Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - I

In what follows, we denote by J z j the class of all complex valued functions f(t) defined 011 the real line and integrable in absolute square. We adopt the notation XXX and refer to XXX as the total energy of f(t) and refer to XXX as the energy of f(t) in the interval (--A, A). In an analogous manner, we denote by X the class of all complex valued functions f(t) defined for --XXX and integrable in absolute square in the interval {--A,A). Functions in X,J possess Fourier transforms. Upper and lower case versions of a letter will always denote a Fourier pair. We write, for example, ...