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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Effective sup-norm bounds on average for cusp forms of even weight
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by J. S. Friedman, J. Jorgenson and J. Kramer PDF
Trans. Amer. Math. Soc. 372 (2019), 7735-7766 Request permission

Abstract:

Let $\Gamma \subset \mathrm {PSL}_{2}(\mathbb {R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb {H}$. Consider the $d_{2k}$-dimensional space of cusp forms $\mathcal {S}_{2k}^{\Gamma }$ of weight $2k$ for $\Gamma$, and let $\{f_{1},\ldots ,f_{d_{2k}}\}$ be an orthonormal basis of $\mathcal {S}_{2k}^{\Gamma }$ with respect to the Petersson inner product. In this paper, we will give effective upper and lower bounds for the supremum of the quantity $S_{2k}^{\Gamma }(z):=\sum _{j=1}^{d_{2k}}\vert f_{j}(z)\vert ^{2} \mathrm {Im}(z)^{2k}$ as $z$ ranges through $\mathbb {H}$.
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Additional Information
  • J. S. Friedman
  • Affiliation: Department of Mathematics and Science, United States Merchant Marine Academy, 300 Steamboat Road, Kings Point, New York 11024
  • MR Author ID: 772419
  • Email: FriedmanJ@usmma.edu
  • J. Jorgenson
  • Affiliation: Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, New York 10031
  • MR Author ID: 292611
  • Email: jjorgenson@mindspring.com
  • J. Kramer
  • Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
  • MR Author ID: 227725
  • Email: kramer@math.hu-berlin.de
  • Received by editor(s): January 22, 2018
  • Received by editor(s) in revised form: August 14, 2018, and January 14, 2019
  • Published electronically: September 9, 2019
  • Additional Notes: The views expressed in this article are the first author’s own and not those of the U.S. Merchant Marine Academy, the Maritime Administration, the Department of Transportation, or the United States Government.
    The second author acknowledges support from numerous PSC-CUNY grants.
    The third author acknowledges support from the DFG Graduate School Berlin Mathematical School.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 7735-7766
  • MSC (2010): Primary 11F11, 11F72, 30F10; Secondary 47A10
  • DOI: https://doi.org/10.1090/tran/7933
  • MathSciNet review: 4029679