Custom Building Finite Frames

After the introduction of frame potentials by Benedetto and Fickus, there was an explosion of new results in frame theory. The generalization of this notation to weighted frame potentials yields a physical interpretation for finite frames along the lines of Coulomb’s law in physics. This allows us to use results from classical mechanics to anticipate new results in frame theory. All of this resulted in the resolution of a large number of important open questions concerning the existence and construction of finite frames. We will put all of this into the framework of a theory as well as adding many new results along the way. As part of the theory, we include the recent “algorithmic constructions” for finite frames. We will also discuss the related important questions which are still open at this time. Introduction Hilbert space frames were introduced by Duffin and Schaeffer in 1952 [DS] while working on some deep questions in non-harmonic Fourier series (see [Y]). These problems grew out of G.D. Birkhoff’s work on Sturm-Liouville functions in 1917; the trigonometric results of Welch in 1921; and results of Paley-Wiener in 1934 (see [Y]). For some reason the results of Duffin and Schaeffer were not continued (except in signal/image processing [G]) until in 1986 Daubechies, Grossman and Meyer [DGS] brought this back to life right at the dawn of the wavelet era. Recently, many new applications of frames have arisen to internet coding (see [GKK] and its references and [CK]); multiple antenna coding (see [HH] and [HM]); Communication Theory; Sampling Theory and more. Benedetto and Fickus [BF] introduced an important tool into frame theory called frame potentials. This gave a geometric interpretation for tight frames which resulted in an explosion of important new results in the field including a physical interpretation for tight frames along the lines of Coulomb’s law in Physics [BF, CF]. As a result, we can now use results from quantum mechanics to anticipate new results on frames. This also led to important advances in the algorithmic construction of frames which is critically important for applications of frames. We will put all of these new advances into the framework of a theory as well as singling out the important open questions at this time. Also, we include may new 1991 Mathematics Subject Classification. Primary 46C05, 46N99; Secondary 42C40. The author was supported by NSF DMS 0102686. c ©0000 (copyright holder) 1