Polynomial and Rational Approximation in a New class of Lipschitz Functions

Polynomial and Rational Approximation in a New class of Lipschitz Functions

Let X and K be compact subsets of C" and K C X. We introduce new classes of Lipschitz algebras, denoted by Lip(X, K, (v) and lip(X, K, Q) and their subalgebras Lipp(X, K, Cy), lipp(X, K, Q). Lip F(X, K. O.) and lip R (X, K, Cy): Lipschitz closure of Po (X, K) and Ro(X, K) in Lip(X, K, Q) and lip(X, K, Q) and lip Rx (X, K, (v): Lipschitz closure of Ro(X) in lip(X, K, Q). We introduce the maximal ideal space of these algebras. At last we show that if S and T be compact subsets of the compact plane set X and the symmetric difference S + T has planar measure zero then R(X, S) = R (X, T), which implies the Hartogʹs-Rosenthal Theorem.