0

I hope to know about the Mercer's expansion on $K(x,y) = \frac{\sin(x-y)}{\pi(x-y)}$, which is the reproducing kernel for a Hilbert space of band-limited functions. By Mercer's theorem, it can be written as $K(x,y) = \sum_{i=1}^\infty \lambda_i \phi_i(x)\phi^*_i(y)$. Could anyone show me the expression of $\lambda_i$ and $\phi_i$?

I have known that $\int_{-\infty}^\infty \frac{\sin(x-y)}{\pi(x-y)} e^{jwy}dy = I(-1 \le w \le 1)e^{jwx}$. But how can I write $\frac{\sin(x-y)}{\pi(x-y)}$ into a countably infinite sum?

Titan_Tong
  • 95
  • 1
    $K f = f \ast h(x)$ where $h(x) = \sin(x) / (\pi x)$ so $K = W \Pi W^$ where $W^$ is the Fourier transform operator and $\Pi$ is the diagonal operator $\Pi[f] (x) = f (x) \hat{h}(x)$. thus if $\Pi = \sum_{i=1}^\infty \lambda_i \varphi_i \varphi_i^$ then $K = \sum_{i=1}^\infty \lambda_i \hat{\varphi}_i \hat{\varphi}_i^$ and vice-versa. – reuns Jan 29 '16 at 19:51
  • 1
    unfortunately, $\Pi$ isn't a compact operator of $L_2$ so neither is $K$ : it doesn't respect the Mercer's conditions – reuns Jan 29 '16 at 19:52

1 Answers1

0

$sinc(\pi(x-y))=S(x-y)=\sum_{n=0}^\infty \phi_n(x)\phi_n(y)$ where $\phi_n(t)$ is a Prolate Spheroidal Wave Function (PSWF). (You must account for the factor of $\pi$.)

Refer to "Sampling with prolate spheroidal wave functions", Walter and Shen. https://www.researchgate.net/publication/268496813_Sampling_with_prolate_spheroidal_wave_functions

The PSWF, $\phi_n(t)$, is defined as the eigenfunctions of the convolution with $S(x-y)$ between $-\tau$ and $\tau$, and normalized under the real line. (See eqns. 1.2, 2.1 and 2.2).

The value of your $\lambda_i$ is unity for all $i$, due to the second half of eq. 2.2.

Section "2.2 Discrete Orthogonality" gives the derivation in the first three equations, ending at eq. 2.4.: $S(t)=\sum_{n=0}^\infty a_n(x)\phi_n(t)$. The value, $a_n$ is $\phi_n(0)$, yielding $S(x)=\sum_{n=0}^\infty \phi_n(0)\phi_n(x)$, and a shift by $y$, yields $S(x-y)=\sum_{n=0}^\infty \phi_n(x)\phi_n(y)$

ALTERNATIVELY, $\sin( \sigma(t-s))/(\pi (t-s)) = (\sigma /\pi) \sum_{n=0}^\infty (2n+1)j_n(\sigma t)j_n(\sigma s)$, where $\ j_n(x)$ is the spherical Bessel function.

Refer to "New method of computing prolate spheroidal wavefunctions and bandlimited extrapolation: A general approach to bandlimited Fredholm kernels" by V. Vaibhav. https://arxiv.org/abs/1804.04713

EEatWork
  • 99