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Orthogonal Polynomials and Fourier Orthogonal Series on a Cone
Journal of Fourier Analysis and Applications ( IF 1.2 ) Pub Date : 2020-04-02 , DOI: 10.1007/s00041-020-09741-x
Yuan Xu

Orthogonal polynomials and the Fourier orthogonal series on a cone in \({{\mathbb {R}}}^{d+1}\) are studied. It is shown that orthogonal polynomials with respect to the weight function \((1-t)^{\gamma }(t^2-\Vert x\Vert ^2)^{\mu -\frac{1}{2}}\) on the cone \({{\mathbb {V}}}^{d+1} = \{(x,t): \Vert x\Vert \le t \le 1\}\) are eigenfunctions of a second order differential operator, with eigenvalues depending only on the degree of the polynomials, and the reproducing kernels of these polynomials satisfy a closed formula that has a one-dimensional characteristic. The latter leads to a convolution structure on the cone, which is then utilized to study the Fourier orthogonal series. This narrative also holds, in part, for more general classes of weight functions. Furthermore, analogous results are also established for orthogonal structure on the surface of the cone.

中文翻译:

圆锥上的正交多项式和傅立叶级数

研究了正交多项式和\({{\ mathbb {R}}} ^ {d + 1} \)中的圆锥上的傅立叶正交级数。结果表明,权重函数\((1-t)^ {\ gamma}(t ^ 2- \ Vert x \ Vert ^ 2)^ {\ mu-\ frac {1} {2} } \)在锥体\({{\ mathbb {V}}} ^ {d + 1} = \ {(x,t):\ Vert x \ Vert \ le t \ le 1 \} \)是二阶微分算子的特征函数,特征值仅取决于多项式的阶数,并且这些多项式的再现核满足具有一维特征的封闭式。后者导致圆锥上的卷积结构,然后将其用于研究傅立叶正交序列。这种叙述在某种程度上也适用于更一般的权重类。此外,对于锥表面上的正交结构也建立了类似的结果。
更新日期:2020-04-02
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