Interpolation by hypercyclic functions for differential operators

We prove that, given a sequence of points in a complex domain Ω without accumulation points, there are functions having prescribed values at the points of the sequence and, simultaneously, having dense orbit in the space of holomorphic functions on Ω. The orbit is taken with respect to any fixed nonscalar differential operator generated by an entire function of subexponential type, thereby extending a recent result about MacLane-hypercyclicity due to Costakis, Vlachou and Niess.     

A note on fractional derivatives and fractional powers of operators

Definitions of fractional derivatives and fractional powers of positive operators are considered. The connection of fractional derivatives with fractional powers of positive operators is presented. The formula for fractional difference derivative is obtained.     

On the iterates of positive linear operators preserving the affine functions

In this note we study the limit behavior of the iterates of a large class of positive linear operators preserving the affine functions and, as a byproduct of our result, we obtain the limit of the iterates of Meyer-König and Zeller operators.     

Two-weight norm inequalities for fractional maximal functions and fractional integral operators on weighted Morrey spaces

In this paper, we consider weighted norm inequalities for fractional maximal operators and fractional integral operators. For suitable weights, we prove the two-weight norm inequalities for both operators on weighted Morrey spaces.     

Weighted norm inequalities for operators of potential type and fractional maximal functions

In this paper, we study two-weight norm inequalities for operators of potential type in homogeneous spaces. We improve some of the results given in [6] and [8] by significantly weakening their hypotheses and by enlarging the class of operators to which they apply. We also show that corresponding results of Carleson type for upper half-spaces can be derived as corollaries of those for homogeneous spaces. As an application, we obtain some necessary and sufficient conditions for a large class of weighted norm inequalities for maximal functions under various assumptions on the measures or spaces involved.Research of the first author was supported in part by NSERC grant A5149.Research of the second author was supported in part by NSF grant DMS93-02991.     

On the iterates of positive linear operators

We have devised a new method for the study of the asymptotic behavior of the iterates of positive linear operators. This technique enlarges the class of operators for which the limit of the iterates can be computed.     

Limits of certain iterates of Post-Widder operators

The paper is a study of the limiting behaviour of the [n t]-th iterates of the well-known Post-Widder operatorsL _{n, x} used in the real inversion of the Laplace transform. It is shown that the limiting operators constitute a semigroup T _t;t0 of class (C ₀) on a family C _,; , >0 of Banach spaces. Applications of the semigroup structure lead to a pointwise saturation theorem forL _{n, x} and a characterization of convex functions inC _, through an inequality involving the action ofL _{n, x}.     

On quasi-periodicity properties of fractional integrals and fractional derivatives of periodic functions

This paper is devoted to the study of quasi-periodic properties of fractional order integrals and derivatives of periodic functions. Considering Riemann–Liouville and Caputo definitions, we discuss when the fractional derivative and when the fractional integral of a certain class of periodic functions satisfies particular properties. We study concepts close to the well known idea of periodic function, such as S-asymptotically periodic, asymptotically periodic or almost periodic function. Boundedness of fractional derivative and fractional integral of a periodic function is also studied.     

Interpolation by neural network operators activated by ramp functions

In this paper, a family of interpolation neural network operators are introduced. Here, ramp functions as well as sigmoidal functions generated by central B-splines are considered as activation functions. The interpolation properties of these operators are proved, together with a uniform approximation theorem with order, for continuous functions defined on bounded intervals. The relations with the theory of neural networks and with the theory of the generalized sampling operators are discussed.     

Asymptotic expansions for Riesz fractional derivatives of Airy functions and applications

Riesz fractional derivatives of a function, (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions, and , via and . Here Bi(x) is the Airy function of the second type. Integral representations are presented for the function A₂(a,b;x)=Ai(x−a)Ai(x−b) with a,b∈R and its Hilbert transform. Combined with the above asymptotic expansions they can be used for computing asymptotics of the Hankel transform of . These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero).     

Stable approximation of fractional derivatives of rough functions

The process of numerical fractional differentiation is well known to be an ill-posed problem, and it has been discussed by many authors, and a large number of different solution methods has been proposed. However, available approaches require a knowledge of a bound of the second or third derivatives of the function under consideration which are not always available. In this paper the following mollification method is suggested for this problem: if the data are given inexactly then we mollify them by elements of well-posedness classes of the problem, namely, by elements of an appropriater-regular multiresolution approximation {V _j } _j ofL ²(). WithinV _j the problem of fractional differentiation is well-posed, and we can find a mollification parameterJ depending on the noise level in the data, such that the error estimation is of Hölder type. The method can be numerically implemented by fundamental results by Beylkin, Coifman and Rokhlin ([2]) on representing differential operators in wavelet bases. It is worthwhile to note that there are very few papers concerning the question of stable numerical fractional differentiation of very rough functions. It is interesting that by our method, in a certain sense, we can approximate the derivatives of very rough functions (functions fromH _s (),s ) which have no derivative in the classical sense, like the hat functions, step functions..., in a stable way. Our method is of interest, since it is local. This means that to approximate the fractional derivative of a function at a point with improperly given data, we need only local information about this function in an appropriate neighbourhood of this point.On leave from Hanoi Institute of Mathematics, supported by the Deutsche Forschungs-gemeinschaft