Hilbert Spaces, Bases And Frames

Abstract

Abstract Hilbert spaces of various types provide the natural setting for much of the sampling theory that we shall be studying.The reader is assumed to be familiar with the foundations of Hilbert and Banach spaces. We review here some of the important facts, mostly concerning Hilbert spaces, that will be needed throughout the text. Of particular importance is the rather more specialized topic of bases in Hilbert space. An elementary account can be found in the book by Higgins (1977). A very readable introduction to the theory of bases is in the book by Young (1980); the book of Marti (1969) has a more advanced treatment, while a vastly comprehensive treatment is to be found in the book of Singer (1970).Apart from the last section this chapter is intended as a review only, and almost no proofs are included. However, ample references to the texts mentioned above are given.