# Conformal transformations and harmonic analysis on the sphere

Consider the $n$-dimensional sphere $S^n$. I'm especially interested in the $n=4$ case. The Hilbert space $L^2(S^n)$ can be decomposed into a direct sum of eigenspaces of the Laplacian, which are finite dimensional. I'm looking for non-isometric conformal transformations

$$f: S^n \to S^n$$

s.t. for some $\lambda, \mu > 0$ if $\psi$ is an eigenvector of the Laplacian with eigenvalue $\alpha < \lambda$ then $f(\psi)$ is a sum of eigenvectors with eigenvalues $< \mu$.

Do such $f$ exist? If so, is it possibly to classify them?

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The only possibility is the trivial one when $\lambda$ is so small that the only eigenfunctions with eigenvalue less than $\lambda$ are constants (eigenvalue zero). Otherwise the eigenfunctions with eigenvalue less than $\lambda$ span the space of polynomials of degree at most $d$ for some positive integer $d$, and then composition with a non-isometric conformal transformation takes it outside the space of polynomials, and thus outside the span of eigenfunctions of eigenvalue less than $\mu$ for any finite $\mu$.
Let me add some remarks... The group of conformal transformations of $S^n$ is generated by isometries, scalings ( $x \to \lambda x$ conjugated by stereographical projection and it's inverse) and spherical inversions $x\to \frac{x}{\|x\|}$. As Noam Elkies indicated in his answer - the eigenfunctions of the Laplace equation (the so called spherical harmonics) are restrictions of harmonic polynomials on $\mathbb{R}^{n+1}$.