关于广义三角多项式的Fejer-Riesz分解
On a Fejer-Riesz factorization of generalized trigonometric polynomials
摘要：
单位圆上的函数理论被证明是统计学、概率论、信号处理文献和应用中一系列问题的关键，在这方面，三角函数和Fejer-Riesz定理占据了一个特殊的位置，该定理认为非负三角多项式可以表示为在单位圆上计算的相同次数的多项式的模。在本说明中，我们考虑一个自然推广的非负三角多项式，矩阵值与指定的非平凡极点（即，而不是在原点或无穷远处）。我们感兴趣的是相应的谱因子，具体而言，我们表明，三角多项式的因式分解可以进行完全类比的Fejer-Riesz定理。与Fejer-Riesz定理的因式分解的亲和力和与经典谱因式分解的对比在于，谱因子的度小于因式分解理论中的标准构造所建议的度。我们提供了两个并列证明这个基本定理，虽然严格的积极性的情况下，一个依赖于解析插值理论，另一个利用经典的因式分解理论的基础上的Yacubovich-Popov-Kalman（YPK）正实引理。
英文摘要：
Function theory on the unit disc proved key to a range of problems in statistics, probability theory, signal processing literature, and applications, and in this, a special place is occupied by trigonometric functions and the Fejer-Riesz theorem that non-negative trigonometric polynomials can be expressed as the modulus of a polynomial of the same degree evaluated on the unit circle. In the present note we consider a natural generalization of non-negative trigonometric polynomials that are matrix-valued with specified non-trivial poles (i.e., other than at the origin or at infinity). We are interested in the corresponding spectral factors and, specifically, we show that the factorization of trigonometric polynomials can be carried out in complete analogy with the Fejer-Riesz theorem. The affinity of the factorization with the Fejer-Riesz theorem and the contrast to classical spectral factorization lies in the fact that the spectral factors have degree smaller than what standard construction in factorization theory would suggest. We provide two juxtaposed proofs of this fundamental theorem, albeit for the case of strict positivity, one that relies on analytic interpolation theory and another that utilizes classical factorization theory based on the Yacubovich-Popov-Kalman (YPK) positive-real lemma.
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