Two-sided polynomial approximation of functions

Conditions are established when the collocation polynomials P_m(x) and P_M(x), m M, constructed respectively using the system of nodes x_j of multiplicities a_j 1, j = O,, n, and the system of nodes x_-r,,x_o,,x_n,,x_n+r1, r O, r₁ O, of multiplicities a_-r,,(a_o + y_o),,(a_n + y_n),,a_n+r1, a_j + y_j 1, are two sided-approximations of the function f on the intervals , x_j[, j = O,...,n + 1, and on unions of any number of these intervals. In this case, the polynomials P_m (x), P_M ^(l) (x) with l a_j are two-sided approximations of the function f⁽¹⁾ in the neighborhood of the node x_j and the integrals of the polynomials P_m(x), P_M(x) over D_j are two-sided approximations of the integral of the function f (over D_j). If the multiplicities a_j a_j + y_j of the nodes x_j are even, then this is also true for integrals over the set _j= _µ ^k D_j µ 1, k n. It is shown that noncollocation polynomials (Fourier polynomials, etc.) do not have these properties.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 31–37, 1989.     

Uniform polynomial approximation of entire functions on arbitrary compact sets in the complex plane

For an arbitrary compact setK⊂ℂ, we relate the order and the type of an entire functionf to the sequenceE _n (f,K) of best polynomial approximations to this function onK. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 355–364, September, 1995.     

On best error bounds for approximation by piecewise polynomial functions

Summary An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.The work presented in this paper was supported by the ERDA Mathematics and Computing Laboratory, Courant Institute of Mathematical Sciences, New York University, under Contract E(11-1)-3077 with the Energy Research and Development Administration.     

Jensen, multi-Jensen and polynomial functions on arbitrary abelian groups

In contrast to Cauchy’s functional equation, the consideration of Jensen’s equation makes no sense for functions f : G → H if G, H are arbitrary abelian groups. This difficulty may be circumvented by implementing a third variable into the equation in such a way that there is no interference with the desired result that the solutions should be of the form “additive function plus constant”. Proceeding analogously with the multi-Jensen equation we can add certain possibilities to define polynomial functions on abelian groups to possibilities given earlier by Laczkovich.     

On pointwise approximation of arbitrary functions by countable families of continuous functions

The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyf _n :n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F?X and every real number ε>0 one can choosen∈ω such that ∥f(x)?f_n(x)∥<ε for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A?X, there exists a continuous mapf _A:X→R ^ω such that A=f _A ^?1 (A). Splitting spaces will be studied systematically.     

Best approximation by holomorphic functions. Application to the best polynomial approximation of classes of holomorphic functions

We establish necessary and sufficient conditions under which a real-valued function from , 1 ≤ p < ∞, is badly approximable by the Hardy subspace H _p ⁰ := {ƒ ∈ H _p : ƒ(0) = 0}. In a number of cases, we obtain the exact values of the best approximations in the mean of functions holomorphic in the unit disk by functions holomorphic outside this disk. We use the obtained results for finding the exact values of the best polynomial approximations and n-widths of some classes of holomorphic functions. We establish necessary and sufficient conditions under which the generalized Bernstein inequality for algebraic polynomials on the unit circle is true. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1047–1067, August, 2007.     

On the degree of weighted polynomial approximation of holomorphic functions

[4] , —. , , , [2] [7].     

The best polynomial approximation of functions analytic in the unit circle

In the Banach space (p,q,), studied by M. I. Gvaradze, we establish a relation between the best polynomial approximation of an entire transcendental function and such important characteristics as its order of growth and its type.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 838–843, June, 1990.     

Accelerated polynomial approximation of finite order entire functions by growth reduction

Let be an entire function of positive order and finite type. The subject of this note is the convergence acceleration of polynomial approximants of by incorporating information about the growth of for . We consider ``near polynomial approximation' on a compact plane set , which should be thought of as a circle or a real interval. Our aim is to find sequences of functions which are the product of a polynomial of degree and an ``easy computable' second factor and such that converges essentially faster to on than the sequence of best approximating polynomials of degree . The resulting method, which we call Reduced Growth method (-method) is introduced in Section 2. In Section 5, numerical examples of the -method applied to the complex error function and to Bessel functions are given.

     

Generalized polynomial approximation

We estimate the rate of covergence to functions in the spacesL ^p [0,1] and C[0,1] by polynomial of the form ∑ _λ α _λ x ^λ, where the λ′s are positive real numbers and 0.     

Uniform and tangential harmonic approximation of continuous functions on arbitrary sets

Necessary and sufficient conditions which must be imposed on a set E are derived, such that functions continuous on E G can be approximated by functions harmonic in a region G (Rⁿ.Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 131–142, February, 1971.In conclusion I wish to thank my scientific director S. N. Mergelyan and also A. A. Gonchar for their valuable advice.     

Multiscale approximation for functions in arbitrary Sobolev spaces by scaled radial basis functions on the unit sphere

In this paper, we prove convergence results for multiscale approximation using compactly supported radial basis functions restricted to the unit sphere, for target functions outside the reproducing kernel Hilbert space of the employed kernel.     

Piecewise polynomial approximation

For best piecewise polynomial approximation _n=_n (f; [0, 1]) of a functionf, which is continuous on the interval [0, 1] and admits a bounded analytic continuation onto the disk K=z:¦z–1¦<, the relation _n=o[ _f (e ^–n )] is valid.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 129–134, February, 1972.     

Uniform estimates for monotonic polynomial approximation

The uniform estimate is established for a monotone polynomial approximation of functions whose smoothness decreases at the ends of a segment.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 38–43, January, 1993.     

Pointwise estimates for monotone polynomial approximation

We prove that iff is increasing on [?1,1], then for eachn=1,2,... there is an increasing algebraic polynomialP _n of degreen such that |f(x)?P _n (x)|≤cω ₂(f,√1?x ²/n), whereω ₂ is the second-order modulus of smoothness. These results complement the classical pointwise estimates of the same type for unconstrained polynomial approximation. Using these results, we characterize the monotone functions in the generalized Lipschitz spaces through their approximation properties.     

Global approximation to arbitrary cost functions: A Bayesian approach with application to US banking

This paper proposes and estimates a globally flexible functional form for the cost function, which we call Neural Cost Function (NCF). The proposed specification imposes a priori and satisfies globally all the properties that economic theory dictates. The functional form can be estimated easily using Markov Chain Monte Carlo (MCMC) techniques or standard iterative SURE. We use a large panel of U.S. banks to illustrate our approach. The results are consistent with previous knowledge about the sector and in accordance with mathematical production theory.     

On the best polynomial approximation of entire transcendental functions of generalized order

We prove a Hadamard-type theorem that associates the generalized order of growth of an entire transcendental function ƒ with the coefficients of its expansion in a Faber series. This theorem is an extension of one result of Balashov to the case of a finite simply connected domain G with boundary γ belonging to the Al'per class Λ*. Using this theorem, we obtain limit equalities that associate with a sequence of the best polynomial approximations of ƒ in certain Banach spaces of functions analytic in G. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1011–1026, August, 2008.