Functional equations for peculiar functions. (English) Zbl 0876.39004
This is a thorough and interesting survey of functional equations which characterize certain everywhere continuous nowhere differentiable functions, strictly monotonic functions with almost everywhere 0 derivatives, replicative functions (cf. D. E. Knuth [The art of computer programming, Vol. I. (1968; Zbl 0191.17903)], etc. It concludes with some open problems and with 122 references.
Reviewer: J.Aczel (Waterloo/Ontario)
MSC:
| 39B22 | Functional equations for real functions |
| 26A27 | Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives |
| 39B62 | Functional inequalities, including subadditivity, convexity, etc. |
| 26A30 | Singular functions, Cantor functions, functions with other special properties |
Keywords:
Continuous nowhere differentiable functions; strictly monotonic functions with almost everywhere 0 derivatives; Replicative functions; Characterizations by functional equations in a single variable; Peculiar functions; SurveyCitations:
Zbl 0191.17903References:
| [1] | Apostol, T. M.,Introduction to Analytic Number Theory. Springer, New York, Heidelberg, Berlin 1976. · Zbl 0335.10001 |
| [2] | Apostol, T. M.,Modular Functions and Dirichlet Series in Number Theory. Springer, New York, Heidelberg, Berlin 1976. · Zbl 0332.10017 |
| [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. · Zbl 0001.28603 |
| [4] | Behrend, F. A.,Some remarks on the construction of continuous nondifferentiable functions. Proc. London Math. Soc. (2)50 (1949), 463–481. · Zbl 0083.28501 · doi:10.1112/plms/s2-50.6.463 |
| [5] | Benz, W.,Remarks on exorbitant equations. Aequationes Math.41 (1991), 283. |
| [6] | Bernstein, B., Erber, T. andKaramolengos, M.,Sneaky plasticity and mesoscopic fractals. Chaos, Solitons & Fractals3 (1993), 269–277. · Zbl 0772.73031 · doi:10.1016/0960-0779(93)90010-X |
| [7] | Billingsley, P.,The singular function of bold play. Amer. Scientist71 (1983), 392–397. |
| [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. · JFM 06.0241.01 |
| [9] | Borwein, J. M. andGirgensohn, R.,Addition theorems and binary expansions. Canad. J. Math.47 (1995), no. 2, 262–273. · Zbl 0830.39018 · doi:10.4153/CJM-1995-013-4 |
| [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. · Zbl 0629.01013 |
| [11] | Cantor, G.,De la puissance des ensembles parfaits des points. Acta. math.4 (1884), 381–392. · JFM 16.0460.01 · doi:10.1007/BF02418423 |
| [12] | Cater, F. S.,On van der Waerden’s nowhere differentiable function. Amer. Math. Monthly91 (1984), 307–308. · Zbl 0598.26015 · doi:10.2307/2322678 |
| [13] | Cellérier, C.,Note sur les principes fondamentaux de l’analyse. Bull. des Sc. Math. (2), vol. XIV première partie (1890), 142–160. · JFM 22.0386.01 |
| [14] | Cesàro, E.,Fonctions continues sans dérivées. Arch. Math. Phys. (3)38 (1906), 57–63. · JFM 37.0412.04 |
| [15] | Cesàro, E.,Fonctions continues sans dérivée formées avec les itérées d’une fraction rationnelle. Ann. Sci. École Norm. Sup48 (1931), 1–14. |
| [16] | Darsow, W. F. andFrank, M. J.,Associative functions and Abel-Schröder systems. Publ. Math. Debrecen30 (1983), 253–272. · Zbl 0542.39001 |
| [17] | Darsow, W. F. andFrank, M. J.,Conjugacy systems. In Proc. 23. ISFE. Centre for Information Theory, Univ. of Waterloo 1985, pp. 8–9. |
| [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. · Zbl 0665.26003 |
| [19] | Denjoy, A.,Sur quelques points de la théorie des fonctions. C. R. Acad. Sci. Paris194 (1932), 44–46. · Zbl 0003.31204 |
| [20] | Denjoy, A.,Sur une fonction de Minkowski. C. R. Acad. Sci. Paris198 (1934), 44–47. · Zbl 0008.20207 |
| [21] | Denjoy, A.,Sur une fonction réelle de Minkowski. J. Math. Pures Appl. (9)17 (1938), 105–151. · Zbl 0018.34602 |
| [22] | Dennler, G.,Bestimmung sämtlicher meromorpher Lösungen der Funktionalgleichung f(z) = (1/k) {\(\Sigma\)} h = 0 k 1 f((z + h)/k). Wiss. Z. Friedrich-Schiller-Univ. Jena Math.-Naturw. Reihe14 (1965). · Zbl 0146.13202 |
| [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. |
| [24] | Dubuc, S.,Une équation fonctionelle pour diverses constructions géométríques. Ann. Sci. Math. Québec9 (1985), no. 2, 151–174. · Zbl 0598.39007 |
| [25] | Dubins, L. E. andSavage, L.,How to Gamble if You Must. Inequalities for stochastic processes. McGraw-Hill 1965. · Zbl 0133.41402 |
| [26] | Duistermaat, J. J.,Selfsimilarity of ’Riemann’s nondifferentiable function’. Nieuw Arch. WiskundeIV9 (1991), 303–337. · Zbl 0760.26009 |
| [27] | Faber, G.,Einfaches Beispiel einer stetigen nirgends differenzierbaren Funktion. Jahresber. Deutsch. Math.-Verein.16 (1907), 538–540. · JFM 38.0417.02 |
| [28] | Faber, G.,Über stetige Funktionen II. Math. Ann.69 (1910), 372–433. · JFM 41.0448.02 · doi:10.1007/BF01456327 |
| [29] | Falconer, K. J.,Fractal Geometry. Mathematical Foundations and Applications. Wiley and Sons, Chichester 1990. · Zbl 0689.28003 |
| [30] | Frank, M. J.,A generalization of the van der Waerden system. Aequationes Math.37 (1989), 67. |
| [31] | Freud, G.,Über trigonometrische Approximation und Fouriersche Reihen. Math. Z.78 (1962), 252–262. · Zbl 0101.04802 · doi:10.1007/BF01195173 |
| [32] | Gelbaum, B. R. andOlmsted, J.,Theorems and Counterexamples in Mathematics. Springer, New York, Berlin, Heidelberg 1990. |
| [33] | Gerver, J.,The differentiability of the Riemann function at certain rational multiples of {\(\pi\)}. Amer. J. Math.92 (1970), 33–55. · Zbl 0203.05904 · doi:10.2307/2373496 |
| [34] | Gerver, J.,More on the differentiability of the Riemann function. Amer. J. Math.93 (1971), 33–41. · Zbl 0228.26008 · doi:10.2307/2373445 |
| [35] | Girgensohn, R.,Funktionalgleichungen für nirgends differenzierbare Funktionen. Clausthal 1992 (Dissertation). · Zbl 0827.39004 |
| [36] | Girgensohn, R.,Functional equations and nowhere differentiable functions. Aequationes Math.48 (1993), 243–256. · Zbl 0788.26007 |
| [37] | Girgensohn, R.,Nowhere differentiable solutions of a system of functional equations. Aequationes Math.47 (1994), 89–99. · Zbl 0791.26005 · doi:10.1007/BF01838143 |
| [38] | Girgensohn, R.,Constructing singular functions via Farey fractions. J. Math. Anal. Appl.203 (1996), 127–141. · Zbl 0866.26003 · doi:10.1006/jmaa.1996.0370 |
| [39] | Hahn, H.,Über stetige Funktionen ohne Ableitung. Jahresber. Deutsch. Math.-Verein.26 (1918), 281–284. · JFM 46.0399.05 |
| [40] | Hardy, G. H.,Weierstrass’s nondifferentiable function. Trans. Amer. Math. Soc.17 (1916), 301–325. · JFM 46.0401.03 |
| [41] | Hartmann, P. andKershner, R.,The structure of monotone functions. Amer. J. Math.59 (1937), 809–822. · Zbl 0018.01202 · doi:10.2307/2371350 |
| [42] | Hata, M.,Singularities of the Weierstrass type functions. J. Analyse Math.51 (1988), 62–90. · Zbl 0663.26004 · doi:10.1007/BF02791119 |
| [43] | Hata, M.,On Weierstrass’s nondifferentiable function. C. R. Acad. Sci. Paris Sér. 1 Math.307 (1988), 119–123. · Zbl 0645.26006 |
| [44] | Hata, M. andYamaguti, M.,The Takagi function and its generalization. Japan. J. Appl. Math.1 (1984), 183–199. · Zbl 0604.26004 · doi:10.1007/BF03167867 |
| [45] | Hewitt, E. andStromberg, K.,Real and Abstract Analysis. Springer, Berlin, Heidelberg, New York 1965. |
| [46] | Hildebrandt, T. H.,A simple continuous function with a finite derivative at no point. Amer. Math. Monthly40 (1933), 547–548. · JFM 59.0285.03 · doi:10.2307/2301999 |
| [47] | Hille, E. andTamarkin, J. D.,Remarks on a known example of a monotone continuous function. Amer. Math. Monthly36 (1929), 255–264. · JFM 55.0142.06 · doi:10.2307/2298506 |
| [48] | Holschneider, M. andTchamitchian, P.,Pointwise analysis of Riemann’s ”nondifferentiable” function. Invent. math.105 (1991), 157–175. · Zbl 0741.26004 · doi:10.1007/BF01232261 |
| [49] | Itatsu, S.,Differentiability of Riemann’s function. Proc. Japan. Acad. Math. Sci. A57 (10) (1981), 492–495. · Zbl 0501.26004 · doi:10.3792/pjaa.57.492 |
| [50] | Jäger, R.,Charakterisierung des Cotangens mit Replikativität. Manuscripta Math.56 (1986), 167–175. · Zbl 0589.39008 · doi:10.1007/BF01172154 |
| [51] | Jäger, R.,Beiträge zur Theorie der Replikativitätsgleichung. Clausthal 1987 (Dissertation). |
| [52] | Kairies, H.-H.,Zur axiomatischen Charakterisierung der Gammafunktion. J. Reine Angew. Math.236 (1969), 103–111. · Zbl 0174.10501 · doi:10.1515/crll.1969.236.103 |
| [53] | Kairies, H.-H.,Multiplikationstheoreme. Braunschweig 1971 (Habilitationsschrift). |
| [54] | Kairies, H.-H.,Definitionen der Bernoulli-Polynome mit Hilfe ihrer Multiplikationstheoreme. Manuscripta Math.8 (1973), 363–369. · Zbl 0248.39003 · doi:10.1007/BF01297668 |
| [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. · Zbl 0286.39005 · doi:10.1007/BF01832863 |
| [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. · Zbl 0445.33002 |
| [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. · Zbl 0431.41047 |
| [58] | Kairies, H.-H.,Charakterisierungen von nirgends differenzierbaren Weierstrass-Funktionen durch Replikativität. Elem. Math.45 (1990), 61–69. · Zbl 0705.26010 |
| [59] | Kairies, H.-H.,Die Gammafunktion als stetige Lösung eines Systems von Gauss-Funktionalgleichungen. Results in Math.26 (1994), 306–315. · Zbl 0831.39011 |
| [60] | Kairies, H.-H.,Zu einem Charakterisierungssatz von E. Artin. Abh. Math. Sem. Univ. Hamburg.65 (1995), 135–146. · Zbl 0858.33001 · doi:10.1007/BF02953320 |
| [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. · Zbl 0842.39009 · doi:10.1007/BF01831144 |
| [62] | Kairies, H.-H. andRose, M.,Funktionalgleichungen und Charakterisierungen für die Riemannsche Funktion. Period. Math. Hung.32 (3) (1996), 199–207. · Zbl 0901.39005 · doi:10.1007/BF02109789 |
| [63] | Kántor, S.,Über die Konstruktion überall stetiger, nirgends differenzierbarer Funktionen I. Publ. Math. Debrecen9 (1962), 111–121. · Zbl 0132.28801 |
| [64] | Katsuura, H.,Continuous nowhere-differentiable functions – an application of contraction mappings. Amer. Math. Monthly98 (1991), 411–416. · Zbl 0752.26005 · doi:10.2307/2323856 |
| [65] | Kiesswetter, K.,Ein einfaches Beispiel für eine Funktion, welche überall stetig und nirgends differenzierbar ist. Math. Phys. Semesterber.13 (1966), 216–221. · Zbl 0143.07304 |
| [66] | Klemmt, H.-J.,Ein maßtheoretischer Satz mit Anwendungen auf Funktionalgleichungen. Arch. Math. (Basel)38 (1981), 537–540. · Zbl 0446.33004 |
| [67] | Knopp, K.,Einfaches Beispiel einer stetigen, nirgends differenzierbarren Funktion. Jahresber. Deutsch. Math.-Verein.26 (1918), 278–280. · JFM 46.0400.02 |
| [68] | Knopp, K.,Ein einfaches Verfahren zur Bildung stetiger nirgends differenzierbarer Funktionen. Math. Z.2 (1918), 1–26. · JFM 46.0400.03 · doi:10.1007/BF01212897 |
| [69] | Knuth, D. E.,The Art of Computer Programming, Vol. I. Addison-Wesley, Reading, Mass. 1969. · Zbl 0191.18001 |
| [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. |
| [71] | Kôno, N.,On generalized Takagi functions. Acta Math. Hung.49 (1987), 315–324. · Zbl 0627.26004 · doi:10.1007/BF01950992 |
| [72] | Kuczma, M.,Functional Equations in a Single Variable. Monografie Math.46. Polish Scientific Publishers, Warszawa 1968. · Zbl 0196.16403 |
| [73] | Kuczma, M., Choczewski, B. andGer, R.,Iterative Functional Equations. Cambridge University Press, Cambridge 1990. · Zbl 0703.39005 |
| [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. |
| [75] | Lebsgue, H.,Leçons sur l’intégration et la recherche des fonctions primitives. Gauthier-Villars, Paris 1904. |
| [76] | Lucht, L.,Zur Charakterisierung der Gamma-Funktion. J. Reine Angew. Math.288 (1976), 77–85. · Zbl 0333.33001 · doi:10.1515/crll.1976.288.77 |
| [77] | Lucht, L.,Zur Gaußschen Funktionalgleichung für die Gammafunktion auf multiplikativen Zahlenmengen. Abh. Math. Sem. Univ. Hamburg49 (1979), 183–188. · doi:10.1007/BF02950656 |
| [78] | Lucht, L. andWolke, D.,Trigonometrische Reihen über multiplikativen Zahlenmengen. Math. Z.149 (1976), 155–167. · doi:10.1007/BF01301572 |
| [79] | Lucht, L. andWolke, D.,Trigonometrische Reihen über multiplikativen Zahlenmengen II. Acta Math. Hungar.33 (3–4) (1979), 263–288. · Zbl 0419.10036 · doi:10.1007/BF01902561 |
| [80] | Luther, W.,The differentiability of Fourier gap series and ”Riemann’s example” of a continuous, nondifferentiable function. J. Approx. Theory48 (1986), 303–321. · Zbl 0626.42008 · doi:10.1016/0021-9045(86)90053-5 |
| [81] | Mayor, G. andTorrens, J.,De Rham systems and the solution of a class of functional equations. Aequationes Math.47 (1994), 43–49. · Zbl 0802.39008 · doi:10.1007/BF01838138 |
| [82] | Metzler, W.,Note on a chaotic map that generates nowhere-differentiability. Math. Semesterber.40 (1993), 87–90. · Zbl 0784.26004 · doi:10.1007/BF03186480 |
| [83] | Mikolás, M.,Construction des familles de fonctions partout continues non dérivables. Acta Sci. Math. (Szeged)17 (1956), 49–62. · Zbl 0070.28202 |
| [84] | Mikolás, M.,Integral formulae of arithmetical characteristics relating to the zeta-function of Hurwitz. Publ. Math. Debrecen5 (1957), 44–53. · Zbl 0081.27403 |
| [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). |
| [86] | Mohr, E.,Wo ist die Riemannsche Funktion {\(\Sigma\)} n = 1 (sinn 2x)/n 2 nichtdifferenzierbar? Ann. Math. Pura Appl. (4)123 (1980), 93–104. · Zbl 0456.26003 · doi:10.1007/BF01796541 |
| [87] | Mordell, L. J.,Integral formula of arthimetical character. J. London Math. Soc. (2)33 (1958), 371–375. · Zbl 0081.27404 · doi:10.1112/jlms/s1-33.3.371 |
| [88] | Neuenschwander, E.,Riemann’s example of a continuous, ”nondifferentiable” function. Math. Intelligencer1 (1978), 40–44. · Zbl 0374.26002 · doi:10.1007/BF03023045 |
| [89] | Nörlund, N. E.,Differenzenrechnung. Springer, Berlin 1924. |
| [90] | Paganoni Marzegalli, S.,One-parameter system of functional equations. Aequationes Math.47 (1994), 50–59. · Zbl 0828.39015 · doi:10.1007/BF01838139 |
| [91] | Peano, G.,Sur une courbe qui remplit toute une aire plane. Math. Ann.36 (1890), 157–160. · JFM 22.0405.01 · doi:10.1007/BF01199438 |
| [92] | Perkins, F. W.,An elementary example of a continuous nondifferentiable function. Amer. Math. Monthly34 (1927), 476–478. · JFM 53.0202.01 · doi:10.2307/2300224 |
| [93] | Perron, O.,Die Lehre von den Kettenbrüchen. 3. Aufl. Teubner, Stuttgart 1954. · Zbl 0056.05901 |
| [94] | Porter, M. B.,Derivativeless continuous functions. Bull. Amer. Math. Soc.25 (1919), 176–180. · JFM 47.0234.02 · doi:10.1090/S0002-9904-1919-03174-7 |
| [95] | Queffelec, H.,Dérivabilité de certaines sommes de séries de Fourier lacunaires. C. R. Acad Sci. Paris Sér. A273 (1971), 291–293. · Zbl 0221.26008 |
| [96] | Remmert, R.,Funktionentheorie I. Springer, Berlin, Heidelberg, New York, Tokyo 1984. |
| [97] | de Rham, G.,Sur certaines équations fonctionelles. École Polytechnique de l’Université de Lausanne, Centenaire (1853–1953), 95–97. Lausanne 1953. |
| [98] | de Rham, G.,Sur quelques courbes définies par des équations fonctionnelles. Rend. Sem. Math. Torino16 (1956), 101–113. |
| [99] | de Rham, G.,Sur une exemple de fonction continue sans dérivée. Enseign. Math.3 (1957), 71–72. · Zbl 0077.06104 |
| [100] | Salem, R.,On singular monotonic functions of the Cantor type. J. Math. Phys.21 (1942), 69–82. · Zbl 0060.18606 |
| [101] | Salem, R.,On some singular monotonic functions which are strictly increasing. Trans. Amer. Math. Soc.53 (1943), 427–439. · Zbl 0060.13709 · doi:10.1090/S0002-9947-1943-0007929-6 |
| [102] | Schnitzer, F. J.,Bemerkungen über stetige, nirgendwo differenzierbare Funktionen. Grazer Math. Berichte No. 326 (1995). · Zbl 0838.26004 |
| [103] | Schroth, P.,On (1, b p)-replicative functions with isolated discontinuities. Aequationes Math.20 (1980), 73–79. · Zbl 0433.39009 · doi:10.1007/BF02190495 |
| [104] | Schroth, P.,Characterization of the Bernoulli-polynomials, exp and {\(\psi\)} by Nörlund’s multiplication formula. Period. Math. Hungar12 (3) (1981), 191–204. · Zbl 0445.39002 · doi:10.1007/BF01849786 |
| [105] | Segal, S. L.,Riemann’s example of a continuous ”nondifferentiable” function continued. Math. Intelligencer1 (1978), 81–82. · Zbl 0396.26003 · doi:10.1007/BF03023065 |
| [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. · JFM 42.0353.02 |
| [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. |
| [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. · JFM 46.0403.04 |
| [109] | Sinai, Y. G. (Editor),Dynamical Systems II. Encyclopaedia of Mathematical Sciences, Vol. 2. Springer, Berlin, Heidelberg, New York 1989. · Zbl 0850.70172 |
| [110] | Singh, A. N.,The Theory and Construction of Nondifferentiable Functions. Lucknow University Studies no. 1 (1935). · JFM 61.1113.06 |
| [111] | Smith, A.,The differentiability of Riemann’s function. Proc. Amer. Math. Soc.34 (1972), 463–468. Correction in ibidem89 (1983), 567–568. · Zbl 0254.26013 |
| [112] | Steinitz, E.,Stetigkeit und Differentialquotient. Math. Ann.52 (1899), 58–69. · JFM 30.0265.02 · doi:10.1007/BF01445347 |
| [113] | Takács, L.,An increasing continuous singular function. Amer. Math. Monthly85 (1978), 35–37. · Zbl 0394.26005 · doi:10.2307/2978047 |
| [114] | Takagi, T.,A simple example of the continuous function without derivative. J. Phys. Math. Soc. Japan1 (1903), 176–177. · JFM 34.0410.05 |
| [115] | Tauber, A.,Über die Weierstrass’sche Funktion. Monatsh. Math. Phys.8 (1897), 330–340. · JFM 28.0343.01 · doi:10.1007/BF01696283 |
| [116] | van der Waerden, B. L.,Ein einfaches Beispiel einer nichtdifferenzierbaren stetigen Funktion. Math. Z.32 (1930), 474–475. · JFM 56.0929.02 · doi:10.1007/BF01194647 |
| [117] | Walter, W.,Old and new approaches to Euler’s trigonometric expansions. Amer. Math. Monthly89 (1982), 225–230. · doi:10.2307/2320218 |
| [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. |
| [119] | Wunderlich, W.,Eine überall stetige und nirgends differenzierbare Funktion. Elem. Math.7 (1952), 73–79. · Zbl 0046.28704 |
| [120] | Wunderlich, W.,Irregular curves and functional equations. Ganita (Proc. Benares Math. Soc.)5 (1954), 215–230. · Zbl 0058.38307 |
| [121] | Yamaguti, M. andHata, M.,Weierstrass’s function and chaos. Hokkaido Math. J.12 (1983), 333–342. |
| [122] | Yoder, M. F.,Continuous replicative functions. Aequationes Math.13 (1975), 251–261. · Zbl 0333.39007 · doi:10.1007/BF01836528 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
