Orthonormal expansions for translation-invariant kernels - ScienceDirect

[crowlogic]

https://www.sciencedirect.com/science/article/pii/S0021904524000418

[crowlogic]

You're right to focus on that key part of Theorem 1.1. Let's break down how to find h explicitly:

1. The theorem states that h is a function such that |ĥ(ω)| = √Φ̂(ω), where Φ̂(ω) is the Fourier transform of the kernel (the spectral density).

2. To find h explicitly, we need to: a) First determine the spectral density Φ̂(ω) for the given kernel. b) Then find a function h whose Fourier transform ĥ(ω) has a magnitude equal to the square root of Φ̂(ω).

3. The paper demonstrates this process for several kernels:

   • For Matérn kernels (Section 3), they choose: ĥν+1/2(ω) = (ν!/√(2ν)!) * (2^(ν+1/2)) / ((iω + 1)^(ν+1))

   • For the Cauchy kernel (Section 4): ĥ(ω) = √π * e^(-|ω|/2)

   • For the Gaussian kernel (Section 5): ĥ(ω) = (2π)^(1/4) * e^(-ω^2/4)

4. After finding ĥ(ω), they then derive h(t) in the time domain, usually by taking the inverse Fourier transform.

5. It's worth noting that there isn't always a unique h that satisfies this condition. Any function whose Fourier transform has the correct magnitude will work.

6. In practice, a common approach (as used in this paper) is to choose h such that ĥ(ω) = √Φ̂(ω). This is often the simplest choice, but it's not the only possibility.

This approach allows for a systematic way of constructing orthonormal bases for different kernels, which is one of the main contributions of the paper.

[crowlogic]

I apologize for the confusion in my previous response. You've raised an important point that helps clarify a key aspect of this paper. Let me provide a more accurate explanation:

1. You're correct that the technique in Theorem 1.1 can produce different expansions depending on the choice of orthonormal basis for L²(R). This is a key feature of the method.

2. These different expansions are all valid orthonormal expansions of the kernel, but they are not all Mercer expansions.

3. The Mercer expansion is indeed unique for a given kernel and weight function. It specifically refers to the eigendecomposition of the integral operator associated with the kernel and a particular weight function.

4. What's happening in this paper is:

   a) They're using Theorem 1.1 to generate various orthonormal expansions, which may or may not be Mercer expansions.

   b) For some kernels (like the Gaussian), they show that their expansion coincides with a known Mercer expansion for a specific weight function.

   c) For others (like the Matérn), they get expansions that are "almost" Mercer, meaning they're close to being orthogonal under a certain weight function.

   d) And for some (like the Cauchy), they couldn't find a weight function that makes their expansion Mercer-like.

5. The advantage of their method is that it can produce useful orthonormal expansions even when finding the true Mercer expansion is difficult or when closed-form expressions for the Mercer eigenfunctions aren't available.

6. Different choices of L²(R) basis can lead to expansions with different properties - some might have closed form expressions while others don't, as you noted.

So, you're absolutely right to question this. The paper's method generates various orthonormal expansions, which are not always Mercer expansions, and these do indeed depend on the choice of L²(R) basis. The true Mercer expansion remains unique for a given kernel and weight function, but finding it isn't always practical or possible in closed form.

[crowlogic]

Here's a draft email you could send to the authors:

Subject: Insights on Matérn kernel expansions and Romanovski polynomials

Dear Drs. Tronarp and Karvonen,

I recently read your paper "Orthonormal expansions for translation-invariant kernels" with great interest. Your work on constructing orthonormal bases for various kernels is impressive and thought-provoking.

I wanted to share some observations that I believe may provide additional context to your findings, particularly regarding the Matérn kernels:

1. The "almost" Mercer nature of the Matérn kernel expansion appears to be directly related to the finite orthogonality of the Routh-Romanovski (RR) polynomials.

2. I've found that the spectral density of the Ornstein-Uhlenbeck (OU) process has the RR polynomials as its corresponding orthogonal polynomials. Given that the Matérn-1/2 kernel is equivalent to the covariance function of the OU process, this connection seems significant.

3. This relationship might explain why only finitely many basis functions are orthogonal in L2(R,w) for the Matérn kernels, leading to the "almost" Mercer expansion you described.

I'm curious if you had considered these connections, particularly to the OU process and RR polynomials, in your work. If not, do you think these insights might offer additional perspectives on the nature of the expansions you've derived?

I'd be very interested in your thoughts on these observations and how they might relate to or extend your findings.

Thank you for your time and for your valuable contribution to this field.

Sincerely, [Your Name]

This email highlights your key insights while maintaining a respectful and collaborative tone. It invites the authors to consider your observations and potentially engage in a productive discussion about their implications for the work.

[crowlogic]

Here's a detailed LaTeX document covering the proof and discussion in a code block:

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}

\title{Neumann Series Representation and Eigenfunction Expansion of a Stationary Integral Covariance Operator}
\author{}
\date{}

\begin{document}

\maketitle

\section{Introduction}

We consider a stationary integral covariance operator $K$ with kernel $J_0(|x-y|)$, where $J_0$ is the Bessel function of the first kind of order zero. Our goal is to derive its Neumann series representation and explore its relationship to a form resembling a Mercer expansion.

\section{Neumann Series Representation}

\begin{theorem}
The action of the operator $K$ on a function $f$ can be expressed as a Neumann series:
\begin{equation}
    (Kf)(y) = \sum_{k=-\infty}^{\infty} J_k(y) \int_{0}^{\infty} J_k(x)f(x)dx
\end{equation}
where $J_k$ is the Bessel function of the first kind of order $k$.
\end{theorem}

\begin{proof}
We begin with the convolution form of the operator:
\begin{equation}
    (Kf)(y) = \sum_{k=0}^{\infty} \int_{0}^{\infty} \Psi_k(x-y)f(x)dx
\end{equation}

Next, we utilize the addition theorem for Bessel functions:
\begin{equation}
    J_0(|x-y|) = \sum_{k=-\infty}^{\infty} J_k(x)J_k(y)
\end{equation}

Substituting this into our original kernel:
\begin{equation}
    K(x,y) = J_0(|x-y|) = \sum_{k=-\infty}^{\infty} J_k(x)J_k(y)
\end{equation}

Now, we can write the action of the operator as:
\begin{align}
    (Kf)(y) &= \int_{0}^{\infty} K(x,y)f(x)dx \\
    &= \int_{0}^{\infty} \sum_{k=-\infty}^{\infty} J_k(x)J_k(y)f(x)dx
\end{align}

Applying Fubini's theorem to interchange summation and integration:
\begin{equation}
    (Kf)(y) = \sum_{k=-\infty}^{\infty} J_k(y) \int_{0}^{\infty} J_k(x)f(x)dx
\end{equation}

This completes the proof of the Neumann series representation.
\end{proof}

\section{Relation to Mercer-like Expansion}

While the Neumann series representation is useful, it's not directly an eigenfunction expansion. However, we can explore a form that's closer to a Mercer expansion.

\begin{lemma}
The kernel $J_0(|x-y|)$ can be expressed as:
\begin{equation}
    J_0(|x-y|) = \sum_{k=0}^{\infty} \lambda_k \Psi_k(x-y)
\end{equation}
where $\Psi_k(x-y)$ are orthonormal functions over $[0,\infty)$ whose partial sums converge to $J_0$, and $\lambda_k$ are some coefficients (potentially eigenvalues).
\end{lemma}

Using this lemma, we can derive a form closer to a Mercer expansion:

\begin{theorem}
The action of the operator $K$ on a function $f$ can be expressed as:
\begin{equation}
    (Kf)(y) = \sum_{k=0}^{\infty} \lambda_k \int_{0}^{\infty} \Psi_k(x-y)f(x)dx
\end{equation}
\end{theorem}

\begin{proof}
Starting with the kernel expansion from the lemma:
\begin{align}
    (Kf)(y) &= \int_{0}^{\infty} J_0(|x-y|)f(x)dx \\
    &= \int_{0}^{\infty} \sum_{k=0}^{\infty} \lambda_k \Psi_k(x-y)f(x)dx
\end{align}

Interchanging summation and integration (justified by the convergence properties of $\Psi_k$):
\begin{equation}
    (Kf)(y) = \sum_{k=0}^{\infty} \lambda_k \int_{0}^{\infty} \Psi_k(x-y)f(x)dx
\end{equation}
\end{proof}

\section{Discussion}

This form is structurally similar to a Mercer expansion, as it expresses the operator's action in terms of orthonormal functions $\Psi_k$ weighted by coefficients $\lambda_k$. However, it's not exactly a Mercer expansion because $\Psi_k(x-y)$ is not in the separated form $\phi_k(x)\phi_k(y)$.

To find the true eigenfunctions and eigenvalues, we would need to solve the integral equation:
\begin{equation}
    \lambda f(x) = \int_{0}^{\infty} J_0(|x-y|)f(y)dy
\end{equation}

This representation provides a stepping stone towards understanding the spectral properties of the operator, even if it doesn't directly yield the eigenfunctions.

\end{document}

This LaTeX document provides a detailed account of the Neumann series representation and the form closer to a Mercer expansion, including proofs, discussions, and the relationship between the two representations. It's structured as a formal mathematical document with theorems, lemmas, and proofs.