For compact sets K ⊂ C d , we introduce a subalgebra A D (K) of A(K), which allows us to obtain Mergelyan type theorems for products of planar compact sets as well as for graphs of functions.

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In this paper, we provide an efficient method for computing the Taylor coefficients of 1 − p n f , where p n denotes the optimal polynomial approximant of degree n to 1/f in a Hilbert space H 2 ω of analytic functions over the unit disc D, and f is a polynomial of degree d with d simple zeros. As a consequence, we show that in many of the spaces H 2 ω , the sequence {1 − p n f } n∈N is uniformly bounded on the closed unit disc and, if f has no zeros inside D, the sequence {1 − p n f } converges uniformly to 0 on compact subsets of the complement of the zeros of f in D, and we obtain precise estimates on the rate of convergence on compacta. We also treat the previously unknown case of a single zero with higher multiplicity.

In this paper, we provide an efficient method for computing the Taylor coefficients of 1 − p n f , where p n denotes the optimal polynomial approximant of degree n to 1/f in a Hilbert space H 2 ω of analytic functions over the unit disc D, and f is a polynomial of degree d with d simple zeros. As a consequence, we show that in many of the spaces H 2 ω , the sequence {1 − p n f } n∈N is uniformly bounded on the closed unit disc and, if f has no zeros inside D, the sequence {1 − p n f } converges uniformly to 0 on compact subsets of the complement of the zeros of f in D, and we obtain precise estimates on the rate of convergence on compacta. We also treat the previously unknown case of a single zero with higher multiplicity.

Let f be a holomorphic function on the unit disc, and (S n k ) be a subsequence of its Taylor polynomials about 0. It is shown that the nontangential limit of f and lim k→∞ S n k agree at almost all points of the unit circle where they simultaneously exist. This result yields new information about the boundary behaviour of universal Taylor series. The key to its proof lies in a convergence theorem for harmonic measures that is of independent interest.