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This paper focuses on integrals and Borel transforms concerning λ$$ \lambda $$‐Bessel and λ$$ \lambda $$‐Tricomi functions, employing an umbral perspective as a central approach. Differintegral techniques, presently utilized in calculus, offer a rich reserve of tools that can be utilized across a broad spectrum of problems encompassing the realm of special functions and beyond. Employing integral transforms akin to the Borel type and the related formalism proves to be a potent approach, facilitating a connection between umbral and operational methodologies. By integrating these perspectives, we establish a novel and effective method for deriving integrals of hybrid special functions and achieving the summation of their corresponding generating functions. [ABSTRACT FROM AUTHOR]

The class of ordinary linear constant coefficient differential equations is naturally embedded into a wider class by associating differential equations to (convergent) Laurent series. It is thought that these more general differential equations can be used as an alternative description of some special functions. [ABSTRACT FROM AUTHOR]

This paper develops a technique to represent linear functionals on spaces of homogeneous polynomials between Banach spaces. This technique applies to cases that are not covered by the classical polynomial Borel transform. We provide applications to represent linear functionals on spaces of approximable, compact, hyper-nuclear and hyper- σ (p) -nuclear polynomials. An unexpected difference between the polynomial and multilinear theories is disclosed. [ABSTRACT FROM AUTHOR]

Differintegral methods, namely those techniques using differential and integral operators on the same footing, currently exploited in calculus, provide a fairly unexhausted source of tools to be applied to a wide class of problems involving the theory of special functions and not only. The use of integral transforms of Borel type and the associated formalism will be shown to be an effective means, allowing a link between umbral and operational methods. We merge these two points of view to get a new and efficient method to obtain integrals of special functions and the summation of the associated generating functions as well. [ABSTRACT FROM AUTHOR]

We consider a nonlinear integral equation of a special type that appears in the inverse spectral theory of integro-differential operators and whose unique solvability in the class of square-integrable functions is known. However, for some applied issues in order to construct effective algorithms for solving equations of this type, it is required to establish their solvability in the class of analytic functions. Assuming the free term of the nonlinear equation under consideration to be an entire function of exponential type, we prove that so is its solution. Leaning on this result we provide a constructive procedure for solving this equation in the class of square-integrable functions, which can be easily implemented numerically. [ABSTRACT FROM AUTHOR]

From 't Hooft's argument, one expects that the analyticity domain of an asymptotically free quantum field theory is horn shaped. In the usual Borel summation, the function is obtained through a Laplace transform and thus has a much larger analyticity domain. However, if the summation process goes through the process called acceleration by Écalle, one obtains such a horn-shaped analyticity domain. We therefore argue that acceleration, which allows to go beyond standard Borel summation, must be an integral part of the toolkit for the study of exactly renormalisable quantum field theories. We sketch how this procedure is working and what are its consequences. [ABSTRACT FROM AUTHOR]

Abstract Let H be a Hilbert space. Suppose A is a positive self-adjoint operator on H and φ ∈ H is a cyclic unit vector. For each λ ∈ R , we can define the rank one perturbation of A by A λ = A + λ 〈 φ , ⋅ 〉 φ. To each A λ we can consider the spectral measure of φ , which we denote by μ λ. This generates a family of measures, { μ λ } , and we analyze the packing dimension of this family. Past results have determined that the Hausdorff dimension of this family can be determined if the limit inferior of a ratio involving μ is constant on a Lebesgue typical set. This ratio is sometimes called the pointwise dimension of μ and is related to the upper derivative of μ. Work has been done to make a similar argument for the packing dimension, but with little success. Using the theory of rank one perturbations and Borel transforms, we introduce the concept of Lebesgue exact dimension for μ , which allows us to determine the packing dimension of spectral measures of almost every rank one perturbation μ λ. If the Lebesgue exact dimension for μ is 1 < α < 2 then the packing dimension of Lebesgue almost every μ λ is 2 − α. As a corollary, we find that this limit condition implies a stronger result: the Hausdorff and packing dimensions are equal for almost every μ λ. [ABSTRACT FROM AUTHOR]

The computation of observables in general interacting theories, be them quantum mechanical, field, gauge or string theories, is a non-trivial problem which in many cases can only be addressed by resorting to perturbative methods. In most physically interesting problems these perturbative expansions result in asymptotic series with zero radius of convergence. These asymptotic series then require the use of resurgence and transseries in order for the associated observables to become nonperturbatively well-defined. Resurgence encodes the complete large-order asymptotic behaviour of the coefficients from a perturbative expansion, generically in terms of (multi) instanton sectors and for each problem in terms of its Stokes constants. Some observables arise from linear problems, and have a finite number of instanton sectors and associated Stokes constants; some other observables arise from nonlinear problems, and have an infinite number of instanton sectors and Stokes constants. By means of two very explicit examples, and with emphasis on a pedagogical style of presentation, this work aims at serving as a primer on the aforementioned resurgent, large-order asymptotics of general perturbative expansions. This includes discussions of transseries, Stokes phenomena, generalized steepest-descent methods, Borel transforms, nonlinear resonance, and alien calculus. Furthermore, resurgent properties of transseries – usually described mathematically via alien calculus – are recast in equivalent physical languages: either a "statistical mechanical" language, as motions in chains and lattices; or a "conformal field theoretical" language, with underlying Virasoro-like algebraic structures. [ABSTRACT FROM AUTHOR]

In this paper we study a beyond all orders phenomenon which appears in the analytic unfoldings of the Hopf-zero singularity. It consists in the breakdown of a two-dimensional heteroclinic surface which exists in the truncated normal form of this singularity at any order. The results in this paper are twofold: on the one hand, we give results for generic unfoldings which lead to sharp exponentially small upper bounds of the difference between these manifolds. On the other hand, we provide asymptotic formulas for this difference by means of the Melnikov function for some non-generic unfoldings. [ABSTRACT FROM AUTHOR]

In this article the class W of quasi-entire functions is introduced and the properties of the Borel transform in this class are studied. The class W includes the quasi-entire functions of exponential type whose module is square-integrable on the whole real axis. For such functions the Borel transform formulae can be represented in the form of integrals of Cauchy type from the jumps of functions that are analytic in the whole complex plane outside some cuts in which the jumps are defined. Due to this fact it is possible to obtain Parseval type equalities in the considered class of quasi-entire functions. These results, as a consequence, lead to the known Paley-Wiener results (in particular, the Paley-Wiener theorem). Therefore, they can be considered as a generalization of similar results for the class W of entire functions of exponential type introduced by them. There are different examples of the Borel transform in the class W of quasi-entire functions. In particular, the examples are chosen to show how the obtained results can be generalized in case the module of the quasi-entire function has at most power growth on the real axis. A similar generalization in the class of entire functions of exponential type is known as the Paley-Wiener-Schwartz theorem. In conclusion, there are simple generalizations for the case when the cut which ensures the single-valuedness of the function Borel-associated with the quasi-entire function of the class W does not necessarily coincide with the real axis. [ABSTRACT FROM AUTHOR]

Starting from the Schwinger-Dyson equation and the renormalization group equation for the massless Wess-Zumino model, we compute the dominant nonperturbative contributions to the anomalous dimension of the theory, which are related by alien calculus to singularities of the Borel transform on integer points. The sum of these dominant contributions has an analytic expression. When applied to the two-point function, this analysis gives a tame evolution in the deep euclidean domain at this approximation level, making doubtful the arguments on the triviality of the quantum field theory with positive $$\beta $$ -function. On the other side, we have a singularity of the propagator for timelike momenta of the order of the renormalization group invariant scale of the theory, which has a nonperturbative relationship with the renormalization point of the theory. All these results do not seem to have an interpretation in terms of semiclassical analysis of a Feynman path integral. [ABSTRACT FROM AUTHOR]

The theory of τ -summing and σ -nuclear linear operators on Banach spaces was developed by Pietsch [20, Chapter 23] . Extending the linear case to the range p > 1 and generalizing all cases to the multilinear setting, in this paper we introduce the concept of σ ( p ) -nuclear linear and multilinear operators. In order to develop the duality theory for the spaces of such operators, we introduce the concept of quasi- τ ( p ) -summing linear/multilinear operators and prove Pietsch-type domination theorems for such operators. The main result of the paper shows that, under usual conditions, linear functionals on the space of σ ( p ) -nuclear n -linear operators are represented, via the Borel transform, by quasi- τ ( p ) -summing n -linear operators. As far as we know, this result is new even in the linear case n = 1 . [ABSTRACT FROM AUTHOR]

In this paper we introduce generating functions of eigenvalues of Daubechies' localization operator, study their analytic properties and give analytic continuation of these eigenvalues. Making use of generating functions, we establish a reconstruction formula of symbol functions of Daubechies' localization operator with rotational invariant symbols.Introducimos funciones generadas por los autovalores del operador de localización de Daubechies, estudiamos sus propiedades analíticas y damos continuación analítica de los autovalores. Haciendo uso de las funciones generadas, establecemos la fórmula de reconstrucción de funciones símbolo del operador de localización de Daubechies con símbolos rotacional invariante.

We map the Schwinger-Dyson equation and the renormalization group equation for the massless Wess-Zumino model in the Borel plane, where the product of functions gets mapped to a convolution product. The two-point function can be expressed as a superposition of general powers of the external momentum. The singularities of the anomalous dimension are shown to lie on the real line in the Borel plane and to be linked to the singularities of the Mellin transform of the one-loop graph. This new approach allows us to enlarge the reach of previous studies on the expansions around those singularities. The asymptotic behavior at infinity of the Borel transform of the solution is beyond the reach of analytical methods and we do a preliminary numerical study, aiming to show that it should remain bounded. [ABSTRACT FROM AUTHOR]

The principal aim of this paper is to establish an iteration method on the space of resurgent functions. We discuss endless continuability of iterated convolution products of resurgent functions associated with rooted trees and derive their estimates. We show the resurgence of formal series solutions of nonlinear differential and difference equations by applying the estimates. [ABSTRACT FROM AUTHOR]

In linear viscoelasticity, the analytically valid Volterra convolution interconversion relationships between the relaxation modulus G and the corresponding creep compliance (retardation) modulus J play a fundamental role. They allow J to be determined from both theoretical and experimental estimates of and conversely. In order to guarantee conservation of energy for the related models of linear viscoelastic flow and deformation processes, some assumption such as the complete monotonicity of G and dJ/dt needs to be invoked. Interesting theoretical questions thereby arise about the analytic and structural properties of G and J. Gros s[Actualités Sci. Ind. 1190, Hermann, Paris, 1953] appears to have been the first to derive analytical expressions for G in terms of J and conversely. However, the regularity invoked only guarantees their validity for a subset of all possible completely monotone functions. The purpose of this paper is an investigation of the extent to which these results extend to all completely monotone functions. This allows issues associated with the effect of perturbations in G on J, and conversely, to be placed on a rigorous footing. In particular, it is shown, among other things, that G (resp., dJ/dt) having absolutely continuous generating measure does not necessarily guarantee the same for dJ/dt (resp., G). Our results have been derived by using the equivalent resolvent kernel equation that comes from a double differentiation of the interconversion equation. Consequently, they will hold more generally for resolvent kernel equations. [ABSTRACT FROM AUTHOR]

By applying the discrete Duhamel product method we calculate the spectral multiplicity of the direct sum of some operators. In particular, we prove that μ(T|Xi ⊕A) = 1+μ(A) and μ(S ⊕A) = 2 for the restriction of the weighted shift operator T|Xi, shift operator S and some appropriate operators A on the Banach spaces. [ABSTRACT FROM AUTHOR]

We calculate the spectral multiplicity of the direct sums J ⊕ A and S ⊕ B, where J is the Volterra integration operator, S is a shift operator and A and B are suitable operators acting on the Banach spaces. [ABSTRACT FROM AUTHOR]

Abstract: Bender–Canfield showed that a plethora of graph counting problems in orientable/non-orientable surfaces involve two constants and for the orientable and the non-orientable case, respectively. T.T.Q. Le and the authors recently discovered a hidden relation between the sequence and a formal power series solution of the Painlevé I equation which, among other things, allows to give exact asymptotic expansion of to all orders in for large g. The paper introduces a formal power series solution of a Riccati equation, gives a non-linear recursion for its coefficients and an exact asymptotic expansion to all orders in g for large g, using the theory of Borel transforms. In addition, we conjecture a precise relation between the sequence and . Our conjecture is motivated by the enumerative aspects of a quartic matrix model for real symmetric matrices, and the analytic properties of its double scaling limit. In particular, the matrix model provides a computation of the number of rooted quadrangulations in the 2-dimensional projective plane. Our conjecture implies analyticity of the - and -types of free energy of an arbitrary closed 3-manifold in a neighborhood of zero. Finally, we give a matrix model calculation of the Stokes constants, pose several problems that can be answered by the Riemann–Hilbert approach, and provide ample numerical evidence for our results. [Copyright &y& Elsevier]

In terms of distribution functions of zeros of an entire function of exponential type, we prove assertions equivalent to the bilogarithmic Levinson condition for the Borel transform of the function. As an application, we present solutions of two problems related to Pavlov-Korevaar-Dixon interpolation. [ABSTRACT FROM AUTHOR]

In this paper, we illustrate with two examples that the computation of Stokes multipliers in problems in which the asymptotic expansions are not of Gevrey order one, is much more complicated and that the singularities in the corresponding Borel plane can be essential singularities. [ABSTRACT FROM AUTHOR]

Abstract: In singularly perturbed vector fields, where the unperturbed vector field has a curve of singularities (a “critical curve”), orbits tend to be attracted towards or repelled away from this curve, depending on the sign of the divergence of the vector field at the curve. When at some point, this sign bifurcates from negative to positive, orbits will typically be repelled away immediately after passing the bifurcation point (“turning point”). Atypical behaviour is nevertheless observed as well, when orbits follow the critical curve for some distance after the turning point, before they repel away from it: a delay in the bifurcation is present. Interesting are systems that have a maximum bifurcation delay, i.e. there is a point on the critical curve beyond which orbits cannot stay close to the critical curve. This behaviour is known to appear in some systems in dimension 3 (see [E. Benoît (Ed.), Dynamic Bifurcations, in: Lecture Notes in Mathematics, vol. 1493, Springer-Verlag, Berlin, 1991]), and it is commonly believed that it is not an issue in (real) planar systems. Beside making the observation that it does appear in non-analytic planar systems, it is shown that whenever bifurcation delay appears, it has no non-trivial maximum for analytic planar vector fields. The proof is based on the notion of family blow-up at the turning point, on formal power series in terms of blow-up variables, the study of their Gevrey properties and analytic continuation of their Borel transform. These results complement existing results concerning the equivalence of local and global canard solutions in [A. Fruchard, R. Schäfke, Overstability and resonance, Ann. Inst. Fourier (Grenoble) 53 (1) (2003) 227–264]. [Copyright &y& Elsevier]

The present paper is devoted to the Cauchy problem of inhomogeneous convolution equations of a fairly general nature. To solve the problems posed here, we apply the operator method proposed in some earlier papers by the author. The solutions of the problems under consideration are found using an effective method in the form of well-convergent vector-valued power series. The proposed method ensures the continuity of the obtained solutions with respect to the initial data and the inhomogeneous term of the equation. [ABSTRACT FROM AUTHOR]

In this paper we study the exponentially small splitting of a heteroclinic connection in a one-parameter family of analytic vector fields in ${\Bbb R}^{3}.$ This family arises from the conservative analytic unfoldings of the so-called Hopf zero singularity (central singularity). The family under consideration can be seen as a small perturbation of an integrable vector field having a heteroclinic orbit between two critical points along the z axis. We prove that, generically, when the whole family is considered, this heteroclinic connection is destroyed. Moreover, we give an asymptotic formula of the distance between the stable and unstable manifolds when they meet the plane z = 0. This distance is exponentially small with respect to the unfolding parameter, and the main term is a suitable version of the Melnikov integral given in terms of the Borel transform of some function depending on the higher-order terms of the family. The results are obtained in a perturbative setting that does not cover the generic unfoldings of the Hopf singularity, which can be obtained as a singular limit of the considered family. To deal with this singular case, other techniques are needed. The reason to study the breakdown of the heteroclinic orbit is that it can lead to the birth of some homoclinic connection to one of the critical points in the unfoldings of the Hopf-zero singularity, producing what is known as a Shilnikov bifurcation. [ABSTRACT FROM AUTHOR]

We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure μ on R and any f∈L1(R,dμ),lim#x03B5;→0(Ffu(E+i#x03B5;)/Fμ(E+i#x03B5;))=f(E) a.e. w.r.t. μsing, where μsing is the part of μ which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, Ffμ(z)=∫(x-z)-1f(x) dμ(x) and Fμ(z)=∫(x-z)-1dμ(x). [Copyright &y& Elsevier]

A singular rank one perturbation % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa % aaleaacqaHXoqyaeqaaOGaeyypa0JaamyqaiabgUcaRiabeg7aHjab % gMYiHlabew9aQjaacYcacqGHflY1cqGHQms8cqaHvpGAaaa!46EB! $$A_\alpha = A + \alpha \langle \varphi , \cdot \rangle \varphi $$ of a self-adjoint operator A in a Hilbert space % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOfdaryqr1ngBPrginfgDObYtUvgaiuGacqWFlecsaaa!4073! $$\mathcal{H}$$ is considered, where % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgc % Mi5kabeg7aHjabgIGiolaahkfacqGHQicYcqGHEisPaaa!3F7E! $$0 \ne \alpha \in {\mathbf{R}} \cup \infty $$ and % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOMaey % icI48efv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWF % GaaiiuGacqGFlecsdaWgaaWcbaGaeyOeI0IaaGOmaaqabaaaaa!4669! $$\varphi \in \mathcal{H}_{ - 2} $$ but % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciGae8x1dO % MaeyycI88efv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuGa % cqGFlecsdaWgaaWcbaGaeyOeI0IaaGymaaqabaGccaGGSaaaaa!4659! $$\varphi \notin \mathcal{H}_{ - 1} ,$$ with % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOfdaryqr1ngBPrginfgDObYtUvgaiuGacqWFlecsdaWgaaWcbaGa % am4CaaqabaGccaGGSaGaaGjbVlaadohacqGHiiIZcaWHsbGaaiilaa % aa!47E5! $$\mathcal{H}_s ,\;s \in {\mathbf{R}},$$ the usual A–scale of Hilbert spaces. A modified version of the Aronszajn-Krein formula is given. It has the form % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa % aaleaacqaHXoqyaeqaaOGaaiikaiaadQhacaGGPaGaeyypa0ZaaSaa % aeaacaWGgbGaaiikaiaadQhacaGGPaGaeyOeI0IaeqySdegabaGaaG % ymaiabgUcaRiabeg7aHjaadAeacaGGOaGaamOEaiaacMcaaaaaaa!480A! $$F_\alpha (z) = \frac{{F(z) - \alpha }} {{1 + \alpha F(z)}}$$ where... [ABSTRACT FROM AUTHOR]

We obtain an estimate of an entire function expressed as the limit of a sequence of polynomials in exponentials in the union of two angles via an estimate of the function and its derivatives on a set lying on a horizontal strip. [ABSTRACT FROM AUTHOR]

We present the Generalized Borel Transform (GBT). This new approach allows one to obtain approximate solutions of Laplace/Mellin transform valid in both, perturbative and non-perturbative regimes. We compare the results provided by the GBT for a solvable model of quantum mechanics with those provided by standard techniques, as the conventional Borel sum, or its modified versions. We found that our approach is very efficient for obtaining both the low and the high energy behavior of the model. [Copyright &y& Elsevier]

We study the growth of an entire function ƒ, whose Borel transform γƒ is holomorphic outside a bounded convex region D with boundary curvature bounded away from 0 and ∞. The function γƒ is assumed to belong to the Dirichlet space, i.e., it satisfies, where dv(ξ) is the area element. It is shown that for γƒ to satisfy the above conditions, it is necessary and sufficient to have and the growth indicatrix h of ƒ satisfies the relation 0< m≤ h″( ϕ)+ h(ϕ)≤M<∞. [ABSTRACT FROM AUTHOR]

The paper establishes the decrease rate of the Kolmogorov diameters of entire functions from the space L 2(0, ∞) in terms of minimal Blaschke products on the singularity sets of Borel transforms. Besides, in C( K) the decrease rate of the Kolmogorov diameters is calculated for entire functions with given finite order. [ABSTRACT FROM AUTHOR]

We study the asymptotic number of certain monotonically labeled increasing trees arising from a generalized evolution process. The main difference between the presented model and the classical model of binary increasing trees is that the same label can appear in distinct branches of the tree. In the course of the analysis we develop a method to extract asymptotic information on the coefficients of purely formal power series. The method is based on an approximate Borel transform (or, more generally, Mittag-Leffler transform) which enables us to quickly guess the exponential growth rate. With this guess the sequence is then rescaled and a singularity analysis of the generating function of the scaled counting sequence yields accurate asymptotics. The actual analysis is based on differential equations and a Tauberian argument. The counting problem for trees of size n exhibits interesting asymptotics involving powers of n with irrational exponents. [ABSTRACT FROM AUTHOR]