Leveraging the Fourier transform's properties and the orthogonality of Chebyshev polynomials to study the spectral properties of the integral covariance operator having J0 as it's translation invariant kernel
Fourier Transform of the Kernel:
Given that the kernel $k(x) = J_0(x)$ is utilized in its translation-invariant form, the Fourier transform of $J_0(x)$, denoted as $\hat{J}_0(s)$, results in $\frac{1}{\sqrt{1-s^2}}$ for $|s| \leq 1$. This expression is critical as it represents the spectral density of the kernel.
Spectral Density and Chebyshev Polynomials:
The Fourier transform of $J_0$ equals the orthogonality measure of the type 1 Chebyshev polynomials, $T_n(x)$. This correspondence indicates that when the type 1 Chebyshev polynomials are Fourier transformed, they correspond to elements of the kernel(null space) of the integral multiplication operator:
Orthogonalization Process:
By applying the Gram-Schmidt process to the Fourier transforms of the Chebyshev polynomials, they are orthogonalized . These orthogonalized Fourier transformed functions are then seen to be the eigenfunctions of the integral covariance operator:
$$(T'f)(y) = \int k(x - y) f(x) dx$$
Given the translation-invariance and the specific structure of $k(x-y)$, the orthogonal functions derived by orthogonalizing the polynomial functions comprising the null space(kernel) of the integral multiplication operator become eigenfunctions of the intevral covariance operator.
Eigenfunctions and Eigenvalues:
The orthogonal functions, now serving as eigenfunctions of the integral covariance operator, when projected onto the kernel by applying the integral multiplication operator, reveal the eigenvalues. The integral of the product of an eigenfunction and the kernel over the domain provides a scalar (eigenvalue) that scales the eigenfunction in the operator equation:
This framework provides a rigorous and quantifiable approach to understanding the spectral properties of the kernel and its implications on the behavior of these two operators. This method shows how the properties of special functions, Fourier transforms, and their orthogonality relations can be harnessed to explore the spectral aspects of integral operators.
crowlogic
commented
4 months ago
Leveraging the Fourier transform's properties and the orthogonality of Chebyshev polynomials to study the spectral properties of the integral covariance operator having J0 as it's translation invariant kernel
Fourier Transform of the Kernel: Given that the kernel $k(x) = J_0(x)$ is utilized in its translation-invariant form, the Fourier transform of $J_0(x)$, denoted as $\hat{J}_0(s)$, results in $\frac{1}{\sqrt{1-s^2}}$ for $|s| \leq 1$. This expression is critical as it represents the spectral density of the kernel.
Spectral Density and Chebyshev Polynomials: The Fourier transform of $J_0$ equals the orthogonality measure of the type 1 Chebyshev polynomials, $T_n(x)$. This correspondence indicates that when the type 1 Chebyshev polynomials are Fourier transformed, they correspond to elements of the kernel(null space) of the integral multiplication operator:
$$(Tf)(y) = \int k(x) f(x) dx = \int J_0(x) f(x) dx = 0$$
$$(T'f)(y) = \int k(x - y) f(x) dx$$
Given the translation-invariance and the specific structure of $k(x-y)$, the orthogonal functions derived by orthogonalizing the polynomial functions comprising the null space(kernel) of the integral multiplication operator become eigenfunctions of the intevral covariance operator.
$$\lambda_n \psi_n(y) = \int k(x-y) \psi_n(x) dx$$
This framework provides a rigorous and quantifiable approach to understanding the spectral properties of the kernel and its implications on the behavior of these two operators. This method shows how the properties of special functions, Fourier transforms, and their orthogonality relations can be harnessed to explore the spectral aspects of integral operators.