to describe the elementary spherical functions on a rank one Riemannian symmetric space $G/K$.

Existence, uniqueness and regularity properties of $\varphi_\lambda$ were investigated in several publications of Heckman and Opdam, later sharpened by Schapira [@Schapira] (where the functions are denoted $F_\lambda$ and their non-symmetric versions $G_\lambda$) and most recently by Narayanan, Pasquale, and Pusti [@Narayanan-Pasquale-Pusti]. Since we do not need to estimate the functions $\varphi_\lambda$ and the associated Harish-Chandra series expansions in the results that follow, we merely refer the reader to [@Narayanan-Pasquale-Pusti Sections 2–4] for the details.

The *Heckman–Opdam* transform of a function $f\in C_c^\infty({\mathfrak{a}})^W$ is defined by $$\mathcal{F}f(\lambda)=\int_{\mathfrak{a}}f(x)\varphi_\lambda(x)\,d\mu(x).$$

Often $\mathcal{F}$ is called the hypergeometric Fourier transform, since the functions $\varphi_\lambda$ can be seen as generalized hypergeometric functions. In some literature, such as [@Schapira], these are denoted by $F_\lambda$. Their non-symmetric version appear as $G_\lambda$, in terms of which one can define a hypergeometric Fourier transform of a function $f\in C_c^\infty({\mathfrak{a}})$ that is not necessarily $W$-invariant. The terminology ‘hypergeometric’ was certainly used by Delorme in [@Delorme-hyper] but the transform itself was studied earlier by Cherednik from a different perspective, and by Opdam in [@Opdam-acta]. When $\mathcal{F}$ acts on functions that might not be $W$-invariant, we therefore talk about the 