# §30.15 Signal Analysis

## §30.15(i) Scaled Spheroidal Wave Functions

Let and be given. Set and define

see §30.11(v).

## §30.15(iii) Fourier Transform

where

30.15.6

Equations (30.15.4) and (30.15.6) show that the functions are -bandlimited, that is, their Fourier transform vanishes outside the interval .

## §30.15(iv) Orthogonality

The sequence , forms an orthonormal basis in the space of -bandlimited functions, and, after normalization, an orthonormal basis in .

## §30.15(v) Extremal Properties

The maximum (or least upper bound) of all numbers

taken over all subject to

for (fixed) , is given by

30.15.11

or equivalently,

30.15.12

The corresponding function is given by

30.15.13

If , then .

For further information see Frieden (1971), Lyman and Edmonson (2001), Papoulis (1977, Chapter 6), Slepian (1983), and Slepian and Pollak (1961).