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We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderón and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions. [ABSTRACT FROM AUTHOR]

We obtain discrete characterizations of wave front sets of Fourier–Lebesgue and quasianalytic type. It is shown that the microlocal properties of an ultradistribution can be obtained by sampling the Fourier transforms of its localizations over a lattice in R d . In particular, we prove the following discrete characterization of the analytic wave front set of a distribution f ∈ D ′ ( Ω ) . Let Λ be a lattice in R d and let U be an open convex neighborhood of the origin such that U ∩ Λ ⁎ = { 0 } . The analytic wave front set W F A ( f ) coincides with the complement in Ω × ( R d ∖ { 0 } ) of the set of points ( x 0 , ξ 0 ) for which there are an open neighborhood V ⊂ Ω ∩ ( x 0 + U ) of x 0 , an open conic neighborhood Γ of ξ 0 , and a bounded sequence ( f p ) p ∈ N in E ′ ( Ω ∩ ( x 0 + U ) ) with f p = f on V such that for some h > 0 sup μ ∈ Γ ∩ Λ ⁡ | f p ˆ ( μ ) | | μ | p ≤ h p + 1 p ! , ∀ p ∈ N . [ABSTRACT FROM AUTHOR]

Abstract: It is shown that if then Lebesgue summability, summability ( ), Abel summability, Riemann summability, and summability ( ) of the series are all equivalent to one another. [Copyright &y& Elsevier]

Abstract: We define and study a summability procedure that is similar to Euler summability but applied in the exterior of a disc, not in the interior. We show that the procedure is well defined and that it actually has many interesting properties. We use these ideas to study some problems of analytic continuation and to give a construction of the convex support of an analytic functional. We also use this exterior summability to study Mittag–Leffler developments. [Copyright &y& Elsevier]

Abstract: We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line and its dual space , namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in . A characterization of boundedness and convergence in is obtained in terms of wavelet coefficients. Our results are then applied to study local and non-local asymptotic properties of Schwartz distributions via wavelet expansions. We provide Abelian and Tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients. [Copyright &y& Elsevier]

Abstract: The jump behavior and symmetric jump behavior of distributions are studied. We give several formulas for the jump of distributions in terms of logarithmic averages, this is done in terms of Cesàro-logarithmic means of decompositions of the Fourier transform and in terms of logarithmic radial and angular local asymptotic behaviors of harmonic conjugate functions. Application to Fourier series are analyzed. In particular, we give formulas for jumps of periodic distributions in terms of Cesàro–Riesz logarithmic means and Abel–Poisson logarithmic means of conjugate Fourier series. [Copyright &y& Elsevier]

We study the family of Fourier-Laplace transforms F α , β (z) = F. p. ∫ 0 ∞ t β exp ⁡ (i t α − i z t) d t , Im z < 0 , for α > 1 and β ∈ C , where Hadamard finite part is used to regularize the integral when Re β ≤ − 1. We prove that each F α , β has analytic continuation to the whole complex plane and determine its asymptotics along any line through the origin. We also apply our ideas to show that some of these functions provide concrete extremal examples for the Wiener-Ikehara theorem and a quantified version of the Ingham-Karamata theorem, supplying new simple and constructive proofs of optimality results for these complex Tauberian theorems. [ABSTRACT FROM AUTHOR]