On Kolmogorov’s ε-Entropy for a Compact Set of Infinitely Differentiable Aperiodic Functions (Babenko’s Problem)

The asymptotics of Kolmogorov’s ε-entropy for a compact set of infinitely differentiable aperiodic functions that are boundedly embedded in the space of continuous functions on a finite interval is calculated.     

Real Taylor shifts and simultaneous approximation

We are interested in the Taylor shift operator acting on the space of infinitely differentiable functions. In particular if we choose a countable family of centers of Taylor expansion, we prove that the associated family of real Taylor shifts fulfills the approximation of any given family of infinitely differentiable functions with a common subsequence of iterates applied on a common vector. We obtain similar conclusions in the context of universal series improving recent statements. Finally we introduce the notion of doubly universal Taylor shift. All these results give new and natural examples of disjoint universality.     

Infinite dimensional Banach spaces of functions with nonlinear properties

The aim of this paper is to show that there exist infinite dimensional Banach spaces of functions that, except for 0, satisfy properties that apparently should be destroyed by the linear combination of two of them. Three of these spaces are: a Banach space of differentiable functions on ?ⁿ failing the Denjoy‐Clarkson property; a Banach space of non Riemann integrable bounded functions, but with antiderivative at each point of an interval; a Banach space of infinitely differentiable functions that vanish at infinity and are not the Fourier transform of any Lebesgue integrable function (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Absolute Ε-Entropy of a Compact Set of Infinitely Differentiable Aperiodic Functions

We calculate asymptotics for the Kolmogorov ε-entropy of the compact set of infinitely differentiable aperiodic functions embedded continuously into the space of continuous functions on a closed finite interval.     

On the Representation of Infinitely Differentiable Functions by Series of Exponentials

We consider the problem of representing elements of a weighted space of infinitely differentiable functions on the real line by series of exponentials.     

Surjectivity of Linear Differential Operators in a Weighted Space of Infinitely Differentiable Functions

We present a sufficient condition for the surjectivity of a linear differential operator of infinite order with constant coefficients in a weighted space of infinitely differentiable functions on the real line.     

Multiresolution Analysis by Spherical Up Functions

A new class of locally supported radial basis functions on the (unit) sphere is introduced by forming an infinite number of convolutions of "isotropic finite elements." The resulting up functions show useful properties: They are locally supported and are infinitely often differentiable. The main properties of these kernels are studied in detail. In particular, the development of a multiresolution analysis is given based on locally supported zonal functions within the reference space of square-integrable functions over the sphere.     

On the Fourier-Laplace transform of functionals on a space of infinitely differentiable functions on a convex compact

Classes of ultradifferentiable functions are classically defined by imposing growth conditions on the derivatives of the functions. Following this approach we consider a Fréchet-Schwartz space of infinitely differentiable functions on a closure of a bounded convex domain of multidimensional real space with uniform bounds on their partial derivatives. Our aim is to obtain Paley-Wiener-Schwartz type theorem connecting properties of linear continuous functionals on this space with the behaviour of their Fourier-Laplace transforms. Very similar problems were considered by M. Neymark, B.A. Taylor, M. Langenbruch, A.V. Abanin.     

On the absolute <Emphasis Type="Italic">ɛ</Emphasis>-entropy of some compact set of infinitely differentiable periodic functions

We calculate the principal term of the asymptotics of the Kolmogorov ɛ-entropy of some compact set of periodic infinitely differentiable functions that is continuously embedded into the space of continuous periodic functions.     

Product integrals in strongly differentiable power associative groupoids

The existence of product integrals in differentiable semigroups with neighborhood of 1 homeomorphic to a Banach space and with multiplication strongly differentiable at (1,1) was shown by Garrett Birkhoff in 1938. This result was then used to transform the multiplication into canonical parameters. We show here a sufficient condition to guarantee the existence of product integrals in differentiable power associative groupoids with multiplication strongly differentiable at (1,1).     

Cauchy’s integral formula via the modified Riemann–Liouville derivative for analytic functions of fractional order

The modified Riemann–Liouville fractional derivative applies to functions which are fractional differentiable but not differentiable, in such a manner that they cannot be analyzed by means of the Djrbashian fractional derivative. It provides a fractional Taylor’s series for functions which are infinitely fractional differentiable, and this result suggests introducing a definition of analytic functions of fractional order. Cauchy’s conditions for fractional differentiability in the complex plane and Cauchy’s integral formula are derived for these kinds of functions.     

Weak and quasi-polynomial tractability of approximation of infinitely differentiable functions

We comment on recent results in the field of information based complexity, which state (in a number of different settings), that the approximation of infinitely differentiable functions is intractable and suffers from the curse of dimensionality. We show that renorming the space of infinitely differentiable functions in a suitable way allows weakly tractable uniform approximation by using only function values. Moreover, the approximating algorithm is based on a simple application of Taylor’s expansion about the center of the unit cube. We discuss also the approximation on the Euclidean ball and the approximation in the _L1$L_1$-norm.     

SPECTRAL APPROXIMATION ORDERS OF MULTIDIMENSIONAL NONSTATIONARY BIORTHOGONAL SEMI-MULTIRESOLUTION ANALYSIS IN SOBOLEV SPACE

Subdivision algorithm （Stationary or Non-stationary） is one of the most active and exciting research topics in wavelet analysis and applied mathematical theory. In multidimensional non-stationary situation, its limit functions are both compactly supported and infinitely differentiable. Also, these limit functions can serve as the scaling functions to generate the multidimensional non-stationary orthogonal or biorthogonal semi-multiresolution analysis （Semi-MRAs）. The spectral approximation property of multidimensional non-stationary biorthogonal Semi-MRAs is considered in this paper. Based on nonstationary subdivision scheme and its limit scaling functions, it is shown that the multidimensional nonstationary biorthogonal Semi-MRAs have spectral approximation order r in Sobolev space H^s（R^d）, for all r ≥ s ≥ 0.     

Monotone transformations and differential properties of functions

The class of all real functions of a single variable which become everywhere differentiable after a certain homeomorphic transformation of coordinate axis is described. Moreover, various examples about differential properties of functions are given (in particular, an elementary construction of a nonconstant continuously differentiable real function of two variables, every value of which is critical-the example of Whitney, is given).Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 859–871, December, 1977.The authors express gratitude to A. L. Brudno, P. L. Ul'yanov, and V. A. Skvortsov for discussion of the results given here.     

Characterization of the speed of convergence of the trapezoidal rule

Summary Our aim is to determine the precise space of functions for which the trapezoidal rule converges with a prescribed rate as the number of nodes tends to infinity. Excluding or controlling odd functions in some way it is possible to establish a correspondence between the speed of convergence and regularity properties of the function to be integrated. In this way we characterize Sobolev spaces, certain spaces of infinitely differentiable functions, of functions holomorphic in a strip, of entire functions of order greater than 1 and of entire functions of exponential type by the speed of convergence.Dedicated to Professor G. Hämmerlin on the occasion of his 60th birthday     

Some Properties of Smooth Convex Functions and Newton’s Method

Doklady Mathematics - New properties of convex infinitely differentiable functions related to extremal problems are established. It is shown that, in a neighborhood of the solution, even if the...     

Injectivity Conditions for Some Classes of Domains. II

New sufficient conditions are found for continuously differentiable mappings in some plane domains to be injective. This result is obtained by using the locally homeomorphic extension method. New univalence criteria for meromorphic functions are given by means of the quasiconformal extension method. These criteria are stated in terms of coefficients of a differential equation related to the Schwarzian derivative of the function under consideration. Bibliography: 13 titles.     

关于多元角形区域上的无穷可微函数的广义准解析性

本文将关于角形闭区域中无穷可微函数类的广义准解析性的结果推广到了多维情形。     

Optimal Gevrey classes for the existence of solution operators for linear partial differential operators in three variables

For a constant coefficient linear partial differential operator acting on all infinitely differentiable functions or ω-ultradifferentiable functions of Beurling type on euclidean 3-space, the existence of a continuous linear solution operator is investigated. It is shown that there is an optimal weight ω in the sense that a solution operator exists for a weight σ if and only if ω=O(σ), provided that such an operator exists for at least one weight. Furthermore, the optimal class is either a Gevrey class of rational exponent or the class of all infinitely differentiable functions.