Estimates for the equiconvergence rate of spectral expansions on an interval

We study the convergence rate of biorthogonal expansions of functions in series in systems of root functions of a broad class of second-order ordinary differential operators on a finite interval. The above-mentioned expansions are compared with the expansions of the same functions in trigonometric Fourier series in an integral or uniform metric on any interior compact set of the basic interval and on the entire interval. We prove the dependence of the equiconvergence rate of the expansions in question on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the presence of infinitely many associated functions in the system of root functions.     

Dependence of estimates of the local convergence rate of spectral expansions on the distance from an interior compact set to the boundary

We consider the problem on the convergence rate of biorthogonal expansions of functions in systems of root functions of a wide class of ordinary second-order differential operators defined on a finite interval. These expansions are compared with expansions of the same functions in Fourier trigonometric series in an integral or uniform metric on any interior compact subset of the main interval. We find the dependence of the equiconvergence rate of resulting expansions on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the presence of infinitely many associated functions in the system of root functions.     

Loaded differential operators: Convergence of spectral expansions

We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a broad class of loaded even-order differential operators defined on a finite interval. These expansions are compared with the Fourier trigonometric series expansions of the same functions in an integral metric on any interior compact set of the main interval or on the entire interval. We obtain estimates for the equiconvergence rate of these expansions.     

Componentwise uniform equiconvergence of expansions in root vector functions of the Dirac operator with the trigonometric expansion

We consider the one-dimensional Dirac operator on a finite interval G = (a, b). We analyze the uniform componentwise equiconvergence of expansions in root vector functions of this operator with the trigonometric Fourier series on a compact set. Theorems on the componentwise equiconvergence on a compact set and the componentwise localization principle are proved.     

On the influence of the potential on the convergence rate of expansions in root functions of the Shrödinger operator

We consider the one-dimensional Schrödinger operator with integrable potential. We analyze the rate of the uniform equiconvergence of the biorthogonal expansion of an absolutely continuous function in the root functions of this operator with its Fourier trigonometric series on a compact set. For this convergence rate, we obtain an estimate depending on the modulus of continuity of the potential. We extract subclasses of absolutely continuous functions on which these estimates can be improved.     

Some properties of root functions of ordinary second-order differential operators in the absence of the carleman condition

On a finite interval G of the real line, we consider the root functions of an ordinary second-order differential operator without any boundary conditions for the case in which the imaginary part of the spectral parameter is unbounded.We refine the estimates for the C-and L _p -norms of a root function and its first derivative on a compact set contained in the interior of G for the case in which the Carleman condition fails.A sufficient condition is obtained for the root functions of an ordinary second-order differential operator to have the Bessel property, assuming that the Carleman condition fails. We show that, under certain conditions, this problem can be reduced to analyzing the Bessel property of systems of exponentials.     

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.     

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method.     

Shift formulas for root functions of odd-order differential operators with nonsmooth coefficients

We obtain shift formulas for the root functions of odd-order differential operators with nonsmooth coefficients (an exact formula for first-order operators and an asymptotic formula for operators of higher odd order); these formulas are needed when studying the convergence of spectral expansions in root functions.     

Biorthogonal expansions in root functions of differential operators

We consider the problem on the convergence of biorthogonal expansions in a system of eigenfunctions and associated functions for a wide class of operators, whose special cases include nonself-adjoint differential operators. We introduce the notion of almost basis property of systems of root functions of a linear operator. We demonstrate the necessity to use a new method, earlier introduced by the authors, for defining associated functions.     

Indefinite Boundary Eigenvalue Problems in a Pontrjagin Space Setting

We study eigenvalue problems Fy = λ Gy consisting of Hamiltonian systems of ordinary differential equations on a compact interval with symmetric λ-linear boundary conditions. The problems we are interested in are non-definite: neither left-nor right-definite. Instead of this, we give some weak condition on one coefficient of the Hamiltonian system which ensures that a hermitian form associated with the operator F has at most finitely many negative squares. This enables us to study the problem by the help of a compact self-adjoint operator in a Pontrjagin space and we obtain as a main result uniformly convergent eigenfunction expansions. In the final section, applications to formally self-adjoint differential equations of higher order are given.     

Equisummability in the sense of M. Riesz of expansions in certain systems of exponential functions

We establish the equisummability in the M. Riesz sense of Fourier series expansions in two systems of exponential functions, these latter being the characteristic functions of boundary-value problems for a first-order differential equation.     

<Emphasis Type="Italic">n</Emphasis>-fold Fourier series expansion in root functions of a differential pencil with <Emphasis Type="Italic">n</Emphasis>-fold multiple characteristic

On a finite interval, we consider a parametric differential pencil of the singular irregular type with an n-fold multiple characteristic and with boundary conditions all of which except for the last are posed at the left end of the interval. We solve the problem on the n-fold expansion of n arbitrary functions in series in Keldysh derived chains of eigenfunctions and associated functions (root functions) of the pencil.     

On the Charshiladze-Lozinski theorem for compact topological groups and homogeneous spaces

This paper is concerned with an extension of the Charshiladze-Lozinski theorem to compact (not necessarily abelian) topological groups G and symmetric compact homogeneous spaces G/H. The proof is based on a generalized Marcinkiewicz — Berman formula. As an application, some divergence theorems for expansions of continuous resp. integrable complex — valued functions on Euclidean spheres and projective spaces in series of polynomial functions on these spaces are established.     

On the convergence of expansions in polyharmonic eigenfunctions

We consider expansions of smooth, nonperiodic functions defined on compact intervals in eigenfunctions of polyharmonic operators equipped with homogeneous Neumann boundary conditions. Having determined asymptotic expressions for both the eigenvalues and eigenfunctions of these operators, we demonstrate how these results can be used in the efficient computation of expansions. Next, we consider the convergence. We establish the key advantage of such expansions over classical Fourier series–namely, both faster and higher-order convergence–and provide a full asymptotic expansion for the error incurred by the truncated expansion. Finally, we derive conditions that completely determine the convergence rate.     

Quasilinear Pursuit Differential Game with a Simple Dynamics under Phase Constraints

On a fixed closed time interval we consider a quasilinear pursuit differential game with a convex compact target set under a phase constraint in the form of a convex closed set. We construct a convex compact guaranteed capture set similar to an alternating Pontryagin sum and define the guaranteed piecewise-programmed strategy of the pursuer ensuring the hitting of the target set by the phase vector satisfying the phase constraint in finite time. Under certain conditions, we prove the convergence of the constructed alternating sum in the Hausdorff metric to a convex compact set, which is an analog of the alternating Pontryagin integral for the differential game.     

The Absolute Ε-Entropy of a Compact Set of Infinitely Differentiable Aperiodic Functions

We calculate asymptotics for the Kolmogorov ε-entropy of the compact set of infinitely differentiable aperiodic functions embedded continuously into the space of continuous functions on a closed finite interval.     

Formal aspects of a theory for pairs of linear ordinary differential operators in the regular case

Summary The ordinary differential equation Su=λTu containing two operators and a parameter λ is considered on a compact interval. Boundary conditions, homogeneous and non-homogeneous problems, expansions in terms of associated eigenfunctions etc. are parallelled with the classical case when T=1. In homage to ProfessorBeniamino Segre Entrata in Redazione il 15 giugno 1973.     

Series expansions for computing Bessel functions of variable order on bounded intervals

Based on a continuity property of the Hadamard product of power series we derive results concerning the rate of convergence of the partial sums of certain polynomial series expansions for Bessel functions. Since these partial sums are easily computable by recursion and since cancellation problems are considerably reduced compared to the corresponding Taylor sections, the expansions may be attractive for numerical purposes. A similar method yields results on series expansions for confluent hypergeometric functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

THE GIBBS PHENOMENON,THE PINSKY PHENOMENON,AND VARIANTS FOR EIGENFUNCTION EXPANSIONS

ABSTRACT

We examine analogues of the Gibbs phenomenon for eigenfunction expansions of functions with singularities across a smooth surface, though of a more general nature than a simple jump. The Gibbs phenomena that arise still have a universal form, but a more general class of “fractional sine integrals” arises, and we study these functions. We also make a uniform analysis of eigenfunction expansions in the presence of the Pinsky phenomenon, and see an analogue of the Gibbs phenomenon there. These analyses are done on three classes of manifolds: strongly scattering manifolds, including Euclidean space; compact manifolds without strongly focusing geodesic flows, including flat tori and quotients of hyperbolic space, and compact manifolds with periodic geodesic flow; including spheres and Zoll surfaces. Finally, we uncover a new divergence phenomenon for eigenfunction expansions of characteristic functions of balls, for a perturbation of the Laplace operator on a sphere of dimension ≥5.