Two Variable Extension and Factorization Problems with Applications to Wavelets

提案编号：DMS-0200219 PI：Jeffrey Geronimo摘要研究将进行到两个变量的正三角多项式的Fejer-Riesz因式分解。最近Hugo Woerdeman在这个问题上得到的结果将被用来研究稳定的二元多项式的性质。一个二元多项式称为稳定的，如果它在双环上和双环内都不为零。 从这些研究中得到的结果将被应用到二元小波的构造、二元滤波器的设计和图像处理中。在二十世纪，一个有着许多应用的重要问题得到了很好的解决，这就是正三角多项式的因式分解问题。x中n次三角多项式是一个多项式，它可以写成频率x的整数倍的西内斯和余弦的线性组合。费耶尔和Riesz证明了这样一个多项式可以写成另一个多项式的平方。这一结果在电路滤波器设计、预测理论以及最近的小波领域有许多有用的应用。

Proposal Number: DMS-0200219PI: Jeffrey GeronimoABSTRACTResearch will be conducted into the Fejer-Riesz factorizationof two variable positive trigonometric polynomials. The resultsobtained recently with Hugo Woerdeman on this problem will beused to investigate the properties of stable two variablepolynomials. A two variable polynomial is said to be stableif it is non vanishing inside and on the bicircle. Resultsobtained from these investigations will then be applied to theconstruction of two variable wavelets, two variable filterdesign, and image processing.An important problem with many applications that was solvedearly inthe twentieth century was the problem of factorizingpositive trigonometric polynomials. A trigonometric polynomialof degree n in x is a polynomial that can be written as linearcombination of sines and cosines of integer multiples of thefrequency x. It was shown by Fejer and Riesz that such apolynomial can be written as the magnitude squared of anotherpolynomial. This result has had many useful applications inthe area of filter design for electrical circuits, predictiontheory, and more recently in wavelets.