Weighted Polynomial Approximation Research Articles

An exchange algorithm for laurent polynomial chebyshev approximation on the unit circle

An exchange algorithmi for weighted Laurent polynomial Chebyshev approximation of continuous functions on the unit circle in the complex plane is presented The algorithm is at least linearly convergent. The weighted real and imaginary parts are approximated separately by weighted trigonometric polynomials using the real Reinez algorithm and a near-best Laurent Polynomial approximation is obtained. Also weighted polynomial approximation of functions analytic in the open and continuous in the closed disc is treated and comparisons CF approximation are made.

Best Approximation and Moduli of Smoothness for Doubling Weights

In this paper we relate the rate of weighted polynomial approximation to some weighted moduli of smoothness for so-called doubling weights. We shall also consider the problem in a more restrictive sense for generalized Jacobi weights with zeros in the interval of approximation. These zeros constitute a special problem that has not been resolved so far in the literature.

On weighted polynomial approximation with monotone weights

We construct an even weight W W monotone on the right half line such that the logarithmic integral of the largest log \log -convex minorant of W W converges and the polynomials are dense in C ( W ) C(W) .

Weighted Polynomial Approximation for Convex External Fields

It is proven that if Q is convex and w(x)= exp(-Q(x)) is the corresponding weight, then every continuous function that vanishes outside the support of the extremal measure associated with w can be uniformly approximated by weighted polynomials of the form w n P n . This solves a problem of P. Borwein and E. B. Saff. Actually, a similar result is true locally for any parts of the extremal support where Q is convex.

K-functional, weighted moduli of smoothness, and best weighted polynomial approximation on a simplex

An inverse theorem for the best weighted polynomial approximation of a function in $$L_{w_\alpha }^p $$ (S) is established. We also investigate Besov spaces generated by Freud weight and their connection with algebraic polynomial approximation in $$L_p (R)_{W_\lambda } $$ , wherew α is a Jacobi-type weight onS, 0<p ≤ ∞,S is a simplex andW λ is a Freud weight. For Ditzian-TotikK-functionals onL p(S), 1 ≤p ≤ ∞, we obtain a new equivalence expression.

Jackson-type Theorems on a Finite Interval with Weights Having Non-symmetric Inner Singularities

We consider Jackson-type theorems for weighted polynomial approximation when the weight has certain type of zeros inside and/or at the end-points of the interval. The approximating polynomials are constructed from truncated convolution integrals.

Smoothness theorems for generalized symmetric Pollaczek weights on (−1,1)

In this note a new characterization of smoothness is obtained for weighted polynomial approximation in L p (1 ⩽ p ⩽ ∞) with respect to a large class of exponential weights in (−1,1) which include the classical Pollaczek weights. Along the way we prove Marchaud inequalities, saturation theorems, existence theorems for derivatives and generalize a theorem of D. Lubinsky.

The Hamburger moment problem and weighted polynomial approximation on discrete subsets of the real line

The Hamburger moment problem and weighted polynomial approximation on discrete subsets of the real line

A characterization of smoothness for Freud weights

We obtain a new characterization of smoothness for weighted polynomial approximation with respect to Freud weights together with an existence theorem for derivatives. Our methods rely heavily on realization functionals.

Polynomial Approximation and Interpolation on the Real Line with Respect to General Classes of Weights

Results on weighted polynomial approximation and interpolation with respect to Freud weights are extended to a more general class of weights. Among others, weights having zeros (Freud-Jacobi type weights) are considered, and a system of nodes is constructed for which the weighted Lebesgue constant of Lagrange interpolation is of optimal order.

Convergence of Weighted Polynomial Approximations to Solutions of Partial Differential Equations with Quasianalytical Coefficients

Sequences of polynomial functions which converge to solutions of partial differential equations with quasianalytical coefficients are constructed. Estimates of the degree of convergence are also given.

The Polynomial Approximation Property in Fock-type Spaces

where dm2 is Lebesgue area measure. We deal with the following question, posed and discussed in [9] (and earlier in [8]). Do the polynomial multiples of an entire function span (in F , r† ˆ r2) all functions which vanish wherever vanishes? We take into account the multiplicities of zeros, and require that all the polynomial (or even exponential polynomial) multiples of belong to F . In other words, the question is whether the polynomials are dense in the Hilbert space F ; of analytic (entire) functions square summable with respect to the measure j z†j2ey jzj†dm2 z†. Let us remark here that the general problem of weighted polynomial approximation for weights vanishing at some inner points of the domain of analyticity is not yet sufficiently investigated. For some classes of we construct for which the answer to our question is negative. The method used is close to that in [2,3]. The idea is to produce an element of F having extremal growth along a sufficiently massive subset of the plane. Then a Phragmen^Lindelo« f type argument shows that if polynomial multiples pn converge to an element in F , then the polynomials pn cannot be large enough to approximate all the elements in F ; . To use this MATH. SCAND. 82 (1998), 256^264

Some Discrepancy Theorems in the Theory of Weighted Polynomial Approximation

We obtain discrepancy theorems for the distribution of the zeros of extremal polynomials arising in the theory of weighted polynomial approximation on the whole real axis.

Weighted polynomial approximation in the complex plane

Given a pair ( G , W ) (G,W) of an open bounded set G G in the complex plane and a weight function W ( z ) W(z) which is analytic and different from zero in G G , we consider the problem of the locally uniform approximation of any function f ( z ) f(z) , which is analytic in G G , by weighted polynomials of the form { W n ( z ) P n ( z ) } n = 0 ∞ \left \{W^{n}(z)P_{n}(z) \right \}^{\infty }_{n=0} , where deg ⁡ P n ≤ n \deg P_{n} \leq n . The main result of this paper is a necessary and sufficient condition for such an approximation to be valid. We also consider a number of applications of this result to various classical weights, which give explicit criteria for these weighted approximations.

A Note on Weighted Polynomial Approximation with Varying Weights

It is shown that if weighted polynomialswnPnwith degPn⩽nconverge uniformly on the support of the extremal measure associated withw, then they converge to 0 everywhere else. It is also shown that uniform approximation on the support can always be characterized by a closed subsetZhaving the property that a function can be approximated if and only if it vanishes onZ.

The role of the endpoint in weighted polynomial approximation with varying weights

For a weight functionw: [a, b]→(0, ∞), we consider weighted polynomials of the formw n Pn where the degree ofP n is at mostn. The class of functions that can be approximated with such polynomials depends on the behavior of the densityv(t) of the extremal measure associated withw. We show that every approximable function must vanish at the endpointa ifv(t) behaves like (t−a) β ast→a with β>−1/2. We also present an analogous result for internal points. Our results solve some open problems posed by V. Totik and disprove a conjecture of G.G. Lorentz on incomplete polynomials.

The Role of the Endpoint in Weighted Polynomial Approximation with Varying Weights

For a weight functionw: [a, b]→(0, ∞), we consider weighted polynomials of the formw n Pn where the degree ofP n is at mostn. The class of functions that can be approximated with such polynomials depends on the behavior of the densityv(t) of the extremal measure associated withw. We show that every approximable function must vanish at the endpointa ifv(t) behaves like (t−a) β ast→a with β>−1/2. We also present an analogous result for internal points. Our results solve some open problems posed by V. Totik and disprove a conjecture of G.G. Lorentz on incomplete polynomials.

Weighted polynomial approximation with Freud weights

We show that ifw(x)=exp(−|x|λ), then in the case λ=1 for every continuousf that vanishes outside the support of the corresponding extremal measure there are polynomialsPn of degree at mostn such thatwnPn uniformly tends tof, and this is not true when λ<1. These are the missing cases concerning approximation by weighted polynomials of the formwnPn wherew is a Freud weight. Our second theorem shows that even if we are only interested in approximation off on the extremal support, the functionf must still vanish at the endpoints, and we actually determine the (sequence of) largest possible intervals where approximation is possible. We also briefly discuss approximation by weighted polynomials of the formW(anx)Pn(x).

Best weighted polynomial approximation on the real line; a functional-analytic approach

The rate of convergence of best approximation by algebraic polynomials in weighted L p( R ) -spaces is investigated by functional-analytic methods. Jackson- and Bernstein-type results as well as those of Stečkin- and Zamansky-type are given. The smoothness of the functions involved is measured in terms of the so-called main part modulus of continuity due to Ditzian and Totik, and a modified K-functional.

Orthogonal expansions and the error of weighted polynomial approximation for erdös weights

Some (C, 1) bounds for orthogonal expansions associated with Erdös weights are established. Roughly speaking, Erdös weights are weights of the form W := e −Q , where Q is even, continuous and has faster than polynomial growth at infinity. Typical examples are or , and so on. We shall use these bounds together with Bohr-type inequalities, to establish L p , 1 ≤ p ≤ ∞, Jackson-Favard inequalities.