Product Reproducing Kernel Hilbert Spaces

• Product reproducing kernel Hilbert spaces are composite function spaces created via tensor products or Cartesian products of RKHSs, preserving key reproducing properties.
• They leverage structured kernels—such as product and sum kernels—to enable robust operator theory analysis and simplify adaptive learning algorithm design.
• These constructions connect functional analysis with practical applications, including multiparameter estimation and the design of efficient, structured learning methods.

A product reproducing kernel Hilbert space arises as a systematic construction blending multiple RKHSs, often via tensor products or Cartesian products, thereby capturing composite function spaces with structure guided by their underlying kernels. This concept illuminates both abstract categorical properties essential in functional analysis and concrete algorithmic strategies in adaptive learning.

1. Hilbert Space Tensor Products and RKHS Structure

Given Hilbert spaces $H_1$ and $H_2$, their algebraic tensor product $H_1 \otimes_a H_2$ consists of finite linear combinations of pure tensors $\xi_1 \otimes \xi_2$. The inner product on $H_1 \otimes_a H_2$ is defined on pure tensors by

$$\langle \xi_1 \otimes \xi_2, \eta_1 \otimes \eta_2 \rangle_{H_1 \otimes H_2} = \langle \xi_1, \eta_1 \rangle_{H_1} \cdot \langle \xi_2, \eta_2 \rangle_{H_2}$$

and extended sesquilinearly, yielding the Hilbert-space tensor product $H_1 \otimes H_2$ upon completion. This tensor product enjoys a universal property: for any Hilbert space $K$, bounded bilinear maps $B : H_1 \times H_2 \to K$ correspond uniquely to bounded linear maps $\widetilde B : H_1 \otimes H_2 \to K$ via $B(\xi, \eta) = \widetilde B(\xi \otimes \eta)$. The construction is functorial with respect to bounded linear maps: if $T_1 \in B(H_1, H_1')$ and $T_2 \in B(H_2, H_2')$, then $T_1 \otimes T_2$ is bounded with operator norm $\leq \|T_1\| \cdot \|T_2\|$.

For $H_1 \subset \mathcal{L}(X_1)$ and $H_2 \subset \mathcal{L}(X_2)$ as RKHSs with reproducing kernels $k_1, k_2$, the tensor product $H_1 \otimes H_2$ comprises functions $F(x_1, x_2) = f(x_1)g(x_2)$, $f \in H_1$, $g \in H_2$, defined on $X_1 \times X_2$. The reproducing kernel is the product kernel

$$K((x_1, x_2), (y_1, y_2)) = k_1(x_1, y_1) \cdot k_2(x_2, y_2)$$

and evaluation functionals are bounded due to the reproducing property in each factor. Therefore, the Hilbert-space tensor product of two RKHSs is again an RKHS, equipped with the product kernel (Giannakis et al., 2 Jan 2024).

2. Cartesian Products and Direct Sums of RKHSs

For a family of RKHSs $(H_q, \langle \cdot, \cdot \rangle_{H_q})$, $q = 1,\ldots, m$, of real-valued functions on a common set $U$ with kernels $K_q$, the Cartesian product

$H = H_1 \times H_2 \times \cdots \times H_m$

consists of $m$-tuples $f = (f_1, \ldots, f_m)$ with $f_q \in H_q$. The inner product is given by

$$\langle f, g \rangle_H = \sum_{q=1}^m \langle f_q, g_q \rangle_{H_q}$$

and the induced norm is $\|f\|_H = \sqrt{\sum_q \|f_q\|_{H_q}^2}$. The associated reproducing kernel is

$K_H(u, v) = \sum_{q=1}^m K_q(u, v)$

with pointwise evaluation satisfying $f_q(u) = \langle f_q, K_q(\cdot, u) \rangle_{H_q}$ for each $q$ (Yukawa, 2014). The Cartesian product is naturally isomorphic to the direct sum when $H_p \cap H_q = \{0\}$ for $p \neq q$, with unique decomposition and identical kernel and norm structures.

The sum-space $H^+ = \{f_1 + \cdots + f_m : f_q \in H_q\}$, equipped with the infimum norm over all decompositions, is also an RKHS with reproducing kernel $K_H(u, v)$. In the direct-sum case, the map $T: H_1 \times \cdots \times H_m \to H^+$ defined by $T((f_1, \ldots, f_m)) = f_1 + \cdots + f_m$ is an isometric isomorphism.

3. Product Structures in Reproducing Kernel Hilbert Algebras

A reproducing kernel Hilbert algebra (RKHA) is an RKHS $H \subset \mathcal{L}(X)$ whose pointwise-diagonal map

$$\Delta : \operatorname{Span}\{k_x : x \in X\} \to H \otimes H, \quad \Delta(k_x) = k_x \otimes k_x$$

extends to a bounded operator $\Delta \in B(H, H \otimes H)$. Its adjoint $\Delta^*$ is a bounded "multiplication" map $m = \Delta^*: H \otimes H \to H$, $m(f \otimes g) = f \cdot g$. If $(H_1, \Delta_1)$ and $(H_2, \Delta_2)$ are RKHAs, then $H_1 \otimes H_2$ with the product kernel on $X_1 \times X_2$ is also an RKHA, carrying a comultiplication

$$\Delta_{H_1 \otimes H_2} = (\operatorname{id} \otimes \tau \otimes \operatorname{id}) \circ (\Delta_1 \otimes \Delta_2)$$

where $\tau$ is the flip operator, and for pure kernel-tensors, $$\Delta(k_{(x_1, x_2)}) = k_{(x_1, x_2)} \otimes k_{(x_1, x_2)}$$. If $H_1$ and $H_2$ are unital, so is the product, with unit $1 \otimes 1$ (Giannakis et al., 2 Jan 2024).

The subcategory of RKHAs is closed under Hilbert-space tensor product, thus forming a monoidal subcategory of (RKHS, $\otimes$).

4. Diagonal Restrictions and Powers of Kernels

Given an RKHS with kernel $k$ (e.g., the analytic Dirichlet space on $\mathbb{D} \subset \mathbb{C}$), one can form its $d$-fold Hilbert-space tensor power $\mathcal{D}^{\otimes d}$, which is a RKHS on $\mathbb{D}^d$ with kernel

$$K_{\otimes}((z_1, \ldots, z_d), (w_1, \ldots, w_d)) = \prod_{j=1}^d k(z_j, w_j)$$

Restricting this space to the diagonal $\{(z, \ldots, z) : z \in \mathbb{D}\}$ yields a RKHS on $\mathbb{D}$ with kernel $k(z, w)^d$. The map from the tensor power restricted to the diagonal to the space with kernel $k^d$ is a unitary isomorphism (Arcozzi et al., 2015). This illustrates a fundamental interplay between tensor product (external) and Hadamard (pointwise power) constructions in the context of RKHSs.

5. Adaptive Learning in Product RKHSs

Adaptive learning algorithms can be formulated in product RKHSs, particularly for estimation problems involving functions with multiple components. The CHYPASS (Cartesian HYPASS) algorithm operates by iterative orthogonal projections in the product space $H_1 \times \cdots \times H_m$, leveraging the sum-kernel structure. The combined dictionary subspace is $M_n = M_{1,n} \oplus \cdots \oplus M_{m,n}$, and the interpolation hyperplane is

$$\Pi_n = \{ f \in M_n : \langle f, \Phi(u_n) \rangle_H = d_n \}$$

where $\Phi(u) = (K_1(\cdot, u), \ldots, K_m(\cdot, u))$ is the concatenated feature map. The explicit update for component $q$ is

$$\varphi_{q, n+1} = \varphi_{q, n} + \lambda_n \frac{d_n - \sum_{\ell=1}^m \varphi_{\ell, n}(u_n)}{\sum_{\ell=1}^m \|P_{M_{\ell, n}} K_\ell(\cdot, u_n)\|_{H_\ell}^2} P_{M_{q, n}} K_q(\cdot, u_n)$$

Selective updating and hyperslab variants reduce computational complexity and allow for approximate projections as the application demands. In the direct-sum case, CHYPASS coincides with HYPASS in the single RKHS with kernel $K_H = \sum_q K_q$ (Yukawa, 2014).

6. Spectrum and Monoidal Functoriality

For unital RKHAs, the spectrum $\operatorname{sp}(H)$, defined as the set of nonzero multiplicative functionals or equivalently the set of nonzero group-like elements of $(H, \Delta)$, forms a compact Hausdorff space in the weak-* topology. The spectrum extends functorially: for $T: H \to K$, one has $$T^*|_{\operatorname{sp}(K)}: \operatorname{sp}(K) \to \operatorname{sp}(H)$$. Moreover, there is a natural homeomorphism

$$\Phi: \operatorname{sp}(H_1) \times \operatorname{sp}(H_2) \to \operatorname{sp}(H_1 \otimes H_2)$$

given by $\Phi(\chi_1, \chi_2) = \chi_1 \otimes \chi_2$, establishing the spectrum as a strict monoidal functor from $(\mathrm{RKHA}_u, \otimes)$ to (Compact Hausdorff spaces, $\times$) (Giannakis et al., 2 Jan 2024).

7. Connections to Operator Theory and Function Spaces

Product RKHSs constructed via tensor powers exhibit rich operator-theoretic properties. For instance, the Hilbert space $H_d$ with kernel $k_d(z, w) = (k(z, w))^d$ for the analytic Dirichlet kernel $k$ on the unit disk has an explicit orthonormal basis, norm estimates, and connections to Hankel-type operators. For $d=2$, $H_2$ corresponds to the Hilbert-Schmidt class Hankel operators, while for higher $d$, it yields multilinear Hankel-type operators whose Hilbert-Schmidt norm is equivalent to the $H_d$ norm. Furthermore, Carleson measure and multiplier criteria, as well as complete Nevanlinna–Pick (CNP) properties, are explicitly determinable based on the associated weighted norm and kernel properties (Arcozzi et al., 2015). This integration with classical function space theory further underscores the structural reach of product RKHS constructions.