Related papers: On Borel summability and analytic functionals

The Euler-MacLaurin summation formula relates a sum of a function to a corresponding integral, with a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series.…

We consider a quartic O(N)-vector model. Using the Loop Vertex Expansion, we prove the Borel summability in 1/N along the real axis of the partition function and of the connected correlations of the model. The Borel summability holds…

A family of formal power series, such that its coefficients satisfy a recursion formula, is characterized in terms of the summability, in the sense of J. P. Ramis, of its elements along certain well chosen directions. We describe a set of…

Borel summation techniques are developed to obtain exact invariants from formal adiabatic invariants (given as divergent series in a small parameter) for a class of differential equations, under assumptions of analyticity of the…

The generating function of the cumulants in random matrix models, as well as the cumulants themselves, can be expanded as asymptotic (divergent) series indexed by maps. While at fixed genus the sums over maps converge, the sums over genera…

Let $f(z) = \sum_{k=0}^\infty a_k z^k$ be an analytic function in a disk $D_R$ of radius $R>0$, and assume that $f$ is $p$-valent in $D_R$, i.e. it takes each value $c\in{\mathbb C}$ at most $p$ times in $D_R$. We consider its Borel…

In this paper we study the exponential uniform strong summability of two-dimensional Vilenkin-Fourier series. In particular, it is proved that the two-dimensional Vilenkin-Fourier series of the continuous function $f$ is uniformly strong…

The method of Fractional Borel Summation is suggested in conjunction with self-similar factor approximants. The method used for extrapolating asymptotic expansions at small variables to large variables, including the variables tending to…

We show that if $\mathcal{F}$ is any "well-behaved" subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on $\pow(\mathbb{R})$ induced by $\mathcal{F}$ turns out to look like the Wadge hierarchy…

We show that there exists a de Branges-Rovnyak space $\mathcal{H}(b)$ on the unit disk containing a function $f$ with the following property: even though $f$ can be approximated by polynomials in $\mathcal{H}(b)$, neither the Taylor partial…

It is proved that the divergent perturbation expansion for the vacuum polarization by an external constant electric field in the pair production sector is Borel summable in the distributional sense.

A classical theorem due to Borel asserts that any formal serie with real coefficients is the Taylor expansion of a germ of $\mathcal{C}^{\infty}- {\rm function}$. We study such a problem in the context of Lie algebras of vector fields or of…

Given a truncated perturbation expansion of a physical quantity, one can, under certain circumstances, obtain lower or upper bounds (or both) to the sum of the full perturbation series by using the Borel transform and a variational…

In this paper I give an evaluation of a functional integral by means of a series in functional derivatives, first of all we propose a differential equation of first order and solve it by iterative methods, to obtain a series for the…

We say a power series $\sum_{n=0}^\infty a_n q^n$ is multiplicative if the function $n\mapsto a_n/a_1$ ($n\ge 1$) is so. In this paper, we consider multiplicative power series $f$ such that $f^2$ is also multiplicative. We find various…

A new summation method is introduced to convert a relatively wide family of infinite sums and local expansions into integrals. The integral representations yield global information such as analytic continuability, position of singularities,…

The theory of summability of divergent series is a major branch of mathematical analysis that has found important applications in engineering and science. It addresses methods of assigning natural values to divergent sums, whose…

Let $\cal R$ be either the Grothendieck semiring (semiring with multiplication) of complex algebraic varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class of the complex affine line. We…

Let f be a sum of exponentials of the form exp(2 pi i N x), where the N are distinct integers. We call f an idempotent trigonometric polynomial (because the convolution of f with itself is f) or, simply, an idempotent. We show that for…

The paper is concerned with the Kontsevich-Zagier formal power series $$ f(q)=\sum_{n=0}^\infty (1-q)... (1-q^n) $$ and its analytic properties. To begin with, we give an explicit formula for the Borel transform of the associated formal…