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Rearrangements of Series

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Abstract

The present work is devoted to the problems related to the rearrangements of series in metrizable topological vector spaces. The idea goes back to Dirichlet (most probably, he was the first who discovered the phenomenon that the sum of a scalar series may depend on the rearrangement of its terms), Riemann, P. Levy, E. Steinitz, Banach, Kolmogorov, and others.

Two main directions in this area, attracting the interest of many researchers, are considered. The first one concerns the problem of the structure of the sum range of conditionally convergent series. The other is the problem of the existence of an almost sure convergent rearrangement of a functional series, including some classical problems on the convergence of Fourier series.

This book consists mainly of material included in the Ph.D. thesis and some recent works of the author and his colleagues.

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References

  1. A. de Acosta, “Exponential moments of vector-valued random series and triangular arrays,” Ann. Probab., 8, No. 2, 381–389 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. de Acosta and J. D. Samur, “Infinitely divisible probability measures and the converse Kolmogorov inequality in Banach spaces,” Stud. Math., 66, No. 2, 143–160 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Banach, “Sur la divergence des séries orthogonales,” Stud. Math., 9, 139–155 (1940).

    Article  MATH  Google Scholar 

  4. W. Banaszczyk, “The Steinitz constant of the plane,” J. Reine Angew. Math., 373, 218–220 (1987).

    MathSciNet  MATH  Google Scholar 

  5. W. Banaszczyk, “The Steinitz theorem on rearrangement of series for nuclear spaces,” J. Reine Angew. Math., 403, 187–200 (1990).

    MathSciNet  MATH  Google Scholar 

  6. W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lect. Notes Math., 1466, Springer-Verlag, Berlin (1991).

  7. W. Banaszczyk, “Balancing vectors and convex bodies,” Stud. Math., 106, No. 1, 93–100 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  8. W. Banaszczyk, “On series of signed vectors and their rearrangements,” Random Struct. Algor., 40, No. 3, 301–316 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  9. I. Barany, “Rearrangements of series in infinite-dimensional spaces,” Mat. Zametki, 46, No. 6, 10–17, 126 (1989).

  10. I. Bárány, “On the power of linear dependencies,” Building Bridges, Bolyai Soc. Math. Stud., 19, Springer, Berlin, (2008), pp. 31–45.

  11. I. Bárány and V. S. Grinberg, “On some combinatorial questions in finite-dimensional spaces,” Linear Algebra Appl., 41, 1–9 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  12. G. Bennett, “Unconditional convergence and almost everywhere convergence,” Z. Wahrscheinlich. und Verw. Gebiete, 34, No. 2, 135–155 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Bessaga and A. Pełczyński, “On a class of B 0-spaces,” Bull. Acad. Polon. Sci. Cl. III., 5, 375–377, XXX (1957).

  14. P. Billard, “Sur la convergence presque partout des séries de Fourier–Walsh des fonctions de l’espace L 2(0, 1),” Stud. Math., 28, 363–388 (1966/1967).

  15. J. Bonet and A. Defant, “The Levy–Steinitz rearrangement theorem for duals of metrizable spaces,” Israel J. Math., 117, 131–156 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Bourgain, “On Kolmogorov’s rearrangement problem for orthogonal systems and Garsia’s conjecture,” in: Geometric aspects of functional analysis (1987–88), Lect. Notes Math., 1376, Springer, Berlin (1989), pp. 209–250.

  17. V. V. Buldygin, The Convergence of Random Elements in Topological Spaces, Naukova Dumka, Kiev (1980).

    MATH  Google Scholar 

  18. L. Carleson, “On convergence and growth of partial sums of Fourier series,” Acta Math., 116, 135–157 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  19. N. L. Carothers, A Short Course on Banach Space Theory, London Math. Soc. Student Texts, 64, Cambridge University Press, Cambridge (2005).

  20. M.-J. Chasco and S. Chobanyan, “On rearrangements of series in locally convex spaces,” Mich. Math. J., 44, No. 3, 607–617 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  21. G. Chelidze, S. Chobanyan, G. Giorgobiani, and V. Kvaratskhelia, “Greedy algorithm fails in compact vector summation,” Bull. Georgian Natl. Acad. Sci., 4, No. 2, 5–7 (2010).

    MathSciNet  MATH  Google Scholar 

  22. G. Chelidze, G. Giorgobiani, and V. Tarieladze, “Sum range of a quaternion series,” J. Math. Sci., New York, 216, No. 4, 519–521 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  23. S. A. Chobanyan, “Convergence of Bernoulli series and the set of sums of a conditionally convergent function series,” Teor. Veroyatn. i Primen., 28, No. 2, 420–429 (1983).

    MathSciNet  Google Scholar 

  24. S. A. Chobanyan, “The structure of a set of sums of a conditionally convergent series in Banach space,” Dokl. Akad. Nauk SSSR, 278, No. 3, 556–559 (1984).

    MathSciNet  MATH  Google Scholar 

  25. S. A. Chobanyan, “The structure of the set of sums of a conditionally convergent series in a normed space,” Mat. Sb., 128(170), No. 1, 50–65, 143 (1985).

  26. S. A. Chobanyan and G. J. Georgobiani, “A problem on rearrangements of summands in normed spaces and Rademacher sums,” in: Probability Theory on Vector Spaces, IV (Łańcut, 1987), Lect. Notes Math., 1391, Springer, Berlin (1989), pp. 33–46.

  27. S. Chobanyan, “Convergence a.s. of rearranged random series in Banach space and associated inequalities,” in: Probability in Banach Spaces, 9 (Sandjberg, 1993), Progr. Probab., 35, Birkhäuser Boston, Boston, MA (1994), pp. 3–29.

  28. L. Chobanyan, S. Chobanyan, and G. Giorgobiani, “A maximum inequality for rearrangements of summands and its applications to orthogonal series and scheduling theory,” Bull. Georgian Natl. Acad. Sci. (N.S.) 5, No. 1, 16–20 (2011).

    MathSciNet  MATH  Google Scholar 

  29. S. A. Chobanyan and G. J. Giorgobiani, “Almost sure permutational convergence of vector random series and Kolmogorov’s problem,” in: New Trends in Probability and Statistics, Vol. 1 (Bakuriani, 1990), VSP, Utrecht, (1991), pp. 93–105.

  30. S. Chobanyan, S. Levental, and V. Mandrekar, “Prokhorov blocks and strong law of large numbers under rearrangements,” J. Theor. Probab., 17, No. 3, 647–672 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  31. S. Chobanyan and H. Salehi, “Exact maximal inequalities for exchangeable systems of random variables,” Teor. Veroyatn. i Primen., 45, No. 3, 555–567 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Chobanyan, G. Giorgobiani, V. Kvaratskhelia, and V. Tarieladze, “A note on the rearrangement theorem in a Banach space,” in: Information and Computer Technologies – Theory and Practice: Proc. Int. Sci. Conf. ICTMC-2010 Devoted to the 80th Anniversary of I. V. Prangishvili, Computer Science, Technology and Applications, Nova Science Publishers (2012), pp. 531–535.

  33. S. Chobanyan, G. Giorgobiani, V. Kvaratskhelia, S. Levental, and V. Tarieladze, “On rearrangement theorems in Banach spaces,” Georgian Math. J., 21, No. 2, 157–163 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  34. S. Chobanyan, G. Giorgobiani, and V. Tarieladze, “Signs and permutations: two problems of the function theory,” Proc. A. Razmadze Math. Inst., 160, 25–34 (2012).

    MathSciNet  MATH  Google Scholar 

  35. M. M. Day, Normed Linear Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 21., Springer-Verlag, New York–Heidelberg (1973).

  36. J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, 43, Cambridge University Press, Cambridge (1995).

  37. V. Drobot, “A note on rearrangements of series,” Studia Math., 35, 177–179 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  38. N. Dunford and J. T. Schwartz, Linear Operators. Part I. General theory, Wiley-Interscience, New York (1988).

    MATH  Google Scholar 

  39. A. Dvoretzky, “Some results on convex bodies and Banach spaces,” in: Proc. Intern. Symp. Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem; Pergamon, Oxford (1961), pp. 123–160.

  40. A. Dvoretzky and C. Hanani, “Sur les changements des signes des termes d’une série à termes complexes,” C. R. Acad. Sci. Paris, 225, 516–518 (1947).

    MathSciNet  MATH  Google Scholar 

  41. A. Dvoretzky and C. A. Rogers, “Absolute and unconditional convergence in normed linear spaces,” Proc. Natl. Acad. Sci. U.S.A., 36, 192–197 (1950).

    Article  MathSciNet  MATH  Google Scholar 

  42. V. P. Fonf, “Conditionally convergent series in a uniformly smooth Banach space,” Mat. Zametki, 11, 209–214 (1972).

    MathSciNet  MATH  Google Scholar 

  43. A. M. Garsia, “Existence of almost everywhere convergent rearrangements for Fourier series of L 2 functions,” Ann. Math. (2), 79, 623–629 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  44. A. M. Garsia, Topics in Almost Everywhere Convergence, Lect. Adv. Math., 4, Markham Publishing Co., Chicago, Ill. (1970).

  45. G. Giorgobiani, “Some remarks about the set of sums of a conditionally convergent series in a Banach space,” Proc. Inst. Comput. Math., Tbilisi, 33, No. 1, 38-44 (1988).

    Google Scholar 

  46. G. Giorgobiani, Some Problems on the Rearrangements of Series in Infinite-Dimensional Spaces, Ph.D. Thesis, Tbilisi (1988).

  47. G. D. Giorgobiani, “Structure of the set of sums of a conditionally converging series in a p-normed space,” Soobshch. Akad. Nauk Gruz. SSR, 130, No. 3, 481–484 (1988).

    MathSciNet  MATH  Google Scholar 

  48. G. D. Giorgobiani, “Convergent almost everywhere rearrangements of expansions in a series in orthogonal systems,” Soobshch. Akad. Nauk Gruz. SSR, 138, No. 2, 257–260 (1990).

    MathSciNet  MATH  Google Scholar 

  49. G. Giorgobiani, “Convergent rearrangements of series of vector-valued functions,” Georgian Math. J., 7, No. 1, 43–51 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  50. G. Giorgobiani and V. Tarieladze, “On complex universal series,” Proc. A. Razmadze Math. Inst., 160, 53–63 (2012).

    MathSciNet  MATH  Google Scholar 

  51. G. Giorgobiani and V. Tarieladze, Special Universal Series, Several Problems of Applied Mathematics and Mechanics, I. Gorgidze et al. (Eds.), Nova Sci. Publ., (2013), pp. 125–130.

  52. Z. G. Gorgadze, V. I. Tarieladze, and S. A. Chobanyan, “Gaussian covariances in Banach sublattices of the space L 0(T,Σ, ν),” Dokl. Akad. Nauk SSSR, 241, No. 3, 528–531 (1978).

    MathSciNet  Google Scholar 

  53. V. S. Grinberg and S. V. Sevast’janov, “The value of the Steinitz constant,” Funkts. Anal. Prilozh., 14, No. 2, 56–57 (1980).

    MathSciNet  Google Scholar 

  54. W. Groß, “Bedingt konvergente Reihen,” Monatsh. Math. Phys., 28, No. 1, 221–237 (1917).

    Article  MathSciNet  MATH  Google Scholar 

  55. A. Haar, “Zur Theorie der orthogonalen Funktionensysteme,” Math. Ann., 69, No. 3, 331–371 (1910).

    Article  MathSciNet  MATH  Google Scholar 

  56. H. Hadwiger, “Eine Bemerkung über Umordnung von Reihen reeller Funktionen,” Tôhoku Math. J., 46, 22–25 (1939).

    MathSciNet  MATH  Google Scholar 

  57. H. Hadwiger, “Über das Umordnungsproblem im Hilbertschen Raum,” Math. Z., 46, 70–79 (1940).

    Article  MathSciNet  MATH  Google Scholar 

  58. H. Hadwiger, “Über die Konvergenzarten unendlicher Reihen im Hilbertschen Raum,” Math. Z., 47, 325–329 (1941).

    Article  MathSciNet  MATH  Google Scholar 

  59. I. Halperin and T. Ando, Bibliography: Series of Vectors and Riemann Sums, Hokkaido University, Research Institute of Applied Electricity, Division of Applied Mathematics, Sapporo (1989).

  60. J. Hoffmann-Jorgensen, “Sums of independent Banach space valued random variables,” Stud. Math., 52, No. 2, 159–186 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  61. S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen, Monogr. Matematyczne 6. Seminarium Matematyczne Uniwersytetu Warszawskiego, Warszawa; Instytut Matematyczny PAN, Warszawa (1935).

  62. M. I. Kadets, “On a property of broken lines in n-dimensional space,” Usp. Mat. Nauk, 8, No. 1(53), 139–143 (1953).

    MathSciNet  MATH  Google Scholar 

  63. M. I. Kadets, “On conditionally convergent series in the space L p,” Usp. Mat. Nauk, 9, No. 1(59), 107–109 (1954).

    MATH  Google Scholar 

  64. M. I. Kadets and V. M. Kadets, Series in Banach Spaces. Conditional and Unconditional Convergence [Transl. from Russian by Andrei Iacob.] Operator Theory: Adv. Appl., 94, Birkhäuser Verlag, Basel (1997).

  65. V. M. Kadets, “B-convexity and the Steinitz lemma,” Izv. Severo-Kavkaz. Nauchn. Tsentra Vyssh. Shkoly Estestv. Nauk., No. 4, 27–29, 101–102 (1984).

  66. V. M. Kadets, “A problem of S. Banach (problem 106 from the “Scottish Book”),” Funkts. Anal. Prilozh., 20, No. 4, 74–75 (1986).

    MathSciNet  MATH  Google Scholar 

  67. M. I. Kadets and K. Woźniakowski, “On series whose permutations have only two sums,” Bull. Pol. Acad. Sci. Math., 37, No. 1–6, 15–21 (1990).

    MathSciNet  MATH  Google Scholar 

  68. J.-P. Kahane, Some Random Series of Functions, Second edition. Cambridge Studies in Adv. Math., 5, Cambridge University Press, Cambridge (1985).

  69. B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Nauka, Moscow (1984).

    MATH  Google Scholar 

  70. Y. Katznelson and O. C. McGehee, “Conditionally convergent series in ℝ∞,” Mich. Math. J., 21, 97–106 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  71. A. Kolmogoroff and D. Menshov, “Une série de Fourier–Lebesgue divergente presque partout,” Fundam. Math., 4, 324–328 (1923).

    Article  MATH  Google Scholar 

  72. A. Kolmogoroff, “Sur la convergence des séries de fonctions orthogonales,” Math. Z., 26, No. 1, 432–441 (1927).

    Article  MathSciNet  MATH  Google Scholar 

  73. A. N. Kolmogorov, Selected Works of A. N. Kolmogorov, Vol. I, Mathematics and Mechanics, Mathematics and Its Applications (Soviet Series), 25, Kluwer Academic, Dordrecht (1991).

  74. S. V. Konyagin, “Rearrangements of trigonometric series and trigonometric polynomials,” Real Anal. Exchange, 29, No. 1, 323–334 (2003–2004).

  75. S. V. Konyagin, “On uniformly converging rearrangements of trigonometric Fourier series,” Sovrem. Mat. Fundam. Napravl., 25, 80–87 (2007).

    Google Scholar 

  76. P. A. Kornilov, “Rearrangements of conditionally convergent functional series,” Mat. Sb., 113(155), No. 4(12), 598–616, 638 (1980).

  77. P. A. Kornilov, “The set of sums of a conditionally converging function series,” Mat. Sb., 137(179), No. 1, 114–127, 144 (1988).

  78. G. Köthe, Topological Vector Spaces I, Springer Verlag, Berlin–Heidelberg–New York (1969).

    MATH  Google Scholar 

  79. B. K. Lahiri and S. K. Bhattacharyya, “A note on rearrangements of series,” Math. Student, 64, No. 1-4, 141–145 (1996).

    MathSciNet  MATH  Google Scholar 

  80. P. Lévy, “Sur les séeries semi-convergentes,” Nouvelles Annales de Mathámatiques, Journal des Candidats aux Écoles Polytechnique et Normale, Sér. 4, 5, 506–511 (1905).

  81. M. Makai, “Reroute sequence planning in telecommunication networks and compact vector summation,” Appl. Math. Comput., 150, No. 3, 785–801 (2004).

    MathSciNet  MATH  Google Scholar 

  82. R. P. Maleev, “On conditionally convergent series in Orlicz spaces L M,” Serdica, 1, No. 2, 178–182 (1975).

    MathSciNet  Google Scholar 

  83. R. P. Maleev, “Conditionally convergent series in some Banach lattices,” C. R. Acad. Bulg. Sci., 32, No. 8, 1015–1018 (1979).

    MathSciNet  MATH  Google Scholar 

  84. B. Mamporia, A. Shangua, and V. Tarieladze, “Permutations and convergence in probability,” Bull. Georgian Acad. Sci., 172, No. 1, 23–25 (2005).

    MathSciNet  Google Scholar 

  85. M. B. Marcus and G. Pisier, Necessary and Sufficient Conditions for the Uniform Convergence of Random Trigonometric Series, Lecture 1977/78. Lecture Notes Ser., 50, Aarhus Universitet, Matematisk Institut, Aarhus (1978).

  86. M. B. Marcus and G. Pisier, “Random Fourier series on locally compact abelian groups,” in: Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78), Lect. Notes Math., 721, Springer, Berlin (1979), pp. 72–89.

  87. M. B. Marcus and G. Pisier, Random Fourier Series with Applications to Harmonic Analysis, Ann. of Math. Stud., 101, Princeton University Press, Princeton (1981).

  88. E. Martín Peinador and A. Rodées Usáan, “Sobre el dominio de sumabilidad débil de una sucesión en un espacio de Banach,” Libro homenaje al Prof. Rafael Cid, Publicaciones de la Universidad de Zaragoza, 137–146 (1987).

  89. B. Maurey, “Type et cotype dans les espaces munis de structures locales inconditionnelles,” in: Séminaire Maurey-Schwartz 1973–1974: Espaces L p , Applications Radonifiantes et Géométrie des Espaces de Banach, Exp. Nos. 24 et 25, Centre Math., École Polytech., Paris (1974).

  90. B. Maurey and G. Pisier, “Remarques sur l’exposé de P. Assouad ‘Espaces p-lisses, réarrangements’,” in: Séminaire Maurey–Schwartz 1974–1975: Espaces L p , Applications Radonifiantes et Géométrie des Espaces de Banach, Exp. No. XVI, CentreMath., École Polytech., Paris (1975); Séminaire Maurey–Schwartz 1974–1975: Espaces L p , Applications Radonifiantes et Géométrie des Espaces de Banach, Annexe, No. 1, Centre Math., École Polytech., Paris (1975).

  91. C. W. McArthur, “On relationships amongst certain spaces of sequences in an arbitrary Banach space,” Can. J. Math., 8, 192–197 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  92. R. M. Megrabian, “On the set of sums of functional series in spaces L φ,” Teor. Veroyatn. Primen., 30, No. 3, 511–523 (1985).

    Google Scholar 

  93. D. Menchoff, “Sur les séries de fonctions orthogonales,” Fundam. Math., 4, No. 1, 82–105 (1923).

    Article  MathSciNet  MATH  Google Scholar 

  94. D. Menchoff, “Sur la sommation des séries de fonctions orthogonales par des méthodes de Cesàro,” Rec. Math. [Mat. Sb.], 8(50), 121–136 (1940).

    MathSciNet  MATH  Google Scholar 

  95. D. Menchoff, “Sur la représentation des fonctions mesurables par des séries trigonométriques,” Dokl. Akad. Nauk SSSR, 26, 214–216 (1940).

    MathSciNet  MATH  Google Scholar 

  96. D. Menchoff, “Sur les sommes partielles des séries de Fourier des fonctions continues,” Rec. Math. [Mat. Sb.], 15(57), 385–432 (1944).

    MathSciNet  MATH  Google Scholar 

  97. F. Móricz and K. Tandori, “An improved Menshov–Rademacher theorem,” Proc. Am. Math. Soc., 124, No. 3, 877–885 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  98. É. Mourier, “Propriétés des caractéristiques d’un élément aléatoire dans un espace de Banach,” C. R. Acad. Sci. Paris, 231, 28–29 (1950).

    MathSciNet  MATH  Google Scholar 

  99. É. Mourier, “Éléments aléatoires dans un espace de Banach,” Ann. Inst. Henri Poincaré, 13, 161–244 (1953).

    MathSciNet  MATH  Google Scholar 

  100. E. M. Nikishin, “The convergence of rearrangements of series of functions,” Mat. Zametki, 1, 129–136 (1967).

    MathSciNet  MATH  Google Scholar 

  101. E. M. Nikishin, “Rearrangements of function series,” Mat. Sb., 85(127), 272–285 (1971).

    MathSciNet  Google Scholar 

  102. E. M. Nikishin, “Rearrangements of series in L p,” Mat. Zametki, 14, No. 1, 31–38 (1973).

    MathSciNet  Google Scholar 

  103. A. M. Olevskǐ, “Divergent series for complete systems in L 2,” Dokl. Akad. Nauk SSSR, 138, 545–548 (1961).

    MathSciNet  Google Scholar 

  104. A. M. Olevskǐ, Fourier Series with Respect to General Orthogonal Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, 86, Springer-Verlag, New York–Heidelberg (1975).

  105. W. Orlicz, “Beiträge zur Theorie der Orthogonalentwicklungen. I, II,” Stud. Math., 1, 1–39, 241–255 (1929).

  106. W. Orlicz, “Über unbedingte Konvergenz in Funktionenräumen. I,” Stud. Math., 4, 33–37 (1933); II. Stud. Math. 4, 41–47 (1933).

  107. W. Orlicz, “Über die Divergenz von allgemeinen Orthogonalreihen,” Stud. Math., 4, 27–32 (1933).

    Article  MATH  Google Scholar 

  108. M. I. Ostrovskiǐ, “Domains of sums of conditionally convergent series in Banach spaces,” Teor. Funkts.–Funkts. Anal. Prilozh., No. 46, 77–85 (1986).

  109. R. E. A. C. Paley and A. Zygmund, “On some series of functions, (1),” Math. Proc. Camb. Philos. Soc., 26, No. 3, 337–357 (1930).

    Article  MATH  Google Scholar 

  110. R. E. A. C. Paley and A. Zygmund, “On some series of functions, (2)” Math. Proc. Camb. Philos. Soc., 26, No. 4, 458–474 (1930).

    Article  MATH  Google Scholar 

  111. R. E. A. C. Paley and A. Zygmund, “On some series of functions, (3),” Math. Proc. Camb. Philos. Soc., 28, No. 2, 190–205 (1932).

    Article  MATH  Google Scholar 

  112. D. V. Pecherskiǐ, “A theorem on projections of rearranged series with terms in L p,” Izv. Akad. Nauk SSSR Ser. Mat., 41, No. 1, 203–214 (1977).

    MathSciNet  Google Scholar 

  113. D. V. Pecherskiǐ, “Rearrangements of series in Banach spaces and arrangements of signs,” Mat. Sb., 135(177), No. 1, 24–35, 142 (1988).

  114. A. Pietsch, Nuclear Locally Convex Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 66, Springer-Verlag, New York–Heidelberg (1972).

  115. H. Rademacher, “Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen,” Math. Ann., 87, No. 1-2, 112–138 (1922).

    Article  MathSciNet  MATH  Google Scholar 

  116. C. J. Read, “Srictly singular operators and the invariant subspace problem,” Stud. Math., 132, No. 3, 203–226 (1999).

    Article  MATH  Google Scholar 

  117. S. Gy. Révész, “Rearrangements of Fourier series,” J. Approx. Theory, 60, No. 1, 101–121 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  118. S. Gy. Révész, “On the convergence of Fourier series of U.A.P. functions,” J. Math. Anal. Appl., 151, No. 2, 308–317 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  119. S. Gy. Révész, “Rearrangement of Fourier series and Fourier series whose terms have random signs,” Acta Math. Hungar., 63, No. 4, 395–402 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  120. S. Rolevicz, Metric Linear Spaces, Poland. P.W.N.–Polish Sci. Publ., Warszawa (1972).

  121. P. Rosenthal, “The remarkable theorem of Lévy and Steinitz,” Amer. Math. Mon., 94, No. 4, 342–351 (1987).

    MATH  Google Scholar 

  122. W. Rudin, “Some theorems on Fourier coefficients,” Proc. Am. Math. Soc., 10, 855–859 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  123. W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York–Düsseldorf–Johannesburg (1973).

  124. R. Salem and A. Zygmund, “Some properties of trigonometric series whose terms have random signs,” Acta Math., 91, 245–301 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  125. H. H. Schaefer, Topological Vector Spaces, Graduate Texts in Math., 3, Springer-Verlag, New York–Berlin (1971).

  126. S. V. Sevast’janov, “On some geometric methods in scheduling theory: A survey,” Discrete Appl. Math., 55, No. 1, 59–82 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  127. D. O. Shklyarskiǐ, “Conditionally convergent series of vectors,” Usp. Mat. Nauk, 10, 51–59 (1944).

    MathSciNet  Google Scholar 

  128. S. B. Stechkin, “On a problem of Ul’yanov,” Uspehi Mar. Nmk., 17, No. 5, 143–144 (1962).

    MATH  Google Scholar 

  129. H. Steinhaus and S. Kaczmarz, Theorie der Orthogonalreihen, Instytut Matematyczny Polskiej Akademi Nauk, Warszawa–Lwów (1936).

    MATH  Google Scholar 

  130. E. Steinitz, “Bedingt konvergente Reihen und konvexe Systeme,” J. Reine Angew. Math., 143, 128–176 (1913).

    Article  MathSciNet  MATH  Google Scholar 

  131. The Scottish Book. Mathematics from the Scottish Café. Including Selected Papers Presented at the Scottish Book Conference Held at North Texas State University, Denton, Tex., May 1979, R. Daniel Mauldin (Ed.), Birkhäuser, Boston, Mass. (1981).

  132. W. Threlfall, “Bedingt konvergente Reihen,” Math. Z., 24, No. 1, 212–214 (1926).

    Article  MathSciNet  MATH  Google Scholar 

  133. S. Trojanski, “Conditionally converging series and certain F-spaces,” Teor. Funkts.–Funkts. Anal. Prilozh., 5, 102–107 (1967).

    MathSciNet  MATH  Google Scholar 

  134. P. L. Ul’yanov, “Divergent Fourier series of class L p (p ≥ 2),” Dokl. Akad. Nauk SSSR, 137 786–789 (1961).

    MathSciNet  Google Scholar 

  135. P. L. Ul’yanov, “Solved and unsolved problems in the theory of trigonometricand orthogonal series,” Usp. Mat. Nauk, 19, No. 1(115), 3–69 (1964).

    Google Scholar 

  136. N. Vakhania, V. Tarieladze, and S. Chobanyan, Probability Distributions on Banach Spaces, Mathematics and its Applications, Springer (1987).

  137. A. Wald, “Vereinfachter beweis des Steinitzschen satzes über vektorenreihen im ℝn,” Ergebnisse math. Kolloquium Wien, 5, 10–13 (1933).

    MATH  Google Scholar 

  138. A. Wald, “Bedingt konvergente Reihen von Vektoren im ℝω,” Ergebnisse math. Kolloquium Wien, 5, 13–14 (1933).

    MATH  Google Scholar 

  139. J. O. Wojtaszczyk, “A series whose sum range is an arbitrary finite set,” Stud. Math., 171, No. 3, 261–281 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  140. G. P. Youvaraj and S. Saeki, “Kolmogorov’s rearrangement problem with respect to some summability methods,” Expo. Math., 15, No. 4, 339–358 (1997).

    MathSciNet  MATH  Google Scholar 

  141. Z. Zahorski, “Une série de Fourier permutée d’une fonction de classe L 2 divergente presque partout,” C. R. Acad. Sci. Paris, 251, 501–503 (1960).

    MathSciNet  MATH  Google Scholar 

  142. A. Zygmund, Trigonometric Series, 2nd ed. Vols. I, II, Cambridge University Press, New York (1959).

    MATH  Google Scholar 

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  1. N. Muskhelishvili Institute of Computational Mathematics of Georgian Technical University, Tbilisi, Georgia

    G. Giorgobiani

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 103, Mathematical Analysis, 2017.

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Giorgobiani, G. Rearrangements of Series. J Math Sci 239, 437–548 (2019). https://doi.org/10.1007/s10958-019-04315-9

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