Sinc Function - Relationship To The Dirac Delta Distribution

Relationship To The Dirac Delta Distribution

The normalized sinc function can be used as a nascent delta function, meaning that the following weak limit holds:

This is not an ordinary limit, since the left side does not converge. Rather, it means that

\lim_{a\rightarrow 0}\int_{-\infty}^\infty \frac{1}{a}\textrm{sinc}(x/a)\varphi(x)\,dx = \varphi(0),

for any smooth function with compact support.

In the above expression, as a approaches zero, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±1/(π a x), and approaches zero for any nonzero value of x. This complicates the informal picture of δ(x) as being zero for all x except at the point x = 0 and illustrates the problem of thinking of the delta function as a function rather than as a distribution. A similar situation is found in the Gibbs phenomenon.

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