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Theory of Computing Systems

, Volume 63, Issue 3, pp 567–586 | Cite as

Algorithmic Randomness and Fourier Analysis

  • Johanna N. Y. Franklin
  • Timothy H. McNichollEmail author
  • Jason Rute
Article
  • 52 Downloads

Abstract

Suppose 1 < p < . Carleson’s Theorem states that the Fourier series of any function in Lp[−π, π] converges almost everywhere. We show that the Schnorr random points are precisely those that satisfy this theorem for every fLp[−π, π] given natural computability conditions on f and p.

Keywords

Computability Fourier analysis Complex analysis Algorithmic randomness Computable analysis 

Notes

Acknowledgements

We are grateful for the helpful comments made by the anonymous referees.

The first author was supported in part by Simons Foundation grant # 420806. Research of the second author was supported in part by Simons Foundation grant #317870.

References

  1. 1.
    Allen, K., Bienvenu, L., Slaman, T.A.: On zeros of Martin-Löf random Brownian motion. J. Log. Anal. 6, Paper 9, 34 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Asarin, E.A., Pokrovskiı̆, A.V.: Application of Kolmogorov complexity to the analysis of the dynamics of controllable systems. Avtomat. i Telemekh. 1, 25–33 (1986)MathSciNetGoogle Scholar
  3. 3.
    Avigad, J.: Uniform distribution and algorithmic randomness. J. Symb. Log. 78(1), 334–344 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bienvenu, L., Day, A.R., Hoyrup, M., Mezhirov, I., Shen, A.: A constructive version of Birkhoff’s ergodic theorem for Martin-Löf random points. Inf. Comput. 210, 21–30 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bienvenu, L., Hölzl, R., Miller, J.S., Nies, A.: Denjoy, Demuth and density. J. Math. Log. 14(1), 1450004, 35 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brattka, V., Weihrauch, K.: Computability on subsets of Euclidean space. I. Closed and compact subsets, vol. 219. In: Computability and Complexity in Analysis (Castle Dagstuhl, 1997), pp 65–93 (1999)Google Scholar
  7. 7.
    Brattka, V., Miller, J.S., Nies, A.: Randomness and differentiability. Trans. Am. Math. Soc. 368(1), 581–605 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Braverman, M., Cook, S: Computing over the reals: foundations for scientific computing. Not. Am. Math. Soc. 53(3), 318–329 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Calvert, W., Franklin, J.N.Y.: Genericity and UD-random reals. J. Log. Anal. 7, Paper 4, 10 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Carleson, L.: On convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Conway, J.B: Functions of One Complex Variable I, Volume 11 of Graduate Texts in Mathematics, 2nd edn. Springer, Berlin (1978)CrossRefGoogle Scholar
  12. 12.
    Cooper, S.B.: Computability Theory. Chapman & Hall/CRC, Boca Raton (2004)zbMATHGoogle Scholar
  13. 13.
    Downey, R.G., Griffiths, E.J.: Schnorr randomness. J. Symb. Log. 69(2), 533–554 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  15. 15.
    Fefferman, C.: Pointwise convergence of Fourier series. Ann. of Math. (2) 98, 551–571 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fefferman, C.L: Erratum: “Pointwise convergence of Fourier series” [Ann. of Math. (2) 98 (1973), no. 3, 551–571; MR0340926 (49 #5676)]. Ann. Math. (2) 146 (1), 239 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Fejér, L.: Sur les fonctions bornées et intégrables. C. R. Acad. Sci. Paris 131, 984–987 (1900)Google Scholar
  18. 18.
    Fouché, W.L.: The descriptive complexity of Brownian motion. Adv. Math. 155(2), 317–343 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Franklin, J.N.Y., Greenberg, N, Miller, J.S., Ng, KM: Martin-Löf random points satisfy Birkhoff’s ergodic theorem for effectively closed sets. Proc. Am. Math. Soc. 140(10), 3623–3628 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Franklin, J.N.Y, Towsner, H.: Randomness and non-ergodic systems. Mosc. Math. J. 14(4), 711–744 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Freer, C., Kjos-Hanssen, B., Nies, A., Stephan, F.: Algorithmic aspects of Lipschitz functions. Computability 3(1), 45–61 (2014)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gács, P., Hoyrup, M., Rojas, C.: Randomness on computable probability spaces—a dynamical point of view. Theory Comput. Syst. 48(3), 465–485 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Galatolo, S., Hoyrup, M., Rojas, C.: Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems. In: Zheng, X., Zhong, N. (eds.) Proceedings Seventh International Conference on Computability and Complexity in Analysis, Zhenjiang, China, 21–25th June 2010, volume 24 of Electronic Proceedings in Theoretical Computer Science, pp. 7–18. Open Publishing Association (2010)Google Scholar
  24. 24.
    Garnett, J.B., Marshall, D.E.: Harmonic Measure, New Mathematical Monographs, vol 2. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  25. 25.
    Grzegorczyk, A.: On the definitions of computable real continuous functions. Fund. Math. 44, 61–71 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hoyrup, M.: Computability of the ergodic decomposition. Ann. Pure Appl. Logic 164(5), 542–549 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hoyrup, M., Rojas, C.: Applications of effective probability theory to Martin-Löf randomness. In: Automata, Languages and Programming. Part I, volume 5555 of Lecture Notes in Computer Science, pp 549–561. Springer, Berlin (2009)Google Scholar
  28. 28.
    Hunt, R.A: On the convergence of Fourier series. In: Orthogonal Expansions and Their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), pp 235–255. Southern Illinois University Press, Carbondale (1968)Google Scholar
  29. 29.
    Kahane, J.-P., Katznelson, Y.: Sur les ensembles de divergence des séries trigonométriques. Studia Math. 26, 305–306 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Katznelson, Y.: Sur les ensembles de divergence des séries trigonométriques. Studia Math. 26, 301–304 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kolmogorov, A.N.: Une série de Fourier-Lebegue divergente presque partout. Fund. Math. Fund. Math. Fund. Math. 4, 324–328 (1923)Google Scholar
  32. 32.
    Lacombe, D.: Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles. I. C. R. Acad. Sci. Paris 240, 2478–2480 (1955)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Lacombe, D.: Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles. II, III. C. R. Acad. Sci. Paris 241, 13–14, 151–153 (1955)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Lebesgue, H.: Recherches sur la convergence des séries de fourier. Math. Ann. 61(2), 251–280 (1905)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Miyabe, K.: L 1-computability, layerwise computability and Solovay reducibility. Computability 2(1), 15–29 (2013)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Miyabe, K., Nies, A., Zhang, J.: “Universal” Schnorr tests, In progressGoogle Scholar
  37. 37.
    Moser, P.: On the convergence of Fourier series of computable Lebesgue integrable functions. MLQ Math. Log. Q. 56(5), 461–469 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Nehari, Z.: Conformal Mapping. McGraw-Hill Book Co., Inc., New York (1952)zbMATHGoogle Scholar
  39. 39.
    Nies, A.: Computability and Randomness. Clarendon Press, Oxford (2009)CrossRefzbMATHGoogle Scholar
  40. 40.
    Nies, A.: Differentiability of polynomial time computable functions. In: 31st International Symposium on Theoretical Aspects of Computer Science, LIPIcs. Leibniz Int. Proc. Inform., vol. 25, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, pp 602–613 (2014)Google Scholar
  41. 41.
    Odifreddi, P.: Classical Recursion Theory. Studies in Logic and the Foundations of Mathematics, no. 125. North-Holland, Amsterdam (1989)Google Scholar
  42. 42.
    Odifreddi, P.: Classical Recursion Theory, Volume ii. Studies in Logic and the Foundations of Mathematics, no. 143. North-Holland, Amsterdam (1999)Google Scholar
  43. 43.
    Pathak, N., Rojas, C., Simpson, S.G.: Schnorr randomness and the Lebesgue differentiation theorem. Proc. Am. Math. Soc. 142(1), 335–349 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin (1989)CrossRefzbMATHGoogle Scholar
  45. 45.
    Rettinger, R., Weihrauch, K.: Products of effective topological spaces and a uniformly computable Tychonoff theorem. Log. Methods Comput. Sci. 9(4), 4:14, 21 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Rute, J.: Topics in algorithmic randomness and computable analysis. PhD thesis, Carnegie Mellon University, August 2013. Available at http://repository.cmu.edu/dissertations/260/
  47. 47.
    Soare, R.I.: Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic. Springer, Berlin (1987)CrossRefGoogle Scholar
  48. 48.
    Schnorr, C.-P.: Zufälligkeit und Wahrscheinlichkeit. Lecture Notes in Mathematics, vol. 218. Springer, Heidelberg (1971)CrossRefGoogle Scholar
  49. 49.
    Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. J. Symb. Log. 14, 145–158 (1949)CrossRefzbMATHGoogle Scholar
  50. 50.
    Turing, A.M: On computable numbers, with an application to the Entscheidungsproblem. A Correction, Proc. London. Math. Soc. S2–43(6), 544 (1937)zbMATHGoogle Scholar
  51. 51.
    V’yugin, V.V.: Effective convergence in probability, and an ergodic theorem for individual random sequences. Teor. Veroyatnost. i Primenen. 42(1), 35–50 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Weihrauch, K.: Computable Analysis, Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2000)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsHofstra UniversityHempsteadUSA
  2. 2.Department of MathematicsIowa State UniversityAmesUSA
  3. 3.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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