Results on Tight Sampling Sets in Polynomial Spaces

Benjamin Aurispa, Texas A&M University

Alex Gittens, University of Houston

Sayaka Olsen, University of Nevada, Reno

Micah Hawkins, University of Washington, St. Louis

Abstract:

The topic of frames is generally studied in Rⁿ, but may, however, also be considered in polynomial spaces. Certain frames in these spaces arise from tight sampling sets. Here, general information will be given concerning what tight sampling sets are and the equations needed to obtain examples of these sets. Specific examples of tight sampling sets will be given as well as properties and conjectures involving the corresponding frame polynomials. It will be shown that if a tight frame is known for a certain interval in a polynomial space, then, by taking dilates and translates of these polynomials, tight frames of the same number of polynomials may be determined for any interval in the same space. The applications of tight sampling sets, such as transmission of data, will also be mentioned.