Homomorphisms with respect to a function

Let X be a realcompact space and H:C(X)?\mathbbR{H:C(X)\rightarrow\mathbb{R}} be an identity and order preserving group homomorphism. It is shown that H is an evaluation at some point of X if and only if there is j ? C(\mathbbR){\varphi\in C(\mathbb{R})} with ${\varphi(r)>\varphi(0)}${\varphi(r)>\varphi(0)} for all r ? \mathbbR-{0}{r\in\mathbb{R}-\{0\}} for which H°j = j°H{H\circ\varphi=\varphi\circ H} . This extends (and unifies) classical results by Hewitt and Shirota.     

Bochner-Riesz means with respect to a rough distance function

The generalized Bochner-Riesz operator may be defined as

where is an appropriate distance function and is the inverse Fourier transform. The behavior of on is described for , a rough distance function. We conjecture that this operator is bounded on when and , and unbounded when . This conjecture is verified for large ranges of .

     

Lusin's theorem for derivatives with respect to a continuous function

For a nowhere constant continuous function on a real interval and for a Borel measure on , we give simple necessary and sufficient conditions guaranteeing, for any Borel function on , the existence of a continuous function on such that the derivative of with respect to is, almost everywhere, equal to .

     

On Riesz means with respect to a cylindric distance function

(1–) ₊ , — R ⁿ =R ^j ×R ^k , ()=max{¦ ₁¦, ¦ ₁¦},=( ₁, ₂), ₁R ^J , ₂R ^k ,j,k1,n=j+k. n=3 , (1–) ₊ [L ¹(R ⁿ )]¹, >1/2; j=4, (1–) ₊ R L ^p (R ⁿ ). .

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The author would like to thank Professor W. Trebels for encouragement and valuable advice.     

On asymptotically equivalent difference sequences with respect to a modulus function

This paper presents new definitions which are a natural combination of the definition for asymptotically equivalence and Δ ^m -lacunary strongly summable with respect to a modulus f. Using this definitions we have proved the (f, Δ ^m )-asymptotically equivalence and Δ ^m -lacunary statistical asymptotically equivalence analogues of theorems of Tripathy and Et (Stud Univ Babe?-Bolyai Math (1):119–130, 2005) and Çolak’s theorems (Filomat 17:9–14, 2003).     

Homomorphisms on some function algebras

In this note we prove, for some classes of real locally convex spacesE including all complete Schwartz spaces, that every non-zero homomorphism on the algebraC (E) ofC -functions onE is given by a point evaluation at some point ofE.     

On the differentiation of a hypergeometric function with respect to the parameter

A new lower bound for integer-valued linear forms of values of certain hypergeometric functions is established.     

Homomorphisms of certain Banach function algebras

In this note, we study homomorphisms with domainD ⁿ(X) orLipα(X, d) of which ranges are certain Banach function algebras and determine in which cases these homomorphisms are compact.     

Properties of fractional calculus with respect to a function and Bernstein type polynomials

This paper is divided into two stages. In the first stage, we investigated a new approach for the $\psi$-Riemann–Liouville fractional integral and the Faa di Bruno formula for the $\psi$-Hilfer fractional derivative. In addition, we discussed other properties involving the $\psi$-Hilfer fractional derivative and the $\psi$-Riemann–Liouville fractional integral. In the second stage, Bernstein polynomials involving the $\psi (·)$ function are investigated and the $\psi$-Riemann–Liouville fractional integral and $\psi$-Hilfer fractional derivative from the Bernstein polynomials are evaluated. We also discussed the relationship between the $\psi$-Hilfer fractional derivative with Laguerre polynomials and hypergeometric functions, and a version of the fractional mean value theorem with respect to a function. Motivated by the Bernstein polynomials, the second stage uses the Bernstein polynomials to approximate the solution of a fractional integro-differential equation with Hilfer fractional derivative and concluding with a numerical approach with its respective graph.     

Fractional differential equations with a Caputo derivative with respect to a Kernel function and their applications

This paper is devoted to the study of the initial value problem of nonlinear fractional differential equations involving a Caputo‐type fractional derivative with respect to another function. Existence and uniqueness results for the problem are established by means of the some standard fixed point theorems. Next, we develop the Picard iteration method for solving numerically the problem and obtain results on the long‐term behavior of solutions. Finally, we analyze a population growth model and a gross domestic product model with governing equations being fractional differential equations that we have introduced in this work.     

Homomorphisms on real-valued little Lipschitz function spaces

We describe the general form of algebra, ring and vector lattice homomorphisms between spaces of real-valued little Lipschitz functions on compact Hölder metric spaces (X,dα) for 0<α<1.     

Systems of linear differential equations symmetric with respect to transformations nonlinear in a function

We show that if a system of first order linear differential equations with constant coefficients admits an operator whose coordinates responsible for the transformation of independent variables depend on functions then it admits an operator whose coordinates responsible for the transformation of functions depend nonlinearly on functions.     

On function classes in P 3 precomplete with respect to a strengthened closure operator

A classical theorem of Post [1] describes five precomplete classes in the set of Boolean functions. In [2], it was shown that there exist 18 precomplete classes of functions of three-valued logic. In [1, 2], the closure of sets of functions with respect to the substitution operator was studied. We consider two closure operators on functions of three-valued logic, which are obtained by supplementing the substitution operator by closures with respect to two identifications of function values, and prove the existence of three precomplete classes for one of these operators and five precomplete classes for the other.     

Choquet integral with respect to a symmetric fuzzy measure of a function on the real line

Some results about the calculation of the Choquet integral of a monotone function are presented. The construction of monotone functions from non-monotone ones that lead to the same Choquet integral is studied.