Differentiability of the composition and quantile operators for regulated and A. E. continuous functions

  • R. M. Dudley
  • R. Norvaiša
Part of the Lecture Notes in Mathematics book series (LNM, volume 1703)


If F is a function defined on the range of a function G, let (FoG)(x)=F(G(x)) for all x. Let (Ω, μ) be a finite measure space. The paper treats differentiability of the two-function composition operator f, g(F+f)o(G+g) into L q (Ω, μ). where g→0 in L s and 1≤q<s. The case where f=0, namely g↦Fo(G+g), for suitable F, G, is a special case of the so-called Nemytskii or superposition operator, which has been extensively studied, as in the book by J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge University Press, 1990, Chapter 3. The remainder R0 in differentiating the two-function composition operator splits as R0=R1+R2, where R1fo(G+g)foG and R2Fo(G+g)FoG(F′oG)·g. Then R2 is the remainder for the Nemytskii operator. Thus, this paper concentrates on R1. For suitable G, the question then is, for what f, and uniformly over what classes of f, is ‖tf○(G+g)-tfg q =o({t{+‖g s ) as {t{+‖g g →0, or equivalently ‖f○(G+g)-fG g =o(1) as ‖g s →0. This is a question of continuity or equicontinuity of Nemytskii operators at points. Previously, for the most part, global continuity had been treated. The individual f are shown to be exactly those which are continuous almost everywhere, suitably measurable, and such that {f(x){/(1+{x{ s/q ) is bounded in x. Large classes of f, called “uniformly Riemann,” are given over which the differentiability is uniform. These give in particular Fréchet differentiability WΦ×L s L q for an arbitrary Φ-variation space WΦ, e.g. any p-variation space Wp. Very similar results are found for the quantile operator g(G+g) for functions G and g from an interval J into ℝ, where H(y)≔inf{xJ:H(x)y}. Also, a theorem is given on composition of Banach-valued functions with supremum norms, where again f need not be differentiable.


Composition Operator Supremum Norm Riemann Function Superposition Operator Finite Measure Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. K. Andersen—Ø. Borgan—R. D. Gill—N. Keiding, Statistical Models Based on Counting Processes, Springer-Verlag, Berlin, 1993.CrossRefzbMATHGoogle Scholar
  2. [2]
    J. Appell, Upper estimates for superposition operators and some applications, Ann. Acad. Sci. Fenn. (=Suomalaisen Tiedeakatemian Helsingfors Toimitsukia) Ser. A I. Math., 8 (1983), pp. 149–159.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    J. Appell, The superposition operator in function spaces—A survey, Expositiones Math., 6 (1988), pp. 209–270.MathSciNetzbMATHGoogle Scholar
  4. [4]
    J. Appell—P. P. Zabrejko, Nonlinear superposition operators, Cambridge University Press, 1990.Google Scholar
  5. [5]
    V. O. Asatiani—Z. A. Chanturia, The modulus of variation of a function and the Banach indicatrix, Acta Sci. Math., 45 (1983), pp. 51–66.MathSciNetzbMATHGoogle Scholar
  6. [6]
    V. I. Averbukh—O. G. Smolyanov, The theory of differentiation in linear topological spaces, Russian Math. Surveys, 22 (1967), no. 6, pp. 201–258=Uspekhi Mat. Nauk, 22 (1967), no. 6, pp. 201–260.CrossRefzbMATHGoogle Scholar
  7. [7]
    V. I. Averbukh—O. G. Smolyanov, The various definitions of the derivative in linear topological spaces, Russian Math. Surveys, 23 (1968), no. 4, pp. 67–113=Uspekhi Mat. Nauk, 23 (1968), no. 4, pp. 67–116.CrossRefzbMATHGoogle Scholar
  8. [8]
    S. K. Berberian, The character space of the algebra of regulated functions, Pacific J. Math., 74 (1978), pp. 15–36.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.zbMATHGoogle Scholar
  10. [10]
    P. Billingsley—F. Topsøe, Uniformity in weak convergence, Z. Wahrscheinlichkeitsth. verw. Geb., 7 (1967), pp. 1–16.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    N. H. Bingham, Fluctuation theory in continuous time, Adv. Appl. Prob. 7 (1975), pp. 705–766.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    N. Bourbaki, Fonctions d'une variable réelle, Hermann, Paris, 1976.zbMATHGoogle Scholar
  13. [13]
    M. Brokate—F. Colonius, Linearizing equations with state-dependent delays, Appl. Math. Optimiz., 21 (1990), pp. 45–52.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    W. Bücher, Differentiabilité de la composition et complétitude de certains espaces fonctionnels, Comm. Math. Helv., 43 (1968), pp. 256–288.CrossRefzbMATHGoogle Scholar
  15. [15]
    Z. A. Chanturia [Čanturija], The modulus of variation of a function and its application in the theory of Fourier series, Dokl. Akad. Nauk SSSR, 214 (1974), pp. 63–66 =Soviet Math. Dokl. 15 (1974), pp. 67–71.MathSciNetGoogle Scholar
  16. [16]
    D. L. Cohn, Measure Theory, Birkhäuser, Boston, 1980.CrossRefzbMATHGoogle Scholar
  17. [17]
    R. B. Darst, A characterization of universally measurable sets, Proc. Camb. Philos. Soc., 65 (1969), pp. 617–618.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    C. Dellacherie—P.-A. Meyer, Probabilities and Potential, Hermann, Paris, 1975; English transl. North-Holland, Amsterdam, 1978.zbMATHGoogle Scholar
  19. [19]
    J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960; Fondements de l'analyse moderne, 1, Gauthier-Villars, Paris, 1963.zbMATHGoogle Scholar
  20. [20]
    R. M. Dudley, Real Analysis and Probability (2d printing, corrected), Chapman and Hall, New York and London, 1993.zbMATHGoogle Scholar
  21. [21]
    R. M. Dudley, Fréchet differentiability, p-variation and uniform Donsker classes, Ann. Probab., 20 (1992), pp. 1968–1982.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    R. M. Dudley, The order of the remainder in derivatives of composition and inverse operators for p-variation norms, Ann. Statist., 22 (1994), pp. 1–20.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    R. M. Dudley, Empirical processes and p-variation, in Festschrift for Lucien Le Cam, Eds. D. Pollard, E. Torgersen, G. L. Yang, Springer-Verlag, New York, 1997, pp. 219–233.CrossRefGoogle Scholar
  24. [24]
    N. Dunford—J. T. Schwartz, Linear Operators, Part I, Interscience, New York, 1958.zbMATHGoogle Scholar
  25. [25]
    W. Esty—R. Gillette—M. Hamilton—D. Taylor, Asymptotic distribution theory of statistical functionals: the compact derivative approach for robust estimators, Ann. Inst. Statist. Math., 37 (1985), pp. 109–129.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    L. T. Fernholz, von Mises calculus for statistical functionals, Lect. Notes in Statist. (Springer-Verlag), 19, 1983.Google Scholar
  27. [27]
    A. Filippova, Mises' theorem on the asymptotic behavior of functionals of empirical distribution functions and its statistical applications, Theory Probab. Appl., 7 (1961), pp. 24–57.CrossRefzbMATHGoogle Scholar
  28. [28]
    M. Fréchet, La notion de différentielle dans l'analyse générale, Ann. Sci. Ecole Norm. Sup. (Sér. 3), 42 (1925), pp. 293–323.zbMATHGoogle Scholar
  29. [29]
    B. V. Gnedenko—A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, 2d ed. Transl. and Ed. by K. L. Chung, Addison-Wesley, Reading, Mass, 1968.Google Scholar
  30. [30]
    C. Goffman—G. Moran—D. Waterman, The structure of regulated functions, Proc. Amer. Math. Soc., 57 (1976), pp. 61–65.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    H. Goldberg—W. Kampowsky—F. Tröltzsch, On Nemytskij operators in Lp-spaces of abstract functions, Math. Nachr., 155, pp. 127–140.Google Scholar
  32. [32]
    A. Gray, Differentiation of composites with respect to a parameter, J. Austral. Math. Soc. (Ser. A), 19 (1975), pp. 121–128.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    T. H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, New York, 1963.zbMATHGoogle Scholar
  34. [34]
    E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier's Series, 1, 3d ed. (1927), repr. Dover, New York, 1957.zbMATHGoogle Scholar
  35. [35]
    M. A. Krasnosel'skiî—P.P. Zabreîko—E.I. Pustyl'nik— P. Sobolevskiî, Integral operators in spaces of summable functions, Nauka, Moscow, 1966; transl. by T. Ando, Noordhoff, Leyden, 1976.zbMATHGoogle Scholar
  36. [36]
    R. Lucchetti—F. Patrone, On Nemytskii's operator and its application to the lower semicontinuity of integral functionals, Indiana Univ. Math. J., 29, pp. 703–713.Google Scholar
  37. [37]
    A. Mukherjea—K. Pothoven, Real and Functional Analysis, Plenum, New York and London, 1978.CrossRefzbMATHGoogle Scholar
  38. [38]
    J. Musielak—W. Orlicz, On generalized variations (I), Studia Math., 18 (1959), pp. 11–41.MathSciNetzbMATHGoogle Scholar
  39. [39]
    E. Nelson, Regular probability measures on function space, Ann. Math., 69 (1959), pp. 630–643.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    S. Perlman, Functions of generalized variation, Fund. Math., 105 (1980), pp. 199–211.MathSciNetzbMATHGoogle Scholar
  41. [41]
    J. A. Reeds III, On the definition of von Mises functionals, Ph. D. Dissertation, Harvard University, 1976.Google Scholar
  42. [42]
    B. Riemann, Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe, Abh. Gesell. Wiss. Göttingen Math. Kl. 13, pp. 87–132; repr. in Bernhard Riemann: Gesammelte mathematische Werke und wissenschaftlicher Nachlass, with commentaries, 2d. ed., ed. Raghavan Narasimhan, Springer-Verlag (Heidelberg) and Teubner (Leipzig), 1990.Google Scholar
  43. [43]
    F. Riesz—B. Sz.-Nagy, Leçons d'analyse fonctionelle, 3d ed., Gauthier-Villars, Paris, 1955; Functional Analysis (transl. by L. F. Boron), Ungar, New York, 1955.zbMATHGoogle Scholar
  44. [44]
    J. Sebastião E Silva, Le calcul différentiel et intégral dans les espaces localement convexes, réels ou complexes I, II, Rend. Accad. Lincei Sci. Fis. Mat. Nat., (Ser. 8) 20 (1956), pp. 743–750, 21 (1956), pp. 40–46.zbMATHGoogle Scholar
  45. [45]
    G. E. Shilov—B. L. Gurevich, Integral, Measure and Derivative: A Unified Approach, Transl. and Ed. by R. A. Silverman, Prentice-Hall, Englewood Cliffs, N.J., 1966.Google Scholar
  46. [46]
    I. V. Shragin, Superposition measurability, Sov. Math. (Iz. Vuz.), 19 (1975), pp. 69–76 =Izv. Vyssh. Uch. Zaved., 1975, no. 1, pp. 82–92.Google Scholar
  47. [47]
    I. V. Shragin, Classes of measurable vector functions and Nemytskii's operators I, II, Russian Math. (Iz. Vuz.), 38 (1994), no. 4, pp. 45–55, no. 5, pp. 70–79, =Izv. Vyssh. Uch. Zaved., 1994, no. 4, pp. 48–58, no. 5, pp. 70–79.MathSciNetGoogle Scholar
  48. [48]
    W. Sierpiński, Sur une propriété des fonctions qui n'ont que des discontinuités de première espèce, Bull. Sect. Scient. Acad Roumaine, 16 (1933), no. 1/3, pp. 1–4. We found these references from secondary sources but have not seen them in the original.zbMATHGoogle Scholar
  49. [49]
    A. V. Skorohod, Limit theorems for stochastic processes with independent increments, Theory Prob. Appl. 2 (1957), pp. 138–171.MathSciNetCrossRefGoogle Scholar
  50. [50]
    R. Taberski, On the power variations and pseudovariations of positive integer orders, Demonstratio Math., 19 (1986), pp. 881–893.MathSciNetzbMATHGoogle Scholar
  51. [51]
    A. E. Taylor, The differential: nineteenth and twentieth century developments, Arch. Hist. Exact Sci., 12 (1974), pp. 355–383.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    O. D. Tsereteli (Cereteli), The metric properties of a function of bounded variation, (in Russian), Akad. Nauk Gruzin. SSR Trudy Tbiliss. Mat. Inst. Razmadze, 26 (1959), pp. 23–64. We found these references from secondary sources but have not seen them in the original.MathSciNetGoogle Scholar
  53. [53]
    O. D. Tsereteli (Cereteli), On the Banach indicatrix and some of its applications, (in Russian), Soobshch. Akad. Gruzin. SSR 25 (1960), pp. 129–136. We found these references from secondary sources but have not seen them in the original.MathSciNetGoogle Scholar
  54. [54]
    M. M. Vaînberg, Variational methods in the study of nonlinear operators, Gostekhizdat, Moscow, 1956; English transl. Holden-Day, San Francisco, 1964.zbMATHGoogle Scholar
  55. [55]
    R. von Mises, Les lois de probabilité pour les fonctions statistiques, Ann. Inst. H. Poincaré, 6 (1936), pp. 185–212.MathSciNetzbMATHGoogle Scholar
  56. [56]
    R. von Mises, On the asymptotic behavior of differentiable statistical functions, Ann. Math. Statist., 18 (1947), pp. 309–348.CrossRefzbMATHGoogle Scholar
  57. [57]
    Wang Sheng-Wang, Differentiability of the Nemyckii operator, Doklady Akad. Nauk SSSR, 150 (1963), pp. 1198–1201 (Russian); Sov. Math. Doklady, 4 (1963), pp. 834–837.MathSciNetGoogle Scholar
  58. [58]
    L. C. Young, General inequalities of Stieltjes integrals and the convergence of Fourier series, Math. Ann., 115(1938), pp. 581–612.MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    W. H. Young, On the distinction of right and left at points of discontinuity, Quarterly J. Pure and Applied Math., 39 (1908), pp. 67–83.zbMATHGoogle Scholar
  60. [60]
    W. H. Young, On the discontinuities of a function of one or more real variables, Proc. London Math. Soc. (Ser. 2) 8 (1909), pp. 117–124.Google Scholar
  61. [61]
    E. Zeidler, Nonlinear Functional Analysis and its Applications Vols I, II/B, Springer-Verlag, Berlin, 1985, 1990.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • R. M. Dudley
  • R. Norvaiša

There are no affiliations available

Personalised recommendations