Research will be conducted into the Fejer-Riesz factorization of two variable positive trigonometric polynomials. The results obtained recently with Hugo Woerdeman on this problem will be used to investigate the properties of stable two variable polynomials. A two variable polynomial is said to be stable if it is non vanishing inside and on the bicircle. Results obtained from these investigations will then be applied to the construction of two variable wavelets, two variable filter design, and image processing.
An important problem with many applications that was solved early inthe twentieth century was the problem of factorizing positive trigonometric polynomials. A trigonometric polynomial of degree n in x is a polynomial that can be written as linear combination of sines and cosines of integer multiples of the frequency x. It was shown by Fejer and Riesz that such a polynomial can be written as the magnitude squared of another polynomial. This result has had many useful applications in the area of filter design for electrical circuits, prediction theory, and more recently in wavelets.
