The Fourier Transform (FT)

**Is orthogonal**: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero**Is invertible**: info-preserving, has inverse function**Is energy-preserving**: via Parseval's theorem**Has discrete counterpart**: Discrete Fourier Transform, $N$ coefficients for length $N$ input**Has discrete counterpart with continuous reconstructor**: sinc interpolation**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 $T$.

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