Conjugate functions of several variables in the class Lip α

It is known that if a function f of a single variable belongs to the class Lip (, C(T)) (0<<1), then its conjugate function also belongs to the same class; in other words, the class Lip (, C(T)) (0<<1) is invariant with respect to the operator of conjugation acting in it. In the two-dimensional case the class Lip (, C(T²)) (0<<1) is no longer invariant with respect to conjugate functions of two variables. Here a final result elucidating the full character of violation of invariance of the class Lip (, C(T^N)) (0<<1) with respect to the multidimensional conjugation operator acting in it is established.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 361–372, March, 1978.     

A conjecture of Erdős and Reddy and the rational approximation of Mittag-Leffler type functions

Пусть \(f(z) = \mathop \sum \limits_{k = 0}^\infty a_k z^k ,a_0 \ne 0, a_k \geqq 0 (k \geqq 0)\) — целая функци я,π _n — класс обыкновен ных алгебраических мног очленов степени не вы ше \(n,a \lambda _n (f) = \mathop {\inf }\limits_{p \in \pi _n } \mathop {\sup }\limits_{x \geqq 0} |1/f(x) - 1/p(x)|\) . П. Эрдеш и А. Редди высказали пр едположение, что еслиf(z) имеет порядок ?ε(0, ∞) и $$\mathop {\lim sup}\limits_{n \to \infty } \lambda _n^{1/n} (f)< 1, TO \mathop {\lim inf}\limits_{n \to \infty } \lambda _n^{1/n} (f) > 0$$ В данной статье показ ано, что для целой функ ции $$E_\omega (z) = \mathop \sum \limits_{n = 0}^\infty \frac{{z^n }}{{\Gamma (1 + n\omega (n))}}$$ , где выполняется $$\lambda _n^{1/n} (E_\omega ) \leqq \exp \left\{ { - \frac{{\omega (n)}}{{e + 1}}} \right\}$$ , т.е. $$\mathop {\lim sup}\limits_{n \to \infty } \lambda _n^{1/n} (E_\omega ) \leqq \exp \left\{ { - \frac{1}{{\rho (e + 1)}}} \right\}< 1, a \mathop {\lim inf}\limits_{n \to \infty } \lambda _n^{1/n} (E_\omega ) = 0$$ . ФункцияE _ω (z) имеет порядок ?.     

Gâteaux differentiability and uniform monotone approximation of convex functions in Banach spaces

Acta Mathematica Hungarica - We prove that if $$X^{*}$$ is strictly convex, a convex function $$f$$ is coercive and b-Lipschitzian iff there exist two convex function sequences...     

Legendre rational approximation on the whole line

The Legendre rational approximation is investigated. Some approximation results are established, which form the mathematical foundation of a new spectral method on the whole line. A model problem is considered. Numerical results show the efficiency of this new approach.     

On the coincidence of some subdifferentials in the class of α(.)-paraconvex functions

In the paper an equivalence of Clarke, Dini, α(.)-subgradients and local α(.)-subgradients for strongly α(.)-paraconvex functions is proved     

Closed convex hull of the family of multivalently close-to-convex functions of order β

In this paper the closed convex hulls of the compact familiesC _β(p), of multivalently close to convex functions of order β andV ₀ ^k (p), of multivalent functions of bounded boundary rotation, have been determined, respectively for β≥1 andk≥2(p+1)/p. Extreme points of these convex hulls are partially characterised. For a fixed pointz ₀∈D={z:|z|<1}, a new familyC _β(p, z₀) is defined through Montel normalisation and its closed convex hull is also foud. Sharp coefficient estimates for functions subordinate to or majorised by some function inC _β(p) orC' _β(p) are discussed for β>0. It is shown that iff is subordinate to some function inC _β(p) then each Taylor coefficient off is dominated by the corresponding coefficient of the function .     

Matrix Padé Approximation of rational functions

In many applications it is of major interest to decide whether a given formal power series with matrix-valued coefficients of arbitrary dimensions results from a matrix-valued rational function. As the main result of this paper we provide an answer to this question in terms of Matrix Padé Approximants of the given power series. Furthermore, given a matrix rational function, the smallest degrees of the matrix polynomials which represent it are not necessarily unique. Therefore we study a certain minimality-type, that is, minimum degrees. We aim to obtain all the minimum degrees for the polynomials which represent the function as equivalents. In addition, given that the rational representation of the function for the same pair of degrees need not be unique, we have obtained conditions to study the uniqueness of said representation. All the results obtained are presented graphically in tables setting out the above information. They lead to a number of properties concerning special structures, staired blocks, in the Padé Table.     

The Collet–Eckmann condition for rational functions on the Riemann sphere

We show that the set of Collet–Eckmann maps has positive Lebesgue measure in the space of rational maps on the Riemann sphere for any fixed degree d ≥ 2.     

Gleason’s problem,rational functions and spaces of left-regular functions: The split-quaternion setting

We study Gleason’s problem, rational functions and spaces of regular functions in the setting of split-quaternions. There are two natural symmetries in the algebra of split-quaternions. The first symmetry allows to define positive matrices with split-quaternionic entries, and also reproducing Hilbert spaces of regular functions. The second leads to reproducing kernel Krein spaces.     

The radius of almost convexity of order α in the class of univalent functions

In this paper the radius of almost convexity of order in the class of functions f(z)=z+₂z²+... analytic and univalent in ¦z¦<1 is found. The solution to a problem of A. Renyi is given in this connection.Translated from Matematicheskii Zametki, Vol. 20, No. 1, pp. 105–112, January, 1976.     

Moreau–Yosida regularization of Lagrangian-dual functions for a class of convex optimization problems

In this paper, we consider the Lagrangian dual problem of a class of convex optimization problems, which originates from multi-stage stochastic convex nonlinear programs. We study the Moreau–Yosida regularization of the Lagrangian-dual function and prove that the regularized function η is piecewise C ², in addition to the known smoothness property. This property is then used to investigate the semismoothness of the gradient mapping of the regularized function. Finally, we show that the Clarke generalized Jacobian of the gradient mapping is BD-regular under some conditions.      

Hölder-continuity of certain generalized convex functions

It is proved that for a rationally s-convex function continuity and local s-Hölder - Continuity are equivalent at each interior point of the domain of definition of the function. Furthermore, it is shown that a rationally s-convex function which is bounded on a nonempty open convex set is s-Hölder-continuous on every compact subset of this set.     

Estimate of the approximation of periodic functions by negative Cesàro means

The problem of approximation of continuous functions by Cesàro (C,α)-means, −1 < α < 0, in terms of L _p and C-modulus of continuity is studied.     

On approximation of functions in terms of Φ-variation

Получен аналог теоре мы Джексона о приближ ении полиномами в метрике Ф-вариации; для некото рых классов функций (в ыпуклых, конечной вариации, с заданным модулем неп рерывности) установл ено соотношение типа сла бой эквивалентности, которое связывает ме жду собой рациональн ые аппроксимации, в этой и равномерной метриках Даны оценкиε-энтропи и в метрике Ф-вариации классов функций с заданным мо дулем Ф-абсолютной непреры вности и с заданным мо дулем непрерывности.     

Are the degrees of best (co)convex and unconstrained polynomial approximation the same?

Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ² the set of convex functions f ∈ ℂ[−,1]. Also, let E _n (f) and E _n ⁽²⁾ (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ² by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ E _n ⁽²⁾ (f), and Lorentz and Zeller proved that the inverse inequality E _n ⁽²⁾ (f) ≦ cE _n (f) is invalid even with the constant c = c(f) which depends on the function f ∈ Δ². In this paper we prove, for every α > 0 and function f ∈ Δ², that