The Busemann-Petty problem via spherical harmonics

The Busemann-Petty problem asks whether symmetric convex bodies in with smaller central hyperplane sections necessarily have smaller n-dimensional volume. The solution has recently been completed, and the answer is affirmative if n?4 and negative if n?5. In this article we present a short proof of the affirmative result and its generalization using the Funk-Hecke formula for spherical harmonics.     

On a spherical code in the space of spherical harmonics

We propose a new method for the construction of new “nice” configurations of vectors on the unit sphere S ^d with the use of spaces of spherical harmonics.     

Representing algebraic integers as linear combinations of units

In this paper we consider representations of algebraic integers of a number field as linear combinations of units with coefficients coming from a fixed small set, and as sums of elements having small norms in absolute value. These theorems can be viewed as results concerning a generalization of the so-called unit sum number problem, as well. Beside these, extending previous related results we give an upper bound for the length of arithmetic progressions of \(t\) -term sums of algebraic integers having small norms in absolute value.     

Representing integers as linear combinations of power products

Let P be a finite set of at least two prime numbers and A the set of positive integers that are products of powers of primes from P. Let F(k) denote the smallest positive integer which cannot be presented as sum of less than k terms of A. In a recent paper Nathanson asked to determine the properties of the function F(k), in particular to estimate its growth rate. In this paper we derive several results on F(k) and on the related function F _±(k) which denotes the smallest positive integer which cannot be presented as sum of less than k terms of A è(-A){A \cup (-A)}.     

A fast transform for spherical harmonics

Spherical harmonics arise on the sphere S² in the same way that the (Fourier) exponential functions {e^ik}k arise on the circle. Spherical harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform (FFT). Without a fast transform, evaluating (or expanding in) spherical harmonic series on the computer is slow—for large computations probibitively slow. This paper provides a fast transform.For a grid ofO(N²) points on the sphere, a direct calculation has computational complexityO(N⁴), but a simple separation of variables and FFT reduce it toO(N³) time. Here we present algorithms with timesO(N^5/2 log N) andO(N²(log N)²). The problem quickly reduces to the fast application of matrices of associated Legendre functions of certain orders. The essential insight is that although these matrices are dense and oscillatory, locally they can be represented efficiently in trigonometric series.     

A fast spherical harmonics transform algorithm

The spectral method with discrete spherical harmonics transform plays an important role in many applications. In spite of its advantages, the spherical harmonics transform has a drawback of high computational complexity, which is determined by that of the associated Legendre transform, and the direct computation requires time of for cut-off frequency . In this paper, we propose a fast approximate algorithm for the associated Legendre transform. Our algorithm evaluates the transform by means of polynomial interpolation accelerated by the Fast Multipole Method (FMM). The divide-and-conquer approach with split Legendre functions gives computational complexity . Experimental results show that our algorithm is stable and is faster than the direct computation for .

     

Vector spherical harmonics in 3-D vector tomography

A method of series expansion in terms of vector spherical harmonics intended for inverting line integrated experimental (Doppler) data is proposed to investigate 3-D vector fields in laboratory plasma in spherical tokamak devices. A number of numerical computations demonstrating 3-D reconstructions of model vector fields have been performed to assess the inversion method proposed.     

The resolution of the Gibbs phenomenon for spherical harmonics

Spherical harmonics have been important tools for solving geophysical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion approximations to functions with discontinuities, causing undesirable oscillations over the entire sphere.

Recently, methods for removing the Gibbs phenomenon for one-dimensional discontinuous functions have been successfully developed by Gottlieb and Shu. They proved that the knowledge of the first expansion coefficients (either Fourier or Gegenbauer) of a piecewise analytic function is enough to recover an exponentially convergent approximation to the point values of in any subinterval in which the function is analytic.

Here we take a similar approach, proving that knowledge of the first spherical harmonic coefficients yield an exponentially convergent approximation to a spherical piecewise smooth function in any subinterval , where the function is analytic. Thus we entirely overcome the Gibbs phenomenon.

     

Explicitly realizing average Siegel theta series as linear combinations of Eisenstein series

We find nice representatives for the 0-dimensional cusps of the degree n Siegel upper half-space under the action of \(\Gamma _0(\mathcal N )\). To each of these, we attach a Siegel Eisenstein series, and then we make explicit a result of Siegel, realizing any integral weight average Siegel theta series of arbitrary level \(\mathcal N \) and Dirichlet character \(\chi _{_L}\) modulo \(\mathcal N \) as a linear combination of Siegel Eisenstein series.     

Irrationality of linear combinations of eigenvectors

A givenn ×n matrix of rational numbers acts onC ^π and onQ ^π. We assume that its characteristic polynomial is irreducible and compare a basis of eigenvectors forC ^π with the standard basis forQ ^π. Subject to a hypothesis on the Galois group we prove that vectors from these two bases are as independent of each other as possible.     

Multipliers of spherical harmonics and energy of measures on the sphere

We consider the operator,f(Δ) for Δ the Laplacian, on spaces of measures on the sphere inR ^d , show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied, and give estimates for theL²-norm off(Δ)μ in terms of the energy of the measure μ. We derive a formula, analogous to the classical formula relating the energy of a measure onR ^d with its Fourier transform, comparing the energy of a measure on the sphere with the size of its spherical harmonics. An application is given to pluriharmonic measures.     

Special linear combinations of orthogonal polynomials

The paper lists a number of problems that motivate consideration of special linear combinations of polynomials, orthogonal with the weight p(x) on the interval (a,b). We study properties of the polynomials, as well as the necessary and sufficient conditions for their orthogonality. The special linear combinations of Chebyshev orthogonal polynomials of four kinds with absolutely constant coefficients hold a distinguished place in the class of such linear combinations.     

Marcinkiewicz–Zygmund inequalities and interpolation by spherical harmonics

We find necessary density conditions for Marcinkiewicz–Zygmund inequalities and interpolation for spaces of spherical harmonics in with respect to the L^p norm. Moreover, we prove that there are no complete interpolation families for p≠2.     

Generalized spherical harmonics and exterior differentiation in weighted sobolev spaces

We generalize spherical harmonics expansions of scalar functions to expansions of alternating differential forms (‘q-forms’). To this end we develop a calculus for the use of spherical co-ordinates for q-forms and determine the eigen-q-forms of the Beltrami-operator on S^N?1 which replace the classical spherical harmonics. We characterize and classify homogeneous q-forms u which satisfy Δu = 0 on ?^N??{0} and determine Fredholm properties, kernel and range of the exterior derivative d acting in weighted L^p-spaces of q-forms (generalizing results of McOwen for the scalar Laplacian). These techniques and results are necessary prerequisites for the discussion of the low-frequency behaviour in exterior boundary value problems for systems occurring in electromagnetism and isotropic elasticity.     

Order of approximation by linear combinations of positive linear operators

Order of uniform approximation is studied for linear combinations due to May and Rathore of Baskakov-type operators and recent methods of Pethe. The order of approximation is estimated in terms of a higher-order modulus of continuity of the function being approximated.