Non-Fourier complete orthogonal basis?

Question

The Fourier Transform (FT)

1. Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero
2. Is invertible: info-preserving, has inverse function
3. Is energy-preserving: via Parseval's theorem
4. Has discrete counterpart: Discrete Fourier Transform, $N$ coefficients for length $N$ input
5. Has discrete counterpart with continuous reconstructor: sinc interpolation
6. Provides conditional reconstruction guarantees: Nyquist-Shannon sampling theorem

Is there any such transform, $T$? Note, FT also has other properties, such as stability and handling of complex inputs -- $T$ only needs to meet above 6 for real-valued.

My motivation is, FT's 6 says if $x[n]$ is a sampling of $x(t)$ with sampling rate above Nyquist, then $x(t)$ can be perfectly recovered from $x[n]$ via sinc interpolation. Most commonly, this interprets as "if we sample below Nyquist, $x(t)$ cannot be recovered perfectly" - but what the theorems really say is, sinc interpolation won't recover it perfectly. If $x$ isn't Fourier-bandlimited, one could imagine it still be $T$-bandlimited, or its "Nyquist frequency" be less in $T$, if $x$ is sparser in $T$.

Answer 1

(First, just for precision, your first point about "orthogonality" is morally correct, but not literally correct, because the exponentials are not in $L^2(\mathbb R)$...)

In fact, there are some situations that are "better" than the Fourier transform situation, in the sense that there is a collection of $L^2$ eigenfunctions for some self-adjoint operator, and these eigenfunctions are well-behaved. In fact, part of the classical approach to Fourier transform, by Wiener and Bochner, used the "Schrodinger operator" $-d^2/dx^2+x^2$ on $\mathbb R$... which had been considered decades earlier, in work of Mehler (google-able) and others. That is, we have the Laplacian plus a "confining potential". This arose in early work on quantum mechanics, too, Dirac and others, with "ladder operators" (raising and lowering), which turned out to be intimiately related to the representation theory of $\mathfrak{sl}_2$.

Abstract situations occur with Riemannian manifolds... For compact ones, the spectrum of the Laplace-Beltrami operator is discrete. For non-compact, there is continuous spectrum.

The only situation I know with usefully-explicit continuous spectrum is the non-compact automorphic quotient case, where various sorts of Eisenstein series, while not in $L^2$, integrate to give parts of $L^2$ not spanned by eigenfunctions for Laplacians (and other parts of the center of the enveloping algebra, etc.)