Equivalent Norms in Hilbert Spaces with Unconditional Bases of Reproducing Kernels
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Journal of Mathematical Sciences, Vol. 250, No. 2, October, 2020
EQUIVALENT NORMS IN HILBERT SPACES WITH UNCONDITIONAL BASES OF REPRODUCING KERNELS K. P. Isaev
∗
Institute of Mathematics, UFRC RAS 112, Chernyshevskii St., Ufa 450008, Russia [email protected]
K. V. Trunov Bashkir State University 32, Zaki Validi St., Ufa 450000, Russia [email protected]
R. S. Yulmukhametov Institute of Mathematics, UFRC RAS 112, Chernyshevskii St., Ufa 450008, Russia Bashkir State Pedagogical University 3-a, Oktyabrskoj revoljucii St., Ufa 450008, Russia [email protected]
UDC 517.95
In a Hilbert space H with an unconditional basis of reproducing kernels, we study the existence of a weighted integral norm with respect to an absolutely continuous measure, which is equivalent to the original H-norm. If the space H is defined via weighted integrals, the problem can be interpreted as restoring the original structure. Bibliography: 16 titles.
1
Introduction
Let H be a Hilbert space of entire functions satisfying the following conditions: (1) H is functional, i.e., the evaluation functionals δz : f → f (z) are continuous for every z ∈ C, (2) H possesses the division property, i.e., F ∈ H and F (z0 ) = 0 imply F (z)(z − z0 )−1 ∈ H. In particular, condition (2) means that the evaluation functionals in H are different from zero and condition (1) means that each functional δz is generated by an element kz (λ) ∈ H in the sense that δz (f ) = (f (λ), kz (λ)). The function k(λ, z) = kz (λ) is called a reproducing kernel of the space H. We denote K(z) = k(z, z). Then the Bergman function for H is defined by ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 105, 2020, pp. 111-120. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2502-0310
310
1
δz H = (K(z)) 2 [1]. A basis {ek }∞ k=1 for a Hilbert space is called unconditional [2] if there are ∞ c, C > 0 such that for any x = xk ek ∈ H k=1
c
∞
∞ ∞ 2 |ck |2 ek 2 ck ek C |ck |2 ek 2 .
j=1
j=1
(1.1)
j=1
The study of unconditional bases of reproducing kernels for Hilbert spaces of entire functions is an actual problem of complex analysis. (Here and below, for the sake of brevity we write “basis of reproducing kernels” instead of “basis consisting of the values of reproducing kernels”) Apparently, the problem in such a formulation was first considered in [3, 4], where the classical interpolation problem was studied for entire functions. Let {k(λ, λi )}∞ i=1 be an unconditional basis for a Hilbert space H satisfying conditions (1) and (2). Then the biorthogonal system consists of functions L(λ) , k = 1, 2, . . . , Lk (λ) = L (λk )(λ − λk ) where L(λ) is an entire function, called the generating function, with simple zeros λk , k = 1, 2, . . . . This system forms an unconditional basis. By properties of biorthogonal systems, Lk 2
1 , K(λk )
k ∈ N.
The expansion of a function F into the series with respect to this basis has the form F (λ) =
∞
F (λk )Lk (λ),
λ ∈ C,
k=1
i.e., the La
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