Reproducing Kernel Hilbert Space vs. Frame Estimates

We consider conditions on a given system $\mathcal{F}$ of vectors in Hilbert space $\mathcal{H}$, forming a frame, which turn $\mathcal{H}$ into a reproducing kernel Hilbert space. It is assumed that the vectors in $\mathcal{F}$ are functions on some set $\Omega$. We then identify conditions on these functions which automatically give $\mathcal{H}$ the structure of a reproducing kernel Hilbert space of functions on $\Omega$. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.


Introduction
A reproducing kernel Hilbert space (RKHS) is a Hilbert space H of functions on a set, say Ω, with the property that f (t) is continuous in f with respect to the norm in H. There is then an associated kernel. It is called reproducing because it reproduces the function values for f in H. Reproducing kernels and their RKHSs arise as inverses of elliptic PDOs, as covariance kernels of stochastic processes, in the study of integral equations, in statistical learning theory, empirical risk minimization, as potential kernels, and as kernels reproducing classes of analytic functions, and in the study of fractals, to mention only some of the current applications. They were first introduced in the beginning of the 20ties century by Stanisaw Zaremba and James Mercer, Gábor Szegö, Stefan Bergman, and Salomon Bochner. The subject was given a global and systematic presentation by Nachman Aronszajn in the early 1950s. The literature is by now vast, and we refer to the following items from the literature, and the papers cited there [4], [1], [16], [12], [15], [8]. Our aim in the present paper is to point out an intriguing use of reproducing kernels in the study of frames in Hilbert space.

An Explicit Isomorphism
Let H be a separable Hilbert space, and let {ϕ n } n∈N be a system of vectors in H. Then we shall study relations of H as a reproducing kernel Hilbert space (RKHS) subject to properties imposed on the system {ϕ n } n∈N . A RKHS is a Hilbert space H of functions on some set Ω such that for all t ∈ Ω, there is Under mild restrictions on {ϕ n } n∈N , it turns out that G defines an unbounded (generally) selfadjoint linear operator Let F denote finitely supported sequence with (2) defined on all finitely supported sequence (c j ) F in l 2 , i.e., (c j ) ∈ F if and only if there exists n ∈ Z + such that c j = 0, for all j ≥ n; but note that n depends the sequences. Denoting δ j the canonical basis in l 2 , δ j (j) = δ i,j , note F = span{δ j |j ∈ N}. Further, note that the RHS in (2) is well defined when j | ϕ k , ϕ j H | 2 < ∞, for all k ∈ N.
Theorem 1 Suppose H, {ϕ n } are given. Assume that (a) each ϕ n is a function on Ω where Ω is a given set (b) {ϕ n } is a frame in H, see (10) and (11), and that (c) {ϕ n (t)} ∈ l 2 , for all t ∈ Ω then H is a reproducing kernel Hilbert space (RKHS) with kernel where l(t) = {ϕ n (t)} ∈ l 2 , and where G is the Gramian of G = ( ϕ n , ϕ m H ). Moreover, G defines selfadjoint operator in l 2 with dense domain, and we get an isometric isomorphism (4) and Lemma 6, the frame operators T and T * are as follows: Given H, {ϕ n }, set to be the two linear operators and adjoint T * as follows:

Lemma 1
We have Do the real case first, then it is easy to extend to complex valued functions. Note that T T * is an operator in l 2 , i.e., It has a matrix-representation as follows Proof By (8), we have which is the desired conclusion (9).
Both T * T and T T * are self-adjoint: If B i , i = 1, 2 are the constants from the frame estimates, then: equivalently We have If B 1 = B 2 = 1, then we say that {ϕ n } n∈N is a Parseval frame. For the theory of frames and some of their applications, see e.g., [7], [6], [5] and the papers cited there.
By the polar-decomposition theorems, see e.g., [11] we conclude that there is a unitary isomorphism u : H → l 2 such that T = u(T * T ) 1/2 = (T T * ) 1/2 u; and so in particular, the two s.a. operators T * T and T T * are unitarily equivalent.
Therefore (T * T ) −1/2 is well defined H → H. Now (6) holds if and only if where Here we used that T * T is a selfadjoint operator in H, and it has a positive spectral lower bound; where {ϕ j } j∈N is assumed to be a frame. holds for all f ∈ H.
Proof We shall apply the Lax-Milgram lemma [11], p. 57 to the sesquilinear form Since {ϕ n } ∞ n=1 is given to be a frame in H, then our frame-bounds B 1 > 0 and B 2 < ∞ such that (11) holds. Introducing B from (17) this into The existence of the operator L as stated in (16) Lemma 5 We have the following: and these functions are in the RKHS of the kernel K G from (3).
Proof Begin with (the frame identity): if and only if (T * T )l(t) = G(l(t)).
Note that (22) is the reproducing property.
Proof Example 1 In the theorem, we assume that the given Hilbert space H has a frame {ϕ n } ⊂ H consisting of functions on a set Ω. So this entails a lower, and an upper frame bound, i.e., 0 < B 1 ≤ B 2 < ∞.
The following example shows that the conclusion in the theorem is false if there is not a positive lower frame-bound.
Moreover, it is immediate by inspection that H = L 2 (0, 1) is not a RKHS.