On the nonexistence of strictly monotonic Hölder continuous functions

2006 ◽  
Vol 2 (1-2) ◽  
pp. 89-91
Author(s):  
Sergio Amat ◽  
Sonia Busquier ◽  
Antonio Escudero
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Simple Analysis ◽  
Hölder Continuous ◽  
Holder Space ◽  
Hölder Space ◽  
Analysis Tools ◽  
Hölder Continuous Functions

This note is devoted to the study of the monotony of the Hölder continuous functions. We prove the nonexistence of a strictly monotonic (increasing or decreasing) hölder continuous function with exponent s ϵ (0, 1) such that it does not belongs for anypoint in a Hölder space with exponent s + ε ε > 0. We use simple analysis’ tools.

Invariant subsets of expanding mappings of the circle

1987 ◽  
Vol 7 (4) ◽  
pp. 627-645 ◽  
Author(s):  
Mariusz Urbański
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Continuous Mapping ◽  
Continuous Functions ◽  
Compact Interval ◽  
Hölder Continuous ◽  
One Dimensional ◽  
Piecewise Monotone ◽  
Hölder Continuous Functions ◽  
Expanding Mapping

AbstractThe continuity of Hausdorff dimension of closed invariant subsetsKof aC2-expanding mappinggof the circle is investigated. Ifg/Ksatisfies the specification property then the equilibrium states of Hölder continuous functions are studied. It is proved that iffis a piecewise monotone continuous mapping of a compact interval and φ a continuous function withP(f,φ)> sup(φ), then the pressureP(f,φ) is attained on one-dimensional ‘Smale's horseshoes’, and some results of Misiurewicz and Szlenk [M−Sz] are extended to the case of pressure.

THE EFFECTS OF THE RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL ON THE BOX DIMENSION OF FRACTAL GRAPHS OF HÖLDER CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050052
Author(s):  
JUNRU WU
Keyword(s):  
Continuous Function ◽  
Fractional Integral ◽  
Closed Interval ◽  
Open Interval ◽  
Continuous Functions ◽  
Box Dimension ◽  
Hölder Continuous ◽  
Positive Order ◽  
Lipschitz Continuous ◽  
Hölder Continuous Functions

In this paper, the linearity of the dimensional-decrease effect of the Riemann–Liouville fractional integral is mainly explored. It is proved that if the Box dimension of the graph of an [Formula: see text]-Hölder continuous function is greater than one and the positive order [Formula: see text] of the Riemann–Liouville fractional integral satisfies [Formula: see text], the upper Box dimension of the Riemann–Liouville fractional integral of the graph of this function will not be greater than [Formula: see text]. Furthermore, it is proved that the Riemann–Liouville fractional integral of a Lipschitz continuous function defined on a closed interval is continuously differentiable on the corresponding open interval.

scholarly journals Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions

2020 ◽  
Vol 25 (6) ◽  
pp. 1079-1106
Author(s):  
Vitalii Makogin ◽  
Yuliya Mishura
Keyword(s):  
Continuous Function ◽  
Integral Equations ◽  
Series Representation ◽  
Continuous Functions ◽  
Fractional Integrals ◽  
Hölder Continuous ◽  
Hölder Continuous Functions ◽  
Linear Integral ◽  
New Formula ◽  
Derivatives Of

In this paper, we find fractional Riemann–Liouville derivatives for the Takagi–Landsberg functions. Moreover, we introduce their generalizations called weighted Takagi–Landsberg functions, which have arbitrary bounded coefficients in the expansion under Schauder basis. The class of weighted Takagi–Landsberg functions of order H > 0 on [0; 1] coincides with the class of H-Hölder continuous functions on [0; 1]. Based on computed fractional integrals and derivatives of the Haar and Schauder functions, we get a new series representation of the fractional derivatives of a Hölder continuous function. This result allows us to get a new formula of a Riemann–Stieltjes integral. The application of such series representation is a new method of numerical solution of the Volterra and linear integral equations driven by a Hölder continuous function.

scholarly journals Continuous solutions to two iterative functional equations

2021 ◽  
Author(s):  
Karol Baron
Keyword(s):  
Probability Measure ◽  
Functional Equations ◽  
Continuous Functions ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Hölder Continuous Functions

AbstractBased on iteration of random-valued functions we study the problem of solvability in the class of continuous and Hölder continuous functions $$\varphi $$ φ of the equations $$\begin{aligned} \varphi (x)=F(x)-\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ),\\ \varphi (x)=F(x)+\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ), \end{aligned}$$ φ ( x ) = F ( x ) - ∫ Ω φ ( f ( x , ω ) ) P ( d ω ) , φ ( x ) = F ( x ) + ∫ Ω φ ( f ( x , ω ) ) P ( d ω ) , where P is a probability measure on a $$\sigma $$ σ -algebra of subsets of $$\Omega $$ Ω .

scholarly journals Yang–Mills Measure on the Two-Dimensional Torus as a Random Distribution

2019 ◽  
Vol 372 (3) ◽  
pp. 1027-1058
Author(s):  
Ilya Chevyrev
Keyword(s):  
Random Distribution ◽  
Random Variable ◽  
Continuous Functions ◽  
Small Scale ◽  
Dimensional Torus ◽  
Hölder Continuous ◽  
Line Segments ◽  
Yang Mills ◽  
Hölder Continuous Functions ◽  
On Line

Abstract We introduce a space of distributional 1-forms $$\Omega ^1_\alpha $$Ωα1 on the torus $$\mathbf {T}^2$$T2 for which holonomies along axis paths are well-defined and induce Hölder continuous functions on line segments. We show that there exists an $$\Omega ^1_\alpha $$Ωα1-valued random variable A for which Wilson loop observables of axis paths coincide in law with the corresponding observables under the Yang–Mills measure in the sense of Lévy (Mem Am Math Soc 166(790), 2003). It holds furthermore that $$\Omega ^1_\alpha $$Ωα1 embeds into the Hölder–Besov space $$\mathcal {C}^{\alpha -1}$$Cα-1 for all $$\alpha \in (0,1)$$α∈(0,1), so that A has the correct small scale regularity expected from perturbation theory. Our method is based on a Landau-type gauge applied to lattice approximations.

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