I am looking for references studying orthonormal systems of functions $\{h\_n\}\_{n\geq0}$ on a sphere $S^d$ ($d=2$ or $d\geq2$) with respect to weights that are not uniform (unlike spherical harmonics). That is, such that
$$
\int\_{S^d} h\_n(x)h\_m(x)w(x)\,\mathrm{d}x=\delta\_{nm}.
$$
I am particularly interested in weights of the form $w(x)=f(x\_1)$, such as the exponential weight $w(x)=\exp(ax\_1)$, and on relatively explicit systems.



Relatively explicit orthogonal systems on the sphere that are not spherical harmonics
This problem immediately reduces to one on orthogonal systems on the interval $[-1,1]$.


Indeed, suppose that $h\_n(x)=H\_n(x\_1)$ for all integers $n\ge0$ and all $x\in S^d$, where $H\_n$ is a polynomial of degree $n$. Then
$$\int\_{S^d}h\_n(x) h\_m(x)\,dx=\int\_{-1}^1 H\_n(x\_1) H\_m(x\_1) W(x\_1)\,dx\_1,$$
where $W:=fp$; $p$ is the p.d.f. of the random variable (r.v.) $R\sqrt B$; $R$ and $B$ are independent r.v.'s; $P(R=1)=P(R=-1)=1/2$ and $B$ has the beta distribution with parameters $1/2,(d-1)/2$. So, it is easy to get a simple, explicit expression for $p(x\_1)$:
$$p(x\_1)=\frac{\left(1-x\_1^2\right){}^{(d-3)/2}}{\text{B}\left(1/2,(d-1)/2\right)}$$
for $x\_1\in(-1,1)$, where $\text{B}(a,b):=\dfrac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$.


Thus, we want the $H\_n$'s to be orthogonal polynomials with respect to the weight function $W$. So, the $H\_n$'s can be obtained, in a standard manner, by the [Gram–Schmidt orthogonalization](https://en.wikipedia.org/wiki/Orthogonal_polynomials#Relation_to_moments), which is easier to do for polynomials and results in a [3-term recurrence relation](https://en.wikipedia.org/wiki/Christoffel%E2%80%93Darboux_formula#Proof).


