Distributional versions of littlewood’s Tauberian theorem

Abstract

Abstract. We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We apply these Tauberian results to deduce a number of Tauberian theorems for power series 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. IntroductionA century ago, Littlewood obtained his celebrated extension of Tauber's theorem [22,14]. Littlewood's Tauberian theorem states that if the series ∞ n=0 c n is Abel summable to the number a, namely, the power series ∞ n=0 c n r n has radius of convergence at least 1 andand if the Tauberian hypothesisis satisfied, then the series is actually convergent, ∞ n=0 c n = a. The result was later strengthened by Hardy and Littlewood in [9,10] to an one-sided version. They showed that the condition (1.2) can be relaxed to the weaker one nc n = O L (1), i.e., there exists C > 0 such that −C < nc n .

On the relation between Lebesgue summability and some other summation methods

“…Under additional assumptions on the growth order of f at ±∞, it is possible to establish a more precise link between the order of summability β and the order of the symmetric point value [20]. We refer to [5,6,7,16,18,20] for studies about the interplay between local behavior of distributions and summability of series and integrals. We now proceed to show our main result.…”

Fourier transform inversion: Bounded variation, polynomial growth, Henstock--Stieltjes integration

“…An anonymous referee has pointed out that: "Theorem 3.1 is essentially contained in Theorem 15 from [6] when applied with any 1 ă p ă 8, since under the author hypothesis the Fourier transform under consideration is Op1{|s|q at infinity. Alternatively, Theorem 3.1 follows at once by combining Theorem 13 in [6] with the distributional version of Littlewood's Tauberian theorem from Theorem 4.1 in [7]. "…”

Multidimensional Tauberian theorems for vector-valued distributions

“…[3,50]). In connection with Example 7.2, see [11,12,14,43] for distributional methods in Tauberian theorems for power and Dirichlet series; see [31,44] for applications in prime number theory. 7.3.…”