Infinitely Differentiable Functions Research Articles

Description of the Duals of Subspaces of Infinitely Differentiable Functions

Description of the Duals of Subspaces of Infinitely Differentiable Functions

On Hypercyclic Operators in Weighted Spaces of Infinitely Differentiable Functions

On Hypercyclic Operators in Weighted Spaces of Infinitely Differentiable Functions

On Hypercyclic Operators in Weighted Spaces of Infinitely Differentiable Functions

Рассматривается инвариантное относительно дифференцирования весовое пространство Фреше ${\mathcal E}(\varphi)$ бесконечно дифференцируемых функций в ${\mathbb R}^n$, порожденное счетным семейством $\varphi$ непрерывных вещественнозначных функций в ${\mathbb R}^n$. При минимальных ограничениях на $\varphi$ показано, что любой линейный непрерывный оператор в пространстве ${\mathcal E}(\varphi)$, коммутирующий с операторами частного дифференцирования и не являющийся скалярным кратным тождественного отображения, является гиперциклическим. Приведены примеры гиперциклических операторов в ${\mathcal E}(\varphi)$ для случаев, когда пространство ${\mathcal E}(\varphi)$ инвариантно относительно сдвига. Библиография: 8 названий.

Approximation of Infinitely Differentiable Functions on the Real Line by Polynomials in Weighted Spaces

By a given family of convex functions on the real axis that grow at infinity faster than any linear function and by a certain logarithmically convex sequence of positive numbers, we construct the space of infinitely differentiable functions on the real line. Under the condition of a logarithmic gap between weight functions, we prove the possibility of approximation by polynomials in this space.

Approximation of Infinitely Differentiable Functions in Unbounded Convex Domains by Polynomials in Weighted Spaces

We examine the problem of polynomial approximation in the space of infinitely differentiable functions in an unbounded convex domain in ℝn that have a prescribed growth rate near the boundary of the domain and at infinity.

Asymptotic Equalities for Best Approximations for Classes of Infinitely Differentiable Functions Defined by the Modulus of Continuity

Получены асимптотические оценки для наилучших приближений тригонометрическими полиномами в метрике пространства $C$ $(L_p)$ классов периодических функций, представимых в виде сверток ядер $\Psi_\beta$ с коэффициентами Фурье, убывающими к нулю быстрее любой степенной последовательности, с функциями $\varphi\in C$ $(\varphi\in L_p)$, модули непрерывности которых не превышают заданной мажоранты $\omega(t)$. Доказано, что в пространствах $C$ и $L_1$ для выпуклых модулей непрерывности $\omega(t)$ полученные оценки являются асимптотически точными. Библиография: 18 названий.

Estimations of the Best Approximations for the Classes of Infinitely Differentiable Functions in Uniform and Integral Metrics

We establish uniform (with respect to the parameter p, 1 ≤ p ≤ ∞) upper estimations of the best approximations by trigonometric polynomials for the classes C , of periodic functions generated by sequences ψ(k) vanishing faster than any power function. The obtained estimations are exact in order and contain constants expressed in the explicit form and depending solely on the function ψ. Similar estimations are obtained for the best approximations of the classes L ,1 in metrics of the spaces L s , 1 ≤ s ≤ ∞.

A Version of Hilbert's 13th Problem for Infinitely Differentiable Functions

In 1957, Kolmogorov and Arnold gave a solution to the 13th problem which had been formulated by Hilbert in 1900. Actually, it is known that there exist many open problems which can be derived from the original problem. From the function-theoretic point of view, Hilbert's 13th problem can be exactly characterized as the superposition representability problem for continuous functions of several variables. In this paper, the solution to the superposition representability problem for infinitely differentiable functions of several variables is given.

Superposition Operator in a Space of Infinitely Differentiable Functions

Superposition Operator in a Space of Infinitely Differentiable Functions

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.

Euler Type Differential Equations in the Class of Infinitely Differentiable Functions

where ak(x) ∈ C∞[0, 1] (k = 1, . . . , n− 1) and f(x) ∈ C∞[0, 1] are given functions, φ(x) ∈ C∞[0, 1] is the unknown function, and n is a positive integer. Here C∞[0, 1] is the class of infinitely differentiable functions on the interval [0, 1]. Equation (1) with f ≡ 0 is said to be homogeneous. If aj(x) ≡ const (j = 0, . . . , n − 1), then this equation coincides with the Euler equation. Since the coefficient of φ(x) in Eq. (1) vanishes for x = 0, it follows that this equation has a singularity at this point. The singularity order of Eq. (1) is equal to n. Differential equations of the form (1) arise in numerous problems of mathematical physics. Such equations include Bessel, Legendre, Chebyshev–Laguerre, and other equations [1]. Some special cases of homogeneous equations of the form (1) have been rather comprehensively studied in the class of analytic functions of the real variable x [1–3] as well as in the class of twice continuously differentiable functions on the interval [0, 1] (see [4, 5]). In [1–5], it was assumed that the coefficients of Eq. (1) are analytic with respect to the real variable x in a neighborhood of zero. The aim of the present paper is to obtain necessary and sufficient conditions for the unique solvability of Eq. (1) in the class C∞[0, 1] and indicate a solution method. By Pn(λ) we denote the polynomial

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.

Endomorphisms of Banach Algebras of Infinitely Differentiable Functions on Compact Plane Sets

Let X be a perfect, compact subset of the complex plane, let (Mn) be a sequence of positive numbers satisfying M0=1 and ([formula])⩽Mm+n/MmMn, and let D(X,M)={f ∈D∞(X): ‖f‖ =∑∞n=0(‖f(n)‖∞/Mn)<∞{. With pointwise addition and multiplication, D(X,M) is a commutative normed algebra. In this note we study the endomorphisms of such algebras. In a previous paper this problem was solved when X=[0,1] for many weights (Mn). Here we investigate the extent to which the methods used previously apply to general X and introduce some new methods. In particular, we obtain many results when X=Δ where Δ is the open unit disc.

Approximation and Extension in Normed Spaces of Infinitely Differentiable Functions

Approximation and Extension in Normed Spaces of Infinitely Differentiable Functions

Construction of Characteristic Functions for Classes of Infinitely Differentiable Functions

Characteristic functions are constructed for classes of infinitely differentiable functions defined on a half-line and for Paley-Wiener classes. A corresponding result is given for normal operators defined in Hilbert space.

Inverse-Closed Carleman Algebras of Infinitely Differentiable Functions

Inverse-Closed Carleman Algebras of Infinitely Differentiable Functions

A Property of Infinitely Differentiable Functions

A Property of Infinitely Differentiable Functions

Factorization and Automatic Continuity for an Algebra of Infinitely Differentiable Functions

Factorization and Automatic Continuity for an Algebra of Infinitely Differentiable Functions

Most Infinitely Differentiable Functions are Nowhere Analytic

We define a natural metric, d, on the space, C∞,, of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, and show that C∞, is complete with respect to this metric. Then we show that the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C∞,.

A Closure Problem for Spaces of Infinitely Differentiable Functions.

A Closure Problem for Spaces of Infinitely Differentiable Functions.