## 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 $$X$$.

### MSC:

 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