# Non-Fourier complete orthogonal basis?

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 $$T$$.

• Your conditions seem more like informal descriptions than formal things one could check. What sort of object $T$ would you want? A single transform? A family? What is the precise definition of (5)? Commented May 15, 2022 at 2:10
• I believe the Mellin transform perhaps meets some but not all of the properties, but I'm not familiar with everything in your question. The paper "Mellin-Fourier series and the classical Mellin transform" might be of interest (see sciencedirect.com/science/article/pii/S0898122100001395 ). Commented May 15, 2022 at 2:11
• Please search the literature for Haar series, Walsh series, wavelet series. Commented May 15, 2022 at 4:45
• @StevenClark Looks promising, thanks Commented May 15, 2022 at 20:10
• The Hankel transform (see mathworld.wolfram.com/HankelTransform.html and en.wikipedia.org/wiki/Hankel_transform) and Fourier Bessel series might also be of interest (see mathworld.wolfram.com/Fourier-BesselSeries.html and en.wikipedia.org/wiki/Fourier%E2%80%93Bessel_series). Commented May 15, 2022 at 20:39

(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 $$\mathfrak{sl}_2$$.
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.)