Frames, their relatives and reproducing kernel Hilbert spaces

Abstract

This paper considers different facets of the interplay between reproducing kernel Hilbert spaces (RKHS) and stable analysis/synthesis processes: First, we analyze the structure of the reproducing kernel of a RKHS using frames and reproducing pairs. Second, we present a new approach to prove the result that finite redundancy of a continuous frame implies atomic structure of the underlying measure space. Our proof uses the RKHS structure of the range of the analysis operator. This in turn implies that all the attempts to extend the notion of Riesz basis to general measure spaces are fruitless since every such family can be identified with a discrete Riesz basis. Finally, we show how the range of the analysis operators of a reproducing pair can be equipped with a RKHS structure.MSC2010: 42C15, 46E22 (primary), 47B32 (secondary)

PIP-Space Valued Reproducing Pairs of Measurable Functions

“…and g = F . In conclusion, (22) holds true and F φ * = g , where · φ * is the dual norm defined in (17). Conversely, every g ∈ H obviously defines a continuous linear functional F by (22)…”

“…The geometry of the Hilbert space imposes a number of constraints that severely limit the existence of continuous families acting as bases or frames. Apart from the case of a separable Hilbert space where orthonormal or Riesz bases cannot be uncountable, there are more general situations where, for instance, Riesz bases cannot be continuous, but they are necessarily discrete, in a certain sense [19][20][21][22]. Nevertheless, this result is essentially of theoretical nature and does not affect the interest for continuous frames in applications.…”

“…In fact, there is more. In a recent paper [22], it is shown that W φ (X, µ) (and thus also W ψ (X, µ), by symmetry), is a reproducing kernel Hilbert space (RKHS) [27]. In order to see this, take an orthonormal…”

“…By [22] (Theorem 23), it follows that H φ K is a reproducing kernel Hilbert space and one has (compare (20)…”

Continuous frames in tensor product Hilbert spaces, localization operators and density operators

“…It does not make sense to address this question in the context of continuous frames, since all continuous Riesz bases are actually discrete, cf. [48].…”