Sinc function ( $sinc (x) = sin x / x$ )

The unnormalized sinc is

sinc (x) = sin x / x

, extended by value one at zero. The normalized form is

sin (π x) / (π x)

. It is the inverse Fourier transform of a frequency interval indicator, so it describes ideal low-pass filtering and the Shannon scaling function.

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- Sinc fourth-power integral Sinc function

Sinc fourth-power integral ( $\int_{0}^{\infty} (sin x / x)^{4} d x = π /3$ )

The triangular function

f (x) = (1 - ∣ x ∣)

on

∣ x ∣ < 1

, zero elsewhere, has Fourier transform

f (k) = [sin (k /2) / (k /2)]^{2}

in the angular-frequency convention. Its squared L2 norm is

2/3

. Parseval's identity therefore gives

\frac{2}{3} = \frac{1}{2 π} \int_{- \infty}^{\infty} (\frac{s i n ( k /2 )}{k /2})^{4} d k = \frac{2}{π} \int_{0}^{\infty} (\frac{s i n u}{u})^{4} d u .

This proves the displayed value. The removable value at zero is one, and the fourth-power decay bounds the tail. The sinc-squared integral follows similarly from a rectangular function rather than a triangle.

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The sinc function is a mathematical function defined in relation to the sine function. There are two commonly used definitions for the sinc function: 1. **Normalized sinc function**: \[ \text{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \quad \text{for } x \neq 0 \] \[ \text{sinc}(0) = 1 \] 2.

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