The Selected Works of Roderick S C Wong

The following sections are included:

• Contents
• Preface
• Photos

The following sections are included:

• Introduction
• General considerations
• Two special cases
• Conclusion

The following sections are included:

• Introduction
• Asymptotic expansions
• Basic integrals and their asymptotic behavior
• Main theorems
• A concluding remark

The existence and uniqueness of a bounded solution are shown for the linear equation in infinite matrix Ax = b where A is strictly diagonally dominant and b is bounded.

Volterra integral equations of the form , are considered, where a(t) ∈ C(O, ∞) ∩ L₁(0, 1). Explicit asymptotic forms are obtained for the solutions, when the kernels a(t) have a specific asymptotic representation.

The following sections are included:

• Introduction
• Existence of solutions
• Uniqueness of solutions
• Approximation and behavior of solutions
• Examples

An explicit expression is derived for the error term associated with the asymptotic expansions of the Hankel transform

A technique is developed which gives explicit expressions for the error terms associated with the asymptotic expansions of the Stieltjes transform of f,

Explicit expressions are derived for the error terms associated with the asymptotic expansions of the convolution integral , where h(t) and f(t) are algebraically dominated at both 0⁺ and + ∞. Examples included are Fourier, Bessel, generalized Stieltjes, Hilbert and “potential” transforms.

Asymptotic expansion of multi-dimensional Fourier transforms is derived. An explicit expression for the remainder term is also given, from which an error bound can readily be obtained.

This is a continuation of work begun in an earlier paper in which we used the theory of distributions to derive explicit expressions for the remainder terms associated with the asymptotic expansions of the Stieltjes transform. In this paper similar results are obtained for the fractional integral of order α defined by

Here f(t) is a locally integrable function on [0, ∞) and satisfies

Asymptotic expansions are obtained for the Hilbert transform

The purpose of this paper is to give an up-to-date account of the error analysis of asymptotic expansions of integral transforms. Particular attention is paid to two new asymptotic tools that have been developed recently, one based on summability methods, the other based on the use of distributions. After a brief discussion of integrals with exponentially decaying kernels, we first turn to the consideration of integral transforms whose kernels are oscillatory. This includes the Fourier, Hankel and multi-dimensional Fourier transforms. Next, we consider the class of integral transforms whose kernels are algebraic functions, e.g., Stieltjes, Hilbert and fractional integral transforms. Also, a quite thorough treatment is given to a Mellin-type convolution integral in which both the function and the kernel are algebraically dominated at t = 0⁺ and t =+∞. Finally, we construct error bounds for the remainders associated with two uniform asymptotic expansions, one for the coalescence of a stationary point and an end point, the other for the coalescence of two stationary points. These bounds are unfortunately difficult to evaluate, and attention is being called to develop completely new methods. Another problem is proposed at the end of the paper, which concerns error bounds for certain asymptotic expansions involving logarithms.

An alternative derivation of the asymptotic expansion of multiple Fourier transforms is presented. The present approach is based on the use of distributions. With some modifications, this method can also be applied to other integral transforms with oscillatory kernels such as the Hankel transform.

In this paper, some of the formal arguments given by Jones and Kline [J. Math. Phys., v. 37, 1958. pp. 1–28] are made rigorous. In particular, the reduction procedure of a multiple oscillatory integral to a one-dimensional Fourier transform is justified, and a Taylor-type theorem with remainder is proved for the Dirac δ-function. The analyticity condition of Jones and Kline is now replaced by infinite differentiability. Connections with the asymptotic expansions of Jeanquartier and Malgrange are also discussed.

An infinite asymptotic expansion is obtained for the Lebesgue constants associated with the polynomial interpolation at the zeros of the Chebyshev polynomials. The error due to truncation is shown to be bounded in absolute value by, and of the same sign as, the first neglected term.

Quadrature formulas are obtained for the Fourier and Bessel transforms which correspond to the well-known Gauss-Laguerre formula for the Laplace transform. These formulas provide effective asymptotic approximations, complete with error bounds. Comparison is also made between the quadrature formulas and the asymptotic expansions of these transforms.

The following sections are included:

• Introduction
• Regularization of infinite integrals
• Generalized Mellin convolution
• Asymptotic expansions of convolutions
• Generalized integral transforms

The following sections are included:

• Introduction
• Some preliminary lemmas
• Proof of the theorem
• Proof of corollary 1
• Proof of corollary 2

An alternative derivation is given for the asymptotic expansion, as s → 0⁺, of the multiple integral

An asymptotic expansion is constructed for the double integral

In 1926, Szegö conjectured that the Lebesgue constants for Legendre series form a monotonically increasing sequence. In this paper, we prove that his conjecture is true. Our method is based on an asymptotic expansion together with an explicit error bound, and makes use of some recent results of Baratella and Gatteschi concerning uniform asymptotic approximations of the Jacobi polynomials.

Two asymptotic expansions are obtained for the Laguerre polynomial for large n and fixed α > −1. These expansions are uniformly valid in two overlapping intervals covering the entire x-axis. The leading terms of both agree with the two asymptotic formulas given by Erdélyi who used the theory of differential equations. Our approach is based on two integral representations for the Laguerre polynomials. The phase function of one of these integrals has two coalescing saddle points, and to this one the cubic transformation introduced by Chester, Friedman, and Ursell is applied. The phase function of the other integral also has two coalescing saddle points, but in addition it has a simple pole. Moreover, the saddle points coalesce onto this pole. In this case a rational transformation is used, which mimics the singular behavior of the phase function. In both cases explicit expressions are given for the remainders associated with the asymptotic expansions.

The existence of a one-to-one analytic transformation z ↔ w is established which takes a function of the form

A necessary and sufficient condition is established for the existence of a nonsingular matrix which simplifies a linear form to a single coordinate and at the same time retains a quadratic form. A version of Morse's lemma is also derived. These results are then used in a rigorous derivation of the asymptotic expansion of the oscillatory integral I(λ) = ∫_Dg(x)e^iλf(x) dx (x ∈ ℝⁿ), where a stationary point of f(x) lies on the boundary of D.

Asymptotic expansions are derived for the double integral

Wong, R. and H. Li, Asymptotic expansions for second-order linear difference equations, Journal of Computational and Applied Mathematics 41 (1992) 65-94.

Formal series solutions are obtained for the difference equation

Infinite asymptotic expansions are derived for the solutions to the second order linear difference equation

An asymptotic expansion is derived for the Fourier integral

Let be the Jacobi polynomial of degree n. For and 0 ≤ θ ≤ π , it is proved that

Asymptotic expansions are derived for the Laplace convolution (f*g)(x) as x → ∞, where f and g have asymptotic power series representation in descending powers of t. Bounds are also constructed for the error terms associated with these expansions. Similar results are given for the convolution integrals

as x → ∞. These results can be used in the study of asymptotic solutions to the renewal equation and the Wiener-Hopf equations.

In 1954, A. Novikoff studied the asymptotic behavior of the Pollaczek polynomials P_n(x; a, b) when , where t > 0 is fixed. He divided the positive t-axis into two regions, 0 < t < (a+b)^½ and t > (a+ b)^½, and derived an asymptotic formula in each of the two regions. Furthermore, he found an asymptotic formula for the zeros of these polynomials. Recently M. E. H. Ismail (1994) reconsidered this problem in an attempt to prove a conjecture of R. A. Askey and obtained a two-term expansion for these zeros. Here we derive an infinite asymptotic expansion for , which holds uniformly for 0 < ∈ t ∈ M < ∞, and show that Ismail's result is incorrect.

A rigorous proof is supplied for the validity of an asymptotic approximation to the integral

In this paper, we investigate the asymptotic behavior of the generalized Bessel polynomials Y_n(z; a). Let z = α/(n + 1). We first derive infinite asymptotic expansions for y_n(z; a) when α lies in various regions of the complex plane, except when α is near ± i. Then we construct uniform asymptotic expansions for Y_n(z; a) in neighborhoods of α = ± i. These expansions involve the Airy function and its derivative. Finally, we use these approximations to study the asymptotic behavior of the zeros of Y_n(z; a) near α = i.

Meixner polynomials m_n(x; β, c) form a postive-definite orthogonal system on the positive real line x > 0 with respect to a distribution step function whose jumps are

Let j_v,k denote the k-th positive zero of the Bessel function j_v,k. In this paper, we prove that for í > 0 and k = 1, 2, 3, … ,

Consider the nonlinear wave equation

This is a continuation of an earlier paper in which we investigated the superasymptotics and hyperasymptotics of the generalized Bessel function

Consider the double integral

Let {p_n(x)}n≥0 be the set of orthonormal polynomials with respect to the exponential weight w(x) = e^−υ(x), where υ(x) = x^2m + … is a monic polynomial of degree 2m with m ≥ 2 and is even. An asymptotic approximation is obtained for p_n(x), as n→∞, which holds uniformly for 0 ≤ x ∞ O(n^½m). As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.

An infinite asymptotic expansion is derived for the Meixner-Pollaczek polynomials M_n (n_α; δ, η) as n → ∞, which holds uniformly for −M ≤ α ≤ α M, where M can be any positive number. This expansion involves the parabolic cylinder function and its derivative. If α_n,s denotes the sth zero of Mn (n_α; δ, η), counted from the right, and if , denotes its sth zero counted from the left, then for each fxed s, three-term asymptotic approximations are obtained for both α_n,s and , as n → ∞.

The development of asymptotic expansions of Stieltjes transforms of exponentially decaying functions has been well established. In this paper, we are concerned with the more difficult case in which the functions decay only algebraically at infinity. By using a Gevrey-type condition, we obtain an exponentially improved asymptotic expansion, and give three representation theorems to show that the Stieltjes transform of algebraically decaying functions can be written as the difference of two integral transforms with exponentially decaying kernels, thus making the asymptotic theory developed for integral transforms with exponentially decaying kernels relevant to Stieltjes transforms of algebraically decaying functions, including the smoothing of the Stokes phenomenon.

The Stokes lines/curves are identified for the Mittag–Leffler function

In this paper, we continue our study of the boundary value problem

A turning-point theory is developed for the second-order difference equation

The Carrier–Pearson equation εü + u² = 1 with boundary conditions u(−1) = u(1) = 0 is studied from a rigorous point of view. Known solutions obtained from the method of matched asymptotics are shown to approximate true solutions within an exponentially small error estimate. The so-called spurious solutions turn out to be approximations of true solutions, when the locations of their “spikes” are properly assigned. An estimate is also given for the maximum number of spikes that these solutions can have.

Asymptotic formulas, as ε → 0⁺, are derived for the solutions of the nonlinear differential equation εu″ + Q(u) = 0 with boundary conditions u(−1) = u(1) = 0 or u′(−1) = u′(1) = 0. The nonlinear term Q(u) behaves like a cubic; it vanishes at s₋, 0, s₊ and nowhere else in [s₋, s₊], where s₋ < 0 < s₊. Furthermore, Q′(s_±) < 0, Q′(0) > 0 and the integral of Q on the interval [s₋, s₊] is zero. Solutions to these boundary-value problems are shown to exhibit internal shock layers, and the error terms in the asymptotic approximations are demonstrated to be exponentially small. Estimates are obtained for the number of internal shocks that a solution can have, and the total numbers of solutions to these problems are also given. All results here are established rigorously in the mathematical sense.

Let be the Krawtchouk polynomials and μ = N/n. An asymptotic expansion is derived for , when x is a fixed number. This expansion holds uniformly for μ in [1,∞), and is given in terms of the confluent hypergeometric functions. Asymptotic approximations are also obtained for the zeros of in various cases depending on the values of p, q and μ.

Let F(z) be an analytic function in |z| < 1. If F(z) has only a finite number of algebraic singularities on the unit circle |z| = 1, then Darboux's method can be used to give an asymptotic expansion for the coefficient of zⁿ in the Maclaurin expansion of F(z). However, the validity of this expansion ceases to hold, when the singularities are allowed to approach each other. A special case of this confluence was studied by Fields in 1968. His results have been considered by others to be too complicated, and desires have been expressed to investigate whether any simplification is feasible. In this paper, we shall show that simplification is indeed possible. In the case of two coalescing algebraic singularities, our expansion involves only two Bessel functions of the first kind.

Two linearly independent asymptotic solutions are constructed for the second-order linear difference equation

An asymptotic expansion is derived for the Jacobi polynomials with varying parameters α_n = −nA + a and β_n = −nB + b, where A > 1,B > 1 and a, b are constants. Our expansion is uniformly valid in the upper half-plane . A corresponding expansion is also given for the lower half-plane . Our approach is based on the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993). The two asymptotic expansions hold, in particular, in regions containing the curve L, which is the support of the equilibrium measure associated with these polynomials. Furthermore, it is shown that the zeros of these polynomials all lie on one side of L, and tend to L as n → ∞.

It has been known for some time that the existing asymptotic methods for integrals and differential equations are not applicable in the case of Stieltjes–Wigert polynomials with degree going to infinity. Using the recently introduced nonlinear steepest descent method for Riemann–Hilbert problems, here we not only derive an asymptotic expansion for these polynomials, but we also show that the result holds uniformly in the complex plane except for a sector containing the real axis from −∞ to 1/4. Furthermore, we give an asymptotic formula for the zeros of these polynomials, which approximates the true values of the zeros closely.

On sait depuis longtemps que les méthodes asymptotiques existantes pour les intégrales et les équations différentielles ne sont pas applicables aux polynômes de Stieltjes–Wigert lorsque le degré tend vers l'infini. En utilisant des méthodes de descente non lineaires, introduites récemment pour les problèmes de Riemann–Hilbert, nous obtenons non seulement un développement asymptotique pour ces polynômes, mais nous montrons également que le résultat est uniforme dans le plan complexe, sauf pour un secteur contenant l'axe réel de −∞ à 1/4. De plus, nous démontrons une formule asymptotique qui donne une bonne approximation des racines de ces polynômes.

This paper is concerned with the positive solutions of the boundary-value problem

Let N_ε denote the maximum number of spikes that a solution to Carrier's problem

Let Γ be a piecewise smooth contour in ℂ, which could be unbounded and may have points of self-intersection. Let V (z,N) be a 2 × 2 matrix-valued function defined on Γ, which depends on a parameter N. Consider a Riemann–Hilbert problem for a matrix-valued analytic function R(z,N) that satisfies a jump condition on the contour Γ with the jump matrix V(z,N). Assume that V(z,N) has an asymptotic expansion, as N → ∞, on Γ. An elementary proof is given for the existence of a similar type of asymptotic expansion for the matrix solution R(z,N), as n → ∞, for z ∈ ℂ\Γ. Our method makes use of only complex analysis.

We consider the connection problem for the sine-Gordon PIII equation

In this paper, we use a new method to derive hyperasymptotic expansions for the modified Bessel function of the third kind of purely imaginary order. Our approach is based on the method of Hadamard-series expansion developed by Paris (Proc. Roy. Soc. Lond. A 457 (2001), 2855–2869). Modifications are made to deal with the presence of an infinite number of saddle points, and numerical computation is used to illustrate the accuracy of our results.

In this paper, we study the asymptotics of polynomials orthogonal with respect to the varying quartic weight ω(x) = e^−nV(x), where . We focus on the critical case t = −2, in the sense that for t ≽ −2, the support of the associated equilibrium measure is a single interval, while for t < −2, the support consists of two intervals. Globally uniform asymptotic expansions are obtained for z in three unbounded regions. These regions together cover the whole complex z-plane. In particular, in the region containing the origin, the expansion involves the ψ function affiliated with the Hastings–McLeod solution of the second Painlevé equation. Our approach is based on a modified version of the steepest-descent method for Riemann–Hilbert problems introduced by Deift and Zhou (Ann. Math. 137 (1993), 295–370).

In this paper, we develop the Riemann–Hilbert approach to study the global asymptotics of discrete orthogonal polynomials with infinite nodes. We illustrate our method by concentrating on the Charlier polynomials . We first construct a Riemann–Hilbert problem Y associated with these polynomials and then establish some technical results to transform Y into a continuous Riemann–Hilbert problem so that the steepest descent method of Deift and Zhou can be applied. Finally, we produce three Airy-type asymptotic expansions for in three different but overlapping regions whose union is the entire complex z-plane. When z is real, our results agree with the ones given in the literature. Although our approach is similar to that used by Baik, Kriecherbauer, McLaughlin and Miller, there are crucial differences in the details. For instance, our expansions hold in much bigger regions. Our results are completely new, and one of them answers a question raised in Bo and Wong. Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials.

Using the steepest descent method for oscillatory Riemann–Hilbert problems introduced by Deift and Zhou [Ann. Math. 137 (1993), 295–368], we derive asymptotic formulas for the Meixner polynomials in two regions of the complex plane separated by the boundary of a rectangle. The asymptotic formula on the boundary of the rectangle is obtained by taking limits from either inside or outside. Our results agree with the ones obtained earlier for z on the positive real line by using the steepest descent method for integrals [Constr. Approx. 14 (1998), 113–150].

We use the Legendre polynomials and the Hermite polynomials as two examples to illustrate a simple and systematic technique on deriving asymptotic formulas for orthogonal polynomials via recurrence relations. Another application of this technique is to provide a solution to a problem recently raised by M. E. H. Ismail.

The discrete Chebyshev polynomials t_n(x, N) are orthogonal with respect to a distribution function, which is a step function with jumps one unit at the points x = 0, 1, …, N − 1, N being a fixed positive integer. By using a double integral representation, we derive two asymptotic expansions for t_n(aN, N + 1) in the double scaling limit, namely, N → ∞ and n/N → b, where b ∈ (0, 1) and a ∈ (−∞, ∞). One expansion involves the confluent hypergeometric function and holds uniformly for , and the other involves the Gamma function and holds uniformly for a ∈(−∞, 0). Both intervals of validity of these two expansions can be extended slightly to include a neighborhood of the origin. Asymptotic expansions for can be obtained via a symmetry relation of t_n(aN, N + 1) with respect to . Asymptotic formulas for small and large zeros of t_n(x, N + 1) are also given.

In this paper, we study the asymptotics of the Hahn polynomials Q_n(x; α, β, N) as the degree n grows to infinity, when the parameters α and β are fixed and the ratio of n/N = c is a constant in the interval (0, 1). Uniform asymptotic formulas in terms of Airy functions and elementary functions are obtained for z in three overlapping regions, which together cover the whole complex plane. Our method is based on a modified version of the Riemann–Hilbert approach introduced by Deift and Zhou.

Asymptotic formulas are derived for the Stieltjes–Wigert polynomials S_n(z;q) in the complex plane as the degree n grows to infinity. One formula holds in any disc centered at the origin, and the other holds outside any smaller disc centered at the origin; the two regions together cover the whole plane. In each region, the q-Airy function A_q(z) is used as the approximant. For real x > 1/4, a limiting relation is also established between the q-Airy function A_q(x) and the ordinary Airy function Ai(x) as q → 1.

The following sections are included:

• List of Publications by Roderick S. C. Wong
• Permissions