Two Variable Extension and Factorization Problems with Applications to Wavelets

小波应用中的两变量扩展和因式分解问题

• 批准号: 0200219
• 负责人: Jeffrey Geronimo
• 金额: $ 11.1万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200219&HistoricalAwards=false
• 关键词: Two; Variable; Extension; Factorization; Problems

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.

项目编号：DMS-0200219PI: Jeffrey geronimo摘要本课题将研究二元正三角多项式的Fejer-Riesz分解。最近与Hugo Woerdeman在这个问题上所得到的结果将用于研究稳定二元多项式的性质。如果一个二元多项式在圆内和圆上不消失，就称它是稳定的。从这些研究中获得的结果将应用于两个可变小波的构造，两个可变滤波器的设计和图像处理。在20世纪，许多应用中都得到了解决的一个重要问题是正三角多项式的因式分解问题。x的n次三角多项式是一个多项式，它可以写成频率为x的整数倍的正弦和余弦的线性组合。Fejer和Riesz证明了这样的多项式可以写成另一个多项式的模的平方。这一结果在电路滤波器设计、预测理论以及最近的小波中有许多有用的应用。