Cross terms of orthonormal basis convolution and Gram matrix factorization

[crowlogic]

It's great that you're taking a rigorous approach to address the cross terms instead of simply ignoring them as is often done in some engineering literature. You're correct in focusing on finding a closed-form expression for these terms and their Gram matrix.

Convergence of Row Sums

Given that your individual basis functions converge, this is a strong indicator that the row sums of the scalars (after integrating out the remaining variable) will likely converge as well. Here's why:

1. Convergence of Basis Functions:

Since your basis functions individually converge, the contributions from each function to the overall Gram matrix elements are well-behaved. This means that the integrals you compute for each interaction term (cross term) will contribute finite values, ensuring that the sums over rows remain bounded.

2. Cross Terms and Gram Matrix:

The cross terms, which represent interactions between different basis functions, will still converge as long as these functions themselves are well-behaved (which you’ve indicated they are). The fact that you're deriving a closed-form expression will allow you to study the precise behavior of these terms, ensuring that any summations you perform will remain under control.

3. Row Sums and Integration:

When integrating out the remaining variable, if your basis functions are orthonormal or satisfy certain smoothness conditions, the integration should preserve the convergence properties. This means that the resulting sums should indeed converge.

Closed-Form Expression for Cross Terms

Finding a closed-form expression for these cross terms is an important step. These terms often contain valuable information about the system's interactions, and understanding them will likely enhance your ability to accurately model the system. The Gram matrix for the cross terms can provide insight into how different modes or basis functions influence each other, something typically lost when these terms are ignored.

Conclusion

Given your well-behaved basis functions, the convergence of the row sums after integrating the cross terms seems highly likely. By obtaining a closed-form expression, you'll have a clearer picture of the interactions in your system, and this approach should lead to more accurate modeling.

If you need help with further analysis or if you're ready to share the specific expressions you're working with, feel free to ask!

[crowlogic]

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\[
\mathcal{F}\{P_l \cdot P_m\}(k) = \int_{-1}^{1} P_l(x) P_m(x) e^{-ikx} \, dx
\]

Expanding the product P_l(x) P_m(x) in terms of a series of Legendre polynomials using coefficients derived from the orthogonality relations:

\[
P_l(x) P_m(x) = \sum_{n=|l-m|}^{l+m} c_n P_n(x)
\]

where c_n are the coefficients.

Thus, the Fourier transform becomes:

\[
\mathcal{F}\{P_l \cdot P_m\}(k) = \sum_{n=|l-m|}^{l+m} c_n \int_{-1}^{1} P_n(x) e^{-ikx} \, dx
\]

Given that the integral of a Legendre polynomial with the Fourier kernel is:

\[
\int_{-1}^{1} P_n(x) e^{-ikx} \, dx = 2\pi i^n j_n(k)
\]

The final expression for the Fourier transform is:

\[
\mathcal{F}\{P_l \cdot P_m\}(k) = \sum_{n=|l-m|}^{l+m} c_n (2\pi i^n j_n(k))
\]

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[crowlogic]

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\section*{Eigenfunction Expansion of a Covariance Operator}

Let K(x, y) be the covariance kernel of a Gaussian process, and let K be the associated covariance operator defined by: [ (K \phi)(x) = \int K(x, y) \phi(y) \, dy ] where \phi(y) is a function in the domain of the operator. Given an orthonormal basis { \phi_n } in a Hilbert space O_1, the basis functions satisfy: [ \langle \phi_n, \phim \rangle = \delta{nm} ] We want to expand the eigenfunctions \psi_i(x) of the covariance operator K in terms of this orthonormal basis.

\subsection*{Eigenfunction Expansion} The eigenfunctions \psi_i(x) can be expanded in the orthonormal basis { \phi_n } as: [ \psii(x) = \sum{n} C_{in} \phin(x) ] where C{in} are the expansion coefficients.

\subsection*{Application of Covariance Operator} Applying the covariance operator K to this expansion: [ (K \psi_i)(x) = \lambda_i \psi_i(x) ] Substituting the expansion for \psi_i(x): [ (K \psi_i)(x) = \sumn C{in} (K \phi_n)(x) ] By the definition of the covariance operator, we have: [ (K \phi_n)(x) = \int K(x, y) \phi_n(y) \, dy ] Thus, the expression becomes: [ \sumn C{in} \int K(x, y) \phi_n(y) \, dy = \lambda_i \sumn C{in} \phi_n(x) ]

\subsection*{Matrix Representation} To solve this, we calculate the matrix elements of the covariance operator in the orthonormal basis { \phin }. The matrix elements K{mn} are given by: [ K_{mn} = \int \int K(x, y) \phi_m(x) \phi_n(y) \, dx \, dy ] Now, we express the eigenvalue problem as a matrix equation: [ \sumn K{mn} C_{in} = \lambdai C{im} ] This is a standard matrix eigenvalue problem, where K{mn} is the covariance matrix in the orthonormal basis, C{in} are the expansion coefficients, and \lambda_i are the eigenvalues.

\subsection*{Reconstructing the Eigenfunctions} Once the matrix eigenvalue problem is solved for the eigenvalues \lambdai and the coefficients C{in}, the eigenfunctions \psi_i(x) can be reconstructed from the expansion: [ \psi_i(x) = \sumn C{in} \phi_n(x) ]

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[crowlogic]

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\section*{Proof}

\textbf{1. Covariance Function:} Let K(t, s) be the covariance function of a band-limited Gaussian process with spectral density S(f) supported on [-B, B]:

[ K(t, s) = \int_{-B}^{B} S(f) e^{2\pi i f (t - s)} \, df ]

\textbf{2. Eigenvalue Equation:} We want to show that the sinc function \text{sinc}(t) satisfies:

[ K \psi(t) = \lambda \psi(t) ]

\textbf{3. Applying the Covariance Operator:}

[ K \text{sinc}(t) = \int K(t, s) \text{sinc}(s) \, ds ]

\textbf{4. Substituting for K(t, s):}

[ K \text{sinc}(t) = \int \left( \int_{-B}^{B} S(f) e^{2\pi i f (t - s)} \, df \right) \text{sinc}(s) \, ds ]

\textbf{5. Change Order of Integration:}

[ K \text{sinc}(t) = \int_{-B}^{B} S(f) \left( \int \text{sinc}(s) e^{2\pi i f (t - s)} \, ds \right) df ]

\textbf{6. Evaluate Inner Integral:}

[ \int \text{sinc}(s) e^{2\pi i f (t - s)} \, ds = e^{2\pi i f t} \int \text{sinc}(s) e^{-2\pi i f s} \, ds ] [ \int \text{sinc}(s) e^{-2\pi i f s} \, ds = \text{rect}(f) ]

\textbf{7. Combine Results:}

[ K \text{sinc}(t) = \int_{-B}^{B} S(f) e^{2\pi i f t} \cdot \text{rect}(f) \, df ]

\textbf{8. Eigenvalue:} Let \lambda be the corresponding eigenvalue. Then:

[ K \text{sinc}(t) = \lambda \text{sinc}(t) ]

\textbf{Conclusion:} The sinc function is an eigenfunction of the covariance operator K for any band-limited Gaussian process.

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[crowlogic]

Given a positive semi-definite kernel \Phi(t - u), we can express it in terms of its spectral density S(\omega):

[ \Phi(t - u) = \mathcal{F}^{-1}(S(\omega)), ]

where S(\omega) is the spectral density.

The function H(t), defined as the inverse Fourier transform of the square root of the spectral density, is given by:

[ H(t) = \mathcal{F}^{-1}(\sqrt{S(\omega)}). ]

The eigenfunctions \phi_n(t) are constructed using H(t) as follows:

[ \phin(t) = \int{-\infty}^{\infty} H(t - u) g_n(u) \, du, ]

where g_n(u) are chosen functions that ensure orthonormality.

The kernel can then be expressed in the Mercer expansion as:

[ \Phi(t, u) = \sum_{n} \lambda_n \phi_n(t) \phi_n(u), ]

where \lambda_n are the corresponding eigenvalues.