Two Variable Extension and Factorization Problems with Applications to Wavelets

建议编号：DMS-0200219PI：Jeffrey Geronimo ABSTRACT研究两个变量正三角多项式的Fejer-Riesz分解。Hugo Woerdeman最近在这个问题上得到的结果将被用来研究稳定的两个变量多项式的性质。一个二元多项式称为稳定的，如果它在双圆内和双圆上都不为零。这些研究的结果将被应用于两个变量小波的构造、两个变量滤波器的设计和图像处理。正三角多项式的因式分解是二十世纪在许多应用中得到解决的一个重要问题。X中n次三角多项式是可以写成频率x的整数倍的正弦和余弦的线性组合的多项式。Fejer和Riesz证明了这种非多项式可以写成另一个多项式的幅值平方。这一结果在电路的滤波器设计、预测理论以及最近的小波领域都有许多有用的应用。

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.