Rectangular Fourier Transform and its Properties

AI Thread Summary
The discussion centers on a transformation using the orthonormal basis defined by s_k(x) = ⌈sin(kx)⌉ and c_k(x) = ⌈cos(kx)⌉, exploring its relation to the Fourier transform and properties such as orthogonality. It questions whether this transformation has a specific name and examines the potential existence of a convolution theorem associated with it. The similarity to the Walsh-transform is noted, but it is pointed out that the functions in question are not orthogonal. The suggestion is made that using sign(sin(kx)) and sign(cos(kx)) may yield better orthogonality. Overall, the discussion seeks to clarify the mathematical framework and implications of this transformation.
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Is there a name for a transformation using the orthonormal base

s_k(x)=\lceil \sin kx \rceil,\: c_k(x) = \lceil \cos kx \rceil \quad ?

So basically a Fourier transform or Fourier series using periodic rectangles. What are the properties? Is there some kind of convolution theorem?
 
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I found some answers. The Walsh-transform looks very similar. I noticed that the functions are not orthogonal so sign(sin(kx)) and sign(cos(kx)) is probably a better choice.
 

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