Now showing 1 - 4 of 4
  • Publication
    Boundary behaviour of Dirichlet series with applications to universal series
    (London Mathematical Society, 2016-10-05) ;
    This paper establishes connections between the boundary behaviour of functions representable as absolutely convergent Dirichlet series in a half-plane and the convergence properties of partial sums of the Dirichlet series on the boundary. This yields insights into the boundary behaviour of Dirichlet series and Taylor series which have universal approximation properties.
      332Scopus© Citations 6
  • Publication
    Ostrowski-type theorems for harmonic functions
    (Elsevier, 2012-07-15)
    Ostrowski showed that there are intimate connections between the gap structure of a Taylor series and the behaviour of its partial sums outside the disk of convergence. This paper investigates the corresponding problem for the homogeneous polynomial expansion of a harmonic function. The results for harmonic functions display new features in the case of higher dimensions.
      426Scopus© Citations 4
  • Publication
    Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains
    A holomorphic function on a planar domain Ω is said to possess a universal Taylor series about a point ζ of Ω if subsequences of the partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compact sets in C∖Ω that have connected complement. In the case where Ω is simply connected, such functions are known to be unbounded and to form a collection that is independent of the choice of ζ. This paper uses tools from potential theory to show that, even for domains Ω of arbitrary connectivity, such functions are unbounded whenever they exist. In the doubly connected case, a further analysis of boundary behaviour reveals that the collection of such functions can depend on the choice of ζ. This phenomenon was previously known only for domains that are at least triply connected. Related results are also established for universal Laurent series.
      412Scopus© Citations 6
  • Publication
    A convergence theorem for harmonic measures with applications to Taylor series
    (American Mathematical Society, 2016-03) ;
    Let $ f$ be a holomorphic function on the unit disc, and let $ (S_{n_{k}})$ be a subsequence of its Taylor polynomials about 0. It is shown that the nontangential limit of $ f$ and lim $ _{k\rightarrow \infty }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.
      444Scopus© Citations 9