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 May 11 revised Cubic diophantine equation in 3 variables $(x+2y)(x-4y+k)(x-4y-k) - 28y^3 = 0$, $x,y,z \neq 0$ added 20 characters in body May 11 revised Cubic diophantine equation in 3 variables $(x+2y)(x-4y+k)(x-4y-k) - 28y^3 = 0$, $x,y,z \neq 0$ added 73 characters in body May 11 comment Cubic diophantine equation in 3 variables $(x+2y)(x-4y+k)(x-4y-k) - 28y^3 = 0$, $x,y,z \neq 0$ Thanks. Please see edited question, my original phrasing lacked details. Sorry. May 11 revised Cubic diophantine equation in 3 variables $(x+2y)(x-4y+k)(x-4y-k) - 28y^3 = 0$, $x,y,z \neq 0$ added 195 characters in body; edited title May 11 asked Cubic diophantine equation in 3 variables $(x+2y)(x-4y+k)(x-4y-k) - 28y^3 = 0$, $x,y,z \neq 0$ Jan 10 comment Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform? Exactly. I don't see how to generalize to noncommutative $T$s but I'll think about it. Jan 10 awarded Commentator Jan 10 comment Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform? I don't understand, are you saying that your $T$s - the last ones you defined - do not commute? Jan 10 comment Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform? Please, see new version of the question :) Jan 10 revised Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform? added 740 characters in body Jan 10 awarded Scholar Jan 10 accepted Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform? Jan 10 comment Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform? I am going to accept the answer because it reallys is an answer to my original question and it has taken you a lot of time. However, these transformations are still not what I am most interested in. In particular, they also satisfy $T^2 = 1$ when $a_n \in \{ 1, -1\}$. It is my fault, the original question was very vague. I am editing once more the original message... Jan 10 awarded Supporter Jan 10 comment Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform? Ok, so it seems I can produce infitely many such transformations making use of some orthonormal set of functions like your $g_i$ above. Sorry to keep going on about this, it might be better if you provided a reference but anyhow. Do you know any $T(x,y)$ - other than Fourier - such that $Tf = \int T(x,y) f(y) dy$ is unitary? Jan 10 revised Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform? added 601 characters in body Jan 10 comment Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform? To be even more specific. These transformations you post here, all satisfy that there exists an $n$ such that $T^n = 1$. Of course, the Fourier transform also has this property. So, are there unitary transformations on $S(\mathbb{R})$ that do not satisfy $T^n = 1$ for some integer $n$? I will edit the original question. Jan 10 comment Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform? To be more specific, I would like to know whether (or how) some standard results from the theory of Lie groups (unitary ones in particular) can be ported to such transforms. The Schwartz space would be analogous to the vector space under which the group element in a specific representation - the analogue of the transform - would act. Jan 10 comment Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform? Thanks. I would like to read general results about such transforms. Could you provide me with a good reference? Jan 10 asked Are there unitary transformations on $S(\mathbb{R})$ other than the Fourier transform?