ABSTRACT
The representation of functions in the Hilbert space in terms of eigenfunctions, or something like eigenfunctions, of an operator L is called the spectral representation for L. This chapter shows that the spectral representation of L depends on the behavior of the inverse of the operator, and determines how to find the appropriate spectral representation. There is an interesting relationship between Green’s functions, eigenvalue expansions and contour integration in the complex plane. The beauty of the formula relating the contour integral of the Green’s function and the delta function is that it gives us an algorithm to generate many different transforms, even those for which the usual eigenfunction expansion techniques do not work. The Fourier Transform theorem is an important example of how spectral theory works when the operator has no discrete spectrum. The Fourier Transform can be used to prove an important fact about the Hermite and Laguerre polynomials.
