Oscillating multipliers for some eigenfunction expansions

Let P be a non-negative, self-adjoint differential operator of degree d on ℝⁿ. Assume that the associated Bochner-Riesz kernel s _R ^δ satisfies the estimate, |s _R ^δ (x, y)| ≤ C R^n/d(1+R^1/d|x - y|^-αδ+β)for some fixed constants a>0 and β. We study L^p boundedness of operators of the form m(P), m coming from the symbol class S _p ^−α . We prove that m(P) is bounded on L^P if . We also study multipliers associated to the Hermite operator H on ℝⁿ and the special Hermite operator L on ℂⁿ given by the symbols . As a special case we obtain L^p boundedness of solutions to the Wave equation associated to H and L.     

Localization and convergence of eigenfunction expansions

We state a localization principle for expansions in eigenfunctions of a self-adjoint second order elliptic operator and we prove an equiconvergence result between eigenfunction expansions and trigonometric expansions. We then study the Gibbs phenomenon for eigenfunction expansions of piecewise smooth functions on two-dimensional manifolds.     

Pointwise fourier inversion and related eigenfunction expansions

Necessary and sufficient conditions are found for the convergence at a pre-assigned point of the spherical partial sums (resp. integrals) of the Fourier series (resp. integral) in the class of piecewise smooth functions on Euclidean space. These results carry over unchanged to spherical harmonic expansions, Fourier transforms on hyperbolic space, and Dirichlet eigenfunction expansions with respect to the Laplace operator on a class of Riemannian manifolds. © 1994 John Wiley & Sons, Inc.     

Transplantation and multiplier theorems for Fourier-Bessel expansions

Proved are weighted transplantation inequalities for Fourier-Bessel expansions. These extend known results on this subject by considering the largest possible range of parameters, allowing more weights and admitting a shift. The results are then used to produce a fairly general multiplier theorem with power weights for considered expansions. Also fractional integral results and conjugate function norm inequalities for these expansions are proved.

     

On uniform and Lp-convergence of eigenfunction expansions for indefinite eigenvalue problems

We show that the classical results of L.Fejér and M. Riesz concerning norm convergence of trigonometric Fourier series can be carried over to eigenfunction expansions arising from regular indefinite boundary eigenvalue problems.     

On summability of eigenfunction expansions of piecewise smooth functions

We consider two forms of eigenfunction expansions associated with an arbitrary elliptic differential operator with constant coefficients and order m, that is the multiple Fourier series and integrals. For the multiple Fourier integrals, we prove the convergence of the Riesz means of order s?>?(N???3)/2 of piecewise smooth functions of N?≥?2 variables. The same result is proved in the case of the N?≥?3 dimensional multiple Fourier series.     

Continuous spectrum eigenfunction expansions associated with the Rayleigh equation

Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 27, No. 3, pp. 87–89, July–September, 1993.     

Summability of eigenfunction expansions of differential and integral operators

Translated from Matematicheskie Zametki, Vol. 54, No. 3, pp. 114–122, September, 1993.