Approximation by harmonic functions

For a compact set we construct a restoring covering for the space of real-valued functions on which can be uniformly approximated by harmonic functions. Functions from restricted to an element of this covering possess some analytic properties. In particular, every nonnegative function , equal to 0 on an open non-void set, is equal to 0 on . Moreover, when , the algebra of complex-valued functions on which can be uniformly approximated by holomorphic functions is analytic. These theorems allow us to prove that if a compact set has a nontrivial Jensen measure, then contains a nontrivial compact set with analytic algebra .

     

Approximation of analytic functions by Hermite functions

We solve the inhomogeneous Hermite equation and apply this result to estimate the error bound occurring when any analytic function is approximated by an appropriate Hermite function.     

Approximation of analytic functions by Laguerre functions

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 Legendre functions

We will solve the inhomogeneous Legendre’s differential equation and apply this result to estimate the error bound occurring when an analytic function is approximated by an appropriate Legendre function.     

Approximation of analytic functions by Legendre functions

We will solve the inhomogeneous Legendre’s differential equation and apply this result to estimate the error bound occurring when an analytic function is approximated by an appropriate Legendre function.     

Approximation of plurisubharmonic functions by multipole Green functions

For a strongly hyperconvex domain we prove that multipole pluricomplex Green functions are dense in the cone in of negative plurisubharmonic functions with zero boundary values.

     

Approximation of smooth functions by S-splines

The paper contains a method for recovering a function from a table of its values. The method ensures smoothing of measured data, can be easily transformed into an algorithm and used in automatic information processing. The reconstruction method described in the paper is related to the approximation of smooth functions by means of some special splines, called S-splines, introduced by one of the authors. Conditions for stability and convergence of S-splines are obtained.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 197–206, 1984.The authors would like to thank Prof. V. M. Tikhomirov for advice and help with this work; thanks are also due to V. A. Gruzdev and E. V. Logacheva who provided their programs and computations.     

Approximation by translates of refinable functions

Summary. The functions are refinable if they are combinations of the rescaled and translated functions . This is very common in scientific computing on a regular mesh. The space of approximating functions with meshwidth is a subspace of with meshwidth . These refinable spaces have refinable basis functions. The accuracy of the computations depends on , the order of approximation, which is determined by the degree of polynomials that lie in . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions are known only through the coefficients in the refinement equation – scalars in the traditional case, matrices for multiwavelets. The scalar "sum rules" that determine are well known. We find the conditions on the matrices that yield approximation of order from . These are equivalent to the Strang–Fix conditions on the Fourier transforms , but for refinable functions they can be explicitly verified from the . Received August 31, 1994 / Revised version received May 2, 1995     

Approximation by Bernstein type rational functions

Acta Mathematica Hungarica -     

Approximation of functions by Fourier-Laguerre sums

Translated from Matematicheskie Zametki, Vol. 57, No. 2, pp. 163–170, February, 1995.     

Approximation of ultra-differentiable functions by polynomials and entire functions

This article deals with ultra-differentiable functions in the sense of A. Beurling. Quasi-analytic classes, in particular the real analytic, are included. A function f is shown to be -ultra-differentiable in the neighbourhood of a compact set K if and only if f can be approximated with a certain speed by holomorphic functions of exponential type n or polynomials of degree n for n. The speed is measured in terms of the weight .     

Approximation of differentiable functions by functions of large smoothness

The order of the quantity \(\delta (L) = \mathop {\sup }\limits_{\chi _{_{_{_1 } } } } \mathop {\inf }\limits_{\chi _2 } \parallel \chi _1 - \chi _2 \parallel L_S [0,2 = ]\) as L → ∞ is studied for the classes of periodic functionsx ₁εW _p ⁿ (I), andx ₂ εW _q ^m (L). Necessary and sufficient conditions under which the inequality $$\parallel x^{(n)} \parallel _{L_p } \leqslant C\parallel x\parallel _{L_q }^x \parallel x^{(m)} \parallel _{L_s }^\beta $$ with the constant independent of x holds for all periodic functions x(t) with \(\int_0^{2\pi } \chi (l)dl = 0\) andx ^(m) (t εL _s [0, 2π] are found.     

Generalized herglotz domains

Let F(θ k, α) be the far field pattern arising from the scattering of a time harmonic plane acoustic wave of wave number k and direction a by a sound-soft cylinder of cross section D. Suppose F has the Fourier expansion where a_n = a_n(k, . Then if ?² is a Dirichlet eigenvalue for D, sufficient conditions are given on D for the existence of a nontrivial sequence |b_n| where the b_n are independent of such that for all directions Domains for which this is true are called generalized Herglotz domains. The conditions for a domain to be a generalized Herglotz domain are given either in terms of the Schwarz function for the analytic boundary ?D or in terms of the Rayleigh hypothesis in acoustic scattering theory and examples are given showing the applicability of these conditions.     

Approximation of measurable periodic functions in measure by step functions

For spaces defined by a function ? of the type of modulus of continuity, we prove direct and inverse Jackson theorems for the approximation by step functions with uniform partition.     

Approximation in the mean by rational functions

In this paper we proved that for a large class of compact subsetsK in the complex plane,R(K) is dense inL ^q () if and only if the set of analytic bounded point evaluations forR ^q (K, ) is empty. As a consequence, we showed that this result is true for allK ifR(K) is replaced byA(K). Our main result includes the corresponding result of James Thomson for polynomials approximation as such a special case thatK is a disk.     

Approximation in the mean by convex functions

We give elementary proofs for the existence and uniqueness of the best L₁-approximation to a continuous function from the class of convex functions on a closed interval, and describe thebest approximation in terms of certain piecewise linear functions.