An Uncertainty Principle for Eigenfunction Expansions

In this paper, we extend a theorem of Hardy’s on Fourier transform pairs to: (a) a noncompact-type Riemannian symmetric space of rank one, with respect to the eigenfunction expansion of the invariant Laplacian; (b) a compact Riemannian manifold with respect to the eigenfunction expansion of a positive elliptic operator; and (c) Rⁿ with respect to Hermite and Laguerre expansions.     

Vector Eigenfunction Expansions for Plane Channel Flows

The full nonlinear initial-boundary value problem for the evolution of disturbances in plane Poiseuille flow is considered. The problem is formulated in vector form using the normal velocity and normal vorticity as components. The solution is presented as an expansion in linear eigenmodes. These modes consist of both Orr-Sommerfeld modes and modes of the normal vorticity (Squire) equation. The case of degenerating eigenmodes is also considered and it is shown that the Benney-Gustavsson normal velocity-normal vorticity resonance is a special case of a degeneracy between the vector eigenmodes. The solution to the nonlinear problem is presented as an expansion in the linear eigenmodes as well as in modes of the self-adjoint part of the linear equation. The full nonlinear solution is further reduced to small systems of coupled amplitude equations using the center manifold theorem.     

关于广义Sublaplacian算子特征函数的展开

本文研究了广义Ｓｕｂｌａｐｌａｃｉａｎ算子特征函数的展开．通过定义广义　－扭曲卷积，我们得到相应的Ｐｌａｎｃｈｅｒｅｌ定理及Ｈａｕｓｄｏｒｆｆ－Ｙｏｕｎｇ不等式，最后，我们还给出了当广义Ｓｕｂｌａｐｌａｃｉａｎ算子的特征值为离散时的一些结果．     

Eigenfunction Expansions by the Delta Function and Contour Integral Method

The formal identity is sometimes usedin applied mathematics and engineering textbooks in conjunctionwith the contour integral method to obtain eigenfunction expansionsassociated with y'+y = 0 with boundary conditions. This papergives a rigorous application of this method to the more generalequation y'+( – q(x))y = 0 in the regular case.     

Singular Matched Eigenfunction Expansions for Stokes Flow around a Corner

A singular matched eigenfunction expansion method is describedfor solving Stokes flow around a corner. The flow region isdecomposed into a number of simpler rectangular subregions;this enables the stream function to be represented by meansof an expansion of Papkovich-Fadle eigenfunctions in each ofthese subregions. The coefficients in these expansions are obtainedby matching them across common interfaces in a weak sense. Theresulting solution is used in a post-processing technique todetermine the coefficients in the known locally convergent expansionof the stream function at reentrant and salient corners. A smallnumber of terms in this expansion is necessary to produce accurateapproximations.     

Eigenfunction Expansions on Geodesic Balls and Rank One Symmetric Spaces of Compact Type

We find sharp conditions for the pointwise convergence ofeigenfunction expansions associated with the Laplace operator and otherrotationally invariant differential operators. Specifically, we considerthis problem for expansions associated with certain radially symmetricoperators and general boundary conditions and the problem in the contextof Jacobi polynomial expansions. The latter has immediate application toFourier series on rank one symmetric spaces of compact type.     

Abel-Summability of Eigenfunction Expansions of Three—Point Boundary Value Problems

We study the Abel-summability of the eigenfunction expansions associated with the differential operator L generated by and splitting three-point boundary conditions. It is shown that there is no straightforward analogy between multipoint and twopoint boundary value problems. Counterexamples show that our main results are best possible. Classification. 34L10, 34B10.     

Convergence of Eigenfunction Expansions of a Differential Operator with Integral Boundary Conditions

For a second-order ordinary differential operator on an interval of the real line with integral boundary conditions, conditions for the unconditional basis property and uniform convergence of the expansion of a function in terms of the eigen- and associated functions of this operator are established. The convergence and equiconvergence rates of this expansion and the equiconvergence rate of the trigonometric Fourier expansion of this function are estimated. The uniform convergence of its expansion in the adjoint system is studied.

     

On Coefficients of Eigenfunction Expansions for Boundary-Value Problems with Parameter in Boundary Conditions

In this paper, we consider boundary-value problems for an nth-order ordinary differential equation with polynomials of the spectral parameter in the equation and in the boundary conditions. For a wide class of such problems, we find coefficients of the eigen and associated function expansions for the Shkalikov linearizer.     

Some Special Tauberian Theorems for Fourier Type Integrals and Eigenfunction Expansions of Sturm-Liouville Equations on the Whole Line

We offer a new proof of a special Tauberian theorem for Fourier type integrals. This Tauberian theorem was already considered by us in the papers [1] and [2]. The idea of our initial proof was simple, but the details were complicated because we used Bochner's definition of generalized Fourier transform for functions of polynomial growth. In the present paper we work with L. Schwartz's generalization. This leads to significant simplification. The paper consists of six sections. In Section 1 we establish an integral representation of functions of polynomial growth (subjected to some Tauberian conditions), in Section 2 we prove our main Tauberian theorems (Theorems 2.1 and 2.2.), using the integral representation of Section 1, in Section 3 we study the asymptotic behavior of M. Riesz's means of functions of polynomial growth, in Sections 4 and 5 we apply our Tauberian theorems to the problem of equiconvergence of eigenfunction expansions of Sturm-Liouville equations and expansion in ordinary Fourier integrals, and in Section 6 we compare our general equiconvergence theorems of Sections 4 and 5 with the well known theorems on eigenfunction expansions in classical orthogonal polynomials. In some sense this paper is a re-made survey of our results obtained during the period 1953-58. Another proof of our Tauberian theorem and some generalization can be found in the papers [3] and [4].     

Conformally Invariant Regularization and Skeleton Expansions in Gauge Theory

We consider a conformally invariant regularization of an Abelian gauge theory in an Euclidean space of even dimension D 4 and regularized skeleton expansions for vertices and higher Green's functions. We set the respective regularized fields and with the scaling dimensions and into correspondence to the gauge field A and Euclidean current j . We postulate special rules for the limiting transition 0. These rules are different for the transversal and longitudinal components of the field and the current . We show that in the limit 0, there appear conformally invariant fields A and j each of which is transformed by a direct sum of two irreducible representations of the conformal group. Removing the regularization, we obtain a well-defined skeleton theory constructed from conformal two- and three-point correlation functions. We consider skeleton equations on the transversal component of the vertex operator and of the spinor propagator in conformal quantum electrodynamics. For simplicity, we restrict the consideration to an Abelian gauge field A , but generalization to a non-Abelian theory is straightforward.     

Asymptotic Expansions of Double and Multiple Integrals Occurring in Diffraction Theory

At present at I.N.S.T.N., Saclay and Faculté des Sciences, Paris, France Asymptotic expansions of double integrals of the type have been derived in terms of thereal parameter k by the method of stationary phase. The resultscan easily be extended to multi-dimensional integrals. In the first part of this paper a rigorous proof of the applicationof the method of stationary phase to double and multiple integralsis established with the aid of neutralizer or unitary functions.It is shown that the principal contributions to U(k) come fromsmall but otherwise arbitrary neighbourhoods of critical pointsof the integral, which may be located in the interior or onthe boundary of the domain of integration. These points areassociated with the phase or amplitude function. An explicitasymptotic series in the parameter k of the principal contributionis exhibited when the amplitude and the phase functions havein the neighbourhood of a critical point (x₁,y₁) a developmentof the form g(x,y) = (x–x₁)^0–1 (y–y₁)^µ0–1g₁(x,y), (x,y) = (x₁,y₁) + a _,0 (x–x₁ [1 + P(x,y) + b_0,(y–y₁[1+Q(x,y)]. The function g₁ is a regular function and P,Q can be developedin power series in the vicinity of the critical point and vanishat this point. The above expansion we shall call normal or canonicaland the critical point a normal or canonical critical pointof the integral. Although the assumption of the normal form expansion of theamplitude and phase functions is too restrictive for the generalcase, nevertheless it is found to be sufficiently broad to includemost of the important and interesting cases which occur in diffraction,scattering and other problems of mathematical physics. In Part II the principal contribution arising from a criticalpoint of normal type has been calculated in the form of a descendingpower series in the parameter k. It is shown, with the use ofmajorant functions, that the contribution due to the remainderpart of the series is of higher order in the parameter thanthat of the last term of the finite part, which proves the asymptoticcharacter of the series in the sense of Poincaré. Theresults derived here are in agreement with that of Part I. However,the new series has a decided advantage over that given in PartI if calculations are desired for even a few terms of the series,since the coefficients entering in the asymptotic expansionof the principal contribution are expressed directly in termsof the original functions g(x,y) and (x,y) and their derivatives,which is not the case in the formulas derived in Part I. In Part III explicit asymptotic expansions of the double integralare derived for several typical critical points associated withthe phase function. These are important in connection with thetheory of diffraction of optical instruments with large aberrationsand scattering problems. On account of their importance, eachcase has been treated in detail. In the appendices we have given an alternative proof of thetheorem announced in Part I and the derivation of the leadingterm due to a boundary stationary point. There will be foundalso a discussion of the more general integral where the parameterk appears implicitly in the phase function and not explicitlyas considered in the text. Integrals of this kind occur in manybranches of physics, especially when dealing with wave propagationin dispersive and absorbing media. Finally, we have concludedon the basis of our results that the Rubinowicz approach todiffraction and the stationary phase application to diffractionintegrals lead to similar mathematical results, although differentphysical interpretations, in diffraction phenomena, the formerleading to Young diffraction phenomena and the latter to Fresneldiffraction phenomena.     

New Perturbation Theory for Quantum Field Theory: Convergent Series Instead of Asymptotic Expansions

We present a new approach to perturbation theory for quantum field theory based on convergent series instead of asymptotic expansions. This approach could be considered as the next step after traditional perturbation theory calculations, which allows more comprehensive use of previously obtained information in finding numerical values with greater accuracy.     

Phase Transformation and Singularity Theory

Phase transformation plays an important role in thermodynamics and materials science. Based on the theory of singularities, a new method to construct phase diagrams is presented. Analysing singularities on base of root sequences, see Tamaschke [16], will help to develop singularity graphs, where workings by H. Whitney, R. Thom, and V. I. Arnold provide fundamentals. The generated singularity graphs build the starting point for singularity phase diagrams. A powerful characteristic of such singularity graphs is that higher-dimensional surfaces can be transformed to a two-dimensional diagram. The attained singularity diagram can be used in materials science for analytical models of temperature-concentrated diagrams. As tools from algebra and analysis build a sound basis for singularity diagrams, it is possible to evolve computer software generating these phase diagrams. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Asymptotic Expansions and Error Bounds in the Derivation of Two-Dimensional Shell Theory

Known results on asymptotic two-dimensional equations for circular cylindrical shells, including the effects of transverse shear and normal stress deformation, are supplemented by upper- and lower-bound determinations of influence coefficients, using minimum-potential and complementary energy principles in conjunction with asymptotic-expansion results. The new bound analysis shows that the consequences of the asymptotic two-dimensional theory are in exact agreement, except for terms which are small of higher order, with the corresponding consequences of three-dimensional theory, for some classes of edge conditions. The analysis also shows the nature of the differences between results of two- and three-dimensional theory, as a function of geometrical and elastic parameters, where this difference is of importance because of the effect of a St. Venant boundary layer.