The function P_k of a real variable are Gegenbauer polynomials.

Following Narcowich, Petrushev, and Ward, we can construct needlets by considering the localized projection kernels by multiplying this kernel by a cutoff function a.

If the universe were known to be a four-dimensional sphere of fixed radius, then the gravitational field equations themselves arise as a consequence of the shape of the universe, and removes gravity as a separate force of nature altogether.

One of the objections to a macroscopic spatial dimensions is that the force law would be 1/r^{D-1} which would be at odds with Newtonian laws, and has been an argument for introducing compact dimensions rather than macroscopic spatial dimensions.

One possible way out of this problem is to consider the gravitational field equations as primary rather than the force law.

We posit that the the physical three dimensional universe is a preferred submanifold of a four dimensional sphere, for which an inverse square force law might hold regardless.

Analysis of functions on the four-sphere can be done as follows.

It is well known that the square-integrable functions on a compact manifold decomposes into finite dimensional orthogonal subspaces of eigenfunctions of the Laplacian.

In the case of the 4-sphere we have a more detailed construction.

The eigenvalues of the Laplacian are k(k+3) and if Y_{kl} are eigenfunction with eigenvalue k(k+1) indexed by l then P_k( xy ) = \sum_l Y_{kl}(x) \bar{Y_{kl}}(y) then these are projection kernels to the eigenspace for general functions.

The idea that the universe has higher than three macroscopic dimensions goes back to the nineteenth century, and Riemann had conceived the notion of hyperspace for example.

We will argue that the universe has exactly four macroscopic spatial dimensions and for this we have, besides the various conceptual arguments, the following concrete argument.

Assume the universe is compact, which is easy to justify on the grounds of parsimony based on the observation that the cosmic background radiation has a uniform lower bound.