Stationary correlation spectra and ULE kernel regularity

Fix a positive integer $n$ and use entrywise Lebesgue integrals over the real line. A stationary matrix correlation is here a function $C:\mathbb{R}\to {\mathbb{C}}^{n\times n}$ of a time difference; stationarity alone imposes no integrability or positivity conclusion in this definition. Suppose every entry of $C$ is integrable. Its spectral density in this convention is $J(\omega )=(2\pi {)}^{-1}{\int }_{\mathbb{R}}C(t)e^{i\omega t}dt$. When Fourier inversion applies, the inverse convention is $C(t)={\int }_{\mathbb{R}}J(\omega )e^{-i\omega t}d\omega$ at the points covered by the inversion theorem. In particular this equality holds for every $t$ if all entries of $C$ are Schwartz functions, assuming countable choice.

For the jump-kernel contract additionally require that $J(\omega )$ is Hermitian positive semidefinite for every real $\omega$: $J(\omega {)}^{\ast }=J(\omega )$ and $v^{\ast }J(\omega )v\ge 0$ for every $v\in {\mathbb{C}}^n$. Define $g(\omega )$ to be the unique Hermitian positive semidefinite matrix whose square is $J(\omega )\mathrm{/}(2\pi )$. Require its entries to be integrable and set $g(t)={\int }_{\mathbb{R}}g(\omega )e^{-i\omega t}d\omega$. For a matrix $A$, put $\mathrm{\parallel }A{\mathrm{\parallel }}_{2,1}={\sum }_{b=1}^n({\sum }_{a=1}^n\mathrm{\mid }{A}_{ab}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. Kernel regularity means $0<{\int }_{\mathbb{R}}\mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt<\mathrm{\infty }$ and ${\int }_{\mathbb{R}}\mathrm{\mid }t\mathrm{\mid }\mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt<\mathrm{\infty }$.

For an integrand $h$ with a singularity at a real point $a$, the symmetric Cauchy principal value is $\mathrm{PV}{\int }_{\mathbb{R}}h(x)dx={\lim }_{\epsilon \downarrow 0}{\int }_{\{\mathrm{\mid }x-a\mathrm{\mid }>\epsilon \}}h(x)dx$, provided those integrals and the limit exist; convergence at infinity is also required. A Dirac mass at $a$ is a measure assigning mass one to sets containing $a$ and zero to other sets. It is not an ordinary density with respect to Lebesgue measure, so its pointwise matrix square root is not defined by the density construction above.

Schwartz space and its seminorms

Integrable real and complex functions, and their integrals

Every at most countable subset of ℝⁿ is Lebesgue null; in particular λ₁(ℚ)=0

A nonnegative integral over a null set vanishes

The entrywise integrability assumptions make the displayed Fourier integrals finite. Hermitian matrices are normal, so [F1] diagonalizes $J(\omega )$ in an orthonormal basis. Its eigenvalues are real and nonnegative by the stated quadratic-form condition. Taking their nonnegative square roots, after division by $2\pi$, gives a Hermitian positive semidefinite square root. It is unique: any such root commutes with its square and preserves each eigenspace of that square; diagonalizing the root on each eigenspace forces its eigenvalues to be the nonnegative square root of the corresponding eigenvalue. Thus the construction is independent of the eigenbasis.

Apply [F2] entrywise to $C$ in the stated Schwartz special case, changing variables from the supplier's $2\pi$ Fourier convention to angular frequency and reversing the sign. The moment conditions explicitly ensure that the denominator used in the kernel correlation-time ratio is finite and nonzero and its numerator is finite. For the Dirac mass at $a$, [F3] gives $\lambda (\{a\})=0$. If a nonnegative measurable function $p$ were a Lebesgue density for that mass, [F4] would give ${\int }_{\{a\}}pd\lambda =0$, whereas the mass of $\{a\}$ is one. This contradiction excludes a Lebesgue density. The principal value is only a definition when the indicated limit exists; no existence follows merely from writing the symbol.

The jump-kernel normalization and mixed matrix norm follow the cited source's definitions. The separate hypotheses above make the domain and convergence requirements explicit; no physical interpretation or microscopic bath assertion is made by this mathematical contract.