Regularity and integrability of rotating shallow-water equations

We consider classical shallow-water equations for a rapidly rotating fluid layer. The Poincaré/Kelvin linear propagator describes fast oscillating waves for the linearized system. We show that solutions of the full nonlinear shallow-water equations can be decomposed as U(t,x₁,x₂) + Ũ(t,x₁,x₂) + W’(t,x₁,x₂) + r, where Ũ is a solution of the quasigeostrophic (QG) equation. Here r is a remainder, which is uniformly estimated from above by a majorant of order 1/f₀. The vector field W’(t,x₁,x₂) describes the rapidly oscillating ageostrophic (AG) component. This component is exactly solved in terms of Poincaré/Kelvin waves with phase shifts explicitly determined from the nonlinear quasigeostrophic equations. The mathematically rigorous control of the error r, based on estimates of small divisors, is used to prove the existence, on a long time interval T^*, of regular solutions to classical shallow-water equations with general initial data (T^* → +∞, as 1/f₀ → 0).     

Norm inequalities for spherical means

The regularity and integrability of spherical means of functions inL ^p ( ⁿ ,n2, are studied. Sharp results are obtained.     

Strong spherical means of multiple Fourier series

The present paper is a survey of some results related to the strong means of spherical partial sums, their integral analogs and the Bochner-Riesz sums of Fourier series of functions defined almost everywhere on the m-dimensional torus.     

Localisation and weighted inequalities for spherical Fourier means

In this work, we establish certain equivalences between the localisation properties with respect to spherical Fourier means of the support of a given Borel measure and the L ²-rate of decay of the Fourier extension operator associated to it. This, in turn, is intimately connected with the property that the X-ray transform of the measure be uniformly bounded. Geometric properties of sets supporting such a measure are studied. The research of this paper has been partially supported by the EU Comission via the network HARP, and by MEC Grant MTM2004-00678. The first author was partially supported by a Leverhulme Study Abroad Fellowship.     

Approximation for spherical functions by Bochner-Riesz means

This paper deals with approximation of functions on the unit sphere. The uniform and almost everywhere approximation properties of Bochner-Riesz means at and below the critical index for spherical harmonic expansions are studied. The results obtained here are analogous to those of multiple Fourier series. Supported by NSFC     

The Gaussian limit for high-dimensional spherical means

We show that the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finite-codimension affine subspace in infinite dimensions.     

On regularity of twisted spherical means and special Hermite expansions

Regularity properties of twisted spherical means are studied in terms of certain Sobolev spaces defined using Laguerre functions. As an application we prove a localisation theorem for special Hermite expansions.     

Approximation of functions of several variables by spherical Riesz means

For even N ≥ 2 and δ 2N-3 (for N-2 or 4 we assume that δ > (N-1)/2) we find asymptotic approximations for the quantity $$E_R^\delta (H_{\rm N}^\omega ) = \mathop {sup}\limits_{f \in H_{\rm N}^\omega } \parallel f(x) - S_R^\omega (x,f)\parallel _ \in (R \to \infty ),$$ , where S _R ^δ (x,f) is the spherical Riesz mean of order δ of the Fourier kernel of the functionf(x), and H _N ^ω is the class of periodic functions of N variables whose moduli of continuity do not exceed a given convex modulus of continuity ω(δ). For N 2 and δ > 1/2 the result is known.     

Injectivity sets for spherical means on the Heisenberg group

We prove that the boundary of a bounded domain is a set of injectivity for the twisted spherical means on ⁿ for a certain class of functions on ⁿ . As a consequence we obtain results about injectivity of the spherical mean operator in the Heisenberg group and the complex Radon transform.     

The removability problem for functions with zero spherical means

We study the functions on the punctured n-dimensional sphere having zero integrals over all admissible “hemispheres.” We find a condition for the point to be a removable set for this class of functions and show that the condition cannot be dropped or substantially improved.     

On functions with given spherical means on symmetric spaces

The inversion problem for a local Pompeiu transformation of rank one on sym- metric spaces X of the noncompact type is studied. The reconstruction of a function defined in the ball B _R ⊂ X by its averages on balls of two fixed radii lying in B _R is obtained.     

On functions with zero spherical means on complex hyperbolic spaces

The class of functions with zero integrals over all balls of one fixed radius on the complex hyperbolic space is described. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 504–512, October, 2000.     

The extension problem for functions with zero weighted spherical means

We study functions on a sphere with a pricked point having zero integrals with a given weight over all admissible “hemispheres”. We find a condition under which the point is a removable set for such a class of functions. We show that this condition cannot be dropped or substantially weakened.     

Asymptotic normality of spherical means of nonlinear functionals of Gaussian random fields

The central limit theorem is proved for the integral-type functionals of nonlinear transformations of two-and three-dimensional uniform isotropic Gaussian random fields. A theorem on convergence of finite-dimensional distributions of these functionals to the corresponding distributions of the Wiener process is also established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 472–480, April, 1993.     

Approximation of continuous periodic functions of many variables by spherical Riesz means

We find in succession exact upper bounds for the magnitudes of the least upper bounds of the deviations of spherical Riesz means on classes of continuous periodic functions of many variables and, in a number of cases, we prove the asymptotic exactness of these estimates.Translated from Matematicheskie Zametki, Vol. 15, No. 5, pp. 821–832, May, 1974.The author thanks S. B. Stechkin for suggesting the topic of this paper.