Approximation by fourier sums and best approximations on classes of analytic functions

We establish asymptotic equalities for upper bounds of approximations by Fourier sums and for the best approximations in the metrics of C and L1 on classes of convolutions of periodic functions that can be regularly extended into a fixed strip of the complex plane.     

Symptotic behavior of the best uniform approximations of individual functions by splines

It is established that in the class W^rH, where (t) is a convex modulus of continuity, that there exists a function for which the error of the best approximation by splines of minimal deficiency (including ones with free nodes) asymptotically coincides with the upper bound of approximation of the functions of class W^rH by these same splines.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 59–64, January, 1980.     

On the zeros of analytic functions in the disk with a given majorant near its boundary

Mathematical Notes - Suppose that λ is an arbitrary positive function from C[0, 1) such that λ(r) → ∞ as r → 1 ? 0 and satisfying some growth regularity...     

Starlike approximations of analytic functions

When an analytic function is not univalent, it is often of interest to approximate it by univalent functions. In this paper we introduce a measure of the non-univalency of a function and we derive a method for constructing the best starlike univalent approximations of analytic functions with respect to it, suitable for both practical problems and numerical implementation.     

Iteration of functions analytic on a disk

Aequationes mathematicae -     

Zeros of Bernoulli-type functions and best approximations

The zero sets of (D+a)ⁿg(t) with in the (t,a)-plane are investigated for and .The results are used to determine entire interpolations to functions , which give representations for the best approximation and best one-sided approximation from the class of functions of exponential type η>0 to .     

Tight uniform algebras and algebras of analytic functions

A uniform algebra A on a compact space X is tight if for each g?C(X), the Hankel-type operator f → gf + A from $\text{A}\text{to}\text{C}\text{A}$ is weakly compact. Two families of uniform algebras are shown to be tight: the algebras such as R(K) that arise in the theory of rational approximation on compact subsets of the complex plane, and algebras of analytic functions on domains in $\text{C}$ⁿ for which a certain $\text{?}$-problem is solvable. A couple of characterizations of tight algebras are given, and one of these is used to show that the property of being tight places severe restrictions on the Gleason parts of A and the measures in A^⊥.     

On best uniform approximations by splines in the presence of restrictions on their derivatives

Translated from Matematicheskie Zametki, Vol. 50, No. 6, pp. 24–30, December, 1991.     

On best rational approximation of analytic functions

This paper contains some theorems related to the best approximation ρ_n(f;E) to a function f in the uniform metric on a compact set by rational functions of degree at most n. We obtain results characterizing the relationship between ρ_n(f;K) and ρ_n(f;E) in the case when complements of compact sets K and E are connected, K is a subset of the interior Ω of E, and f is analytic in Ω and continuous on E.     

On l p -multipliers of functions analytic in the disk

We consider bounded analytic functions in domains generated by sets with the Littlewood-Paley property. We show that each such function is an l ^p -multiplier.     

The prime-counting function and its analytic approximations

The paper describes a systematic computational study of the prime counting function π(x) and three of its analytic approximations: the logarithmic integral \({\text{li}}{\left( x \right)}: = {\int_0^x {\frac{{dt}}{{\log \,t}}} }\), \({\text{li}}{\left( x \right)} - \frac{1}{2}{\text{li}}{\left( {{\sqrt x }} \right)}\), and \(R{\left( x \right)}: = {\sum\nolimits_{k = 1}^\infty {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} \mathord{\left/ {\vphantom {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} k}} \right. \kern-\nulldelimiterspace} k} }\), where μ is the Möbius function. The results show that π(x)x) for 2≤x≤10¹⁴, and also seem to support several conjectures on the maximal and average errors of the three approximations, most importantly \({\left| {\pi {\left( x \right)} - {\text{li}}{\left( x \right)}} \right|} < x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}\) and \( - \frac{2}{5}x^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} < {\int_2^x {{\left( {\pi {\left( u \right)} - {\text{li}}{\left( u \right)}} \right)}du < 0} }\) for all x>2. The paper concludes with a short discussion of prospects for further computational progress.