The Möbius inversion technique is applied to the Poisson summation formula. This results in expressions for the remainder term in the Fourier coefficient asymptotic expansion as an infinite series. Each element of this series is a remainder term in the corresponding Euler-Maclaurin summation formula, and the series has specified convergence properties. These expressions may be used as the basis...

—Pointwise-supported generalised wavelets are introduced, based on Dirac, doublet and further derivatives of (t). A generalised biorthogonal analysis leads to standard Taylor series and new Dual-Taylor series that may be interpreted as Laurent Schwartz distributions. A Parseval-like identity is also derived for Taylor series, showing that Taylor series support an energy theorem. New representa...

We study boundary properties of universal Taylor series. We prove that if f is a universal Taylor series on the open unit disk, then there exists a residual subset G of the unit circle such that f is unbounded on all radii with endpoints in G. We also study the effect of summability methods on universal Taylor series. In particular, we show that a Taylor series is universal if and only if its C...

and Applied Analysis 3 in an infinite number of variables x1, x2, . . ., defined by the series expansion 1 k! (∑ m≥1 xm t m! )k ∑ n≥k Bn,k t n! , k 0, 1, 2, . . . . 2.2 Their explicit expressions are given by the formula Bn,k x1, x2, . . . , xn−k 1 ∑ n! a1! 1! a2! 2! a2 · · · x1 1x2 a2 . . . , 2.3 where the summation takes place over all nonnegative integers a1, a2, . . ., such that a1 2a2 · · ...