Completeness and Basis Properties of Sets of Special Functions
Cambridge University Press, 3 juin 2004 - 148 pages
This tract presents an exposition of methods for testing sets of special functions for completeness and basis properties, mostly in L2 and L2 spaces. The first chapter contains the theoretical background to the subject, largely in a general Hilbert space setting, and theorems in which the structure of Hilbert space is revealed by properties of its bases are dealt with. Later parts of the book deal with methods: for example, the Vitali criterion, together with its generalisations and applications, is discussed in some detail, and there is an introduction to the theory of stability of bases. The last chapter deals with complete sets as eigenfunctions of differential and a table of a wide variety of bases and complete sets of special functions. Dr Higgins' account will be useful to graduate students of mathematics and professional mathematicians, especially Banach spaces. The emphasis on methods of testing and their applications will also interest scientists and engineers engaged in fields such as the sampling theory of signals in electrical engineering and boundary value problems in mathematical physics.
Autres éditions - Tout afficher
already Appendix applies arise associated Banach space basis boundary value problems bounded calculation called closed coefficients complete sequence CON sequence condition constant contains continuous converges defined definition denote dense differential eigenfunctions eigenvalues element equal equation equivalent example exists expansion extend fact forms a complete formula Fourier functions further give given hence Hilbert space implies important inequality inner product integral isometric L²(R lemma linear mapping measure method metric norm normal null obtained operator orthogonal polynomials positive powers problem proof prove provides reader region relation respect result satisfied scalars sense separable sequence in H Show singular solution stability subset sufficient term theorem theory transform unique verified Vitali criterion yields zero