Let M be a smooth compact oriented Riemannian manifold, and let M be the Laplace-Beltrami operator on M. Say 0 = f ∈ S(R + ), and that f (0) = 0. For t > 0, let K t (x, y) denote the kernel of f (t 2 M ). We show that K t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f (t 2 ) on R n . We define continuous S-wavelets on M, in such a manner that K t (x, y) satisfies this definition, because of its localization near the diagonal. Continuous S-wavelets on M are analogous to continuous wavelets on R n in S(R n ). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous S-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus T 2 or the sphere S 2 , and f (s) = se −s (the "Mexican hat" situation), we obtain two explicit approximate formulas for K t , one to be used when t is large, and one to be used when t is small.

Let M be a smooth compact oriented Riemannian manifold of dimension n without boundary, and let be the Laplace-Beltrami operator on M. Say 0 = f ∈ S(R + ), and that f (0) = 0. For t > 0, let K t (x, y) denote the kernel of f (t 2 ). Suppose f satisfies Daubechies' criterion, and b > 0. For each j, write M as a disjoint union of measurable sets E j,k with diameter at most ba j , and measure comparable toform a frame for (I − P)L 2 (M), for b sufficiently small (here P is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for b sufficiently small). Moreover, based upon how well-localized a function F ∈ (I − P)L 2 is in space and in frequency, we can describe which terms in the summation F ∼ S F = j k F, φ j,k φ j,k are so small that they can be neglected. If n = 2 and M is the torus or the sphere, and f (s) = se −s (the "Mexican hat" situation), we obtain two explicit approximate formulas for the φ j,k , one to be used when t is large, and one to be used when t is small.

We show that one can characterize the Besov spaces on a smooth compact oriented Riemannian manifold, for the full range of indices, through a knowledge of the size of frame coefficients, using the smooth, nearly tight frames we have constructed in [8].

Abstract. In this paper we study subsets E of Z d p such that any function f : E → C can be written as a linear combination of characters orthogonal with respect to E. We shall refer to such sets as spectral. In this context, we prove the Fuglede Conjecture in Z 2 p which says that E ⊂ Z 2 p is spectral if and only if E tiles Z 2 p by translation. Arithmetic properties of the finite field Fourier transform, elementary Galois theory and combinatorial geometric properties of direction sets play the key role in the proof.

Let G be a stratified Lie group and L be the sub-Laplacian on G. Let 0 = f ∈ S(R + ). We show that Lf (L)δ, the distribution kernel of the operator Lf (L), is an admissible function on G. We also show that, if ξf (ξ) satisfies Daubechies' criterion, then Lf (L)δ generates a frame for any sufficiently fine lattice subgroup of G.