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A Bergman space is an example of a reproducing kernel Hilbert space, which is a Hilbert space of functions along with a kernel K(ζ, z) that verifies a reproducing property analogous to this one. The Hardy space H 2 ( D ) also admits a reproducing kernel, known as the Szegő kernel . [ 37 ]
The first three functions in the sequence () = on [,].As converges weakly to =.. The Hilbert space [,] is the space of the square-integrable functions on the interval [,] equipped with the inner product defined by
The simplest example of a reproducing kernel Hilbert space is the space (,) where is a set and is the counting measure on . For x ∈ X {\displaystyle x\in X} , the reproducing kernel K x {\displaystyle K_{x}} is the indicator function of the one point set { x } ⊂ X {\displaystyle \{x\}\subset X} .
where H(D) is the space of holomorphic functions in D. Then L 2,h (D) is a Hilbert space: it is a closed linear subspace of L 2 (D), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ƒ in D
Most theorems on integration and differentiation of scalar functions can be generalized to vector-valued functions, often using essentially the same proofs.Perhaps the most important exception is that absolutely continuous functions need not equal the integrals of their (a.e.) derivatives (unless, for example, is a Hilbert space); see Radon–Nikodym theorem
In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). The Hilbert transform is given by the Cauchy principal value of the convolution with the function 1 / ( π t ) {\displaystyle 1/(\pi t)} (see ...
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of manifolds to infinite-dimensional setting.
The single point denoted in this space is represented by the set of functions () where and represents an index set. In quantum field theory , it is expected that the Hilbert space is also the L 2 {\displaystyle L^{2}} space on the configuration space of the field, which is infinite dimensional, with respect to some Borel measure naturally defined.