The Finite Fractional Hilbert Transform

In the present paper, we introduce the finite fractional Hilbert transform. Parseval-type identities concerning finite fractional Hilbert transform are proved. Moreover, we obtain inequality for finite fractional Hilbert transform of β— Hölder continuous functions. Applications for some functions are also provided.

Banach manifold structure and infinite-dimensional analysis for causal fermion systems

A mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold endowed with a canonical Fréchet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating derivatives of functions on Banach spaces which are differentiable only in certain directions. A chain rule is proven for Hölder continuous functions which are differentiable on expedient subspaces. These results are made applicable to causal fermion systems by proving that the causal Lagrangian is Hölder continuous. Moreover, Hölder continuity is analyzed for the integrated causal Lagrangian.

Continuous solutions to two iterative functional equations

Based 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 .

A new approach with new solutions to the Matkowski and Wesołowski problem

Based on a result of de Rham, we give a family of functions solving the Matkowski and Wesołowski problem. This family consists of Hölder continuous functions, and it coincides with the whole family of solutions to the Matkowski and Wesołowski problem found earlier by a different method. Moreover, applying some results due to Hata and Yamaguti and due to Berg and Krüppel, we prove that there are functions solving the Matkowski and Wesołowski problem that are not Hölder continuous.

Deep Network with Approximation Error Being Reciprocal of Width to Power of Square Root of Depth

A new network with super-approximation power is introduced. This network is built with Floor ([Formula: see text]) or ReLU ([Formula: see text]) activation function in each neuron; hence, we call such networks Floor-ReLU networks. For any hyperparameters [Formula: see text] and [Formula: see text], we show that Floor-ReLU networks with width [Formula: see text] and depth [Formula: see text] can uniformly approximate a Hölder function [Formula: see text] on [Formula: see text] with an approximation error [Formula: see text], where [Formula: see text] and [Formula: see text] are the Hölder order and constant, respectively. More generally for an arbitrary continuous function [Formula: see text] on [Formula: see text] with a modulus of continuity [Formula: see text], the constructive approximation rate is [Formula: see text]. As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of [Formula: see text] as [Formula: see text] is moderate (e.g., [Formula: see text] for Hölder continuous functions), since the major term to be considered in our approximation rate is essentially [Formula: see text] times a function of [Formula: see text] and [Formula: see text] independent of [Formula: see text] within the modulus of continuity.

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

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.

On a quantitative theory of limits: Estimating the speed of convergence

The classical “ε-δ” definition of limits is of little use to quantitative purposes, as is needed, for instance, for computational and applied mathematics. Things change whenever a realistic and computable estimate of the function δ(ε) is available. This may be the case for Lipschitz continuous and Hölder continuous functions, or more generally for functions admitting of a modulus of continuity. This, provided that estimates for first derivatives, fractional derivatives, or the modulus of continuity might be obtained. Some examples are given.

Trapezoid type inequalities for generalized Riemann-Liouville fractional integrals of functions with bounded variation

In this paper we establish some trapezoid type inequalities for the Riemann-Liouville fractional integrals of functions of bounded variation and of Hölder continuous functions. Applications for the g-mean of two numbers are provided as well. Some particular cases for Hadamard fractional integrals are also provided.