多变量一次对称多项式的自回归滤波问题
The autoregressive filter problem for multivariable degree one symmetric polynomials
摘要：
多变量自回归滤波器问题要求基于其谱密度函数$1/|p(z)|^2$的规定傅里叶系数在封闭的$d$圆盘中无根的多项式$p(z)=p(z_1, \ldots , z_d)$。本文导出的构造一阶对称多项式的条件揭示了最多两个变量的情况与三个或更多个变量的情况之间的主要区别。后者涉及多变量椭圆函数，而前者（由于[J.S. Geronimo和H. J. Woerdeman，Ann. of Math.（2），160（3）：839- 906，2004]）仅涉及多项式。三个变量的情况下，更详细地处理，并需要超几何函数。在此过程中，我们发现了$_2F_1(\frac13,\frac23;1;z)$和$_2F_1(\frac12,\frac12;1;\widetilde{z})$之间看似新的关系。
英文摘要：
The multivariable autoregressive filter problem asks for a polynomial $p(z)=p(z_1, \ldots , z_d)$ without roots in the closed $d$-disk based on prescribed Fourier coefficients of its spectral density function $1/|p(z)|^2$. The conditions derived in this paper for the construction of a degree one symmetric polynomial reveal a major divide between the case of at most two variables vs. the the case of three or more variables. The latter involves multivariable elliptic functions, while the former (due to [J. S. Geronimo and H. J. Woerdeman, Ann. of Math. (2), 160(3):839--906, 2004]) only involve polynomials. The three variable case is treated with more detail, and entails hypergeometric functions. Along the way, we identify a seemingly new relation between $_2F_1(\frac13,\frac23;1;z)$ and $_2F_1(\frac12,\frac12;1;\widetilde{z})$.
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