On a problem of L. Leindler concerning strong approximation by Fourier series

Л. Лейндлер поставил з адачу о том, следует ли при 0<р<1 из условия $$\mathop {\max }\limits_x \sum\limits_{k = 0}^\infty {\left| {S_k (x) - f(x)} \right|^p< \infty } $$ принадлежность функ цииf классу Lip 1 (здесьS _k (x) — сумма Фурье порядкаk функц ииf). В работе дан положите льный ответ на этот во прос. Рассматриваются так же различные обобщен ия этой задачи.     

On a problem of W. J. LeVeque concerning metric diophantine approximation

We consider the diophantine approximation problem

where is a fixed function satisfying suitable assumptions. Suppose that is randomly chosen in the unit interval. In a series of papers that appeared in earlier issues of this journal, LeVeque raised the question of whether or not the central limit theorem holds for the solution set of the above inequality (compare also with some work of Erdos). Here, we are going to extend and solve LeVeque's problem.

     

On rational approximation of convex functions with a given modulus of continuity

Доказывается, что для наименьших равномер ных рациональных уклоне нийR _n(f) выпуклой на [0,1] функции с модулем непрерывно сти, не превосходящемω(δ), сп раведлива оценка $$R_n (f) \leqq c\frac{{\ln ^2 n}}{{n^2 }}\mathop {\max }\limits_{e^{ - n} \leqq \theta< 1} \left\{ {\omega (\theta )\ln \frac{1}{\theta }} \right\},$$ гдес — абсолютная по стоянная.     

On a problem of L. Leindler concerning strong approximation by Fourier series and Lipschitz classes

Пусть? — возрастающа я непрерывная фцнкци я на [0,π],?(0)=0 и $$\mathop \smallint \limits_0^h \frac{{\varphi \left( t \right)}}{t}dt = O\left( {\varphi \left( h \right)} \right){\text{ }}\left( {h \to 0} \right).$$ Положим $$\psi \left( h \right) = h\mathop \smallint \limits_h^\pi \frac{{\varphi \left( t \right)}}{{t^2 }}dt \left( {h \in (0, \pi ]} \right).$$ Доказывается следую щая теорема.Пусть f∈ С[?π, π], ω(f, δ)=О(?(δ))) и $$\mathop {\lim }\limits_{h \to 0} \frac{1}{{\varphi \left( {\left| h \right|} \right)}}\left| {f\left( {x + h} \right) - f\left( x \right)} \right| = 0$$ для x∈E?[?π, π], ¦E¦>0. Тогда д ля сопряженной функц ии f почти всюду на E выполн яется соотношение $$\mathop {\lim }\limits_{h \to 0} \frac{1}{{\psi \left( {\left| h \right|} \right)}}\left| {\tilde f\left( {x + h} \right) - \tilde f\left( x \right)} \right| = 0.$$ Из этой теоремы вытек ает положительное ре шение одной задачи Л. Лейндлера.     

On the continuity of functions in connection with a problem of V. G. Krotov

В РАБОтЕ ДАЕтсь ОтВЕт НА ОДИН ВОпРОс, пОстАВ лЕННыИ В. г. кРОтОВыМ. УстАНОВлЕН О, ЧтО ЕслИ Ф(х) — МОНОтОННО ВО жРАстАУЩАь ФУНкцИь,Ф (0)=0, Ф(2х)≦кФ(х), х[0, ∞), тО $$\left\{ {f:\left\| {\sum\limits_{k = 1}^\infty {\mu _k \Phi (\lambda _k \left| {S_k - f} \right|)} } \right\|_c< \infty } \right\} \subseteqq C \Leftrightarrow \sum\limits_{k = 1}^\infty {\mu _k } \Phi (\lambda _k ) = \infty $$ Дль пРОИжВОльНых НЕО тРИцАтЕльНых ЧИслОВ ых пОслЕДОВАтЕльНОстЕ И {Μ_k} И {λ_k}. (жДЕсьS _k ОБОжНАЧАЕт ЧАстНУУ с УММУ пОРьДкАk РьДА ФУ РьЕ ФУНкцИИf). УстАНОВлЕН О тАкжЕ, ЧтО ВО МНОгИх слУЧАьх $$\left\{ {f:\left\| {\sum\limits_{k = 1}^\infty {\mu _k \Phi (\lambda _k \left| {\tilde S_k - \tilde f} \right|)} } \right\|_c< \infty } \right\} \subseteqq C \Leftrightarrow \sum\limits_{k = 1}^\infty {\frac{1}{{k\lambda _k }}} \Phi ^{ - 1} \left( {\frac{1}{{k\mu _k }}} \right)< \infty .$$      

On the modulus of continuity of an operator of best approximation

In this paper we characterize spaces with an operator of best approximation uniformly continuous on a class of subspaces.     

On the uniform modulus of continuity of certain discrete approximation operators

For a certain class of discrete approximation operators B_n^f defined on an interval I and including, e.g., the Bernstein polynomials, we prove that for all f ε C(I), the ordinary moduli of continuity of B_n^f and f satisfy ω(B_n^f; h) cω(f; h), N = 1,2,…, 0 < h < ∞, with a universal constant c > 0. A similar result is shown to hold for a different modulus of continuity which is suitable for functions of polynomial growth on unbounded intervals. Some special operators are discussed in this connection.     

A Note on a Theorem of J. Szabados

In this note, we establish a companion result to the theorem of J. Szabados on the maximum of fundamental functions of Lagrange interpolation based on Chebyshev nodes.     

On a problem of B. Jónsson concerning subalgebras and endomorphisms

Presented by the R. W. Quackenbush.     

On the modulus of continuity of the operator of best approximation in the space of continuous functions

In the paper we establish estimates for the strong and the weak moduli of continuity of the operator of best approximation in the space of continuous functions.Translated from Matematicheskie Zametki, Vol. 10, No. 6, pp. 601–613, December, 1971.I wish to express my thanks to S. B. Stechkin for suggesting the problem and for his constant interest in the paper.     

On the modulus of continuity of the solution to the Dirichlet problem at a regular boundary point

A linear elliptic equation of second order with coefficients satisfying a Dini condition is considered in the paper. The modulus of continuity of a solution at a regular boundary point is investigated. An estimate for the modulus of continuity in terms of the Wiener capacity is obtained.Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 67–72, July, 1972.     

On a generalization of W.A.J. Luxemburg's asymptotic problem concerning the Laplace transform

In this paper we prove a more general case of Luxemburg's asymptotic problem concerning the Laplace transform: The problem deals with the conservation of a certain asymptotic behavior of a function at infinity, under analytic transformation of its Laplace transform. The theory of commutative Banach algebras tells us that the problem is equivalent to a family of special cases of the original problem, viz. a set of convolution integral equations, parametrized by a complex variable λ. For ∥ λ ∥ large enough, we may use Luxemburg's original result, and for other λ we modify the integral equations, and apply a modification of Luxemburg's result.     

On a conjecture concerning some automatic continuity theorems

Let A and B be commutative locally convex algebras with unit. A is assumed to be a uniform topological algebra. Let Φ be an injective homomorphism from A to B. Under additional assumptions, we characterize the continuity of the homomorphism Φ^?1/Im?Φ by the fact that the radical (or strong radical) of the closure of Im?Φ has only zero as a common point with Im?Φ. This gives an answer to a conjecture concerning some automatic continuity theorems on uniform topological algebras.