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Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science pp 603–651Cite as

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A Harmonic Analysis of Directed Graphs from Arithmetic Functions and Primes

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Abstract

In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By fixing a prime p and a graph G, we establish a noncommutative free probabilistic structure embedded in the algebra of all arithmetic functions. We act the additive group \((\mathbb{R},+),\) the flow, on the free probability space dependent both on p and on G, and construct uncountable families of free probability spaces determined by the flow. We study fundamental properties of such a family.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, St. Ambrose University, 421 Ambrose Hall, 518 W. Locust St., Davenport, IA, 52803, USA

    Ilwoo Cho

  2. Department of Mathematics, University of Iowa, 14 McLean Hall, Iowa City, IA, 52242, USA

    Palle E. T. Jorgensen

Authors
  1. Ilwoo Cho
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  2. Palle E. T. Jorgensen
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Corresponding author

Correspondence to Ilwoo Cho .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, Pennsylvania, USA

    Isaac Pesenson

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia

    Quoc Thong Le Gia

  3. Department of Mathematics, The Graduate Center, CUNY, New York, New York, USA

    Azita Mayeli

  4. Institute of Mathematical Sciences, Claremont Graduate University, Claremont, California, USA

    Hrushikesh Mhaskar

  5. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Ding-Xuan Zhou

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Cho, I., Jorgensen, P.E.T. (2017). A Harmonic Analysis of Directed Graphs from Arithmetic Functions and Primes. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55556-0_7

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