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Consider a set of linearly independent functions $\{(x\in\mathbb{R},y\in\mathbb{R})\mapsto f_i(x,y)\in\mathbb{R}\}$ with the property that for any given $\theta\in\mathbb{R}$ and any given $\{a_i\in\mathbb{R}\}$,

$$\exists\{b_i\in\mathbb{R}\} | \sum_i a_if_i(x,y)=\sum_i b_i f_i(x\cos\theta-y\sin\theta,y\cos\theta+x\sin\theta)$$

That is to say $\{f_i\}$ is a basis with a span that is closed under rotations in the domain coordinate system.


For example, $\{1,x,y\}$, considered as functions of $(x,y)$, form a basis that contains a description for any non-vertical plane $z=ax+by+c$. And this span is closed under rotations in the $(x,y)$ coordinate system because any non-vertical plane rotated about the vertical axis remains a non-vertical plane.

Likewise for any $N\in\mathbb{N}$, the span of $\{(x,y)\mapsto x^i y^j|i,j\in\mathbb{N}_0,i+j\le N\}$ is closed under rotations in the $(x,y)$ coordinate system.


Are there other finite bases of bivariate basis functions that have this property? And if so, can anything be said about necessary conditions, that might aid in designing such bases?

(excluding the trivial case of functions of the form $(x,y)\mapsto h(x^2+y^2)$ which have only surfaces of revolution in their span)


Edit:

If we restrict support to a circle about the origin, I feel the resonant modes on a circular membrane may form such a basis. Am I onto something?

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