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I seek a transform $T$ that operates on real-valued $x(t)$, that

  1. Is perfectly invertible
  2. Has discrete counterpart with continuous reconstructor
  3. Provides conditional reconstruction guarantees

Qualifying example: The Fourier Transform's Nyquist-Shannon Sampling Theorem roughly states, if $x(t)$ has bandwidth $B$, then sampling it at a rate above $2B$ allows recovering $x(t)$ perfectly from $x[n]$ via sinc interpolation. Put differently, if $X(\omega)=\mathcal{F\{x(t)\}}$ has nonzero values over some interval $(\omega_0, \omega_1)$, then $B=|\omega_1 - \omega_0|$, and we have a clear criterion on the sampling rate required to not lose any information and recover $x(t)$ for $t \in (-\infty, \infty)$.

Non-qualifying example: The Wavelet Transform is perfectly invertible and has a discrete counterpart. However, as far as I know, latter can only invert over the finite interval of transformation, $t_0$ to $t_1$. It lacks FT's equivalent of "if bandlimited, then inversion over $(-\infty, \infty)$ is this" (or, it's identical to FT's in simply duplicating the inversion). More importantly, it doesn't pose its own sampling criteria or guarantees, instead it builds directly off of Fourier theory.

Clarifying further, 3) and $x[n]$ are the core of the question. There are no requirements on allowed behavior of $x(t)$, only that we allow $x[n]$ that is real-valued and finite in length (# of samples) and value. Given some function, our question is "what sampling rate is sufficient to recover it perfectly?", and currently the only player in town seems to be the FT. Is there any other, and if not, is there proof that FT is the only such transform, or that any other transform necessarily yields sampling criteria identical to FT's?

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  • $\begingroup$ I've relaxed some of this question's restrictions and clarified my intent further $\endgroup$ May 15, 2022 at 20:17
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    $\begingroup$ There are different wavelet transforms. I wouldn't bother with the non-baseband sampling generalization and just pose the question in the context of baseband sampling. $\endgroup$ May 16, 2022 at 5:10
  • $\begingroup$ @robertbristow-johnson But do any of them reconstruct differently than sinc interpolation, from finite samples? And the problem with "baseband" is "band", the very concept to which I seek an alternative: another $T$'s coefficient support for a given signal may differ from Fourier's. $\endgroup$ May 16, 2022 at 16:59

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