Eigenfunction and Eigenvalue of the Sine Kernel

The sine kernel is a significant element in the study of Gaussian processes and random matrices. It is defined by the integral operator with kernel: $$K (x, y) = \frac{\sin (x - y)}{\pi (x - y)}$$

Eigenfunction and Eigenvalue

An important eigenfunction of the sine kernel is $\sin (x)$, which satisfies the integral equation: $$\int_{- \infty}^{\infty} \frac{\sin (x - y)}{\pi (x - y)} \sin (y) \hspace{0.17em} dy = \sin (x)$$ This shows that $\sin (x)$ is an eigenfunction of the integral operator with the sine kernel, and its corresponding eigenvalue is 1.

Identity Validation

The identity involving the sine kernel can be verified through: $$\int_{- \infty}^{\infty} \frac{\sin (x - y) }{\pi(x-y)}\frac{\sin y}{ \pi y} \hspace{0.17em} dy = \frac{\sin x}{x \pi}$$ This integral represents a convolution of the sine function under the sine kernel, emphasizing the role of the sine function and its spectral properties in the context of the sine kernel.