ABSTRACT
The space of continuous functions on Sd is denoted by C(Sd), and as usual it is endowed with the supremum norm
|f(x)|.
Let us denote by L2(Sd) the Hilbert space of square-integrable functions on Sd with the inner product
f(x) g(x) dωd(x)
and the induced norm ‖f‖L2(Sd) := (f, f) 1 2
L2(Sd) , where dωd is the
Lebesgue surface measure on Sd. The surface area of Sd is denoted by ωd,
ωd = |Sd| = 2π d+1 2
Γ (
) . The L2(Sd)-orthogonal projection Tn : L2(Sd) → Pn(Sd) is then
uniquely defined by
Tnf ∈ Pn(Sd) ∀f ∈ L2(Sd) and
(Tnf, p)L2(Sd) = (f, p)L2(Sd) ∀f ∈ L2(Sd), ∀p ∈ Pn(Sd).
