Approximation in Lipschitz algebras of infinitely differentiable functions. (English) Zbl 0951.46025

Summary: We introduce Lipschitz algebras of differentiable functions on a perfect compact plane set \(X\) and then extend the definition to Lipschitz algebras of infinitely differentiable functions on \(X\). Then we define the subalgebras generated by polynomials, rational functions, and analytic functions in some neighbourhood of \(X\), and determine the maximal ideal spaces of some of these algebras. We investigate the polynomial and rational approximation problems on certain compact sets \(X\).


46J10 Banach algebras of continuous functions, function algebras
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46J20 Ideals, maximal ideals, boundaries
41A10 Approximation by polynomials
41A20 Approximation by rational functions