Uniform Approximation Of Functions Research Articles

On Dirichlet problem and uniform approximation by solutions of second-order elliptic systems in [formula omitted

We consider the Dirichlet problem for solutions of second-order elliptic systems in the plane that correspond to homogeneous elliptic equations with constant complex coefficients. We prove that any Jordan domain G⊂C with C1,α-smooth boundary, 0<α<1, is not regular with respect to the Dirichlet problem for any non-strongly elliptic equation under consideration, which means that there always exists a continuous complex valued function on ∂G that can not be continuously extended to G to a function satisfying the corresponding equation therein. Since there exists a Jordan domain with Lipschitz boundary, which is regular with respect to the Dirichlet problem for bianalytic functions, the result obtained is near to be sharp. We also consider the problem on uniform approximation of functions by polynomial solutions of homogeneous second-order elliptic equations with constant complex coefficients and give a new proof of the approximation criterion in this problem on Carathéodory compact sets.

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It is considered a problem of the best uniform approximation of functions by polynomial splines with fixed knots. It is proposed an approach based on evolutionary algorithms — a powerful class of stochastic search optimization methods — for its solution. To find a spline of the best uniform approximation, a differential evolution algorithm is adapted. It is one of the best evolutionary algorithms that consistently finds a global optimum of a target function (optimization criterion) in a minimal time. An evolutionary process in the algorithm begins with a generation of random vectors, coordinates of which are possible values of spline coefficients. Further, the vectors are constantly modified by mutation, crossover and selection operations in order to reduce a value of the target function (spline approximation error). The algorithm is completed if a specified maximum number of populations is exhausted or a stagnation of the evolutionary process takes place. The differential evolution algorithm is simple in program realization and using (it contains few varied parameters that need to be selected). It is easily paralleled. Recommendations for choosing optimal values of main parameters of the algorithm such as a population size, a mutation factor, a crossover probability are developed. A comparison of the approximation errors obtained by the stochastic differential evolution algorithm and by other (deterministic) algorithms is made for a series of test functions. Results of the comparison showed that an accuracy of the functions approximation by splines using the differential evolution is not worse than using much more complicated deterministic algorithms of the best uniform approximation. This testifies about the effectiveness of the differential evolution algorithm. It can be used as an alternative for known deterministic algorithms of spline approximation

Approximation of functions by some exponential operators of max-product type

Abstract In this paper we study the uniform approximation of functions by a generalization of the Picard and Gauss-Weierstrass operators of max-product type in exponential weighted spaces. We estimate the rate of approximation in terms of a suitable modulus of continuity. We extend and improve previous results.

Two problems on approximation by solutions of elliptic systems on compact sets in the plane

Motivated by recent results by M. Mazalov about uniform approximation of functions by solutions of elliptic equations with constant complex coefficients we study two problems on approximation of functions by solutions of general homogeneous elliptic second-order systems of partial differential equations. The approximation is considered in spaces of continuous and -functions on compact sets in the complex plane.

Weighted Approximation of Functions by Meyer–König and Zeller Operators of Max-Product Type

ABSTRACTIn this paper, we study the uniform approximation of functions by Meyer–König and Zeller operators of max-product type in some weighted spaces. We estimate the rate of approximation in terms of a suitable modulus of continuity.

Uniform Approximations of Functions of Lipschitz Class by Threeharmonic Poisson Integrals

Uniform Approximations of Functions of Lipschitz Class by Threeharmonic Poisson Integrals

On the Sundberg approximation theorem

Let H∞ be the Banach algebra of bounded holomorphic functions on the open unit disk D⊂C. We extend Sundberg's theorem on uniform approximation of functions in BMOA by H∞ functions to other classes of holomorphic functions on D. In our proofs we use a new characterization of meromorphic functions on D that extend to continuous maps of the maximal ideal space of H∞ to the Riemann sphere.

Uniform approximation of functions by Bernstein-Stancu operators

For the class of bounded and continuous functions on (0, 1) we give a characterization of the functions which can be uniformly approximated by Bernstein-Stancu operators. We also study the possibility of uniform approximation of unbounded functions by Bernstein-Stancu operators in weighted spaces with Jacobi weights.

A note on uniform approximation of functions having a double pole

AbstractWe consider the classical problem of finding the best uniform approximation by polynomials of$1/(x-a)^2,$where$a&gt;1$is given, on the interval$[-\! 1,1]$. First, using symbolic computation tools we derive the explicit expressions of the polynomials of best approximation of low degrees and then give a parametric solution of the problem in terms of elliptic functions. Symbolic computation is invoked then once more to derive a recurrence relation for the coefficients of the polynomials of best uniform approximation based on a Pell-type equation satisfied by the solutions.

Approximation of functions by universal sums of series in subsystems of hermite polynomials

The uniform approximation of functions from a certain class on compact sets was performed first by means of polynomials, then by partial sums of universal functional series, and further by special sums associated with functional series. In this paper we construct universal sums associated with Hermite polynomials (any function from this class on compact sets is approximated by these sums). We apply the matrix transformation method that was used earlier for constructing special sums for other functional series.

Uniform approximation of functions by Meyer-König and Zeller operators

We give a characterization of functions which can be uniformly approximated in some weighted spaces by the Meyer-König and Zeller operators. We also express the rate of convergence in terms of the usual modulus of continuity.

On uniform approximation of abstract functions by aggregates of summatory type on a rectangle

Uniform approximations of functions of two variables with the values in a real Banach space are studied by means of aggregates of summatory type. The accuracy of approximation is estimated in terms of the moduli of continuity of the second order. Bibliography: 7 titles.

Comparison of the best uniform approximations of analytic functions in the disk and on its boundary

Denote by C A the set of functions that are analytic in the disk |z| < 1 and continuous on its closure |z| ≤ 1; let ℜ n , n = 0, 1, 2, ..., be the set of rational functions of degree at most n. Denote by R n (f) (R n (f) A ) the best uniform approximation of a function f ∈ C A on the circle |z| = 1 (in the disk |z| ≤ 1) by the set ℜ n . The following equality is proved for any n ≥ 1: sup{R n (f) A /R n (f): f ∈ C A ∖ ℜ n } = 2. We also consider a similar problem of comparing the best approximations of functions in C A by polynomials and trigonometric polynomials.

Uniform approximation by polynomial solutions of second-order elliptic equations, and the corresponding Dirichlet problem

Conditions for the uniform approximability of functions by polynomial solutions of second-order elliptic equations with constant complex coefficients on compact sets of special form in ℝ2 are studied. The results obtained are of analytic character. Conditions of solvability and uniqueness for the corresponding Dirichlet problem are also studied. It is proved that the polynomial approximability on the boundary of a domain is not generally equivalent to the solvability of the corresponding Dirichlet problem.

On Uniform Approximation in \mathbb R2of Continuous Functions of Two Variables with Bounded Variation in the Sense of Hardy

Introduce the notation: $$\mathbb{Z}$$ is the set of integers, $$\bar {\mathbb{Z}}={\mathbb{Z}} \cup \{-\infty, +\infty\},{\mathbb{R}}_+^2 =\{x=(x_1,x_2) \in {\mathbb{R}}^2; x_1>0,x_2>0\}$$ , $$g_{k,m} (x,\alpha,h)= \int\limits_0^1 {g_1 (\frac{(k+u)h_1 - x_1}{\alpha_1})g_2(\frac{(m+u)h_2 - x_2}{\alpha_2})}du$$ , where $$g_i :\mathbb{R} \to \mathbb{R},x \in \mathbb{R}^2 ,\alpha ,h \in \mathbb{R}_ + ^2 $$ . Under certain conditions on the functions g 1, g 2, we prove that the system of functions $$g_{k,m} (x,\alpha^(n), h^(n)) (k,m \in \bar {\mathbb{Z}})$$ , where $$\alpha ^{\left( n \right)} ,h^{\left( n \right)} \in \mathbb{R}_ + ^2 $$ are arbitrary infinitesimal sequences, is complete in the space C $$\mathbb{R}^2 $$ of uniformly continuous bounded functions f equipped with the norm $$||f|| = \mathop {\sup }\limits_{x \in \mathbb{R}^2 } |f(x)|$$ . Starting with the functions g k,m , it is possible to construct a method for uniform approximating in $$\mathbb{R}^2 $$ any continuous function of bounded variation in the sense of Hardy. An error estimate is derived in terms of the second order moduli of continuity. Based on the obtained results, we discuss in detail the accuracy of uniform approximation of functions of several variables by linear functions. The error estimates are derived by using second order moduli of continuity. We pay a particular attention to sharpness of constants. Bibliography: 8 titles.

Weighted Approximation of Functions with Inner Singularities by Exponential Weights in [−1, 1

We study the weighted uniform approximation of functions with inner singularities by exponential-type weights in [−1, 1]. Convergence theorems and pointwise and uniform approximation error estimates not possible by polynomials are proved.

Uniform Approximation of Functions by Polynomial Solutions to Second-Order Elliptic Equations on Compact Sets in ℝ2

In this paper, we study necessary and sufficient conditions for functions to be approximated uniformly on plane compact sets by polynomial solutions to second-order homogeneous elliptic equations with constant coefficients. Sufficient conditions for approximability are of reductive character, i.e., the possibility of approximating on some (simpler) parts of the compact set implies approximability on the entire compact set.

Uniform Approximability of Functions by Polynomials of Special Classes on Compact Sets in \(\mathbb{R}^2 \)

We derive several sufficient conditions for the uniform approximability of functions by polynomial solutions of homogeneous elliptic equations of second order with constant coefficients on Caratheodory compact sets in \(\mathbb{R}^2 \).

Uniform Approximation of Functions with Discrete Approximation Functionals

Hoffman, Kouri, and collaborators have calculated nonrelativistic quantum scattering amplitudes by numerically evaluating Feynman path integrals. They observed that the errors introduced by their numerical scheme were uniform in coordinate space, implying that their scheme accurately reproduces both the shape and the phase of functions. Furthermore, they observed that the size and the uniform nature of the errors were preserved when the functions were allowed to evolve in time under the action of the kinetic energy operator. In this paper it is established that these observed properties of the errors are not numerical artifacts but follow from analytical properties of a general class of approximations that include those of Hoffman, Kouri, and collaborators as a special case.

Uniform approximation with linear combinations of reproducing kernels

We show several theorems on uniform approximation of functions. Each of them is based on the choice of a special reproducing kernel in an appropriate Hilbert space. The theorems have a common generalization whose proof is founded on the idea of the Kaczma