Model subspaces techniques to study Fourier expansions in \(L^2\) spaces associated to singular measures. (English) Zbl 1471.42007
Let \(\{f_n\}_{n\geq0}\) be a sequence in a separable Hilbert space \(H\) . This sequence \(\{f_n\}\) is called Parseval frame for \(H\) when
\[
||f||_H=\sum_{n\geq0}|<f,f_n>_H|^2
\]
for every \(f\in H\). Also \(\{\phi\}_{n\geq0}\), a complete sequence of unit vector in \(H\), is called effective, if when it is defined that
\[ \xi_0=<\xi,\phi_0>\]
\[\xi_n=\xi_{n-1}+<\xi-\xi_{n-1},\phi_n>\phi_n\ \ \ (n=1,2,\dots)\]
for every \(\xi\in H\), \[ \lim_{n\rightarrow\infty}||\xi-\xi_n||=0. \] Now let \(\mu\) be a Borel measure in the torus \(\mathcal{T}\) that is singular with respect to the Lebesgue measure. The aim of this paper is to study of Fourier representation for \(L^2\) spaces of singular measure by using the theory of model subspaces of the Hardy space \(\mathcal{H}^2\), whose idea already appears in the references [R. Haller and R. Szwarc, Stud. Math. 169, No. 2, 123–132 (2005; Zbl 1102.41031); J. E. Herr et al., J. Anal. Math. 138, No. 1, 209–234 (2019; Zbl 1428.30044); S. V. Khrushchev et al., Lect. Notes Math. 864, 214–335 (1981; Zbl 0466.46018)], etc. The authors study the Parseval frame associate to the effective sequence of monomials \(\{z^n\}_{n\geq0}\) in \(L^2(\mathcal{T},\mu)\), and characterize the Paseval frame as the boundary values of the Parseval frame obtained by projecting the monomials onto a convenient model space. Moreover, they show the related some results.
for every \(\xi\in H\), \[ \lim_{n\rightarrow\infty}||\xi-\xi_n||=0. \] Now let \(\mu\) be a Borel measure in the torus \(\mathcal{T}\) that is singular with respect to the Lebesgue measure. The aim of this paper is to study of Fourier representation for \(L^2\) spaces of singular measure by using the theory of model subspaces of the Hardy space \(\mathcal{H}^2\), whose idea already appears in the references [R. Haller and R. Szwarc, Stud. Math. 169, No. 2, 123–132 (2005; Zbl 1102.41031); J. E. Herr et al., J. Anal. Math. 138, No. 1, 209–234 (2019; Zbl 1428.30044); S. V. Khrushchev et al., Lect. Notes Math. 864, 214–335 (1981; Zbl 0466.46018)], etc. The authors study the Parseval frame associate to the effective sequence of monomials \(\{z^n\}_{n\geq0}\) in \(L^2(\mathcal{T},\mu)\), and characterize the Paseval frame as the boundary values of the Parseval frame obtained by projecting the monomials onto a convenient model space. Moreover, they show the related some results.
Reviewer: Enji Sato (Yamagata)
MSC:
| 42A16 | Fourier coefficients, Fourier series of functions with special properties, special Fourier series |
| 42C15 | General harmonic expansions, frames |
| 30H10 | Hardy spaces |
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