aequationes mathematicae

, Volume 53, Issue 1–2, pp 207–241 | Cite as

Functional equations for peculiar functions

  • Hans-Heinrich Kairies
Survey Paper
Received:
Accepted:

Summary

We establish functional equations for peculiar functionsf:I → ℝ,I ⊂ ℝ an interval, such as
  1. (1)

    continuous, nowhere differentiable functions of various types (Weierstrass, Takagi, Knopp, Wunderlich),

     
  2. (2)

    Riemann's function, which is nondifferentiable except on certain rational points,

     
  3. (3)

    singular functions of various types (Cantor, Minkowski, de Rham).

     
All these functional equations take the form
$$+\sum\limits_{v = 1}^n {f[h_v (x)] = \alpha f(x) + g(x),} x \in I,$$
(F)
where thehv are given rational functions,g is a given function andα is a given real constant. For each of the above mentioned peculiar functions we discuss the essential non-differentiability properties and present at least one characterization by means of functional equations of type (F). Solutionsf:I → ℝ of the system of replicativity equations
$$\frac{1}{p}\sum\limits_{k = 0}^{p - 1} {f\left( {\frac{{x + k}}{p}} \right) = (p)f(x), p \in S \subset \mathbb{N}} $$
(all of type (F)) exhibit a particularly rich structure and are discussed in some detail.

AMS (1991) subject classification

Primary 39B22, 39B62 Secondary 26A27, 26A30 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Apostol, T. M.,Introduction to Analytic Number Theory. Springer, New York, Heidelberg, Berlin 1976.Google Scholar
  2. [2]
    Apostol, T. M.,Modular Functions and Dirichlet Series in Number Theory. Springer, New York, Heidelberg, Berlin 1976.Google Scholar
  3. [3]
    Artin, E.,Einführung in die Theorie der Gammafunktion. Hamb. Math. Einzelschriften, Leipzig 1931. English translation:The Gamma function. Holt, Rinehart and Winston, New York 1964.Google Scholar
  4. [4]
    Behrend, F. A.,Some remarks on the construction of continuous nondifferentiable functions. Proc. London Math. Soc. (2)50 (1949), 463–481.Google Scholar
  5. [5]
    Benz, W.,Remarks on exorbitant equations. Aequationes Math.41 (1991), 283.Google Scholar
  6. [6]
    Bernstein, B., Erber, T. andKaramolengos, M.,Sneaky plasticity and mesoscopic fractals. Chaos, Solitons & Fractals3 (1993), 269–277.Google Scholar
  7. [7]
    Billingsley, P.,The singular function of bold play. Amer. Scientist71 (1983), 392–397.Google Scholar
  8. [8]
    Du Bois-Reymond, P.,Versuch einer Classifikation der willkürlichen Funktionen reeller Argumente nach ihren Änderungen in den kleinsten Intervallen. J. Reine Angew. Math.79 (1875), 21–37.Google Scholar
  9. [9]
    Borwein, J. M. andGirgensohn, R.,Addition theorems and binary expansions. Canad. J. Math.47 (1995), no. 2, 262–273.Google Scholar
  10. [10]
    Butzer, P. L. andStark, E. L.,“Riemann's example” of a continuous nondifferentiable function in the light of two letters (1865) of Christoffel to Prym. Bull. Soc. Math. Belg., Sér. A38 (1986), 45–73.Google Scholar
  11. [11]
    Cantor, G.,De la puissance des ensembles parfaits des points. Acta. math.4 (1884), 381–392.Google Scholar
  12. [12]
    Cater, F. S.,On van der Waerden's nowhere differentiable function. Amer. Math. Monthly91 (1984), 307–308.Google Scholar
  13. [13]
    Cellérier, C.,Note sur les principes fondamentaux de l'analyse. Bull. des Sc. Math. (2), vol. XIV première partie (1890), 142–160.Google Scholar
  14. [14]
    Cesàro, E.,Fonctions continues sans dérivées. Arch. Math. Phys. (3)38 (1906), 57–63.Google Scholar
  15. [15]
    Cesàro, E.,Fonctions continues sans dérivée formées avec les itérées d'une fraction rationnelle. Ann. Sci. École Norm. Sup48 (1931), 1–14.Google Scholar
  16. [16]
    Darsow, W. F. andFrank, M. J.,Associative functions and Abel-Schröder systems. Publ. Math. Debrecen30 (1983), 253–272.Google Scholar
  17. [17]
    Darsow, W. F. andFrank, M. J.,Conjugacy systems. In Proc. 23. ISFE. Centre for Information Theory, Univ. of Waterloo 1985, pp. 8–9.Google Scholar
  18. [18]
    Darsow, W. F., Frank, M. J. andKairies, H.-H.,Functional equations for a function of van der Waerden type. Rad. Mat.4 (1988), 361–374. Correction of misprints in ibidem5 (1989), 179–180.Google Scholar
  19. [19]
    Denjoy, A.,Sur quelques points de la théorie des fonctions. C. R. Acad. Sci. Paris194 (1932), 44–46.Google Scholar
  20. [20]
    Denjoy, A.,Sur une fonction de Minkowski. C. R. Acad. Sci. Paris198 (1934), 44–47.Google Scholar
  21. [21]
    Denjoy, A.,Sur une fonction réelle de Minkowski. J. Math. Pures Appl. (9)17 (1938), 105–151.Google Scholar
  22. [22]
    Dennler, G.,Bestimmung sämtlicher meromorpher Lösungen der Funktionalgleichung f(z) = (1/k) Σ h = 0k − 1f((z + h)/k). Wiss. Z. Friedrich-Schiller-Univ. Jena Math.-Naturw. Reihe14 (1965).Google Scholar
  23. [23]
    Dubuc, S.,Functional equations connected with peculiar curves. In: Liedl, R., Reich, L., Targonski, Gy. (Eds.),Iteration theory and its functional equations. Lecture Notes in Math. no. 1163. Springer, Berlin, Heidelberg, New York, Tokyo 1984, pp. 33–40.Google Scholar
  24. [24]
    Dubuc, S.,Une équation fonctionelle pour diverses constructions géométríques. Ann. Sci. Math. Québec9 (1985), no. 2, 151–174.Google Scholar
  25. [25]
    Dubins, L. E. andSavage, L.,How to Gamble if You Must. Inequalities for stochastic processes. McGraw-Hill 1965.Google Scholar
  26. [26]
    Duistermaat, J. J.,Selfsimilarity of ‘Riemann's nondifferentiable function’. Nieuw Arch. WiskundeIV9 (1991), 303–337.Google Scholar
  27. [27]
    Faber, G.,Einfaches Beispiel einer stetigen nirgends differenzierbaren Funktion. Jahresber. Deutsch. Math.-Verein.16 (1907), 538–540.Google Scholar
  28. [28]
    Faber, G.,Über stetige Funktionen II. Math. Ann.69 (1910), 372–433.CrossRefMathSciNetGoogle Scholar
  29. [20]
    Falconer, K. J.,Fractal Geometry. Mathematical Foundations and Applications. Wiley and Sons, Chichester 1990.Google Scholar
  30. [30]
    Frank, M. J.,A generalization of the van der Waerden system. Aequationes Math.37 (1989), 67.Google Scholar
  31. [31]
    Freud, G.,Über trigonometrische Approximation und Fouriersche Reihen. Math. Z.78 (1962), 252–262.CrossRefGoogle Scholar
  32. [32]
    Gelbaum, B. R. andOlmsted, J.,Theorems and Counterexamples in Mathematics. Springer, New York, Berlin, Heidelberg 1990.Google Scholar
  33. [33]
    Gerver, J.,The differentiability of the Riemann function at certain rational multiples of π. Amer. J. Math.92 (1970), 33–55.Google Scholar
  34. [34]
    Gerver, J.,More on the differentiability of the Riemann function. Amer. J. Math.93 (1971), 33–41.Google Scholar
  35. [35]
    Girgensohn, R.,Funktionalgleichungen für nirgends differenzierbare Funktionen. Clausthal 1992 (Dissertation).Google Scholar
  36. [36]
    Girgensohn, R.,Functional equations and nowhere differentiable functions. Aequationes Math.48 (1993), 243–256.Google Scholar
  37. [37]
    Girgensohn, R.,Nowhere differentiable solutions of a system of functional equations. Aequationes Math.47 (1994), 89–99.CrossRefGoogle Scholar
  38. [38]
    Girgensohn, R.,Constructing singular functions via Farey fractions. J. Math. Anal. Appl.203 (1996), 127–141.CrossRefGoogle Scholar
  39. [39]
    Hahn, H.,Über stetige Funktionen ohne Ableitung. Jahresber. Deutsch. Math.-Verein.26 (1918), 281–284.Google Scholar
  40. [40]
    Hardy, G. H.,Weierstrass's nondifferentiable function. Trans. Amer. Math. Soc.17 (1916), 301–325.MathSciNetGoogle Scholar
  41. [41]
    Hartmann, P. andKershner, R.,The structure of monotone functions. Amer. J. Math.59 (1937), 809–822.Google Scholar
  42. [42]
    Hata, M.,Singularities of the Weierstrass type functions. J. Analyse Math.51 (1988), 62–90.Google Scholar
  43. [43]
    Hata, M.,On Weierstrass's nondifferentiable function. C. R. Acad. Sci. Paris Sér. 1 Math.307 (1988), 119–123.Google Scholar
  44. [44]
    Hata, M. andYamaguti, M.,The Takagi function and its generalization. Japan. J. Appl. Math.1 (1984), 183–199.Google Scholar
  45. [45]
    Hewitt, E. andStromberg, K.,Real and Abstract Analysis. Springer, Berlin, Heidelberg, New York 1965.Google Scholar
  46. [46]
    Hildebrandt, T. H.,A simple continuous function with a finite derivative at no point. Amer. Math. Monthly40 (1933), 547–548.MathSciNetGoogle Scholar
  47. [47]
    Hille, E. andTamarkin, J. D.,Remarks on a known example of a monotone continuous function. Amer. Math. Monthly36 (1929), 255–264.MathSciNetGoogle Scholar
  48. [48]
    Holschneider, M. andTchamitchian, P.,Pointwise analysis of Riemann's “nondifferentiable” function. Invent. math.105 (1991), 157–175.CrossRefGoogle Scholar
  49. [49]
    Itatsu, S.,Differentiability of Riemann's function. Proc. Japan. Acad. Math. Sci. A57 (10) (1981), 492–495.Google Scholar
  50. [50]
    Jäger, R.,Charakterisierung des Cotangens mit Replikativität. Manuscripta Math.56 (1986), 167–175.CrossRefGoogle Scholar
  51. [51]
    Jäger, R.,Beiträge zur Theorie der Replikativitätsgleichung. Clausthal 1987 (Dissertation).Google Scholar
  52. [52]
    Kairies, H.-H.,Zur axiomatischen Charakterisierung der Gammafunktion. J. Reine Angew. Math.236 (1969), 103–111.Google Scholar
  53. [53]
    Kairies, H.-H.,Multiplikationstheoreme. Braunschweig 1971 (Habilitationsschrift).Google Scholar
  54. [54]
    Kairies, H.-H.,Definitionen der Bernoulli-Polynome mit Hilfe ihrer Multiplikationstheoreme. Manuscripta Math.8 (1973), 363–369.CrossRefGoogle Scholar
  55. [55]
    Kairies, H.-H.,Ein Existenz- und Eindeutigkeitssatz für die Lösungen einer Funktionalgleichung aus der Nörlundschen Theorie der Differenzengleichungen. Aequationes Math.10 (1974), 278–285.CrossRefGoogle Scholar
  56. [56]
    Kairies, H.-H.,Convexity in the theory of the Gamma function. In: Beckenbach, E. F. (Ed.), General Inequalities 1, Internat. Ser. Numer. Math.41 (1978), 49–62, Birkhäuser, Basel.Google Scholar
  57. [57]
    Kairies, H.-H.,Mean values and function spaces. In: Beckenbach, E. F. (Ed.), General Inequalities 2, Internat. Ser. Numer. Math.47 (1980), 287–295, Birkhäuser, Basel.Google Scholar
  58. [58]
    Kairies, H.-H.,Charakterisierungen von nirgends differenzierbaren Weierstrass-Funktionen durch Replikativität. Elem. Math.45 (1990), 61–69.Google Scholar
  59. [59]
    Kairies, H.-H.,Die Gammafunktion als stetige Lösung eines Systems von Gauss-Funktionalgleichungen. Results in Math.26 (1994), 306–315.Google Scholar
  60. [60]
    Kairies, H.-H.,Zu einem Charakterisierungssatz von E. Artin. Abh. Math. Sem. Univ. Hamburg.65 (1995), 135–146.Google Scholar
  61. [61]
    Kairies, H.-H.,On the optimality of a characterization theorem for the gamma function using the multiplication formula. Aequationes Math.51 (1996), 115–128.CrossRefGoogle Scholar
  62. [62]
    Kairies, H.-H. andRose, M.,Funktionalgleichungen und Charakterisierungen für die Riemannsche Funktion. Period. Math. Hung.32 (3) (1996), 199–207.CrossRefGoogle Scholar
  63. [63]
    Kántor, S.,Über die Konstruktion überall stetiger, nirgends differenzierbarer Funktionen I. Publ. Math. Debrecen9 (1962), 111–121.Google Scholar
  64. [64]
    Katsuura, H.,Continuous nowhere-differentiable functions — an application of contraction mappings. Amer. Math. Monthly98 (1991), 411–416.Google Scholar
  65. [65]
    Kiesswetter, K.,Ein einfaches Beispiel für eine Funktion, welche überall stetig und nirgends differenzierbar ist. Math. Phys. Semesterber.13 (1966), 216–221.Google Scholar
  66. [66]
    Klemmt, H.-J.,Ein maßtheoretischer Satz mit Anwendungen auf Funktionalgleichungen. Arch. Math. (Basel)38 (1981), 537–540.Google Scholar
  67. [67]
    Knopp, K.,Einfaches Beispiel einer stetigen, nirgends differenzierbarren Funktion. Jahresber. Deutsch. Math.-Verein.26 (1918), 278–280.Google Scholar
  68. [68]
    Knopp, K.,Ein einfaches Verfahren zur Bildung stetiger nirgends differenzierbarer Funktionen. Math. Z.2 (1918), 1–26.CrossRefMathSciNetGoogle Scholar
  69. [69]
    Knuth, D. E.,The Art of Computer Programming, Vol. I. Addison-Wesley, Reading, Mass. 1969.Google Scholar
  70. [70]
    von Koch, H.,Sur une courbe continue sans tangente obtenue par une construction géométrique élémentaire. Arkiv för Mathematik, Astronomie och Fysik1 (1904), 681–704.Google Scholar
  71. [71]
    Kôno, N.,On generalized Takagi functions. Acta Math. Hung.49 (1987), 315–324.CrossRefGoogle Scholar
  72. [72]
    Kuczma, M.,Functional Equations in a Single Variable. Monografie Math.46. Polish Scientific Publishers, Warszawa 1968.Google Scholar
  73. [73]
    Kuczma, M., Choczewski, B. andGer, R.,Iterative Functional Equations. Cambridge University Press, Cambridge 1990.Google Scholar
  74. [74]
    Kuwagaki, A.,Sur quelques équations fonctionelles et leurs solutions charactéristiques I. Mem. of the Coll. of Science, Univ. of Kyoto, Ser. A26 (1950), 271–277.Google Scholar
  75. [75]
    Lebsgue, H.,Leçons sur l'intégration et la recherche des fonctions primitives. Gauthier-Villars, Paris 1904.Google Scholar
  76. [76]
    Lucht, L.,Zur Charakterisierung der Gamma-Funktion. J. Reine Angew. Math.288 (1976), 77–85.Google Scholar
  77. [77]
    Lucht, L.,Zur Gaußschen Funktionalgleichung für die Gammafunktion auf multiplikativen Zahlenmengen. Abh. Math. Sem. Univ. Hamburg49 (1979), 183–188.Google Scholar
  78. [78]
    Lucht, L. andWolke, D.,Trigonometrische Reihen über multiplikativen Zahlenmengen. Math. Z.149 (1976), 155–167.CrossRefGoogle Scholar
  79. [79]
    Lucht, L. andWolke, D.,Trigonometrische Reihen über multiplikativen Zahlenmengen II. Acta Math. Hungar.33 (3–4) (1979), 263–288.CrossRefGoogle Scholar
  80. [80]
    Luther, W.,The differentiability of Fourier gap series and “Riemann's example” of a continuous, nondifferentiable function. J. Approx. Theory48 (1986), 303–321.CrossRefGoogle Scholar
  81. [81]
    Mayor, G. andTorrens, J.,De Rham systems and the solution of a class of functional equations. Aequationes Math.47 (1994), 43–49.CrossRefGoogle Scholar
  82. [82]
    Metzler, W.,Note on a chaotic map that generates nowhere-differentiability. Math. Semesterber.40 (1993), 87–90.Google Scholar
  83. [83]
    Mikolás, M.,Construction des familles de fonctions partout continues non dérivables. Acta Sci. Math. (Szeged)17 (1956), 49–62.Google Scholar
  84. [84]
    Mikolás, M.,Integral formulae of arithmetical characteristics relating to the zeta-function of Hurwitz. Publ. Math. Debrecen5 (1957), 44–53.Google Scholar
  85. [85]
    Minkowski, H.,Zur Geometrie der Zahlen. Verhandlungen des III. Internationalen Mathematiker-Kongresses in Heidelberg 1904, pp. 164–173. (Gesammelte Abhandlungen von Hermann Minkowski, Bd. II. B. G. Teubner, Leipzig 1911, pp. 43–52. Reprinted by Chelsea, New York 1967).Google Scholar
  86. [86]
    Mohr, E.,Wo ist die Riemannsche Funktion Σ n = 1∞ (sinn 2x)/n 2 nichtdifferenzierbar? Ann. Math. Pura Appl. (4)123 (1980), 93–104.CrossRefGoogle Scholar
  87. [87]
    Mordell, L. J.,Integral formula of arthimetical character. J. London Math. Soc. (2)33 (1958), 371–375.Google Scholar
  88. [88]
    Neuenschwander, E.,Riemann's example of a continuous, “nondifferentiable” function. Math. Intelligencer1 (1978), 40–44.Google Scholar
  89. [89]
    Nörlund, N. E.,Differenzenrechnung. Springer, Berlin 1924.Google Scholar
  90. [90]
    Paganoni Marzegalli, S.,One-parameter system of functional equations. Aequationes Math.47 (1994), 50–59.CrossRefGoogle Scholar
  91. [91]
    Peano, G.,Sur une courbe qui remplit toute une aire plane. Math. Ann.36 (1890), 157–160.CrossRefMathSciNetGoogle Scholar
  92. [92]
    Perkins, F. W.,An elementary example of a continuous nondifferentiable function. Amer. Math. Monthly34 (1927), 476–478.MathSciNetGoogle Scholar
  93. [93]
    Perron, O.,Die Lehre von den Kettenbrüchen. 3. Aufl. Teubner, Stuttgart 1954.Google Scholar
  94. [94]
    Porter, M. B.,Derivativeless continuous functions. Bull. Amer. Math. Soc.25 (1919), 176–180.Google Scholar
  95. [95]
    Queffelec, H.,Dérivabilité de certaines sommes de séries de Fourier lacunaires. C. R. Acad Sci. Paris Sér. A273 (1971), 291–293.Google Scholar
  96. [96]
    Remmert, R.,Funktionentheorie I. Springer, Berlin, Heidelberg, New York, Tokyo 1984.Google Scholar
  97. [97]
    de Rham, G.,Sur certaines équations fonctionelles. École Polytechnique de l'Université de Lausanne, Centenaire (1853–1953), 95–97. Lausanne 1953.Google Scholar
  98. [98]
    de Rham, G.,Sur quelques courbes définies par des équations fonctionnelles. Rend. Sem. Math. Torino16 (1956), 101–113.Google Scholar
  99. [99]
    de Rham, G.,Sur une exemple de fonction continue sans dérivée. Enseign. Math.3 (1957), 71–72.Google Scholar
  100. [100]
    Salem, R.,On singular monotonic functions of the Cantor type. J. Math. Phys.21 (1942), 69–82.Google Scholar
  101. [101]
    Salem, R.,On some singular monotonic functions which are strictly increasing. Trans. Amer. Math. Soc.53 (1943), 427–439.Google Scholar
  102. [102]
    Schnitzer, F. J.,Bemerkungen über stetige, nirgendwo differenzierbare Funktionen. Grazer Math. Berichte No. 326 (1995).Google Scholar
  103. [103]
    Schroth, P.,On (1, b p)-replicative functions with isolated discontinuities. Aequationes Math.20 (1980), 73–79.CrossRefGoogle Scholar
  104. [104]
    Schroth, P.,Characterization of the Bernoulli-polynomials, exp and ψ by Nörlund's multiplication formula. Period. Math. Hungar12 (3) (1981), 191–204.CrossRefGoogle Scholar
  105. [105]
    Segal, S. L.,Riemann's example of a continuous “nondifferentiable” function continued. Math. Intelligencer1 (1978), 81–82.Google Scholar
  106. [106]
    Sierpiński, W.,Sur un système d'équations fonctionnelles définissant une fonction avec un ensemble dense d'intervalles d'invariabilité. Bull. Intern. Acad. Sci. Cracovie, Cl. Sci. Math. Nat. Sér. A (1911), 577–582.Google Scholar
  107. [107]
    Sierpiński, W.,Sur deux problèmes de la théorie des fonctions non dérivables. Bull. Intern. Acad. Sci. Cracovie, Cl. Sci. Math. Nat. Sér. A (1914), 162–182.Google Scholar
  108. [108]
    Sierpiński, W.,Un exemple élémentaire d'une fonction croissante qui a presque partout une dérivée nulle. Giornale di mat. di Battaglini54 (3) (1916), 314–344.Google Scholar
  109. [109]
    Sinai, Y. G. (Editor),Dynamical Systems II. Encyclopaedia of Mathematical Sciences, Vol. 2. Springer, Berlin, Heidelberg, New York 1989.Google Scholar
  110. [110]
    Singh, A. N.,The Theory and Construction of Nondifferentiable Functions. Lucknow University Studies no. 1 (1935).Google Scholar
  111. [111]
    Smith, A.,The differentiability of Riemann's function. Proc. Amer. Math. Soc.34 (1972), 463–468. Correction in ibidem89 (1983), 567–568.Google Scholar
  112. [112]
    Steinitz, E.,Stetigkeit und Differentialquotient. Math. Ann.52 (1899), 58–69.CrossRefMathSciNetGoogle Scholar
  113. [113]
    Takács, L.,An increasing continuous singular function. Amer. Math. Monthly85 (1978), 35–37.Google Scholar
  114. [114]
    Takagi, T.,A simple example of the continuous function without derivative. J. Phys. Math. Soc. Japan1 (1903), 176–177.Google Scholar
  115. [115]
    Tauber, A.,Über die Weierstrass'sche Funktion. Monatsh. Math. Phys.8 (1897), 330–340.CrossRefGoogle Scholar
  116. [116]
    van der Waerden, B. L.,Ein einfaches Beispiel einer nichtdifferenzierbaren stetigen Funktion. Math. Z.32 (1930), 474–475.CrossRefGoogle Scholar
  117. [117]
    Walter, W.,Old and new approaches to Euler's trigonometric expansions. Amer. Math. Monthly89 (1982), 225–230.Google Scholar
  118. [118]
    Weierstrass, K.,Über continuierliche Functionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen. Mathematische Werke II,32 (1872), 71–74. Mayer u. Müller, Berlin 1885.Google Scholar
  119. [119]
    Wunderlich, W.,Eine überall stetige und nirgends differenzierbare Funktion. Elem. Math.7 (1952), 73–79.Google Scholar
  120. [120]
    Wunderlich, W.,Irregular curves and functional equations. Ganita (Proc. Benares Math. Soc.)5 (1954), 215–230.Google Scholar
  121. [121]
    Yamaguti, M. andHata, M.,Weierstrass's function and chaos. Hokkaido Math. J.12 (1983), 333–342.Google Scholar
  122. [122]
    Yoder, M. F.,Continuous replicative functions. Aequationes Math.13 (1975), 251–261.Google Scholar

Copyright information

© Birkhäuser Verlag 1997

Authors and Affiliations

  • Hans-Heinrich Kairies
    • 1
  1. 1.Institut für MathematikTechnische Universität ClausthalClausthal-ZellerfeldGermany

Personalised recommendations