### Notes on constructing approximate embeddings

Here are some brief notes on how to construct approximate equivariant isometric
embeddings. Refer to *On Equivariant Isometric Embeddings* for notation
and the rest of the proof.
L^{2}(M) decomposes into a direct sum of finite-dimensional eigenspaces of D, and G
acts on each eigenspace, so each eigenspace decomposes into irreducible unitary representations.
(I am now looking at complex functions, as these will correspond to functions into euclidean space.)
Let { f_{ja} } be a basis for the vector space of the j-th irreducible representation.

Approximate g by Σ_{i} dy^{i} dy^{i} in the C^{k} topology

Let
z^{i} = Σ_{ja} c_{i}ja f_{ja}
approximate y^{i} in the C^{k+1} topology, and let
X_{iabr} = Σ N_{j}^{-1/2} c_{ija} f_{jb}
where the sum is over all j such that the j-th representation is equivalent to r.

Proposition 1. X is equivariant

2. z^{i} = Σ_{ar} N_{r}^{1/2} X_{iaar}

3. F(X) = P(Σ_{i} dz^{i} dz^{i})

It follows from (2) that X is an embedding if z is, which it will be if it approximates y and y is.
(3) implies that F(X) approximates g in the C^{k} topology.