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Let's start from a system $f_n(x) = e^{inx}$.

I can rotate one element: $f_m(x + x_0) = e^{imx_0} e^{imx}$ , which is still orthogonal to all but one element of system.

Same holds for $n$-dimensional trigonometric systems on a torus: being shifted, a function is still orthogonal to all other elements.

When considering real-valued trigonometric systems, $\sin m(x + x_0)$ is orthogonal to all elements but $\sin m x$, $\cos m x$.

Question: do spherical functions enjoy same property? Say, if I rotate one of the spherical functions $Y^l_m(\theta, \phi)$, will it be orthogonal to all other functions (for the exception of finite subset?)

P.S. I've checked this for $l=0, l=1$.

Alleo
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  • What do you mean by "rotating" the function? Certainly, there is a transformation which takes $e^{inx}$ to the normalized sum of two such funcitons, which does not preserve orthogonality. – Adam Hughes Jun 07 '16 at 17:22
  • @AdamHughes I define a spherical function as a function on the sphere. So, rotation of a function is a mapping $g(x) = f(Ax)$, where $A$ is an orthogonal matrix. (obviously, I can't write arbitrary rotation simply in spherical coordinates) – Alleo Jun 07 '16 at 17:25
  • Well then even your original one isn't going to work, there are plenty of orthogonal transformations which take one basis element to something not orthogonal to the original basis. Now if you change this to demand that all image elements are still orthogonal as a new basis, then you have a case. – Adam Hughes Jun 07 '16 at 17:27
  • @AdamHughes never mind, I've found out. Spherical functions $Y^{l1}{m1}, Y^{l2}{m2} $ are orthogonal after any rotation of one of them if $l1 \neq l2$ – Alleo Jun 07 '16 at 17:29

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