Fourier and Spectral Analysis

Fourier analysis converts translation into multiplication. For a specified self-adjoint or normal operator, the spectral theorem resolves point and continuous spectral parts. A general nonnormal operator can also have residual spectrum and need not admit an orthogonal spectral decomposition. None of these structures is fixed by a differential expression alone. This record fixes the transform normalization, operator data, analytic continuation, and checks needed for wave and stability calculations. The full black-hole construction is developed in Green Functions, Scattering, and Retarded Response.

Helpful background. Calculus and Linear Algebra for Relativity supplies inner products and eigenproblems; Differential Equations for Relativity supplies boundary and singular-point analysis; The Wave Equation and Characteristics supplies causal propagation.

Fourier and Laplace pairs

With the site’s time dependence $e^{-i\omega t}$, use

$$\widetilde f(\omega)=\int_{-\infty}^{\infty} e^{+i\omega t}f(t)\,dt, \qquad f(t)=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi} e^{-i\omega t}\widetilde f(\omega).$$

For square-integrable $f$ and $g$, the matching Parseval identity is

$$\int_{-\infty}^{\infty}f^*(t)g(t)\,dt =\int_{-\infty}^{\infty}\frac{d\omega}{2\pi} \widetilde f^*(\omega)\widetilde g(\omega).$$

The fast normalization check is $\int e^{i(\omega-\omega')t}dt=2\pi\delta(\omega-\omega')$. NIST states the equivalent symmetrically normalized transform, inversion, and Parseval formulas in NIST DLMF 2026, § 1.14(i), Eqs. (1.14.1)–(1.14.8).

For a causal function, define

$$F(\omega)=\int_0^\infty e^{i\omega t}f(t)\,dt, \qquad f(t)=\frac{1}{2\pi}\int_{-\infty+i\gamma}^{\infty+i\gamma} e^{-i\omega t}F(\omega)\,d\omega,$$

where the horizontal line lies in the convergence half-plane and above every singularity. This is the Laplace inversion formula after $s=-i\omega$; NIST DLMF 2026, § 1.14(iii), Eqs. (1.14.17)–(1.14.20) states its hypotheses. For elapsed time $\tau\lt0$, close upward; for $\tau\gt0$, close downward. Analyticity and decay on the closing arc are assumptions, not consequences of drawing the contour.

Spectrum means operator plus domain

A Sturm–Liouville expression can be written

$$Lu=\rho^{-1}\left[-(pu')'+qu\right], \qquad \langle u,v\rangle=\int_a^b u^*v\,\rho\,dx,$$

For the regular real-coefficient case on a finite interval, take $p\in C^1([a,b];\mathbb R)$, $q,\rho\in C([a,b];\mathbb R)$, and $p,\rho\gt0$. Lagrange’s identity then gives formal symmetry. A realization is symmetric when the boundary form

$$\left[p\left(u^*v'-u'^*v\right)\right]_a^b$$

vanishes for every pair $u,v$ in its domain; it is self-adjoint only when that domain equals its adjoint domain. Changing the endpoint conditions can therefore change the operator and its spectrum. The coefficient, boundary, and domain hypotheses and examples of discrete, continuous, and mixed spectra are summarized in NIST DLMF 2026, §§ 1.18(iii)–(viii), especially Eqs. (1.18.25)–(1.18.28).

Setting	Data that must be stated	Safe conclusion
Regular self-adjoint problem on a finite interval	Weight, inner product, endpoint conditions, and domain	The spectrum is real and, under the standard regular hypotheses, discrete.
Self-adjoint problem on a noncompact domain	Domain, measure, behavior at infinity, and spectral measure	Point and continuous parts may coexist; generalized modes are not ordinary normalized eigenvectors.
Radiative resonance problem	Ingoing/outgoing conditions, transform convention, continuation, sheet, and cut	A resonance is a pole of the continued resolvent, not automatically an $L^2$ eigenvalue.
Spectral discretization	Basis, map, endpoint rows, resolution, precision, and residual norm	Matrix eigenvalues are candidates until continuum convergence and boundary behavior are checked.

For $R(z)=(L-z)^{-1}$, a large resolvent may signal sensitivity even away from a pole. A pole list alone does not reconstruct initial-value response; continuous spectral measure, branch cuts, source projections, and prompt propagation can all contribute.

Causal response and numerical spectra

For Schwarzschild modes, the shared time factor makes $e^{-i\omega r_*}$ future-horizon ingoing and $e^{+i\omega r_*}$ infinity outgoing. In Kerr the horizon phase uses $\omega-m\Omega_H$. Stable retarded poles satisfy $\operatorname{Im}\omega\lt0$. Their contribution depends on the chosen analytic sheet and on whether the inverse contour crosses them.

The separated radial kernel and its Wronskian are organized in Mode Green Functions and Wronskians; Analytic Structure: Poles, Cuts, and Thresholds fixes sheet data; and Inverse Transforms and Causal Time-Domain Response performs the contour deformation. The distinction among prompt, pole, and branch-cut terms is explicit in Leaver 1986, pp. 384–408. On curved spacetime the contour represents a globally selected causal inverse; local construction alone does not supply that global result, as emphasized in Friedlander 1975, chs. 5–6, pp. 116–227.

Numerical collocation is a different use of “spectral.” Require resolution and precision sweeps, a changed basis or domain map, the original boundary conditions, an off-grid differential residual, and an independent method. Basis construction, multidomain matching, compactification, and convergence tests are reviewed in Grandclément and Novak 2009, §§ 2.1–2.6, § 3.1.2, especially Eq. (89), and § 4. A tiny matrix residual can coexist with spectral pollution or a wrongly regularized endpoint.

References

• Friedlander, F. G. (1975). The Wave Equation on a Curved Space-Time. Cambridge University Press. Publisher record.
• Grandclément, P., and Novak, J. (2009). “Spectral Methods for Numerical Relativity.” Living Reviews in Relativity 12, 1. DOI.
• Leaver, E. W. (1986). “Spectral Decomposition of the Perturbation Response of the Schwarzschild Geometry.” Physical Review D 34, 384–408. DOI.
• National Institute of Standards and Technology. Digital Library of Mathematical Functions. Version 1.2.7, 15 June 2026. § 1.14 and § 1.18.