Approximation of Functions of a Complex Variable

the branch of complex analysis concerned with the approximate representation of functions of a complex variable by means of analytic functions of certain classes. Its central concern is approximation by polynomials and rational functions. The chief problems it investigates are the possibility of approximation, the rate of convergence, and the approximation properties of various methods of representing functions. The methods of representation include interpolation sequences and series, series of orthogonal polynomials and Faber polynomials, and continued fraction expansions. Approximation theory is closely connected with other branches of complex analysis, such as the theory of conformal mappings, integral representations, and potential theory. Many theorems formulated in terms of approximation theory are actually fundamental results on the properties of analytic functions and the nature of analyticity.

An early result concerning polynomial approximations was Runge’s theorem, according to which any function that is holomorphic in a simply connected region of the complex z-plane can be uniformly approximated on compact subsets of the region by means of polynomials in z. The general question of the possibility of uniform approximation by polynomials can be formulated in the following way: for which compacta K in the complex plane does a function f which is continuous on K and holomorphic in its interior, admit of a uniform approximation on K (with arbitrary degree of accuracy) by means of polynomials in z. A necessary and sufficient condition for such an approximation to be possible is the connectedness of the complement of compactum K. This theorem was proved by M. A. Lavrent’ev in 1934 for compacta without interior points, by M. V. Keldysh in 1945 for closed regions, and by S. N. Mergelian in 1951 for the general case.

Let E_n = E_n (f, K) be the best approximation to a function f on a compactum K by polynomials in z of degree up to n (relative to the uniform metric). If K is a compactum with a connected complement and the function f is holomorphic on K, then the sequence {E_n} converges to zero faster than some geometric progression: E_n < qⁿ, where 0 < q = q (f) < 1 and n > N. If f is continuous on K and holomorphic in its interior, then the rate of convergence of its polynomial approximation depends both on the properties of f on the boundary of K (modulus of continuity, differentiability) and on the geometric properties of the boundary of K.

Other areas of study include uniform and best approximations by rational functions, approximations by entire functions, weighted approximations by polynomials, and approximations by polynomials and rational functions relative to integral metrics. Great attention is also paid to approximation of functions of several complex variables.