Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds

Introduction

In the last decade, methods based on spherical wavelets have found applications in virtually all areas where analysis of spherical data is required, including cosmology, weather prediction and geodesy (see , , , and the references therein). In particular, they have become an important tool for the analysis of Cosmic Microwave Background (CMB) temperature data (, , , , , , , , , , , , and many other articles). In analyzing CMB temperature data, one seeks precise estimates of several parameters of the greatest interest for Cosmology and Theoretical Physics, as well as information on possible regions of non-Gaussianity, and other information as well.

In the past few years, a new kind of wavelet has found many fruitful applications in the analysis of CMB temperature data, the so-called spherical needlets, which form a Parseval (= normalized tight) frame on the sphere (see , , for information about Parseval frames on Euclidean spaces). Spherical needlets were introduced in , , and then used for rigorous statistical analysis of spherical random fields in , , and other articles. This analysis was particularly effective in extracting the desired consequences from CMB temperature data.

The interest in needlets on spheres can be explained by their nearly optimal space-frequency localization properties. These properties of needlets (and other localized bases, such as the “Mexican needlets” of ) allow one to perform frequency analysis of signals (functions), even when one only has partial information about them.

For example, the CMB models are best analyzed in the frequency domain, where the behavior at different multipoles can be investigated separately; on the other hand, partial sky coverage and other missing observations make the evaluation of spherical harmonic transforms impossible.

A recent advance in this area was the development of spin needlets on the sphere ,,,, for the purpose of statistical analysis of CMB polarization, which is also expected to have very significant consequences in physics.

In a different direction, nearly tight frames, which were smooth and highly localized in both space and frequency, were developed on general smooth compact manifolds without boundary, in -. These frames, as in the case of spherical needlets, were constructed from the kernels of certain functions of the Laplace-Beltrami operator. (An analogous construction had been carried out earlier for stratified Lie groups with lattice subgroups, in .)

In this article, we will show that on compact homogeneous manifolds, one can do better – one can arrange for the frames arising from these methods to actually be Parseval (and, of course, highly localized in space and frequency). We will also show that one can characterize Besov spaces through a knowledge of the size of frame coefficients, thereby generalizing results of for the sphere. (Our results on frame characterizations of Besov spaces are closely related to those in .)

Our frames are a natural generalization of spherical needlets. They can be also be regarded as analogous to the well-known $\varphi$-transform .

In addition to the fact that one can find Parseval frames, we offer the following motivations for specializing to the case of homogeneous manifolds. First, on such manifolds, there is the possibility of finding exact formulas for the frame elements. Secondly, in theoretical physics – where many manifolds are considered – symmetry is of capital importance. Third, on homogeneous manifolds, one has the advantage that all of the frame elements at a particular scale can be obtained from each other through the group action, in the same manner as standard wavelets at a particular scale on the real line can be obtained from each other by translation.

Our frames will be band-limited, and hence smooth. One should understand that the notion of band-limitedness on a compact manifold is not canonical. Consider a (connected) compact smooth Riemannian manifold $\bf{M}$, and a smooth elliptic self-adjoint positive differential operator $A$ on it. It is known that the spectrum of $A$, as an operator in the corresponding space $L_{2}(\bf{M})$, is discrete, nonnegative, and accumulates at infinity. Call the eigenvalues $\lambda_{0}\leq\lambda_{1}\leq....$, where we repeat eigenvalues according to their multiplicities. The space $L_{2}(\bf{M})$ has an orthonormal basis $\ u_{\lambda_{0}},u_{\lambda_{1}},...$ consisting of eigenfunctions of $A$.

For a fixed operator $A$ we understand the space of $\omega$-band-limited functions ${\mathbf{E}}_{\omega}(A)$ to be the span of all eigenfunctions $u_{\lambda_{j}}$ such that $\lambda_{j}\leq\omega$.

Formally, then, there is great freedom in the notion of band-limitedness. However, for the purposes of this article, all these spaces of band-limited functions are essentially equivalent, in the sense that they give rise to the same Besov spaces $B^{\alpha q}_{p}(\bf{M})$, at least if $\alpha>0$ and $1\leq p\leq\infty$ (see Theorem 7.5 below).

The plan of the paper is as follows. In section 2, we review some basic facts about compact homogeneous manifolds. In section 3, we discuss properties of band-limited functions associated with a second-order smooth positive elliptic differential operator ${\mathcal{L}}$. It is shown in particular that if $\bf{M}$ is equivariantly embedded into Euclidean space then the span of eigenfunctions of the operator $\mathcal{L}$ is exactly the set of restrictions to $\bf{M}$ of all polynomials in the ambient space. In this section we also give several equivalent definitions of Besov spaces on the manifold. In one of the definitions, we use a global modulus of continuity, constructed through use of certain vector fields on $\bf{M}$. This definition is similar to the original definition of Besov spaces on Euclidean spaces and uses just the notion of smoothness. Later in the article, in Theorems 7.5 and 8.1, we describe the same spaces in terms of approximations by band-limited functions. Thus, as one of the consequences of our results, we obtain a new development of one of the oldest topics of classical harmonic analysis: the relationships between smoothness and rate of approximations by band-limited functions. Specifically, we show that there exists a complete balance between smoothness expressed in terms of modulus of continuity, and the rate of approximation by band-limited functions in all spaces $L_{p}(\bf{M}),$ $\ 1\leq p\leq\infty$, as well as other equivalent definitions of Besov spaces on such manifolds.

In section 4, we describe Plancherel-Polya inequalities (Corollary 4.4) and in section 5 we obtain cubature formulas with desirable properties (Theorem 5.3). In these two sections, we do not use any special properties of homogeneous manifolds or the second-order positive elliptic differential operator ${\mathcal{L}}$; the results hold for any smooth compact manifold, and for any ${\mathcal{L}}$.

In section 6, we use the homogeneous manifold structure in an essential way to prove a crucial fact, namely that, for a particular ${\mathcal{L}}$, the product of two band-limited functions of the same bandwidth $\omega$ is also a band-limited function, with a certain bandwidth $C\omega$, where $C$ is independent of $\omega$. On the sphere, this property is familiar for spherical harmonics; then one may take ${\mathcal{L}}$ to be the spherical Laplacian, and one may take $C=2$. This property of spherical harmonics was used crucially in the construction of spherical needlets in . The generalization to homogeneous manifolds is similarly needed in our construction of band-limited Parseval frames. For more general manifolds, it is not clear how to verify this property, or even if it is true. It was conjectured in that this ”product” property holds for Laplace-Beltrami operators on analytic compact manifolds. If ${\bf M}=G/K$ is a homogeneous manifold, we specifically take ${\mathcal{L}}$ to be the image of the Casimir operator under the differential of the quasiregular representation of $G$ in $L_{2}(\bf{M})$ (see section 2). The operator $-{\mathcal{L}}$ is a sum of squares of certain vector fields on $\bf{M}$. In some common cases, such as compact symmetric spaces of rank one and all compact Lie groups, this operator $\mathcal{L}$ coincides with the corresponding Laplace-Beltrami operator.

A number of the results stated, and methods used, in sections 3-6, are from the articles -.

In section 7, we review some of the results of - , where the Laplace-Beltrami operator was used to construct nearly tight frames, which were then used to characterize Besov spaces. (The Besov space results in used, in addition to results from and , methods of Frazier-Jawerth and Seeger-Sogge .) We argue that the results of - continue to hold if one uses a general ${\mathcal{L}}$ in place of the Laplace-Beltrami operator. The arguments of section 7 do not use any special properties of homogeneous manifolds. However, the point is that, if we are on a homogeneous manifold, we are free to use the ${\mathcal{L}}$ of section 2 in place of the Laplace-Beltrami operator.

Finally, in section 8, by using the results of sections 5 and 6, we construct our Parseval frames on homogeneous manifolds. By using the results of section 7, we show that they are highly localized, and that one can use them to characterize Besov spaces, for the full range of the indices. It is only in the construction of our Parseval frames that we use the results of section 6.

Compact homogeneous manifolds

We review some very basic notions of harmonic analysis on compact homogeneous manifolds , Ch. II. More details on this subject can be found, for example, in , .

Let ${\bf M},\ dim{\bf M}=n,$ be a compact connected $C^{\infty}$-manifold. One says that a compact Lie group $G$ effectively acts on ${\bf M}$ as a group of diffeomorphisms if:

1) every element $g\in G$ can be identified with a diffeomorphism

where $g_{1}g_{2}$ is the product in $G$ and $g\cdot x$ is the image of $x$ under $g$,

2) the identity $e\in G$ corresponds to the trivial diffeomorphism

3) for every $g\in G,\ g\neq e,$ there exists a point $x\in{\bf M}$ such that $g\cdot x\neq x$.

A group $G$ acts on ${\bf M}$ transitively if in addition to 1)- 3) the following property holds:

4) for any two points $x,y\in{\bf M}$ there exists a diffeomorphism $g\in G$ such that

A homogeneous compact manifold ${\bf M}$ is a $C^{\infty}$-compact manifold on which a compact Lie group $G$ acts transitively. In this case ${\bf M}$ is necessary of the form $G/K$, where $K$ is a closed subgroup of $G$. The notation $L_{p}({\bf M}),1\leq p\leq\infty,$ is used for the usual Banach spaces $L_{p}({\bf M},dx),1\leq p\leq\infty$, where $dx$ is an invariant measure.

Every element $X$ of the (real) Lie algebra of $G$ generates a vector field on ${\bf M}$, which we will denote by the same letter $X$. Namely, for a smooth function $f$ on ${\bf M}$ one has

for every $x\in{\bf M}$. In the future we will consider on ${\bf M}$ only such vector fields. The translations along integral curves of such vector fields $X$ on ${\bf M}$ can be identified with a one-parameter group of diffeomorphisms of ${\bf M}$, which is usually denoted as $\exp tX,-\infty<t<\infty$. At the same time, the one-parameter group $\exp tX,-\infty<t<\infty,$ can be treated as a strongly continuous one-parameter group of operators acting on the space $L_{p}({\bf M}),1\leq p\leq\infty$. These operators act on functions according to the formula

The generator of this one-parameter group will be denoted by $D_{X,p}$, and the group itself will be denoted by

According to the general theory of one-parameter groups in Banach spaces , Ch. I, the operator $D_{X,p}$ is a closed operator on every $L_{p}({\bf M}),1\leq p\leq\infty$. In order to simplify notation, we will often write $D_{X}$ in place of $D_{X,p}$.

If g is the Lie algebra of a compact Lie group $G$ then (, Ch. II, Proposition 6.6,) it is a direct sum $\textbf{g}=\textbf{a}+[\textbf{g},\textbf{g}]$, where a is the center of g, and $[\textbf{g},\textbf{g}]$ is a semi-simple algebra. Let $Q$ be a positive-definite quadratic form on g which, on $[\textbf{g},\textbf{g}]$, is opposite to the Killing form. Let $X_{1},...,X_{d}$ be a basis of g, which is orthonormal with respect to $Q$. Since the form $Q$ is $Ad(G)$-invariant, the operator

is a bi-invariant operator on $G$. This implies in particular that the corresponding operator on $L_{p}({\bf M}),\ 1\leq p\leq\infty,$

commutes with all operators $D_{j}=D_{X_{j}}$. This operator $\mathcal{L}$, which is usually called the Laplace operator, is elliptic, and is involved in most of the constructions and results of our paper. However, as we discussed in the introduction, in many of the results prior to section 6, one could use other second order elliptic differential operators.

In some situations the operator $\mathcal{L}$ is essentially the Laplace-Beltrami operator ($-d^{*}d$) of an invariant metric on ${\bf M}$. This happens for example in the following cases.

1) If ${\bf M}$ is a $d$-dimensional torus, and $-\mathcal{L}$ is the sum of squares of partial derivatives.

2) If the manifold ${\bf M}$ is itself a group $G$ which is compact and semi-simple, then $\mathcal{L}$ is exactly the Laplace-Beltrami operator of an invariant metric on $G$ (, Ch. II, Exercise A4).

3) If ${\bf M}=G/K$ is a compact symmetric space of rank one, then the operator $\mathcal{L}$ is proportional to the Laplace-Beltrami operator of an invariant metric on $G/K$. This follows from the fact that, in the rank one case, every second-order operator which commutes with all isometries $x\rightarrow g\cdot x,\>\>\>x\in{\bf M},\>\>\>g\in G,$ is proportional to the Laplace-Beltrami operator (, Ch. II, Theorem 4.11).

Let us stress one more time that in the present paper we use only the properties that the operator $\mathcal{L}$ has the form (2.2) and commutes with all isometries $g:{\bf M}\rightarrow g\cdot{\bf M},\>\>\>g\in G,\>\>\>$ of ${\bf M}$, and we do not explore its relation to the Laplace-Beltrami operator of the invariant metric.

Function spaces on compact homogeneous manifolds

The operator $\mathcal{L}$ is an elliptic differential operator which is defined on $C^{\infty}({\bf M})$, and we will use the same notation $\mathcal{L}$ for its closure from $C^{\infty}({\bf M})$ in $L_{p}({\bf M}),1\leq p\leq\infty$. In the case $p=2$ this closure is a self-adjoint positive definite operator on the space $L_{2}({\bf M})$. The spectrum of this operator is discrete and goes to infinity $0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq...$ . Let $u_{0},u_{1},u_{2},...$ be a corresponding complete system of real-valued orthonormal eigenfunctions, and let $\textbf{E}_{\omega}(\mathcal{L}),\ \omega>0,$ be the span of all eigenfunctions of $\mathcal{L}$, whose corresponding eigenvalues are not greater than $\omega$.

Since $\mathcal{L}$ on the space $L_{2}(M)$ is self-adjoint and positive-definite, there exists a unique positive square root $\mathcal{L}^{1/2}$. Thus the last inequality is equivalent to the inequality

which means that if the Bernstein-type inequalities (3.1) are satisfied for a single $1\leq p\leq\infty$, then they are satisfied for all $1\leq p\leq\infty$.

These embeddings obviously imply the equality

which means that a function on ${\bf M}$ satisfies a Bernstein inequality (3.1) in a norm of $L_{p}(M),1\leq p\leq\infty,$ if and only if it is a linear combination of eigenfunctions of $\mathcal{L}$. As a consequence we have the following Bernstein-Nikolski inequality: for every $\varphi\in\textbf{E}_{\omega}(\mathcal{L})$ and $1\leq p\leq q\leq\infty,$

for a certain constant $C({\bf M})$ which depends only on the manifold.

Let $B(x,r)$ be a metric ball on ${\bf M}$ whose center is $x$ and radius is $r$. The following important Lemma can be found in , .

For any Riemannian manifold of bounded geometry ${\bf M}$ there exists a natural number $N_{{\bf M}}$, such that for any sufficiently small $\rho>0$ there exists a set of points $\{y_{\nu}\}$ such that:

the balls $B(y_{\nu},\rho/4)$ are disjoint,

the balls $B(y_{\nu},\rho/2)$ form a cover of ${\bf M}$,

the multiplicity of the cover by balls $B(y_{\nu},\rho)$ is not greater than $N_{{\bf M}}.$

Any set of points $M_{\rho}=\{y_{\nu}\}$ which is as described in Lemma 3.1 will be called a metric $\rho$-lattice.

(Here we use the usual conventions if $p$ or $q$ is $\infty$. The definition of $B_{p}^{\alpha q}(\bf{R}^{n})$ is independent of the choices of $\Phi,\varphi_{\nu}$ (, page 49). Moreover, $B_{p}^{\alpha q}(\bf{R}^{n})$ is a quasi-Banach space, and the inclusion $B_{p}^{\alpha q}\subseteq{\mathcal{S}}^{\prime}$ is continuous (, page 48). In particular the space $B^{\alpha}_{\infty,\infty}(\bf{R}^{n})={\mathcal{C}}^{\alpha}(\bf{R}^{n})$, which is the usual Hölder space if $0<\alpha<1$, or in general a Hölder-Zygmund space for $\alpha>0$ (, page 51). It is not hard to see, by using the definition and the Fourier transform, that if $K\subseteq\bf{R}^{n}$ is compact, and if $N$ is sufficiently large, then

where the inclusion map is continuous if we regard the left side as a subspace of $C^{N}$.

If $\eta:\bf{R}^{n}\rightarrow\bf{R}^{n}$ is a diffeomorphism which equals the identity outside a compact set, then one can define $F\circ\eta$ for $F\in B_{p}^{\alpha q}(\bf{R}^{n})$, and the map $F\rightarrow F\circ\eta$ is bounded on the Besov spaces (, chapter 2.10). These facts then enable one to define $B_{p}^{\alpha q}({\bf M})$: let $(W_{i},\chi_{i})$ be a finite atlas on ${\bf M}$ with charts $\chi_{i}$ mapping $W_{i}$ into the unit ball on $\bf{R}^{n}$, and suppose $\{\zeta_{i}\}$ is a partition of unity subordinate to the $W_{i}$. Then one defines $B_{p}^{\alpha q}({\bf M})$ to be the space of distributions $F$ on ${\bf M}$ for which

This definition does not depend on the choice of charts or partition of unity ().

Using the closed graph theorem and the fact that each $D_{i}$ is a closed operator in $L_{p}({\bf M}),\ 1\leq p\leq\infty$, it is easy to show that the norm (3.12) is equivalent to the norm

For the same operators as above ($D_{1},...,D_{d},\ d=dim\ G$), let $T_{1},...,T_{d}$ be the corresponding one-parameter groups of translation along integral curves of the corresponding vector fields i.e.

with the usual modifications for $q=\infty$.

The following Theorem follows from general results of the second author about interpolation in spaces of representations of Lie groups -:

When $p=2$ the first of these norms can be changed to

where $c_{j}=\left<f,u_{\lambda_{j}}\right>$ are the Fourier coefficients of $f$.

A description of Besov spaces $B_{2}^{\alpha q}({\bf M}),\ \alpha>0,\ 1\leq q\leq\infty,$ in terms of the best approximation $\mathcal{E}(f,\omega)$ was given in , Theorems 1.1 and 1.2. We will obtain a generalization of these results in Theorem 7.5 below.

Plancherel-Polya (==Marcinkiewicz-Zygmund) inequalities

In this section, we again consider a compact homogeneous Riemannian manifold ${\bf M}$, and the elliptic self-adjoint positive definite operator $\mathcal{L}$ on $L_{2}({\bf M})$, which was introduced in (2.2). However, the results of this section hold for general ${\bf M}$ and ${\mathcal{L}}$ (if at least ${\bf M}$ is smooth, compact and ${\mathcal{L}}$ is a positive elliptic self-adjoint second-order differential operator on ${\bf M}$).

Since the operator $\mathcal{L}$ is of order two, the dimension $\mathcal{N}_{\omega}$ of the space ${\mathbf{E}}_{\omega}(\mathcal{L})$ is given asymptotically by Weyl’s formula

The next two theorems were proved in , , for a Laplace-Beltrami operator on a Riemannian manifold of bounded geometry, but their proofs go through for any $C^{\infty}$-bounded uniformly elliptic self-adjoint positive definite differential operator on ${\bf M}$. In what follows the notation $n=dim\ {\bf M}$ is used.

There exist constants $\>\>C_{1}=C_{1}({\bf M},\mathcal{L})>0\>\>$ and $\>\>\>\rho_{0}({\bf M},\mathcal{L})>0,$ such that for any natural number $m>n/2$, any $0<\rho<\rho_{0}({\bf M},\mathcal{L})$, and any $\rho$-lattice $M_{\rho}=\{x_{k}\}$, the following inequality holds:

There exist constants $C_{2}=C_{2}({\bf M},\mathcal{L})>0,$ and $\>\>\>\rho_{0}({\bf M},\mathcal{L})>0,$ such that for any natural $m>n/2$, any $0<\rho<\rho_{0}({\bf M},\mathcal{L})$, and any $\rho$-lattice $M_{\rho}=\{x_{k}\}$ the following inequality holds

Using the constant $C_{2}({\bf M},\mathcal{L})$ from this Theorem, we define another constant

The previous Theorem and the Bernstein inequality imply the following Plancherel-Polya-type inequalities. Such inequalities are also known as Marcinkewicz-Zygmund inequalities.

There exists a constant $\>\>\>c_{2}=c_{2}({\bf M},\>\>\mathcal{L})$ such that for any $\omega>0$, and for every metric $\rho$-lattice $M_{\rho}=\{x_{k}\}$ with $\rho=c_{0}\omega^{-1/2}$, the following inequalities hold:

for all $f\in{\mathbf{E}}_{\omega}(\mathcal{L})$ and $c_{0}$ defined in (4.3). Moreover, for the same lattice there exists a constant $\>\>c_{1}=c_{1}({\bf M},\>\>\mathcal{L},\>\>\omega)\>\>$ such that

Indeed, if $f\in{\mathbf{E}}_{\omega}({\bf M})$, and $\rho$ in (4.2) is given by $\rho=c_{0}\omega^{-1/2}$ where $c_{0}$ was defined in (4.3), then by the Bernstein inequality

with $c_{2}=2C_{2}({\bf M},\mathcal{L})$ and $C_{2}({\bf M},\mathcal{L})$ is the same as in (4.2). To prove (4.5) we apply elliptic regularity of $\mathcal{L}$ to obtain

By choosing $m_{0}=1+n/2$ for $\>\>m\>\>$ and using Theorem 4.1, we obtain (4.5) with

There exist constants $\>\>\>c_{1}=c_{1}({\bf M},\mathcal{L})>0,\>\>$ $c_{2}=c_{2}({\bf M},\mathcal{L})>0,$ and $\>\>c_{0}=c_{0}({\bf M},\mathcal{L})>0,$ such that for any $\omega>0$, and for every metric $\rho$-lattice $M_{\rho}=\{x_{k}\}$ with $\rho=c_{0}\omega^{-1/2}$, the following Plancherel-Polya inequalities hold:

for all $f\in{\mathbf{E}}_{\omega}(\mathcal{L})$ and $n=\dim\ {\bf M}$.

The following Theorem shows that our lattices (appearing in the previous Theorems) always produce sampling sets with essentially the optimal number of sampling points (see also ,).

If the constant $c_{0}({\bf M},\mathcal{L})>0$ is the same as above, then for any $\omega>0$ and $\rho=c_{0}\omega^{-1/2}$, there exist $C_{1}({\bf M},\mathcal{L}),C_{2}({\bf M},\mathcal{L})$ such that the number of points in any $\rho$-lattice $M_{\rho}$ satisfies the following inequalities

According to the definition of a lattice $M_{\rho}$ we have

Since for certain $c_{1}({\bf M}),c_{2}({\bf M})$, all $x\in{\bf M}$ and all sufficiently small $\rho>0$, one has a double inequality

and since $\rho=c_{0}\omega^{-1/2},$ we obtain that for certain $C_{1}({\bf M},\mathcal{L}),C_{2}({\bf M},\mathcal{L})$ and all $\omega>0$

Since the inequalities (4.10) are in an agreement with Weyl’s formula (4.1), the Theorem shows that if $\omega>0$ is large enough, every uniqueness set $M_{\rho}$ for ${\mathbf{E}}_{\omega}(\mathcal{L})$ contains essentially the ”correct” number of points.

Cubature formulas

Again we work on a compact homogeneous Riemannian manifold ${\bf M}$, and use the operator ${\mathcal{L}}$ of (2.2). However, the results of this section hold for general ${\bf M}$ and ${\mathcal{L}}$ (if at least ${\bf M}$ is smooth and compact, and ${\mathcal{L}}$ is a positive elliptic self-adjoint second-order differential operator on ${\bf M}$).

Corollary 4.4 shows that if $\vartheta_{k}$ is the orthogonal projection of the Dirac measure $\delta_{x_{k}}$ on the space ${\mathbf{E}}_{\omega}(\mathcal{L})$ (in a Hilbert space $H^{-n/2-\varepsilon}({\bf M}),\>\>\varepsilon>0$, which can be defined as the domain of the operator $\mathcal{L}^{-n/4-\varepsilon/2}$) then there exist constants $c_{1}=c_{1}({\bf M},\mathcal{L},\omega)>0,\>\>c_{2}=c_{2}({\bf M}mat,\mathcal{L})>0,$ such that the following frame inequality holds

for all $f\in{\mathbf{E}}_{\omega}(\mathcal{L})$.

Let $M_{\rho}=\{x_{k}\},\ k=1,...,N(M_{\rho}),$ be a $\rho$-lattice on ${\bf M}$ (see Lemma 3.1). We construct the Voronoi partition of ${\bf M}$ associated to the set $M_{\rho}=\{x_{k}\},\ k=1,...,N(M_{\rho})$. Elements of this partition will be denoted as $\mathcal{M}_{k,\rho}$. Let us recall that the distance from each point in $\mathcal{M}_{j,\rho}$ to $x_{j}$ is less than or equal to its distance to any other point of the family $M_{\rho}=\{x_{k}\},\ k=1,...,N(M_{\rho})$. Some properties of this cover of ${\bf M}$ are summarized in the following Lemma. which follows easily from the definitions.

The sets $\mathcal{M}_{k,\rho},\ k=1,...,N(M_{\rho}),$ have the following properties:

4) there exist positive $a_{1},\ a_{2}$, independent of $\rho$ and the lattice $M_{\rho}=\{x_{k}\}$, such that

Our next goal is to prove the following fact.

where $C(K)$ is independent of $\rho$ and the $\rho$-lattice $M_{\rho}$.

We are going to use the following inequality, which easily fellows from Lemma 6.19 in , and which is essentially the Sobolev imbedding theorem:

where $m>n/p,$ and the functions $\{\psi_{\nu}\}$ form the partition of unity which we used to define the Sobolev norm in (3.9). Using (5.5) for $p=1$ we obtain that the following inequality

for some $C(n,m)\geq 0$. Since, by the Schwarz inequality,

we obtain the following estimate, which holds for small $\rho$:

Next, using the Schwarz inequality and the assumption that $m>n=dim\ {\bf M},\ |\alpha|=m,$ we obtain

We square this inequality, and integrate both sides of it over the ball $B(x_{k},\rho/2)$, using the spherical coordinate system $(\tau,\theta).$ We find

where $\tau=\|x-x_{k}\|\leq\rho/2,\ m=|\alpha|>n.$ Let $\left\{\mathcal{M}_{k,\rho}\right\}$ be the Voronoi cover of ${\bf M}$ which is associated with a $\rho$-lattice $M_{\rho}$ (see Lemma 5.1). From here we obtain

where $\ m>n.$ Using the definition of the Sobolev norm and elliptic regularity of the operator $I+\mathcal{L}$, where $I$ is the identity operator on $L_{2}({\bf M})$, we obtain the inequality (5.3). ∎

Now we are going to prove existence of cubature formulas which are exact on ${\mathbf{E}}_{\omega}({\bf M})$, and have positive coefficients of the ”right” size.

There exists a positive constant $a_{0}$, such that if $\rho=a_{0}(\omega+1)^{-1/2}$, then for any $\rho$-lattice $M_{\rho}$, there exist strictly positive coefficients $\lambda_{x_{k}}>0,\ x_{k}\in M_{\rho}$, for which the following equality holds for all functions in ${\mathbf{E}}_{\omega}({\bf M})$:

Moreover, there exists constants $\ c_{1},\ c_{2},$ such that the following inequalities hold:

By using the Bernstein inequality, and our Plancherel-Polya inequalities (4.9), and assuming that

we obtain from (5.3) the following inequality:

where $C_{2}$ is independent of $\rho\in\left(0,(2\sqrt{\omega+1}\right)^{-1})$ and the $\rho$-lattice $M_{\rho}$.

Let $R_{\omega}(\mathcal{L})$ denote the space of real-valued functions in ${\mathbf{E}}_{\omega}(\mathcal{L})$. Since the eigenfunctions of ${\mathcal{L}}$ may be taken to be real, we have ${\mathbf{E}}_{\omega}(\mathcal{L})=R_{\omega}(\mathcal{L})+iR_{\omega}(\mathcal{L})$, so it is enough to show that (5.11) holds for all $f\in R_{\omega}(\mathcal{L})$.

where $P$ is the orthogonal projection onto $V$. Accordingly, if $z$ is the real vector $v-Pw$, then

for all $f\in R_{\omega}(\mathcal{L})$, and hence for all $f\in{\mathbf{E}}_{\omega}(\mathcal{L})$, as desired.

On the product of eigenfunctions of the Casimir operator ℒℒ\mathcal{L} on compact homogeneous manifolds

In this section, we will use the assumption that ${\bf M}$ is a compact homogeneous manifold, and that ${\mathcal{L}}$ is the operator of (2.2), in an essential way.

The following Theorem 6.1 plays a crucial role in our construction of Parseval frames in section 8. Note that some parts of the proof of this Theorem can be found in the papers , .

If ${\bf M}=G/K$ is a compact homogeneous manifold and $\mathcal{L}$ is defined as in (2.2), then for any $f$ and $g$ belonging to ${\mathbf{E}}_{\omega}(\mathcal{L})$, their product $fg$ belongs to ${\mathbf{E}}_{4d\omega}(\mathcal{L})$, where $d$ is the dimension of the group $G$.

First, we are going to show that a function $f\in L_{2}({\bf M})$ belongs to the space ${\mathbf{E}}_{\omega}(\mathcal{L})$ if and only if there exists a constant $C(f,\omega)$ such that the following Bernstein inequality is satisfied for all natural $k$

The fact that the above Bernstein inequality holds true for any $f\in{\mathbf{E}}_{\omega}(\mathcal{L})$ with $C(f,\omega)=1$ is obvious. Conversely, assume that

If a vector $f$ belongs to the space $\mathbf{E}_{\omega}(\mathcal{L})$ and the Fourier series

In the last inequality the fraction $\omega/\lambda_{m+1}$ is strictly less than $1$ and $k$ can be any natural number. This shows that the series (6.2) does not contain terms with $j\geq m+1$, i.e. the function $f$ belongs to $\textbf{E}_{\omega}(\mathcal{L})$.

Now, since every smooth vector field on ${\bf M}$ is a differentiation of the algebra $C^{\infty}({\bf M})$, one has that for every operator $D_{j},1\leq j\leq d,$ the following equality holds for any two smooth functions $f$ and $g$ on ${\bf M}$:

Thus, the function $\mathcal{L}^{k}\left(fg\right)$ is a sum of $(4d)^{k}$ terms of the form

Let us show that the following inequalities hold:

for all $f,g\in{\mathbf{E}}_{\omega}(\mathcal{L})$. First, we note that the operator

commutes with every $D_{j}$ (see the explanation before the formula (2.2) ). The same is true for $\mathcal{L}^{1/2}$. But then

Thus, for $f,g\in{\mathbf{E}}_{\omega}(\mathcal{L})$ we obtain the estimate

Now, by using (6.5) we arrive at the following estimate:

According to previous steps of the proof, this implies that the product $fg$ belongs to ${\mathbf{E}}_{4d\omega}(\mathcal{L})$. The Theorem is proved. ∎

The last part of the Theorem can be proved without referring to the paper . Indeed, the formula (6.5) along with the formula (6.9) imply the estimate

Using the Sobolev embedding Theorem and elliptic regularity of $\mathcal{L}$, we obtain for every $s>\frac{dim{\bf M}}{2}$

where $H^{s}({\bf M})$ is the Sobolev space of $s$-regular functions on ${\bf M}$. Since the operator $\mathcal{L}$ commutes with each of the operators $D_{j}$, the estimate (6.9) gives the following inequality:

which leads to the same result that was obtained above.

Results on General Manifolds

In this section, we explain some general results on compact manifolds. We start afresh in our notation.

Let $({\bf M},g)$ be a smooth, connected, compact Riemannian manifold without boundary with () Riemannian measure $\mu$. Let $L$ be a smooth, positive, second order elliptic differential operator on ${\bf M}$, whose principal symbol $\sigma_{2}(L)(x,\xi)$ is positive on $\{(x,\xi)\in T^{*}{\bf M}:\ \xi\neq 0\}$. For $x,y\in{\bf M}$, let $d(x,y)$ denote the geodesic distance from $x$ to $y$.

for all $t>0$ and all $x,y\in{\bf M}$. (b) For general $f$, the estimate (7.1) at least holds for $0<t\leq 1$.

This was proved in section 4 of , in the special case in which $L=\Delta$, the Laplace-Beltrami operator on ${\bf M}$. (A similar result to part (a) had been proved earlier in and in the special case where ${\bf M}$ was a sphere and $f$ had compact support away from the origin.) The arguments in used certain properties of $\Delta$, which we shall now argue are shared by general $L$. Once this is observed, the proofs in go through just the same as in , and will not be repeated here.

Let us then list the properties of $L$ which were used in section 4 of in the special case $L=\Delta$, and verify that they hold for general $L$.

For $\lambda>0$, let $N(\lambda)$ denote the number of eigenvalues of $L$ which are less than or equal to $\lambda$ (counted with respect to multiplicity). Then for some $c>0$, $N(\lambda)=c\lambda^{n/2}+O(\lambda^{(n-1)/2})$.

$\sqrt{L}$ is a positive elliptic pseudodifferential operator on ${\bf M}$ of order $1$.

If $p(\xi)\in S^{m}_{1}(\bf{R})$ (an ordinary symbol of order $m$ on $\bf{R}$, depending only on the “dual variable” $\xi$), then $p(\sqrt{L})\in OPS^{m}_{1,0}({\bf M})$.

Say $h\in\mathcal{S}(\bf{R})$ is even, and satisfies $\ supp\ \hat{h}\subseteq(-1,1)$, and let $K_{t}^{h}(x,y)$ be the kernel of $h(t\sqrt{L})$. Then for some $C>0$, if $d(x,y)>C|t|$, then $K_{t}^{h}(x,y)=0$.

#1 is a sharp form of Weyl’s theorem, which is true for any second order elliptic differential operator on ${\bf M}$ whose principal symbol is positive on $\{(x,\xi)\in T^{*}{\bf M}:\ \xi\neq 0\}$. (, Corollary 4.2.2). (Actually, weaker forms of Weyl’s theorem would suffice for the arguments in .)

#2 was used implicitly in (specifically, in the use of #3). It follows from Theorem 2 of Seeley , as Seeley himself pointed out in that article. That theorem tells us, in particular, that if $S$ is a classical positive invertible elliptic pseudodifferential operator of order $k>0$ on ${\bf M}$, whose principal symbol is positive on $\{(x,\xi)\in T^{*}{\bf M}:\ \xi\neq 0\}$, then $\sqrt{S}$ is a classical positive elliptic pseudodifferential operator on ${\bf M}$ of order $k/2$. To apply this theorem to obtain #2, one lets $P$ be the projection onto the null space of $L$, which is a finite-dimensional space of smooth functions. Thus $P$ has a smooth kernel. Then one notes that $\sqrt{L}=\sqrt{L+P}-P$.

#3 is an immediate consequence of the main theorem of Strichartz . In fact, that theorem tells us, that if $S$ is a self-adjoint elliptic operator in $OPS^{1}_{1,0}({\bf M})$, then $p(S)\in OPS^{m}_{1,0}({\bf M})$.

#4 is a consequence of the finite speed of propagation property of the wave equation. With no claim of originality, we now explain this in some detail. In this discussion, all differential operators and functions will be taken to be smooth, without further comment.

Suppose that $L_{1}$ is a second-order differential operator on an open set $V$ in $\bf{R}^{n}$, that $L_{1}$ is elliptic, and in fact that, for some $c>0$, its principal symbol $\sigma_{2}(L_{1})(x,\xi)\geq c^{2}|\xi|^{2}$, for all $(x,\xi)\in V\times\bf{R}^{n}$. Suppose that $U\subseteq\bf{R}^{n}$ is open, and that $\overline{U}\subseteq V$. Then if $\ suppF,G\subseteq K\subseteq U$, where $K$ is compact, then any solution $u$ of

on $U$ satisfies $\ supp\ u(t,\cdot)\subseteq\{x:$ dist $(x,K)\leq|t|/c\}$. (This is a special case of Theorem 4.5 (iii) of . In that reference, $V=\bf{R}^{n}$. But we can always extend $L_{1}$ from $U$ to an operator on all of $\bf{R}^{n}$ satisfying the hypotheses, by letting $L_{1}^{\prime}=\psi L_{1}+c^{2}(1-\psi)\Delta$ for a cutoff function $\psi\in C_{c}^{\infty}(V)$ which equals $1$ in a neighborhood of $\overline{U}$.)

It is an easy consequence of this that a similar result holds on manifolds. With $L$ as before, let us look at the problem

on ${\bf M}$. The first thing to note is that the problem has a unique solution in any open $t$-interval about zero. Namely, if $F=\sum_{k}a_{k}\varphi_{k}$ and $G=\sum_{k}b_{k}\varphi_{k}$, where the $\varphi_{k}$ are an orthonormal basis of eigenfunctions of $L$, with corresponding eigenvalues $\lambda_{k}$, then the solution is

where we interpret $\frac{\sin(\sqrt{\lambda_{k}}t)}{\sqrt{\lambda_{k}}}$ as $t$ if $\lambda_{k}=0$. Note also that

We then claim that there is a $C>0$, depending only on ${\bf M}$ and $L$, such that if $\ supp\ F,G\subseteq K\subseteq{\bf M}$, then the solution $u$ satisfies $\ supp\ u(t,\cdot)\subseteq\{x:d(x,K)\leq C|t|\}$, where now $d$ is geodesic distance. This is proved as follows:

It is enough to show that, for some $\delta>0$, the result is true whenever $|t|<\delta$. For, suppose that this is known. It suffices then to show that if, for some $T>0$, the result is true whenever $|t|<T$, then it is also true whenever $|t|<T+\delta$. For this, say $T\leq t<T+\delta$, and select $t_{0}<T$ with $t-t_{0}<\delta$. By assumption, $\ supp\ u(t_{0},\cdot)\subseteq K^{\prime}:=\{x:d(x,K)\leq Ct_{0}\}$, and thus also $\ supp\ u_{t}(t_{0},\cdot)\subseteq K^{\prime}$. We clearly have that $u(t,x)=v(t-t_{0},x)$, where $v$ is the solution of

as claimed. Similarly if $-T\geq t\geq-T-\delta$.

It suffices to show that, for some $\delta,\epsilon>0$, the result is true whenever $|t|<\delta$, and the supports of $F$ and $G$ are both contained in an open ball $B$ of radius $\epsilon$. For, we could then cover ${\bf M}$ by a finite number of such open balls, and choose a partition of unity $\{\zeta_{j}\}$ subordinate to this covering. If we let $(f_{j},g_{j})=(\zeta_{j}f,\zeta_{j}g)$, and if we let $u_{j}$ be the solution with data $f_{j},g_{j}$ in place of $f,g$, then surely $u=\sum_{j}u_{j}$. Then surely $\ supp\ u(t,\cdot)\subseteq\{x:d(x,K)\leq C|t|\}$ as desired.

To prove #4, it suffices to write (for some $c$)

for any $F\in C^{\infty}({\bf M})$. (This is easily verified by using the eigenfunction expansion of $F$ and the Fourier inversion formula.) #4 follows at once from (7.8) and the “claim”.

Thus we have Theorem 7.1 for general $L$.

We turn now to Besov spaces. For the rest of this section, we fix $a>1$. We also fix $\alpha,p,q$ with $-\infty<\alpha<\infty$ and $0<p,q\leq\infty$. We let $B_{p}^{\alpha q}$ be the Besov space of section 3.

We fix a finite set ${\mathcal{P}}$ of real $C^{\infty}$ vector fields on ${\bf M}$, whose elements span the tangent space at each point. We also fix a spanning set of the differential operators on ${\bf M}$ of degree less than or equal to $J$ (for any fixed $J$):

The following results were obtained in Lemmas 2.4, 3.2 and 3.3 of , again in the special case $L=\Delta$. In the present article, as we shall see, the technical restrictions on $l$ and $M$ in Lemmas 7.3 and 7.4 below will end up playing no role, so the reader is advised not to pay undue attention to them.

Say $l,M$ are integers with $l\geq 0$ and $M>n$. Then there exists $C>0$ as follows.

Say $\sigma,\nu\in\bf{R}$ with $\sigma\geq\nu$.

Say $x_{0}\in{\bf M}$, and suppose that $\varphi_{1}=L^{l}\Phi$, where $\Phi\in C^{2l}({\bf M})$ satisfies:

Also suppose $x_{1}\in{\bf M}$, that $\varphi_{2}\in C^{2l}({\bf M})$, and that for all $y\in{\bf M}$,

Fix $b>0$. Also fix an integer $l\geq 1$ with

where here $x_{+}=\max(x,0)$. Fix $M$ with $(M-2l-n)p>n+1$ if $0<p<1$, $M-2l-n>n+1$ otherwise.

Suppose that, for each $j\geq 0$, and each $k$,

Then, for every $F$ in the inhomogeneous Besov space $B_{p}^{\alpha q}({\bf M})$, if we let

Fix $b>0$. Also fix an integer $l\geq 1$ with

where here $x_{+}=\max(x,0)$. Fix $M$ with $(M-n)p>n+1$ if $0<p<1$, $M-n>n+1$ otherwise.

If $0<p<1$, we also fix a number $\rho>0$. Then there exists $C>0$ as follows.

Again, these three results were proved in in the special case in which $L=\Delta$. The arguments in used certain properties of $\Delta$, which we shall now argue are shared by general $L$. Once this is observed, the proofs in go through just the same as in , and will not be repeated here.

Let us then list the properties of $L$ which were used in the proofs of these lemmas in in the special case $L=\Delta$, and verify that they hold for general $L$.

In the proof of Lemma 7.2 in the special case $L=\Delta$ (which was Lemma 2.4 in ), the only property used of $L$ was that it was a smooth, second-order partial differential operator, which satisfied $\langle LF,G\rangle=\langle F,LG\rangle$ for all $F,G\in C^{2}({\bf M})$.

In the proof of Lemma 7.3 in the special case $L=\Delta$ (which was Lemma 3.2 in ), the only properties of $L$ that were used was that it was a smooth, second-order partial differential operator, and that Lemma 7.2 above holds.

In the proof of Lemma 7.4 in the special case $L=\Delta$ (which was Lemma 3.3 in ), again these properties of $L$ were used: it is a smooth, second-order partial differential operator, and Lemma 7.2 above holds. In addition, the following result of Seeger-Sogge was used:

Choose $\beta_{0}\in C_{c}^{\infty}((1/4,16))$, with the property that for any $s>0$, $\sum_{\nu=-\infty}^{\infty}\beta^{2}_{0}(2^{-2\nu}s)=1$. For $\nu\geq 1$, define $\beta_{\nu}\in C_{c}^{\infty}((2^{2\nu-2},2^{2\nu+4}))$, by $\beta_{\nu}(s)=\beta_{0}(2^{-2\nu}s)$. Also, for $s>0$, define the smooth function $\beta_{-1}(s)$ by $\beta_{-1}(s)=\sum_{\nu=-\infty}^{-1}\beta(2^{-2\nu}s)$. (Note that $\beta_{-1}(s)=0$ for $s>4$.) Then (), for $F\in C^{\infty}({\bf M})$, $\|F\|_{B_{p}^{\alpha q}}$ is equivalent to the $l^{q}$ norm (actually a quasi-norm if $0<q<1$) of the sequence $\{2^{\nu\alpha}\|\beta_{\nu}(L)F\|_{p}:-1\leq\nu\leq\infty\}$.

(Note that the notation of is slightly different from that of ; what calls $\beta_{k-1}(s^{2})$, is called $\beta_{k}(s)$ in .) By Theorem 4.1 of , the result does hold for general $L$, and in fact would hold if we only knew that $L=P^{2}$ for some first-order elliptic, positive, classical pseudodifferential operator on ${\bf M}$. Of course, we do know that our $L$ satisfies this condition (see our comments on Seeley’s work above, in our discussion of point #2, following Theorem 7.1).

Thus we do indeed have Lemmas 7.3 and 7.4 for general $L$. In the next section, we will put this to use in the case where ${\bf M}$ is a compact homogeneous manifold.

To conclude this section, we shall continue to work on our general ${\bf M}$, and show how Theorem 7.1 and the Seeger-Sogge characterization of Besov spaces can be used to obtain a description of Besov spaces in terms of best approximations by band-limited functions. This result gives a generalization of a part of Theorem 1.1 of , where such a description was given in the case $p=2$ for manifolds of bounded geometry. Our arguments are analogous to those of , Proposition 5.3, where the case in which ${\bf M}$ is the sphere was dealt with.

We need to make a few observations first. In the situation of Theorem 7.1 (a), it is easy to see from eigenfunction expansions that $f(t^{2}L)$ maps distributions on ${\bf M}$ to distributions on ${\bf M}$. We have:

Indeed, say that $K_{t}$ is the kernel of $f(t^{2}L)$; it suffices to observe that for some $C>0$, $\int|K_{t}(x,y)|dx\leq C$ for all $y$, and $\int|K_{t}(x,y)|dy\leq C$ for all $x$. This however is evident from (7.1) with $j=k=0$, since by (21) of , for any $N>n$ there is a $C_{N}$ such that $\int[1+d(x,y)/t]^{-N}dy\leq C_{N}t^{n}$ for all $x$.

Suppose next that $\alpha>0$ and $1<p\leq\infty$, $0<q<\infty$. Then, on ${\bf M}$,

The argument that $\sum_{\nu=0}^{\infty}\|\check{\varphi}_{\nu}*F\|_{L_{p}}$ converges absolutely may be adapted to ${\bf M}$. Let the $\beta_{nu}$ be as in the Seeger-Sogge result described above. A similar argument, using their characterization of Besov spaces, shows that, assuming $\alpha>0$ and $1<p\leq\infty$, $0<q<\infty$, one has:

We let ${\bf E}_{\omega}(L)$ denote the span of all eigenfunctions of $L$ with eigenvalue less than or equal to $\omega$. Let ${\mathcal{D}}^{\prime}$ denote the space of distributions on ${\bf M}$. We note:

Indeed, the convergence of the series in ${\mathcal{D}}^{\prime}$ follows from an examination of the eigenfunction expansion of a smooth function. Next, let $G=F-\sum_{\nu=j+1}^{\infty}\beta^{2}_{\nu}(L)F$. By the properties of the $\beta_{\nu}$, note that one has that $\sum_{\nu=j+1}^{\infty}\beta^{2}_{\nu}(s)=1$ for $s\geq 2^{2j+4}$. If $u$ is an eigenfunction of $L$ with eigenvalue $\lambda$, we then see that $G(u)=0$ if $\lambda\geq 2^{2j+4}$. Let $\{u_{i}\}$ be an orthonormal basis for ${\bf E}_{2^{2j+4}}(L)$, consisting of real-valued eigenfunctions, and say $G(u_{i})=a_{i}$. Then $G-\sum_{i}a_{i}u_{i}$ annihilates all eigenfunctions of $L$, so it must be zero, as needed.

For $1\leq p\leq\infty$, if $F\in L_{p}$, we let

Say $\alpha>0$, $1\leq p\leq\infty$, and $0<q<\infty$. Then $F\in B^{\alpha q}_{p}$ if and only if $F\in L_{p}$ and

We first show, for $F\in B^{\alpha q}_{p}$, that $\|F\|^{A}_{B^{\alpha q}_{p}}\leq C\|F\|_{B^{\alpha q}_{p}}$. Because of (7.23), it is enough to show that

But by (7.25), (7.22), and (7.24), for $j\geq 2$ we have

For the converse, say $\nu\geq 0$. We simply note that if $G\in E_{2^{2\nu-2}}(L)$, then $\beta_{\nu}(L)G=0$. Thus, by (7.22), if $F\in L^{p}$, then $\|\beta_{\nu}(L)(F)\|_{p}=\|\beta_{\nu}(L)(F-G)\|_{p}\leq C\|F-G\|_{p}$. Accordingly,

From this, we find at once that $\|F\|_{B^{\alpha q}_{p}}\leq C\|F\|^{A}_{B^{\alpha q}_{p}}$. $\Box$

Parseval frames and Besov spaces

We now revert to the notation of sections 1 through 6. We modify the construction of “needlets” in , to produce a Parseval frame on ${\bf M}$.

Say $a>1$. Choose a function $f\in C_{c}^{\infty}$, supported in the interval $[a^{-2},a^{4}]$ such that