Problem set 6, Harmonic Analysis

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Due March 15

Problem 1

1. A polynomial $latex p\in\mathcal P_k$ if and only if $latex \displaystyle \sum_{j=1}^d x_j \frac{\partial p}{\partial x_j} = k p.$
2. In fact, if $latex f$ is a differentiable function on $latex \mathbb R^d$ that is homogeneous of degree $latex \kappa\in\mathbb R$, i. e., $latex f(tx) = t^\kappa f(x)$ for any $latex t>0$ and $latex x\in\mathbb R^d$, then $latex \displaystyle \sum_{j=1}^d x_j \frac{\partial f}{\partial x_j} = \kappa f.$

Problem 2

If $latex p\in\mathcal H_k$, then $latex \dfrac{p(x)}{|x|^{2k+d-2}}$ is harmonic in $latex \mathbb R^d\setminus\{0\}.$

Problem 3

If $latex R$ is any rotation on $latex \mathbb R^d$, then $latex Z_k^{R\zeta}(R\xi) = Z_k^\zeta(\xi).$

Problem 4

1. Use Parseval's identity to verify $latex ||Z_k^\zeta||^2 = d_k/\omega_d$.
2. Use Cauchy-Schwarz inequality to verify $latex ||Z_k^\zeta||_u = d_k/\omega_d$.

Problem 5

The Gegenbauer polynomials $latex C_k^{\alpha}(t)$ are defined by the generating function

$latex \displaystyle \sum_{k=0}^\infty C_k^{\alpha}(t)r^k = \frac{1}{(1 - 2rt + r^2)^\alpha}.$

1. The polynomials $latex C_k^\alpha(t)$ satisfy the recurrence relations $latex C_0^\alpha(t) = 1$, $latex C_1^\alpha(t) = 2\alpha t$, and
   $latex k C_k^\alpha(t) = 2t(k + \alpha - 1) C_{k-1}^\alpha(t) - (k + 2\alpha - 2) C_{k-2}^\alpha(t).$
2. The polynomials $latex F_k(t)$ are given by $latex \displaystyle F_k(t) = \frac{d-2+2k}{(d-2)\omega_d} C_k^{(d-2)/2}(t).$