Abstract
We establish exact conditions for non triviality of all subspaces of the standard Hardy space in the upper half plane, that consist of the character automorphic functions with respect to the action of a discrete subgroup of \(SL_2({\mathbb {R}})\). Such spaces are the natural objects in the context of the spectral theory of almost periodic differential operators and in the asymptotics of the approximations by entire functions. A naive idea: it should be completely parallel to the celebrated Widom characterization for Hardy spaces on Riemann surfaces with a minor modification, namely, one has to substitute the Green function of the domain with the Martin function. Basically, this is correct, but…
Dedicated to our teacher Prof. V. E. Katsnelson 1 on the occasion of his 75-th birthday
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Notes
- 1.
Following the Russian tradition, we use the initials V.E. for Victor Emmanuilovich. V.E. was the most unusual and therefore the most attractive person among professors in the math department for students of our generation. The method he used to bring us in mathematics was also very much unusual for our time and our country: V.E. took us (four first year Master students) to a REAL mathematical conference. Actually, it was a school, but definitely of the highest conference level. At this school, we were learning the J-theory for 2 weeks at least 6 h a day. V.E. was one of the lecturers, highly enthusiastic. We studied the theory together with the most prominent professors of our department. At the lunchtime, during a ski trip, or at the night lectures we were able to meet the authors of practically all popular textbooks of that time. V.E. was our guide to this new world. If indeed a personality is completely determined by the first 3 years of our life, our mathematical personalities definitely were determined by these first 2 weeks of our mathematical childhood.
- 2.
Here \(\mathbb C_+\) is considered as a subset of \(\mathbb C\setminus E\).
- 3.
If E is regular, then Γ is of the convergent type. That is, the orbit of every point in \(\mathbb C_+\) satisfies the Blaschke condition.
- 4.
This case reduces to the standard Dominated Convergence by applying \((-\log )\) to the products.
- 5.
Same explanation as in the previous footnote.
- 6.
This case reduces to the standard Dominated Convergence by applying \((-\log )\) to the products.
- 7.
Same explanation as in the previous footnote.
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Acknowledgment
P. Yuditskii was supported by the Austrian Science Fund FWF, project no: P29363-N32.
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Kheifets, A., Yuditskii, P. (2020). Martin Functions of Fuchsian Groups and Character Automorphic Subspaces of the Hardy Space in the Upper Half Plane. In: Alpay, D., Fritzsche, B., Kirstein, B. (eds) Complex Function Theory, Operator Theory, Schur Analysis and Systems Theory. Operator Theory: Advances and Applications(), vol 280. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-44819-6_17
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