A quick distributional way to the prime number theorem

We use distribution theory (generalized functions) to show the prime number theorem. No tauberian results are employed.     

Structural theorems for quasiasymptotics of distributions at the origin

An open question concerning the quasiasymptotic behavior of distributions at the origin is solved. The question is the following: Suppose that a tempered distribution has quasiasymptotic at the origin in S ′(?), then the tempered distribution has quasiasymptotic in D ′(?), does the converse implication hold? The second purpose of this article is to give complete structural theorems for quasiasymptotics at the origin. For this purpose, asymptotically homogeneous functions with respect to slowly varying functions are introduced and analyzed (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the order of summability of the Fourier inversion formula

In this article we show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesàro summable to the distributional point value of order k+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesàro summable of order k, then the distribution is the (k+1)-th derivative of a locally integrable function, and the distribution has a distributional point value of order k+2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.     

pH‐controlled reaction divergence of decarboxylation versus fragmentation in reactions of dihydroxyfumarate with glyoxylate and formaldehyde: parallels to biological pathways

The reactions of dihydroxyfumarate with glyoxylate and formaldehyde exhibit a unique pH‐controlled mechanistic divergence leading to different product suites by two distinct pathways. The divergent reactions proceed via a central intermediate (2,3‐dihydroxy‐oxalosuccinate, 3 , in the reaction with glyoxylate and 2‐hydroxy‐2‐hydroxymethyl‐3‐oxosuccinate, 14 , in the reaction with formaldehyde). At pH 7–8, products ( 7 , 8 , and 15 ) exclusively from a decarboxylation of the intermediate are observed, while at pH 13–14, products ( 9 , 10 , and 16 ) solely derived from a hydroxide‐promoted fragmentation of the intermediate are formed. The decarboxylative and fragmentation pathways are mutually exclusive and do not appear to coexist under the range of pH (7–14) conditions investigated. Herein, we employ a combination of quantitative ¹³C NMR measurements and density functional theory calculations to provide a rationale for this pH‐driven reaction divergence. These rationalizations also hold true for the reactions of dihydroxyfumarate produced in situ by the catalytic cyanide‐mediated dimerization of glyoxylate. In addition, the non‐enzymatic decarboxylation and fragmentation transformations of these central intermediates ( 3 and 14 ) appear to have intriguing parallels to the enzymatic reactions of oxalosuccinate and formation of glyceric acid derivatives in extant metabolism – the high and low pH mimicking the precise control exerted by the enzymes over reaction pathways. Copyright © 2016 John Wiley & Sons, Ltd.     

On the jump behavior of distributions and logarithmic averages

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.     

Ultradistributional boundary values of harmonic functions on the sphere

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.     

Distributional versions of littlewood’s Tauberian theorem

We provide several general versions of Littlewood’s Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We employ two types of Tauberian hypotheses; the first kind involves distributional boundedness, while the second type imposes a one-sided assumption on the Cesàro behavior of the distribution. We apply these Tauberian results to deduce a number of Tauberian theorems for power series and Stieltjes integrals where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.     

A tauberian theorem for distributional point values

We give a tauberian theorem for boundary values of analytic functions. We prove that if is the distributional limit of the analytic function F defined in a region of the form (a, b) × (0, R), if F (x ₀ + iy)→ γ as y → 0⁺, and if f is distributionally bounded at x = x ₀, then f (x ₀) = γ distributionally. As a consequence of our tauberian theorem, we obtain a new proof of a tauberian theorem of Hardy and Littlewood. Received: 10 December 2007