Imagine the functions
$$f_n(x) = \text{sgn}(\sin(nx))\cdot\lim_{k\to\infty}|\sin(nx)|^{1/k}\\g_n(x) = \text{sgn}(\cos(nx))\cdot\lim_{k\to\infty}|\cos(nx)|^{1/k}$$
They would be in some sense discrete/step counterparts of sin and cos which simultaneously can be very readily expressed as linear functions of the Haar transform scaling and wavelet functions.
Will this build an ON basis $\{f_1,g_1,\cdots,f_k,g_k,\cdots\}$ for some smooth space of functions in say, $L_2(2\pi)$ or are they too discontinous?
Own work : I have made some premature experiments and found that such a transform can have some benefits in different norm-senses, but that it also produces a kind of inverse Gibbs phenomenon, producing "ringings" at smooth turns of functions (as opposed to at discontinuities which is what happens for the Fourier Transform.)
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