The Equivalence Relations Between Hilbert Frames, Frame Measure Function And The Overcompleteness Of Frames

Frames were first introduced by Duffin and Schaeffer in the context of nonharmonic Fourier series.In 1986,Daubechies,Grossman and.Meyer found that the frames provides basis-like representations of the function of L~2(R).Until that moment,many Mathematicians just started concerning it.And today frames play important roles in many applications in mathematics,science,and engineering.Frames are sequences that have basis-like properties but which need not be bases.A frame that is not Riesz basis is called redundancy frame.The essential feature of frames that are not bases is the overcompleteness of its elements.Central, both theoretically and practically,to the interest in frames has been their over-complete nature.Recently many works from Radu.Balan,Peter.Caszza,Christopher. Heil,Zeph.Landau examined and explored the notion of overcompleteness of a frame.This paper is concerned with the equivalence relations and distances between Hilbert frames indexed by the same set,and we introduced the notion of frame measure function and explored its properties.This paper is consist of five chapters. The first chapter are introduction and maincontent.In this part,we introduce the origin of frame,progress and application,otherwise outline the main content of this paper.The second chapter are preliminaries.In this part,we give some fundamental definitions and properties about frames,superframe and ultrafilters.In the third chapter,we list the main results concerning the equivalence and distances between Hilbert frames which is explored by Balan,and extend part of his works.In the fourth chapter,we define ultrafilter frames equivalence,partial order and frame measure function.Moreover,we compare the standard equivalence relations to ultrafilter frames equivalence.Finally,we study the excess of frames,which can be used to quantities the overcompleteness of frames.