Approximation Of Analytic Functions Research Articles

Approximations of Analytic Functions via Generalized Power Product Expansions

Let f(x)=1+∑n=1∞anxn be a formal power series. For a fixed set of nonzero complex numbers {rk}k=1∞ we convert f(x) into the formal product ∏k=1∞(1+gkxk)rk, namely the Generalized Power Product Expansion. We provide estimates on the domain of absolute convergence of the infinite product when f(x)=1+∑n=1∞anxn is absolutely convergent. This makes it possible to use the truncated Generalized Power Product Expansions ∏n=1M(1+gkxk)rk as approximations to the analytic function f(x). The results are made possible by certain intriguing algebraic properties characteristic of the expansions for the case of rk≥1. An asymptotic formula for the gk associated with the majorizing power series is provided. A combinatorial interpretation of the Generalized Power Product Expansion with {rk}k=1∞ being integers is also given.

A discrete universality theorem for the Hurwitz zeta-function

In the paper, a discrete universality theorem on the approximation of analytic functions by discrete shifts of the Hurwitz zeta-function is proved. The parameter of the Hurwitz zeta-function and the step of discrete shifts are related by a certain linear independence relation. Also, discrete universality theorems for composite functions are discussed.

Approximation of Analytic Functions by Generalized Favard–Szász–Mirakjan–Faber Operators in Compact Sets

Let \(a_{n}>0, b_{n}>0, n\in \mathbb {N}\), be two sequences with the property that \(\frac{b_{n}}{a_{n}}\searrow 0\) as fast as we want. In this paper we construct a generalized Favard–Szasz–Mirakjan–Faber operator attached to analytic functions of exponential growth in a continuum in \(\mathbb {C}\) , for which the approximation order \(\frac{b_{n}}{a_{n}}\) is obtained. Also, several concrete examples are presented.

Discrete universality theorems for the Hurwitz zeta-function

In the paper, we obtain a discrete universality theorem on approximation of analytic functions by discrete shifts of the Hurwitz zeta-function under a new hypothesis on the parameter generalizing the transcendence condition.

A MIXED JOINT UNIVERSALITY THEOREM FOR ZETA-FUNCTIONS. II

In the paper, a joint universality theorem on the approximation of analytic functions for zeta-function of a normalized Hecke eigen cusp form and a collection of periodic Hurwitz zeta-functions with algebraically independent parameters is obtained.

Approximation of analytic functions in Korobov spaces

We study multivariate L2-approximation for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. The weights are defined, in particular, in terms of two sequences a={aj} and b={bj} of positive real numbers bounded away from zero. We study the minimal worst-case error eL2−app,Λ(n,s) of all algorithms that use n information evaluations from the class Λ in the s-variate case. We consider two classes Λ in this paper: the class Λall of all linear functionals and the class Λstd of only function evaluations.We study exponential convergence of the minimal worst-case error, which means that eL2−app,Λ(n,s) converges to zero exponentially fast with increasing n. Furthermore, we consider how the error depends on the dimension s. To this end, we define the notions of weak, polynomial and strong polynomial tractability. In particular, polynomial tractability means that we need a polynomial number of information evaluations in s and 1+logε−1 to compute an ε-approximation. We derive necessary and sufficient conditions on the sequences a and b for obtaining exponential error convergence, and also for obtaining the various notions of tractability. The results are the same for both classes Λ. They are also constructive with the exception of one particular sub-case for which we provide a semi-constructive algorithm.

Approximation of Analytic Functions by Special Functions

‎We survey the recent results concerning the applications of power‎ ‎series method to the study of Hyers-Ulam stability of differential‎ ‎equations‎.

On the Rational Approximation of Analytic Functions Having Generalized Types of Rate of Growth

The present paper is concerned with the rational approximation of functions holomorphic on a domainG⊂C, having generalized types of rates of growth. Moreover, we obtain the characterization of the rate of decay of product of the best approximation errors for functionsfhaving fast and slow rates of growth of the maximum modulus.

Voronovskaja-Type Results in Compact Disks for Quaternion q-Bernstein Operators, q ≥ 1

Attaching to a compact disk \({\overline{\mathbb{D}_{r}}}\) in the quaternion field \({\mathbb{H}}\) and to some analytic function in Weierstrass sense on \({\overline{\mathbb{D}_{r}}}\) the so-called q-Bernstein operators with q ≥ 1, Voronovskaja-type results with quantitative upper estimates are proved. As applications, the exact orders of approximation in \({\overline{\mathbb{D}_{r}}}\) for these operators, namely \({\frac{1}{n}}\) if q = 1 and \({\frac{1}{q^{n}}}\) if q > 1, are obtained. The results extend those in the case of approximation of analytic functions of a complex variable in disks by q-Bernstein operators of complex variable in Gal (Mediterr J Math 5(3):253–272, 2008) and complete the upper estimates obtained for q-Bernstein operators of quaternionic variable in Gal (Approximation by Complex Bernstein and Convolution-Type Operators, 2009; Adv Appl Clifford Alg, doi:10.1007/s00006-011-0310-8, 2011).

Nikishin Systems Are Perfect

K. Mahler introduced the concept of perfect systems in the general theory he developed for the simultaneous Hermite–Padé approximation of analytic functions. We prove that Nikishin systems are perfect, providing by far the largest class of systems of functions for which this important property holds. As consequences, in the context of Nikishin systems, we obtain: an extension of Markov’s theorem to simultaneous Hermite–Padé approximation, a general result on the convergence of simultaneous quadrature rules of Gauss–Jacobi type, the logarithmic asymptotics of general sequences of multiple orthogonal polynomials, and an extension of the Denisov–Rakhmanov theorem for the ratio asymptotics of mixed type multiple orthogonal polynomials.

Approximation by Quaternion q-Bernstein Polynomials, q > 1

Defining for q > 1 the q-Bernstein polynomials of degree n of a quaternion variable, attached to a function f defined on a ball in the field of quaternions, the order of approximation \({\frac{1} {q^n}}\) is obtained when f is in some classes of analytic functions in the sense of Weierstrass. The result extends that in the case of approximation of analytic functions of a complex variable in disks, by q-Bernstein polynomials of complex variable.

Nikishin systems are perfect. The case of unbounded and touching supports

K. Mahler introduced the concept of perfect systems in the theory of simultaneous Hermite–Padé approximation of analytic functions. Recently, we proved that Nikishin systems, generated by measures with bounded support and non-intersecting consecutive supports contained on the real line, are perfect. Here, we prove that they are also perfect when the supports of the generating measures are unbounded or touch at one point. As an application, we give a version of the Stieltjes theorem in the context of simultaneous Hermite–Padé approximation.

Voronovskaja’s theorem, shape preserving properties and iterations for complex q-Bernstein polynomials

In this paper, first we prove Voronovskaja’s convergence theorem for complex q-Bernstein polynomials, 0 < q < 1, attached to analytic functions in compact disks in ℂ centered at origin, with quantitative estimate of this convergence. As an application, we obtain the exact order in approximation of analytic functions by the complex q-Bernstein polynomials on compact disks. Finally, we study the approximation properties of their iterates for any q > 0 and we prove that the complex qn-Bernstein polynomials with 0 < qn < 1 and qn → 1, preserve in the unit disk (beginning with an index) the starlikeness, convexity and spiral-likeness.

Approximation of analytic functions by Laguerre functions

Abstract We will solve the inhomogeneous Laguerre differential equation and apply this result to prove that if a function can be represented by a power series whose radius of convergence is larger than 1, then the function can be approximated, on the interval [0, 1), by a Laguerre function with an error bound C x for some constant C > 0.

Approximation of Analytic Functions by Bessel′s Functions of Fractional Order

We will solve the inhomogeneous Bessel′s differential equation , where ν is a positive nonintegral number and apply this result for approximating analytic functions of a special type by the Bessel functions of fractional order.

Approximation of Analytic Functions by Chebyshev Functions

We solve the inhomogeneous Chebyshev′s differential equation and apply this result for approximating analytic functions by the Chebyshev functions.

Approximation of analytic functions without exponential growth conditions by complex Favard-Szász-Mirakjan operators

In this paper we obtain quantitative estimate in the Voronovskaja’s theorem and the exact orders in the approximation of analytic functions without exponential growth conditions by complex Favard-Szasz-Mirakjan operators and their derivatives on compact disks.

A MIXED JOINT UNIVERSALITY THEOREM FOR ZETA‐FUNCTIONS

In the paper, a joint universality theorem for the Riemann zeta‐function and a collection of periodic Hurwitz zeta‐functions on approximation of analytic functions is obtained.

Approximation of Analytic Functions by Kummer Functions

We solve the inhomogeneous Kummer differential equation of the form and apply this result to the proof of a local Hyers-Ulam stability of the Kummer differential equation in a special class of analytic functions.

On Approximate Euler Differential Equations

We solve the inhomogeneous Euler differential equations of the formx2y′′(x)+αxy′(x)+βy(x)=∑m=0∞amxmand apply this result to the approximation of analytic functions of a special type by the solutions of Euler differential equations.