An Introduction to the Theory of Reproducing Kernel Hilbert Spaces

Paulsen, Vern I.; Raghupathi, Mrinal

doi:10.1017/cbo9781316219232

Cited by 405 publications

(333 citation statements)

References 6 publications

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“…In an analogous way as in the previous section we can define the space of multipliers M (H) for an arbitrary Hilbert space of analytic functions H. If H is a reproducing kernel Hilbert space, then M (H) is a unital Banach subalgebra of B(H), which is closed in the weak operator topology [27]. It is clear that unlike …”

Section: Banach Algebrasmentioning

confidence: 97%

“…First, note that the evaluation functional E λ : H −→ C, f E λ → f (λ) is bounded for every λ ∈ Ω, since it is a multiplicative functional on a Banach algebra H and hence E λ ≤ 1 (see [3], §16, Proposition 3, p. 77), so H is a reproducing kernel Hilbert space (see [2], [27]). Let k z denote the reproducing kernel of H at z ∈ Ω.…”

Section: Theorem 3 Let H Be a Hilbert Space Of Complex-valued Functimentioning

confidence: 99%

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Multipliers of Hilbert spaces of analytic functions on the complex half-plane

Kucik¹

2017

Operators and Matrices

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It follows, from a generalised version of Paley-Wiener theorem, that the Laplace transform is an isometry between certain spaces of weighted L 2 functions defined on (0, ∞) and (Hilbert) spaces of analytic functions on the right complex half-plane (for example Hardy, Bergman or Dirichlet spaces). We can use this fact to investigate properties of multipliers and multiplication operators on the latter type of spaces. In this paper we present a full characterisation of multipliers in terms of a generalised concept of a Carleson measure. Under certain conditions, these spaces of analytic functions are not only Hilbert spaces but also Banach algebras, and are therefore contained within their spaces of multipliers. We provide some necessary as well as sufficient conditions for this to happen and look at its consequences. Mathematics Subject Classification (2010). Primary 30H50, 46J15, 47B99; Secondary 46E22, 46J20.

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Section: Banach Algebrasmentioning

confidence: 97%

Section: Theorem 3 Let H Be a Hilbert Space Of Complex-valued Functimentioning

confidence: 99%

Multipliers of Hilbert spaces of analytic functions on the complex half-plane

Kucik¹

2017

Operators and Matrices

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“…By the abstract theory of reproducing kernel Hilbert spaces [21], it follows that there is a reproducing kernel Hilbert space of C n -valued functions on C \ T with reproducing kernel K w (z). This RKHS is denoted by L (b) and is called the Herglotz space corresponding to b.…”

Section: Herglotz Spacesmentioning

confidence: 99%

“…* is a positive M n (C)-valued kernel function on C \ T, and so it follows from the general theory of reproducing kernel Hilbert spaces that W (z) is a contractive multiplier of L (Θ) into L (Φ) [21,Theorem 10.20]. Theorem 8.1 now implies that Z Θ q Z Φ whenever Θ divides Φ.…”

Section: Partial Orders and Multipliersmentioning

confidence: 99%

Partial orders on partial isometries

Ross¹,

Garcia²,

W.³

2016

J. Operator Theory

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Abstract. This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spaces of analytic functions. For large classes of partial isometries these spaces can be realized as the well-known model subspaces and deBrangesRovnyak spaces. This characterization is applied to investigate properties of these pre-orders and the equivalence classes they generate.

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“…Some complex kernel which properties have been studied are the Szego and Bregman kernels [30]. Another interesting kernel is complex Gaussian kernel k(z, w) = exp −…”

Section: Complex Rkhsmentioning

confidence: 99%

Principal Component Analysis With Complex Kernel: The Widely Linear Model

Papaioannou

Zafeiriou

2014

IEEE Trans. Neural Netw. Learning Syst.

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Non-linear complex representations, via the use of complex kernels, can be applied to model and capture the nonlinearities of complex data. Even though, the theoretical tools of Complex Reproducing Kernel Hilbert Spaces (CRKHS) have been recently successfully applied to the design of digital filters and regression and classification frameworks there is limited research on component analysis and dimensionality reduction in CRKHS. The aim of this paper is to properly formulate the most popular component analysis methodology, i.e. Principal Component Analysis (PCA), in CRKHS. In particular, we define a general Widely Linear Kernel Complex PCA (WLCKPCA) framework. Furthermore, we show how to efficiently perform Widely Linear PCA (WLPCA) in small sample sized problems. Finally, we show the usefulness of the proposed framework in robust reconstruction using Euler data representation.

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An Introduction to the Theory of Reproducing Kernel Hilbert Spaces

Cited by 405 publications

References 6 publications

Multipliers of Hilbert spaces of analytic functions on the complex half-plane

Multipliers of Hilbert spaces of analytic functions on the complex half-plane

Partial orders on partial isometries

Principal Component Analysis With Complex Kernel: The Widely Linear Model

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