an interval centered at this point and Hölder continuous in <inline-formula><tex-math id="M1">\begin{document}$ E $\end{document}</tex-math></inline-formula> on this interval. What's more, if the coupling number of the potential is large, then the Lyapunov exponent is always positive for all irrational frequencies and Hölder continuous in <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula> for all finite Liouville frequencies. For the Schrödinger cocycles, a special case of the Jacobi ones, its Lyapunov exponent is also Hölder continuous in the frequency and the lengths of the intervals where the Hölder condition of the Lyapunov exponent holds only depend on the coupling number.</p>

Constructive description of Hölder classes on some multidimensional compact sets

We give a constructive description of Hölder classes of functions on certain compacts in Rm (m > 3) in terms of a rate of approximation by harmonic functions in shrinking neighborhoods of these compacts. The considered compacts are a generalization to the higher dimensions of compacts that are subsets of a chord-arc curve in R3. The size of the neighborhood is directly related to the rate of approximation it shrinks when the approximation becomes more accurate. In addition to being harmonic in the neighborhood of the compact the approximation functions have a property that looks similar to Hölder condition. It consists in the fact that the difference in values at two points is estimated in terms of the size of the neighborhood, if the distance between these points is commensurate with the size of the neighborhood (and therefore it is estimated in terms of the distance between the points).

ESTIMATION OF FRACTAL DIMENSION OF FRACTIONAL CALCULUS OF THE HÖLDER CONTINUOUS FUNCTIONS

In the present paper, fractal dimension and properties of fractional calculus of certain continuous functions have been investigated. Upper Box dimension of the Riemann–Liouville fractional integral of continuous functions satisfying the Hölder condition of certain positive orders has been proved to be decreasing linearly. If sum of order of the Riemann–Liouville fractional integral and the Hölder condition equals to one, the Riemann–Liouville fractional integral of the function will be Lipschitz continuous. If the corresponding sum is strictly larger than one, the Riemann–Liouville fractional integral of the function is differentiable. Estimation of fractal dimension of the derivative function has also been discussed. Finally, the Riemann–Liouville fractional derivative of continuous functions satisfying the Hölder condition exists when order of the Riemann–Liouville fractional derivative is smaller than order of the Hölder condition. Upper Box dimension of the function has been proved to be increasing at most linearly.

ESTIMATION OF FRACTAL DIMENSION OF WEYL FRACTIONAL INTEGRAL OF CERTAIN CONTINUOUS FUNCTIONS

In this work, we consider fractal dimension such as Box dimension, of Weyl fractional integral of certain continuous functions. Upper Box dimension of Weyl fractional integral of continuous functions satisfying [Formula: see text]-order Hölder condition of positive order [Formula: see text] is no more than both [Formula: see text] and [Formula: see text]. Furthermore, it is no more than [Formula: see text] which means strictly less than [Formula: see text]. Meanwhile, [Formula: see text], Box dimension of Weyl fractional integral of continuous functions satisfying [Formula: see text]-order Hölder condition must be one.

Estimation of the multifractional function and the stability index of linear multifractional stable processes

In this paper we are interested in multifractional stable processes where the self-similarity index H becomes time-dependent, while the stability index α remains constant. Using β- negative power variations ( − 1∕2 < β < 0), we propose estimators for the value at a fixed time of the multifractional function H which satisfies an η-Hölder condition and for α in two cases: multifractional Brownian motion (α = 2) and linear multifractional stable motion (0 < α < 2). We get the consistency of our estimates for the underlying processes together with the rate of convergence.