Some spherical uniqueness theorems for multiple trigonometric series

Some trigonometric integrals and the Fourier transform of a spherically symmetric exponential function

We calculate the Fourier transform of a spherically symmetric exponential function. Our evaluation is much simpler than the known one. We use the polar coordinates and reduce the Fourier transform to the integral of a rational function of trigonometric functions. Its evaluation turns out to be much easier than expected because of homogeneity and a hidden symmetry. Relationship with a Fourier integral representation formula for harmonic functions is explained.

Cantor uniqueness and multiplicity along subsequences

We construct a trigonometric series converging to zero everywhere on a subsequence, with coefficients tending to zero. We show that any such series must satisfy that the subsequence is very sparse, and that the support of the related distribution is quite large.

幂级数、三角级数与切比雪夫多项式级数的一致有理逼近

Consistent rational approximations of power series, trigonometric series and series of Chebyshev polynomials

本文研究三角级数与切比雪夫多项式级数的一致有理逼近，建立三角 Hermite-Padé 多项式的存在唯一性准则并给出显式形式，同时构造例子说明 Hermite-Jacobi 逼近与 Hermite-Padé 逼近可以不同。

Heisenberg uniqueness pairs for some algebraic curves in the plane

A Heisenberg uniqueness pair is a pair $\left(Γ, Λ\right)$, where $Γ$ is a curve and $Λ$ is a set in $\mathbb R^2$ such that whenever a finite Borel measure $μ$ having support on $Γ$ which is absolutely continuous with respect to the arc length on $Γ$ satisfies $\hatμ\vert_Λ=0,$ then it is identically $0.$ In this article, we investigate the Heisenberg uniqueness pairs corresponding to the spiral, hyperbola, circle and certain exponential curves. Further, we work out a characterization of the Heisenberg uniqueness pairs corresponding to four parallel lines. In the latter case, we observe a phe…

Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

Due to the isotropy of $d$-dimensional hyperspherical space, one expects there to exist a spherically symmetric opposite antipodal fundamental solution for its corresponding Laplace-Beltrami operator. The $R$-radius hypersphere ${\mathbf S}_R^d$ with $R>0$, represents a Riemannian manifold with positive-constant sectional curvature. We obtain a spherically symmetric opposite antipodal fundamental solution of Laplace's equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers…

Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order

Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or octahedral monodromy. The dihedral Legendre functions are expressed in terms of Jacobi polynomials. For the last two monodromy types, an underlying `octahedral' polynomial, indexed by the degree and order and having a non-classical kind of orthogonality, is identified, and recurrences for it a…