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Essays in Mathematics and its Applications pp 37–48Cite as

The Fox–Li Operator as a Test and a Spur for Wiener–Hopf Theory

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Abstract

The paper is a concise survey of some rigorous results on the Fox–Li operator. This operator may be interpreted as a large truncation of a Wiener–Hopf operator with an oscillating symbol. Employing theorems from Wiener–Hopf theory one can therefore derive remarkable properties of the Fox–Li operator in a fairly comfortable way, but it turns out that Wiener–Hopf theory is unequal to the task of answering the crucial questions on the Fox–Li operator.

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Notes

  1. 1.

    According to [12], this conjecture comes from “using a distinctly physical approach based on wave-guide theory”, but we admit that we have not been able to follow the argument of [23]. Moreover, numerical computations do not support the conjecture.

  2. 2.

    The reader might enjoy knowing the following, which is cited from [1]: “Harold Widom grew up in Brooklyn, New York. He went to Stuyvesant High School where he was captain of the math team. Coincidentally, the captain of the rival team at the Bronx High School of Science was Henry Landau …”.

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Acknowledgements

The second author acknowledges support by CONACYT grants 80503 and 102800.

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Authors and Affiliations

  1. Faculty of Mathematics, TU Chemnitz, 09107, Chemnitz, Germany

    Albrecht Böttcher

  2. Department of Mathematics, CINVESTAV, Mexico-City, Mexico

    Sergei Grudsky

  3. Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WA, UK

    Arieh Iserles

Authors
  1. Albrecht Böttcher
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  2. Sergei Grudsky
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  3. Arieh Iserles
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Correspondence to Arieh Iserles .

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Editors and Affiliations

  1. , Department of Industrial & Systems Engin, University of Florida, Weil Hall 401, Gainesville, 32611, Florida, USA

    Panos M. Pardalos

  2. , Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens, 15780, Greece

    Themistocles M. Rassias

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Dedicated to the 80th Anniversary of Professor Stephen Smale

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Böttcher, A., Grudsky, S., Iserles, A. (2012). The Fox–Li Operator as a Test and a Spur for Wiener–Hopf Theory. In: Pardalos, P., Rassias, T. (eds) Essays in Mathematics and its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28821-0_3

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