WEIGHTED EIGENFUNCTION ESTIMATES WITH APPLICATIONS TO COMPRESSED SENSING

Using tools from semiclassical analysis, we give weighted L-infinity estimates for eigenfunctions of strictly convex surfaces of revolution. These estimates give rise to new sampling techniques and provide improved bounds on the number of samples necessary for recovering sparse eigenfunction expansions on surfaces of revolution. On the sphere, our estimates imply that any function can be efficiently recovered to within a factor of its best s-term approximation in the first N spherical harmonics from its values at m greater than or similar to sN(1/6) log(4)(N) sampling points, improving on the previous bound of m greater than or similar to sN(1/4) log(4)(N) necessary sampling points. In particular, any function having an s-sparse expansion can be recovered exactly from such undersampled measurements.