Multivariable Polynomial Approximations

Abstract

In Chapter 3 we introduced the approximation of univariate functions by polynomials and trigonometric functions (see Sections 3.4–3.7). In this chapter we extend those ideas to multivariable functions and multivariable polynomials. In the univariate case there are only three types of approximation domains, namely [a, b], [a,∞), and (–∞,∞), with a and b finite. In contrast, there are many types of approximation domains in the multivariable case. This chapter is only an introduction to this area, and to make it more accessible, we emphasize planar problems. In particular, we consider the unit disk