Dimensionality of Crosstalk Functions

The class of functions bandlimited to the interval (-W, W) and considered over the interval ( - T , T) has long been held to have essentially [2WT] degrees of freedom. This goes back at least as far as the discovery of the sampling (or cardinal) series, since exactly this number of terms in the series is available with knowledge of the function over the interval ( -- T, T).1 The notion was made precise and validated by Landau and Pollak."' The fundamental quantity in their approach was the energy (or L 2 -norm) of the bandlimited function over ( - T , T). The energy is computed as a quadratic form of the function and to this there corresponds a positive definite, compact operator. We shall call this the energy operator. The distribution of the eigenvalues for the energy operator, i.e., the energy eigenvalues, determine the approximate number of degrees of freedom or dimensionality of this class of functions. The idea is that energy eigenfunctions with small enough eigenvalues (or energy) can contribute only minimally to the energy in the interval; hence, they can be disregarded. They find that [2WT] energy eigenfunctions span this space of functions within an error bound which they compute. To be more definite, let D T denote the operator which acts on square integrable functions as follows: 1347