Convolution quadrature and discretized operational calculus. II

Summary Operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations: singular integrals (power and logarithmic singularities, finite part integrals), multiple timescale convolution, Volterra integral equations, Wiener-Hopf integral equations. Frequency domain conditions, which determine, the stability of such equations, can be carried over to the discretization.This is Part II to the article with the same title (Part I), which was published in Volume 52, No. 2, pp. 129–145 (1988)     

On the asymptotic stability ofθ-methods for delay differential equations

Stability regions of -methods for the linear delay differential test equations

Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points

We consider first the initial-boundary value problem for the parabolic equation

Truncation error bounds for modified continued fractions with applications to special functions

Much recent work has been done to investigate convergence of modified continued fractions (MCF's), following the proof by Thron and Waadeland [35] in 1980 that a limit-periodic MCFK(a _n , 1;x ₁), with andnth approximant

Numerical procedure for solving a minimization eigenvalue problem

A numerical procedure for solving the left eigenvalue problem

On a generalization of compound Newton-Cotes quadrature formulas

We consider quadrature formulas defined by piecewise polynomial interpolation at equidistant nodes, admitting the nodes of adjacent polynomials to overlap, which generalizes the interpolation scheme of the compound Newton-Cotes quadrature formulas. The error constantse _,n in the estimate

Elements of functional calculus andL 2 regularity for some classes of fourier integral operators

We employ the method of slices to develop a rudimentary calculus describing the nature of operators T*T (respectively, TT*), where T are Fourier integral operators with one-sided right (respectively, left) singularities; this idea has its roots in the work of Greenleaf and Seeger. Such a result allows us to reduce the L² regularity problem for operators in n dimensions with one-sided singularities to the L² regularity problem for operators with two-sided singularities in n − 1 dimensions. As a consequence we deduce almost sharp L²-Sobolev estimates for operators in three-dimensions; an interesting special case is provided by certain restricted X-ray transforms associated to line complexes which are not well curved. We also provide a proof of almost-sharpness by looking at a restricted X-ray transform associated to the line complex generated by the curve t → (t, t^k). Appropriate notions of singularity, strong singularity, and type are also developed.     

Truncated interpolation processes on unbounded intervals and their applications

The authors describe the major results they have recently obtained on the truncation of interpolation processes on unbounded intervals, and on their applications to the numerical evaluation of corresponding integrals and to the resolution of a class of integral equations. Work partially supported by Progetto Innovativo GNCS “Trattamento numerico di equazioni integrali e connessi problemi di approssimazione e quadratura”.     

Singular solutions of a transmission problem in plane linear elasticity for wedge-shaped regions

In plane elasticity, when two different wedge-shaped elastic materials (isotropic, homogeneous) are bonded together along a common edge and subject to tractions on the boundary, the stress field will become infinite at the apex. In fact, asymptotically, the displacementu satisfies

Ratio asymptotics and quadrature formulas

A technique to find the asymptotic behavior of the ratio between a polynomialss _n and thenth orthonormal polynomial with respect to a positive measureμ is shown. Using it, some new results are found and a very simple proof for other classics is given.     

Complex Gaussian quadrature of oscillatory integrals

We construct and analyze Gauss-type quadrature rules with complex- valued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially decaying behaviour, suitable for the application of Gaussian rules with non-standard weight functions. The results differ from those in previous research in the sense that the constructed rules are asymptotically optimal, i.e., among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence, relative to the required number of evaluations of the integrand.     

The Weierstrass mean

Theorem.Let the sequences {e _i ⁽ⁿ⁾ },i=1, 2, 3,n=0, 1, 2, ...be defined by where the e ⁽⁰⁾ s satisfy and where all square roots are taken positive. Then where the convergence is quadratic and monotone and where The discussions of convergence are entirely elementary. However, although the determination of the limits can be made in an elementary way, an acquaintance with elliptic objects is desirable for real understanding.     

On the error of compound quadrature formulas forr-convex functions

LetC _m be a compound quadrature formula, i.e.C _m is obtained by dividing the interval of integration [a, b] intom subintervals of equal length, and applying the same quadrature formulaQ _n to every subinterval. LetR _m be the corresponding error functional. Iff ^(r) > 0 impliesR _m [f] > 0 (orR _m [f] < 0),=" then=" we=" say=">C _m is positive definite (or negative definite, respectively) of orderr. This is the case for most of the well-known quadrature formulas. The assumption thatf ^(r) > 0 may be weakened to the requirement that all divided differences of orderr off are non-negative. Thenf is calledr-convex. Now letC _m be positive definite or negative definite of orderr, and letf be continuous andr-convex. We prove the following direct and inverse theorems for the errorR _m [f], where , denotes the modulus of continuity of orderr:

Functional calculus for groups and applications to evolution equations

Let –iA be the generator of a C ₀-group U on a Banach space X. Via a transference principle we obtain results of the form

Power series equivalent to rational functions: A shifting-origin kronecker type theorem,and normality of Padé tables

Letf(z) be a function analytic in a neighbourhood of zero. For each pair of non-negative integers (m, n), form then byn Toeplitz determinantD(m/n) whose entries are the Maclaurin series coefficients off, namely,

Geometric conditions for interpolation in weighted spaces of entire functions

We use L² estimates for the equation to find geometric conditions on discrete interpolating varieties for weighted spaces A_p(ℂ) of entire functions such that |f(z)|≤Ae^Bp(z) for some A, B>0. In particular, we give a characterization when p(z)=e^|z| and more generally, when In p(e^r) is convex andIn p(r) is concave. Acknowledgements and Notes. The author wishes to thank X. Massaneda for useful talks and remarks.     

Cyclotomic units over finite fields

Let ζ be a primitivesp-th root of unity for a primep>2, and consider the group Ω(ζ) of cyclotomic units in the ringR(ζ)=ℒ[ζ+ζ^-1]. This paper deals with the image of Ω(ζ) in the unit group ofR(ζ)/qR(ζ), whereq is a prime ≠p. In particular, it obtains criteria for this image to be essentially everything, and a lower bound on the density of primesp (withq fixed) for which it cannot be. These results have a direct bearing on previous work about units in integral group rings for cyclic groups of orderpq. Work supported in part by an operating grant from NSERC (Canada).     

The orbit method and Gelfand pairs, associated with nilpotent Lie groups

Let K be a compact Lie group acting by automorphisms on a nilpotent Lie group N. One calls (K, N) a Gelfand pair when the integrable K-invariant functions on N form a commutative algebra under convolution. We prove that in this case the coadjoint orbits for G:= K × N which meet the annihilator of the Lie algebra of K do so in single K-orbits. This generalizes a result of the authors and R. Lipsman concerning Gelfand pairs associated with Heisenberg groups.     

An improvement of the Ivanov bound

Let be a distance-regular graph of diameterd, valencyk andr=max{i|(c _i ,b _i )=(c ₁,b ₁)}. In this paper, we prove that