Weighted Maximum Over Minimum Modulus of Polynomials, Applied to Ray Sequences of Padé Approximants

Abstract

Let a≥ 0 , ɛ >0 . We use potential theory to obtain a sharp lower bound for the linear Lebesgue measure of the set . Here P is an arbitrary polynomial of degree ≤ n . We then apply this to diagonal and ray Padé sequences for functions analytic (or meromorphic) in the unit ball. For example, we show that the diagonal \left{ [n/n]\right} _n=1 ^∞ sequence provides good approximation on almost one-eighth of the circles centre 0 , and the \left{ [2n/n]\right} _n=1 ^∞ sequence on almost one-quarter of such circles.