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Reproducing Kernel Hilbert Algebras on Compact Abelian Groups

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Reproducing Kernel Hilbert Algebras on Compact Abelian Groups REPRODUCING KERNEL HILBERT ALGEBRAS ON COMPACT ABELIAN GROUPS SUDDHASATTWA DAS AND DIMITRIOS GIANNAKIS Abstract. We describe the construction of a nested family of reproducing kernel Hilbert spaces ∗ Hτ , τ > 0, on a compact abelian group G, which also have a unital Banach -algebra structure under the pointwise product and complex conjugation of functions. For any τ > 0, Hτ is dense in the space of continuous functions on G, and has an explicit orthonormal basis consisting of suitably scaled characters, satisfying a convolution criterion. In the case of the circle, G = S1, these spaces are induced from a fractional diffusion semigroup, generated by the p/2-th power of the Laplacian with 0 <p< 1. In higher-dimensional tori, the reproducing kernels of Hτ have the structure of product kernels built from the one-dimensional case. Aspects of the spectra and state spaces of Hτ are also studied. In particular, the spectrum of Hτ is shown to be homeomorphic to G, analogously ∗ to the corresponding property of the C -algebra of continuous functions on G. 1. Introduction and statement of main results In many areas of mathematics and statistics, such as functional analysis, harmonic analysis, analysis on manifolds, and ergodic theory, one is interested in the analysis and approximation of linear operators on functions. In these and many other contexts, it is desirable that the function space employed be endowed with properties, such as a norm, inner product, and algebraic product, which are compatible with the structure of the underlying space and the codomain. For example, if (X, Σ,µ) is a measure space it is natural to consider the Lp(µ) spaces of equivalence classes of complex-valued functions on X. These spaces are Banach spaces for any p [1, ], and in the case p = they become abelian C∗-algebras under the product inherited from∈ multiplication∞ of complex numbers∞ (the codomain). The case p = 2 is a Hilbert space, which allows for the important operation of orthogonal projection, but in general does not have an algebra structure under function multiplication. Moreover, aside from special cases, there is no notion of pointwise evaluation of elements of Lp(µ). If X is a locally compact Hausdorff space, it is natural to consider the space C0(X) of bounded continuous functions vanishing at infinity, equipped with the uniform norm. This space has the structure of an abelian C∗-algebra of functions with a well-defined notion of pointwise evaluation, but lacks the inner-product structure of L2 spaces. If X is a compact arXiv:1912.11664v6 [math.FA] 17 Nov 2020 Cr manifold, the Cr(X) space of r-times continuously differentiable functions on X is a Banach ∗-algebra (though not a C∗-algebra) containing elements of higher regularity than C(X). If X is a locally compact Hausdorff group, then there are different Banach and C∗-algebras constructed by completion of the convolution algebra of compactly supported functions with respect to the Haar measure, e.g., the L1(X) algebra and the group C∗-algebra. Meanwhile, a bridge between L2 and C-spaces is the class of reproducing kernel Hilbert spaces (RKHSs) of complex-valued functions. Every RKHS on a set X has Hilbert space structure, analogously to L2, induced by a unique, positive-definiteH kernel function k : X X C. The kernel is also related to the pointwise evaluation at x X through the reproducing× property,→ ∈ f(x)= k(x, ),f , f , h · iH ∀ ∈H 2010 Mathematics Subject Classification. 46J10, 46E22, 22B99. Key words and phrases. Abelian groups, Banach algebras, reproducing kernel Hilbert spaces. 1 2 SUDDHASATTWADASANDDIMITRIOSGIANNAKIS and the evaluation functional f f(x) is bounded analogously to C-spaces. If X is a topological space, or a Cr manifold, equipped7→ with a Borel measure µ, there is a rich theory on embeddings of RKHSs into Lp(µ) and/or Cr(X) spaces [1], which in some ways parallels the classical embedding theorems for Sobolev spaces. RKHSs have found widespread use in the mathematical theory of learning [2, 3], and more recently in ergodic theory [4]. Yet, despite their attractive properties, RKHSs generally lack the algebra structure enjoyed by Cr and L∞ spaces. Intuitively, an RKHS induces a notion of regularity of functions (depending on the kernel), which need not be preserved under products of functions. While there are important examples of Hilbert spaces of functions with an algebra structure, such as Sobolev algebras of sufficient regularity on Rn [5], these spaces rely on the existence of an underlying differentiable structure. Other examples of Sobolev algebras include Banach spaces induced from abstract “Carr´e du Champ” operators [6], or positive characters on Lie groups [7]. More generally, in situations where an RKHS fails to be closed under multiplication, one can employ weak product spaces with a Banach algebra structure [8, 9]. The main contribution of this work is to construct a class of function spaces which are simulta- neously RKHSs and unital Banach ∗-algebras under the pointwise product of functions. We refer to such spaces as reproducing kernel Hilbert algebras (RKHAs). Our focus is on the setting where X has the structure of a compact abelian group, which is relevant in a variety of signal processing and ergodic theory applications such as analysis of periodic or quasiperiodic signals. However, some of our results also hold for more general locally compact abelian groups. We now state more precisely the algebraic properties required of an RKHA, starting from the notion of a Hilbert ∗-algebra, or H∗-algebra, introduced by Ambrose [10]. Definition 1. An H∗-algebra is a Hilbert space H over the complex numbers, equipped with a continuous binary multiplication operation making it a Banach algebra. That is, we require that fg C f g , f, g H, k kH ≤ k kH k kH ∀ ∈ where C is a constant and H the norm induced by the inner product , H of H. Additionally, we require that for every f k·kH there exists an adjoint element f ∗ H,h· such·i that ∈ ∈ fg,h = g,f ∗h = f, hg∗ , g, h H. (1) h iH h iH h iH ∀ ∈ Equation (1) implies that for each f H, the operations of left and right multiplication by f is a norm-preserving, ∗-homomorphism∈ between H and B(H), the C∗-algebra of bounded linear maps on H. Note that this notion of a Hilbert algebra is somewhat different from other definitions [11, 12], mainly by the requirement of a Banach algebra property. A classical example of a Hilbert ∗-algebra is the L2(X) convolution algebra of a compact topological group X equipped with a Haar measure. In what follows, we will work with a weaker notion than the H∗-algebras in Definition 1, obtained by relaxing the adjoint condition in (1), while requiring an isometric antilinear involution as in Banach ∗-algebras. We refer to the resulting algebraic structure as a semi-H∗-algebra. Definition 2. A semi-H∗-algebra is a Hilbert space H over the complex numbers, equipped with a continuous binary multiplication operation and an antilinear, norm-preserving involution ∗ : H H, making it a Banach ∗-algebra. That is, we require that → fg C f g , f ∗ = f , (fg)∗ = g∗f ∗, f, g H, (2) k kH ≤ k kHk kH k kH k kH ∀ ∈ where C is a constant and the norm of H. k·kH The RKHAs constructed in this work are unital semi-H∗-algebras of complex-valued functions. Moreover, we employ pointwise function multiplication and complex conjugation as the algebraic product and involution, respectively, akin to C∗-algebras of functions. 3 Definition 3. An RKHS of complex-valued functions on a set X is said to be a reproduc- ing kernel Hilbert algebra (RKHA)H if it is a (commutative) semi-H∗-algebra when endowed with the pointwise product and complex conjugation of functions as the multiplication operation and antilinear involution, respectively. If is unital, we will use the symbol 1 to denote the unit of , i.e., the function equal to 1 at H X H every point in X. Note that we do not require that the norm of 1X is equal to 1. Remark 1. In this paper, we have opted to state the Banach algebra condition fg C f g k kH ≤ k kHk kH allowing a general constant C, as opposed to the more conventional definition fg H f H g H. This choice does not affect any of the results presented below, as C can be absorbedk k in≤ a redefinition k k k k of the reproducing kernel k of to a scaled kernel k˜ := C2k. The corresponding RKHA, ˜, has the same elements as , and satisfiesH fg f g , so we can view as an equivalentH H k kH˜ ≤ k kH˜ k kH˜ k·kH˜ norm to H. In the present work, it is natural to allow a general C, as this enables a direct identificationk·k the reproducing kernels of certain RKHAs with Markov transition kernels without having to employ additional normalization factors. That being said, it should be kept in mind that a number of standard results on Banach algebras require appropriate modification when C 1. For example, the fact that a state on a unital Banach ∗-algebra with isometric involution has≥ unit operator norm does not necessarily hold when C 1. ≥ Locally compact abelian groups. For the rest of the paper, G X will be assumed to be a locally compact abelian group (LCA), equipped with a Haar measure≡ µ. If G is compact, we will assume that µ is normalized to a probability measure. We let Gˆ denote the dual group of G, i.e., the abelian group of continuous homomorphisms γ : G S1, equipped with its dual measure,µ ˆ → [13, 14].
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