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Let ψ be an orientation-reversing homeomorphism of the unit circle onto itself. We consider approximation in certain Sobolev norms by functions of the form f(z) + g(ψ), where / and g are polynomials. The methods involve conf ormal welding and Hardy space theory. We construct a Jordan arc of positive continuous analytic capacity such that the harmonic measures for the two complementary domains are mutually absolutely continuous. Let C(S) denote the space of continuous, complex-valued functions on the unit circle S. For feC(S), let P(f) denote the space of all polynomials in / with complex coefficients. Let z denote the identity function. Browder and Wermer proved the following theorem [3, p. 551]. Let ψ be a direction-reve rsing homeomorphism of S onto S. Then the vector space sum P(z) + P(ψ) is uniformly dense in C(S). In the first part of this paper we prove a result that partially extends this theorem to the C1 norm. We say that a direction-reve rsing homeomorphism ψ:S—*S is an involution if ψoψ = z. Let C^S) denote the space of continuously-differentiable, complex-valued functions on S, with the norm

Sobolev approximation by a sum of subalgebras on the circle