与Fourier-Legendre展开有关的全纯扩张

On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre expansions

Motivated by our previous work on hypergeometric functions and the parbelos constant, we perform a deeper investigation on the interplay among generalized complete elliptic integrals, Fourier-Legendre (FL) series expansions, and ${}_p F_q$ series. We produce new hypergeometric transformations and closed-form evaluations for new series involving harmonic numbers, through the use of the integration method outlined as follows: Letting $K$ denote the complete elliptic integral of the first kind, for a suitable function $g$ we evaluate integrals such as $$ \int_{0}^{1} K\left( \sqrt{x} \right) g(x)…

Uniform Fourier Restriction Estimate for Simple Curves of Bounded Frequency

In this paper we prove a uniform Fourier restriction estimate over the class of simple curves where the last coordinate function can be extended to a holomorphic function of bounded frequency in a sufficiently large disc. The proof is based on a decomposition scheme for this class of functions.

little $q$-Legendre 多项式的调和分析

Harmonic analysis of little $q$-Legendre polynomials

本文研究 little $q$-Legendre 多项式的调和分析，利用 Fourier 变换与连分数等工具证明其 $L^1$-代数具有弱顺从性与右特征顺从性但不顺从，提供了首个此类多项式超群实例。

A fast algorithm for the inversion of Abel's transform

We present a new algorithm for the computation of the inverse Abel transform, a problem which emerges in many areas of physics and engineering. We prove that the Legendre coefficients of a given function coincide with the Fourier coefficients of a suitable periodic function associated with its Abel transform. This allows us to compute the Legendre coefficients of the inverse Abel transform in an easy, fast and accurate way by means of a single Fast Fourier Transform. The algorithm is thus appropriate also for the inversion of Abel integrals given in terms of samples representing noisy measurem…

多重调和方程对数基本解的Fourier展开

Fourier expansions for a logarithmic fundamental solution of the polyharmonic equation

在偶数维欧几里得空间中，拉普拉斯算子的整数次幂大于或等于维数除以2，多重调和方程的基本解具有对数行为。我们给出了两种方法来开发这个对数基本解的傅立叶展开。第一种方法是代数的，依赖于双参数多项式的构造。我们描述了这些多项式的一些性质，并使用它们来推导多调和方程的对数基本解的傅立叶展开式。第二种方法依赖于对多重调和方程的幂律基本解的傅里叶级数的参数导数的计算。由此产生的傅里叶级数给出的第一类相关勒让德函数的总和。最后，我们比较了这两种方法，并给出了旋转不变坐标系中多重调和方程的对数基本解的方位傅里叶级数。

Peanut harmonic expansion for a fundamental solution of Laplace's equation in flat-ring coordinates

We derive an expansion for the fundamental solution of Laplace's equation in flat-ring cyclide coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and exterior of coordinate surfaces which are peanut shaped and orthogonal to surfaces which are flat-rings. These internal and external peanut harmonic functions are expressed in terms of Lamé-Wangerin functions. Using the expansion for the fundamental solution, we derive an addition theorem for the azimuthal Fourier component in terms of the odd-half-integer…