This work is concerned with two different problems in harmonic analysis, one on the Heisenberg group and other on $\mathbb{R}^n$, as 
 described in the following two paragraphs respectively. Let $\mathbb{H}^n$ be the $(2n+1)$-dimensional Heisenberg group, and let $K$ be a compact 
 subgroup of $U(n)$ such that $(K,\mathbb{H}^n)$ is a Gelfand pair. Also assume that the $K$-action on $\mathbb{C}^n$ is polar. We prove a Hecke-
 Bochner identity associated to the Gelfand pair $(K,\mathbb{H}^n)$. For the special case $K=U(n)$, this was proved by Geller, giving a formula for 
 the Weyl transform of a function $f$ of the type $f=Pg$, where $g$ is a radial function, and $P$ a bigraded solid $U(n)$-harmonic polynomial. Using 
 our general Hecke-Bochner identity we also characterize (under some conditions) joint eigenfunctions of all differential operators on $\mathbb{H}^n$ 
 that are invariant under the action of $K$ and the left action of $\mathbb{H}^n$.We consider convolution equations of the type $f*T=g$, where $f,g\in
 L^p(\mathbb{R}^n)$ and $T$ is a compactly supported distribution. Under natural assumptions on the zero set of the Fourier transform of $T$, we show 
 that $f$ is compactly supported, provided $g$ is.