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## Park City Mathematics InstituteUndergraduate Summer School 2018

### Introduction to Harmonic Analysis

1. A polynomial $p\in\mathscr P_k$ if and only if $\sum x_j \partial_j p = kp.$

2. $\dim \mathscr P_k = \binom{k+d-1}{k}$

1. Let $p\in\mathscr P_k$, and consider its orthogonal projection $|x|^2q$ onto the space $|x|^2\mathscr P_{k-2}$ with respect to the inner product seen in class. Then $p - |x|^2q$ is harmonic. (Hint: Prove $\langle r,\Delta(p - |x|^2q\rangle = 0$ for every $r\in\mathscr P_{k-2}$.)

2. If $p\in\mathscr H_k$, then $\dfrac{p}{|x|^{2k+d-2}}$ is harmonic on $\R^d\setminus\{0\}$.

3. The spaces are invariant under rotations , i.e. if $f\in H_k$ then $f\circ R\in H_k$ for any rotation $R$.