Multivariate approximation by polynomial and generalized rational functions

Multivariate n-term rational and piecewise polynomial approximation

We study nonlinear approximation in Lp(R) (0 < p < ∞, d > 1) from (a) n-term rational functions, and (b) piecewise polynomials generated by different anisotropic dyadic partitions of Rd. To characterize the rates of each such piecewise polynomial approximation we introduce a family of smoothness spaces (B-spaces) which can be viewed as an anisotropic variation of Besov spaces. We use the B-spac...

The best uniform polynomial approximation of two classes of rational functions

In this paper we obtain the explicit form of the best uniform polynomial approximations out of Pn of two classes of rational functions using properties of Chebyshev polynomials. In this way we present some new theorems and lemmas. Some examples will be given to support the results.

Approximation by Rational Functions

Approximation by Rational Functions

Making use of the Hardy-Littlewood maximal function, we give a new proof of the following theorem of Pekarski: If f' is in L log L on a finite interval, then f can be approximated in the uniform norm by rational functions of degree n to an error 0(1/n) on that interval. It is well known that approximation by rational functions of degree n can produce a dramatically smaller error than that for p...

Multivariate Rational Approximation

We estimate the error in approximating a function f by rational functions of degree n in the norm of Lq(Q), Q := [0,1]d. Among other things, we prove that if f is in the Sobolev space Wk(Q) and if k/d l/p + l/q > O, then f can be approximated by rational functions of degree n to an order O(n-k/d).