The coefficients that appear in uniform asymptotic expansions for integrals are typically very complicated. In the existing literature the majority of the work only give the first two coefficients. In a limited number of papers where more coefficients are given the evaluation of the coefficients near the coalescence points is normally highly numerically unstable. In this paper, we illustrate how well-known Cauchy type integral representations can be used to compute the coefficients in a very stable and efficient manner. We discuss the cases: (i) two coalescing saddles, (ii) two saddles coalesce with two branch points, (iii) a saddle point near an endpoint of the interval of integration. As a special case of (ii) we give a new uniform asymptotic expansion for Jacobi polynomials $P_n^{(α,β)}(z)$ in terms of Laguerre polynomials $L_n^{(α)}(x)$ as $n\to\infty$ that holds uniformly for $z$ near $1$. Several numerical illustrations are included.
A representation for the Riemann zeta function valid for arbitrary complex $s=σ+it$ is $ζ(s)=\sum_{n=0}^\infty A(n,s)$, where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left(\!\begin{array}{c}n\\k\end{array}\!\right) \frac{(-)^k}{(k+1)^s}.\] In this note we examine the asymptotics of $A(n,s)$ as $n\to\infty$ when $t=an$, where $a>0$ is a fixed parameter, by application of the method of steepest descents to an integral representation. Numerical results are presented to illustrate the accuracy of the expansion obtained.
In this paper, we shall prove the uniform sharp $L^p$ decay estimates for a class of oscillatory integral operators with polynomial phases. By this one-dimensional result, we can use the rotation method to obtain uniform sharp $L^p$ estimates of certain higher-dimensional oscillatory integral operators.
We consider the problem on uniform estimates for an oscillatory integrals with the smooth phase functions having singularities $D_{\infty} $. The estimate is sharp and analogy to estimates of the work of V.N.Karpushkin.
We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson's lemma, Laplace's method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn's book {\em Asymptotic Methods in Analysis}. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are u…
This paper establishes the optimal decay rate for scalar oscillatory integrals in $n$ variables which satisfy a nondegeneracy condition on the third derivatives. The estimates proved are stable under small linear perturbations, as encountered when computing the Fourier transform of surface-carried measures. The main idea of the proof is to construct a nonisotropic family of balls which locally capture the scales and directions in which cancellation occurs.
We find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of $2π$-periodic functions, representable in the form of convolutions of functions $φ$, which belong to unit balls of spaces $L_{p}$, with generalized Poisson kernels. For obtained asymptotic equalities we introduce the estimates of remainder, which are expressed in the explicit form via the parameters of the problem.
We study oscillatory integrals in several variables with analytic, smooth, or $C^k$ phases satisfying a nondegeneracy condition attributed to Varchenko. With only real analytic methods, Varchenko's estimates are rediscovered and generalized. The same methods are pushed further to obtain full asymptotic expansions of such integrals with analytic and smooth phases, and finite expansions with error assuming the phase is only $C^k$. The Newton polyhedron appears naturally in the estimates; in particular, we show precisely how the exponents appearing in the asymptotic expansions depend only on the …
We examine the relation between oscillatory integral estimates and sublevel set estimates associated to convex functions. Whilst the former implies the latter in many cases, the reverse requires additional assumptions. Under finite (line) type assumptions, Bruna, Nagel & Wainger were able to demonstrate a very precise control of oscillatory integrals with convex phases via their sublevel sets. Without the finite type assumption, certain erratic behaviour can force this precise control to fail (Bak, McMichael, Vance & Wainger). We establish the same precise control under an alternative qualitat…
We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vallée Poussin sums on the sets C^{q}_βH_ωof Poisson integrals of functions from the class H_ωgenerated by convex upwards moduli of continuity ω(t) which satisfy the condition ω(t)/t\to\infty as t\to 0. As an implication, a solution of the Kolmogorov-Nikol'skii problem for de la Vallée Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes H^α, 0<α<1, is obtained
We consider a class of multiparameter singular Radon integral operators on the Heisenberg group ${\mathbb H}^1$ where the underlying variety is the graph of a polynomial. A remarkable difference with the euclidean case, where Heisenberg convolution is replaced by euclidean convolution, is that the operators on the Heisenberg group are always $L^2$ bounded. This is not the case in the euclidean setting where $L^2$ boundedness depends on the polynomial defining the underlying surface. Here we uncover some new, interesting phenomena. For example, although the Heisenberg group operators are always…
Oscillatory integrals arise in many situations where it is important to obtain decay estimates which are stable under certain perturbations of the phase. Examining the structural problems underpinning these estimates leads one to consider sublevel set estimates, which behave nicely under certain algebraic operations such as composition with a polynomial. This motivates us to ask how oscillatory integral estimates behave under such transformations of the phase, and under some natural higher order convexity assumptions we obtain stable estimates under composition with polynomial phases in one di…
The paper is an investigation of the analytic properties of a new class of special functions that appear in the kernels of a class of integral operators underlying the dynamics of matter relaxation processes in attractive fields. These functions, recently introduced by the second author, generate the kernels of the principal parts of these operators and play an important role in understanding their spectral characteristics. We reveal the representations of these functions in terms of the Gauss and Clausen hypergeometric functions and present differential-difference and differential equations t…
We find uniform with respect to parameter $p \ (1\leq p\leq\infty)$ upper estimations of best approximations by trigonometric polynomials of classes $C^ψ_{β,p}$ of periodic functions generated by sequences $ψ(k)$, that decrease to nought faster than any power function. Obtained estimations are exact for order and have constants that are written in explicit form and depend on function $ψ$ only. We obtain analogical estimations for best approximations of classes $L^ψ_{β,1}$ in metrics of spaces $L_{s}$, $1\leq s\leq \infty$.
Using the Calderón-Zygmund decomposition, we give a novel and simple proof that $L^2$ bounded dyadic shifts admit a domination by positive sparse forms with linear growth in the complexity of the shift. Our estimate, coupled with Hytönen's dyadic representation theorem, upgrades to a positive sparse domination of the class $\mathcal U$ of singular integrals satisfying the assumptions of the classical $T(1)$-theorem of David and Journé, with logarithmic-Dini type smoothness of the integral kernel. Furthermore, our proof extends rather easily to the $\mathbb R^n$-valued case, yielding as a corol…
We study AAK as well as Padé approximants to functions f, where f is a sum of a Cauchy transform of a complex measure μsupported on a real interval included in (-1,1), whose Radon-Nikodym derivative with respect to the arcsine distribution on its support is Dini-continuous, non-vanishing and with and argument of bounded variation, and of a rational function with no poles on the support of μ. It is shown that the approximants converge to f locally uniformly in the domain of holomorphy of f, intersected with the unit disk in the case of AAK approximants. In the case of Padé approximants we need …
We obtain the rigorous uniform asymptotics of a particular integral where a stationary point is close to an endpoint. There exists a general method introduced by Bleistein for obtaining uniform asymptotics in this situation. However, this method does not provide rigorous estimates for the error. Indeed, the method of Bleistein starts with a change of variables, which implies that the parameter governing how close the stationary point is to the endpoint appears in several parts of the integrand, and this means that one cannot obtain general error bounds. By adapting the above method to our part…
This is the full version of "Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results" appearing on this server and also published in IMRN 2003, no. 15, pp. 821-858.
We prove a sharp asymptotic formula for certain oscillatory integrals that may be approached using the stationary phase method. The estimates are uniform in terms of auxiliary parameters, which is crucial for application in analytic number theory.
This paper contains some estimates for the {\it integral-uniform} norm and the uniform norm of a wide class of random polynomials. The family of integral-uniform norms introduced by Kasin and Tzafriri is a natural generalization of the maximum norm taken over a net. We prove some properties of the integral-uniform norms. The given application of the established estimates demonstrates that the integral-uniform norms may be useful whenever one is interested in the properties of a function distribution. Key words: integral-uniform norm; random polynomials with respect to a general function system…
We describe a method to evaluate integrals that arise in the asymptotic analysis when two saddle points may be close together. These integrals, which appear in problems from optics, acoustics or quantum mechanics as well as in a wide class of special functions, can be transformed into Airy-type integrals and we use the trapezoidal rule to compute these integrals numerically. The quadrature method, which remains valid when two saddle points coalesce, is illustrated with numerical examples.
We consider the question of when it is possible to force a degenerate scalar oscillatory integral to decay as fast as a nondegenerate one by restricting the support to the region where the Hessian determinant of the phase is bounded below. We show in two dimensions that the desired outcome is not always possible, but does occur for a broad class of phases which may be described in terms of the Newton polygon. The estimates obtained are uniform with respect to linear perturbation of the phase and uniform in the cutoff value of the Hessian determinant. In the course of the proof, we investigate …
An Abelian integral is the integral over the level curves of a Hamiltonian $H$ of an algebraic form $ω$. The infinitesimal Hilbert sixteenth problem calls for the study of the number of zeros of Abelian integrals in terms of the degrees $H$ and $ω$. Petrov and Khovanskii have shown that this number grows at most linearly with the degree of $ω$, but gave a purely existential bound. Binyamini, Novikov and Yakovenko have given an \emph{explicit} bound growing doubly-exponentially with the degree. We combine the techniques used in the proofs of these two results, to obtain an explicit bound on the…
