Skip to main content
SpringerLink
Log in

Series with respect to the Walsh system and their generalizations

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Literature cited

  1. G. N. Agaev, “A Wiener-type theorem for series on Walsh functions,” Dokl. Akad. Nauk SSSR,142, No. 4, 751–753 (1962).

    Google Scholar 

  2. G. N. Agaev and G. N. Dzhafarli, “A class of multiplicative systems of functions,” Izv. Akad. Nauk Azerb. SSR, Ser. Fiz.-Matem. i Tekh. Nauk, No. 2, 27–36 (1963).

    Google Scholar 

  3. G. Aleksits, Convergence Problems for Orthogonal Series [Russian translation], IL, Moscow (1963), 359 pages.

    Google Scholar 

  4. F. G. Arutyunyan, “Reconstruction of the coefficients of series with respect to the Haar and Walsh systems convergent to Denjoy-integrable functions,” Izv. Akad. Nauk SSSR, Ser. Matem.,30, No. 2, 325–344 (1966).

    Google Scholar 

  5. F. G. Arutyunyan and A. A. Talalyan, “Uniqueness of series with respect to the Haar and Walsh systems,” Izv. Akad. Nauk SSSR, Ser. Matem.,28, No. 6, 1391–1408 (1964).

    Google Scholar 

  6. L. A. Balashov, “On series with gaps,” Izv. Akad. Nauk SSSR, Ser. Matem.,29, No. 3, 631–644 (1965).

    Google Scholar 

  7. L. A. Balashov, “On series with respect to the Walsh system with monotone coefficients,” Sibirsk. Matem. Zh.,12, No. 1 (1971).

  8. N. K. Bari, Trigonometric Series [in Russian], Fizmatgiz, Moscow (1961), 935 pages [English translation: Pergamon, New York (1964)].

    Google Scholar 

  9. S. L. Blyumin, “Certain properties of a class of multiplicative systems and problems of approximation of functions by polynomials with respect to those systems,” Izv. VUZov, Matematika, No. 4, 13–22 (1968).

    Google Scholar 

  10. S. L. Blyumin, “Linear methods of summation of Fourier series with respect to multiplicative systems,” Sibirsk. Matem. Zh.,9, No. 2, 449–455 (1968).

    Google Scholar 

  11. S. V. Bochkarev, “Fourier coefficients of functions of class Lipα with respect to complete orthonormal systems,” Matem. Zametki,7, No. 4, 397–402 (1970).

    Google Scholar 

  12. S. V. Bochkarev, “On the Walsh-Fourier coefficients,” Izv. Akad. Nauk SSSR, Ser. Matem.,34, No. 1, 203–208 (1970).

    Google Scholar 

  13. N. Ya. Vilenkin, “A class of complete orthonormal systems,” Izv. Akad. Nauk SSSR, Ser. Matem.,11, 363–400 (1947).

    Google Scholar 

  14. N. Ya. Vilenkin, “On the theory of orthogonal systems with gaps,” Izv. Akad. Nauk SSSR, Ser. Matem.,13, 245–252 (1949).

    Google Scholar 

  15. N. Ya. Vilenkin, G. N. Agaev, and G. M. Dzhafarli, “Theory of multiplicative orthonormal systems of functions,” Dokl. Akad. Nauk Azerb. SSR,18, No. 9, 3–7 (1962).

    Google Scholar 

  16. V. F. Gaposhkin, “Series with gaps and independent functions,” Usp. Matem. Nauk,21, No. 6, 3–82 (1966).

    Google Scholar 

  17. V. F. Gaposhkin, “On asymptotically best systems with gaps,” Izv. Akad. Nauk SSSR, Ser. Matem.,31, No. 5, 1011–1026 (1967).

    Google Scholar 

  18. V. F. Gaposhkin, “On the absolute convergence of series with gaps,” Izv. Akad. Nauk SSSR, Ser. Matem.,31, No. 6, 1271–1288 (1967).

    Google Scholar 

  19. G. M. Dzhafarli, “Convergence of Fourier series with respect to a class of multiplicative orthonormal systems,” Izv. Akad. Nauk Azerb. SSR, Ser. Fiz.-Matem. i Tekh. Nauk, No. 4, 17–36 (1962).

    Google Scholar 

  20. A. V. Efimov, “Approximation of continuous periodic functions by Fourier sums,” Izv. Akad. Nauk SSSR, Ser. Matem.,24, No. 2, 243–296 (1960).

    Google Scholar 

  21. A. V. Efimov, “Some approximative properties of periodic multiplicative orthonormal systems,” Matem. Sborn.,69, No. 3, 354–370 (1966).

    Google Scholar 

  22. A. V. Efimov, “On the upper bounds of the Walsh-Fourier coefficients,” Matem. Zametki,6, No. 6, 725–736 (1969).

    Google Scholar 

  23. A. Zygmund, Trigonometric Series [Russian translation], Vols. 1 and 2, Mir, Moscow (1965) [2nd English ed.: Cambridge Univ. Press, New York (1959)].

    Google Scholar 

  24. A. M. Zubakin, “An analog of Steinhaus's theorem,” Sibirsk. Matem. Zh.,9, No. 1, 206–210 (1968).

    Google Scholar 

  25. A. M. Zubakin, “On the ‘adjustment’ theorems of Men'shov for a class of multiplicative orthonormal systems of functions,” Izv. VUZov, Matematika, No. 12, 34–46 (1969).

    Google Scholar 

  26. A. M. Zubakin, “On the theory of complete multiplicative periodic orthonormal systems,” Usp. Matem. Nauk,24, No. 6, 187–188 (1969).

    Google Scholar 

  27. S. Kaczmarz and H. Steinhaus, Theory of Orthogonal Series [Russian translation], Fizmatgiz, Moscow (1958) [German ed.: Warsaw-Lvov (1935)].

    Google Scholar 

  28. V. M. Kokilashvili, “A function space and the Fourier coefficients,” Soobshch. Akad. Nauk Gruz. SSR,35, No. 3, 523–530 (1964).

    Google Scholar 

  29. V. M. Kokilashvili, “Best approximations of functions by Walsh polynomials; the Walsh-Fourier coefficients,” Bull. Akad. Pol. Sci., Ser. Sci. Math. Astron. Phys.,13, No. 6, 405–410 (1965).

    Google Scholar 

  30. V. M. Kokilashvili, “Approximation of functions by the averages of Walsh-Fourier series,” Soobshch. Akad. Nauk Gruz. SSR,45, No. 2, 305–310 (1967).

    Google Scholar 

  31. A. A. Konyushkov, “Best approximations by trigonometric polynomials; the Fourier coefficients,” Matem. Sborn.,44, No. 1, 53–84 (1958).

    Google Scholar 

  32. B. D. Kotlyar, “Walsh series and Men'shov's theorem on the ‘adjustment’ of functions,” Izd. Akad. Nauk SSSR, Ser. Matem.,30, No. 5, 1193–1200 (1966).

    Google Scholar 

  33. Li Luang-tang, “The Walsh system of orthogonal functions,” Acta Sci. Natur. Univ. Pekinensis,3, No. 4, 403–422 (1957).

    Google Scholar 

  34. B. S. Mityagin, “Absolute convergence of a series of Fourier coefficients,” Dokl. Akad. Nauk SSSR,157, No. 5, 1047–1050 (1964).

    Google Scholar 

  35. A. A. Muromskii, “Functions representable by series with respect to the Rademacher system,” Izv. VUZov, Matematika, No. 11, 54–66 (1967).

    Google Scholar 

  36. A. M. Olevskii, “Order of growth of the Lebesgue functions of bounded orthonormal systems,” Dokl. Akad. Nauk SSSR,176, No. 6, 1247–1250 (1967).

    Google Scholar 

  37. R. I. Osipov, “Convergence of series with respect to the Walsh system,” Izv. Akad. Nauk Arm. SSR, Matematika,1, No. 4, 270–283 (1966).

    Google Scholar 

  38. B. T. Polyak and Yu. A. Shreider, “Application of Walsh polynomials in approximate calculations,” in: Aspects of the Theory of Mathematical Machines [in Russian], Vol. 2, Fizmatgiz, Moscow (1962), pp. 174–190.

    Google Scholar 

  39. L. S. Pontryagin, Continuous Groups [in Russian], Gostekhizdat, Moscow (1954).

    Google Scholar 

  40. V. I. Prokhorenko, “Divergent Fourier series,” Matem. Sborn.,75, No. 2, 185–198 (1968).

    Google Scholar 

  41. A. A. Protsenko, “Construction of the best approximation by Walsh functions for a given number of terms in the expansion,” Trudy Leningrad. Polytekh. Inst., No. 275-5-11 (1967).

  42. A. I. Rubinshtein, “On series with gaps,” Izv. VUZov, Matematika, No. 3, 137–148 (1963).

    Google Scholar 

  43. A. I. Rubinshtein, “The A-integral and series with respect to the Walsh system,” Usp. Matem. Nauk,18, No. 3, 191–197 (1963).

    Google Scholar 

  44. A. I. Rubinshtein, “Onω-lacunary series and functions of classes Hω,” Matem. Sbom.,65, No. 2, 239–271 (1964).

    Google Scholar 

  45. A. I. Rubinshtein, “Certain properties of a class of multiplicative periodic complete orthonormal systems,” Matem. Zametki,2, No. 3, 128–138 (1966).

    Google Scholar 

  46. A. I. Rubinshtein, “Series with respect to the Walsh system with monotone coefficients,” Matem. Zametki,2, No. 3, 279–388 (1967).

    Google Scholar 

  47. A. I. Rubinshtein, “On Walsh-Paley series with gaps,” Interinst. Conf. Application of Functional Analysis in the Theory of Approximations [in Russian], Kalinin (1970), pp. 139–142.

  48. S. B. Stechkin, “Approximation of periodic functions by Fejér sums,” Trudy Matem. Inst. Akad. Nauk SSSR,62, 48–60 (1961).

    Google Scholar 

  49. S. B. Stechkin and P. L. Ul'yanov, “On uniqueness sets,” Izv. Akad. Nauk SSSR, Ser. Matem.,26, No. 2, 211–222 (1962).

    Google Scholar 

  50. A. A. Talalyan and F. G. Arutyunyan, “Convergence to +∞ of series with respect to the Haar system,” Matem. Sborn.,66, No. 2, 240–247 (1965).

    Google Scholar 

  51. M. F. Timan, “Best approximation of functions and linear methods of summation of Fourier series,” Izv. Akad. Nauk SSSR, Ser. Matem.,29, No. 3, 587–604 (1965).

    Google Scholar 

  52. P. L. Ul'yanov, “Divergent Fourier series,” Usp. Matem. Nauk,16, No. 3, 61–142 (1961).

    Google Scholar 

  53. P. L. Ul'yanov, “Application of A-integration to a class of trigonometric series,” Matem. Sborn.,35, No. 3, 469–490 (1954).

    Google Scholar 

  54. P. L. Ul'yanov, “On the approximation of functions,” Sibirsk. Matem. Zh.,5, No. 2, 418–437 (1964).

    Google Scholar 

  55. G. A. Fomin, “Certain properties of orthogonal series,” Interinst. Conf. Application of Functional Analysis in the Theory of Approximations [in Russian], Kalinin (1970), pp. 165–169.

  56. P. R. Halmos, Measure Theory [Russian translation], LL, Moscow (1953), [English ed.: Van Nostrand, New York (1950)].

    Google Scholar 

  57. F. I. Kharshiladze, “On Walsh-Fourier series,” Trudy Gruz. Politekh. Inst., No. 1(81), 47–53 (1962).

    Google Scholar 

  58. N. P. Khoroshko, Dissertation [in Russian], Dnepropetrovsk (1969).

  59. N. P. Khoroshko, “Upper bound of Fourier coefficients on certain classes of functions with respect to the Haar, Rademacher, and Walsh systems,” in: Dnepropetrovsk University Graduate Student Publications [in Russian], Dnepropetrovsk (1970), pp. 212–215.

  60. A. A. Shneider, “On series with respect to Walsh functions with monotone coefficients,” Izd. Akad. Nauk SSSR, Ser. Matem.,12, 179–192 (1948).

    Google Scholar 

  61. A. A. Shneider, “Uniqueness of expansions with respect to the Walsh system of functions,” Matem. Sborn.,24, 279–300 (1949).

    Google Scholar 

  62. A. A. Shneider, “Convergence of subsequences of partial sums of Fourier series with respect to Walsh functions,” Dokl. Akad. Nauk SSSR,50, No. 6, 969–971 (1950).

    Google Scholar 

  63. A. A. Shneider, “Convergence of Fourier series with respect to Walsh functions,” Matem. Sborn.,34, No. 3, 441–472 (1954).

    Google Scholar 

  64. M. A. Yastrebova, “Approximation of functions satisfying the Lipschitz condition by the arithmetical means of their Walsh-Fourier series,” Matem. Sborn.,71, No. 2, 214–226 (1966).

    Google Scholar 

  65. G. Andreoli, “Sistemi ortogonali de Walsh; matrici di Hadamard; transformazioni di sistemi ortogonali,” Ricerca Napoli,6, No. 1, 3–17 (1955).

    Google Scholar 

  66. W. A. Beyer, “Hausdorff dimensions of level sets of some Rademacher series,” Pacific J. Math.,12, No. 1, 35–46 (1962).

    Google Scholar 

  67. P. Billard, “Sur la convergence presque partout des séries de Fourier-Walsh des fonctions de l'espace L2(0, 1),” Studia Math.,28, No. 3, 363–388 (1967).

    Google Scholar 

  68. A. Bonami, “A propos d'un théorme de Paley sur les fonctions de Walsh,” Compt. Rend.,266, No. 13, A655-A657 (1968).

    Google Scholar 

  69. J. S. Byrnes and D. A. Swick, “Instant Walsh functions,” SIAM Rev.,12, No. 1, 131 (1970).

    Google Scholar 

  70. W. C. Carefoot and T. M. Flett, “A note on Rademacher series,” J. London Math. Soc.,42, No. 3, 542–544 (1967).

    Google Scholar 

  71. L. Carleson, “On convergence and growth of partial sums of Fourier series,” Acta Math.,116, Nos. 1–2, 135–157 (1966).

    Google Scholar 

  72. H. E. Chrestenson, “A class of generalized Walsh functions,” Pacific J. Math.,5, No. 1, 17–31 (1955).

    Google Scholar 

  73. J. E. Coury, “A class of Walsh M-sets of measure zero,” J. Math. Anal. Appl.,31, No. 2, 318–320 (1970).

    Google Scholar 

  74. R. B. Crittenden, A Theorem on the Uniqueness of the (C, 1) Summability of Walsh Series, Doctoral Dissertation, Univ. Oregon (1964); Dissert. Abstr.,25, No. 5, 2997 (1964).

  75. J. B. Crittenden and V. L. Shapiro, “Sets of uniqueness on the group 2ω,” Ann. Math.,81, No. 3, 550–564 (1965).

    Google Scholar 

  76. N. I. Fine, “On the Walsh functions,” Trans. Amer. Math. Soc.,65, No. 3, 372–414 (1949).

    Google Scholar 

  77. N. I. Fine, “The generalized Walsh functions,” Trans. Amer. Math. Soc.,69, 66–77 (1950).

    Google Scholar 

  78. N. I. Fine, “Cesaro summability of Walsh-Fourier series,” Proc. Nat. Acad. Sci. USA,41, No. 8, 588–591 (1955).

    Google Scholar 

  79. L. H. Harper, “Capacities of sets and harmonic analysis on the group 2ω,” Trans. Amer. Math. Soc.,126, No. 2, 301–315 (1967).

    Google Scholar 

  80. I. I. Hirschman, “The decomposition of Walsh and Fourier series,” Mem. Amer. Math. Soc., No. 15, 65 (1955).

    Google Scholar 

  81. S. Igari, “Sur les facteurs de convergence des séries de Walsh-Fourier,” Proc. Japan. Acad.,40, No. 4, 250–252 (1964).

    Google Scholar 

  82. S. Igari, “Sur les facteurs de convergence des séries de Walsh-Fourier,” J. Analyse Math.,15, 389–401 (1965).

    Google Scholar 

  83. S. Kaczmarz, “Über ein Orthogonalsystem,” Compt. Rend. 1-e Congr. Math. des Pays Slaves, Warsaw (1929), pp. 189–192.

  84. S. Kaczmarz and H. Steinhaus, “Le system orthogonale de M. Rademacher,” Studia Math., No. 2, 231–247 (1930).

    Google Scholar 

  85. Y. Katznelson, “Trigonometric series with positive partial sums,” Bull. Amer. Math. Soc.,71, 718–719 (1965).

    Google Scholar 

  86. A. Khinchin [Khintchine] and A. Kolmogorov [Kolmogoroff], Über Konvergenz von Reihen, deren Glieder durch den Zufall bestimmt werden,” Matem. Sborn.,32, 668–677 (1925).

    Google Scholar 

  87. Kobayashi Masako, “Capacities of sets and harmonic analysis on the group 2ω,” Tohoku Math. J.,21, No. 3, 419–433 (1969); correction: Tohoku Math. J.,21, No. 4, 676 (1969).

    Google Scholar 

  88. P. Levy, “Sur une generalisation des fonctions orthogonales de M. Rademacher,” Comment. Math. Helv.,16, 146–152 (1944).

    Google Scholar 

  89. R. Liedl, “Über gewisse Functionale in Raum C(v)[0, 1] und Walsh-Fourierkoeffizienten,” Monat. Math.,72, No. 1, 38–44 (1968).

    Google Scholar 

  90. J. R. McLaughlin, “Functions represented by Rademacher series,” Pacific J. Math.,27, No. 2, 373–378 (1968).

    Google Scholar 

  91. J. R. McLaughlin, “Functions represented by integrated Rademacher series,” Colloq. Math.,20, No. 2, 277–286 (1969).

    Google Scholar 

  92. J. R. McLaughlin, “Rademacher series with nondifferentiable sums,” Proc. Amer. Math. Soc.,23, No. 1, 140–143 (1969).

    Google Scholar 

  93. G. Morgenthaler, “Walsh-Fourier series,” Trans. Amer. Math. Soc.,84, No. 2, 472–507 (1957).

    Google Scholar 

  94. C. Mutafian and J. Reyrière, “Sur les procédés de sommation des séries de Rademacher,” Bull. Sci. Math.,93, Nos. 3–4, 181–185 (1969–1970).

    Google Scholar 

  95. Okuyama Yasuo, “On contraction of Walsh-Fourier series,” Tohoku Math. J.,19, No. 2, 156–157 (1967).

    Google Scholar 

  96. C. W. Onneweer, “On uniform convergence for Walsh-Fourier series,” Pacific J. Math.,34, No. 1, 117–122 (1970).

    Google Scholar 

  97. L. G. Pal, “On the absolute and uniform convergence of Walsh-Fourier expansions of Faber functions,” Ann. Univ. Sci. Budapest, Sec. Math.,9, 109–113 (1966).

    Google Scholar 

  98. L. G. Pal, “On the role of the structural properties of functions in the theory of their Walsh-Fourier expansions,” Ann. Univ. Sci. Budapest, Sec. Math.,10, 155–166 (1967).

    Google Scholar 

  99. R. Paley, “A remarkable system of orthogonal functions,” Proc. London Math. Soc.,34, 241–279 (1932).

    Google Scholar 

  100. J. J. Price, “Walsh series and adjustment of functions on small sets,” Illinois J. Math.,13, No. 1, 131–136 (1969).

    Google Scholar 

  101. H. Rademacher, “Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen,” Math. Ann.,87, 112–138 (1922).

    Google Scholar 

  102. P. Révész and M. Wschebor, “On the statistical properties of the Walsh functions,” Magyar. Tud. Akad. Mat. Kutato Int. Közl.,9, No. 3, 543–554 (1964–1965).

    Google Scholar 

  103. F. Schipp, “Constructions of a continuous function whose Walsh series diverges at a prescribed point,” Ann. Univ. Sci. Budapest, Sec. Math.,9, 103–108 (1966).

    Google Scholar 

  104. F. Schipp, “Über die Grössenordnung der Partialsummen der Entwicklung integrierbarer Funktionen nach W-Systemen,” Acta Sci. Math.,28, Nos. 1–2, 123–134 (1967).

    Google Scholar 

  105. F. Schipp, “Über die starke summation von Walsh-Fourierreihen,” Acta Sci. Math.,30, Nos. 1–2, 77–87 (1969).

    Google Scholar 

  106. F. Schipp, “Bemerkung zur starken summation der Walsh-Fourierreihe,” Acta Math. Acad. Sci. Hung.,20, Nos. 3–4, 263–274 (1969).

    Google Scholar 

  107. F. Schipp, “Über Walsh-Fourierreihen mit nichtnegativen Partialsummen,” Ann. Univ. Sci. Budapest, Sec. Math.,12, 43–48 (1969).

    Google Scholar 

  108. F. Schipp, “Über die Divergenz der Walsh-Fourierreihen,” Ann. Univ. Sci. Budapest, Sec, Math.,12, 49–62 (1969).

    Google Scholar 

  109. F. Schipp, “Walsh-Fourier-sorok erös approximaciojapol,” Magyar. Tud. Akad. Mat. és Fis. Tud. Oszt. KözL.,19, Nos. 1–2, 101–111 (1970).

    Google Scholar 

  110. F. Schipp, “Über Walsh-Fourierreihen mit nichtnegativen Partialsummen,” Ann. Univ. Sci. Budapest, Sec. Math.,10, 113–116 (1967).

    Google Scholar 

  111. V. Shapiro, “U(ε)-sets for Walsh series,” Proc. Amer. Math. Soc.,16, No. 5, 867–870 (1965).

    Google Scholar 

  112. A. Siddiqi, “On the summability of a sequence on Walsh functions,” J. Austral. Math. Soc.,10, Nos. 3–4, 385–394 (1969).

    Google Scholar 

  113. S. Sidon, “Über orthogonal Entwicklungen,” Acta Sci. Math.,10, 206–253 (1941–1943).

    Google Scholar 

  114. P. Sjölin, “An inequality of Paley and convergence a.e. of Walsh-Fourier series,” Ark. Mat.,7, No. 6, 551–570 (1969).

    Google Scholar 

  115. E. M. Stein, “On limits of sequences of operators,” Ann. Math.,74, No. 1, 140–170 (1961).

    Google Scholar 

  116. H. Steinhaus, “Zur Konvergenzfrage bei dem Rademacherschen Orthogonalsystem,” Matem. Sborn.,35, No. 1, 312–315 (1928).

    Google Scholar 

  117. Sunouchi Gen-ichiro, “On the Walsh-Kaczmarz series,” Proc. Amer. Math. Soc.,2, No. 1, 5–11 (1951).

    Google Scholar 

  118. Sunouchi Gen-ichiro, “Strong summability of Walsh-Fourier series,” Tohoku Math. J.,16, No. 2, 228–237 (1964).

    Google Scholar 

  119. Tandori Karoly, “Über die Divergenz der Walshschen Reihen,” Acta Sci. Math.,27, Nos. 3–4, 261–263 (1966).

    Google Scholar 

  120. Tandori Karoly, “Bemerkung zur starken summation der Fourierreihen,” Acta Math. Acad. Sci., Hung.,19, Nos. 3–4, 271–285 (1968).

    Google Scholar 

  121. Tateoka Jun, “On almost-everywhere convergence of Walsh-Fourier series,” Proc. Japan. Acad.,44, No. 7, 647–650 (1968).

    Google Scholar 

  122. Tsuchikura Tamotsu, “Absolute summability of Rademacher series,” Tohoku Math. J.,10, No. 1, 49–59 (1958).

    Google Scholar 

  123. W. R. Wade, “A uniqueness theorem for Haar and Walsh series,” Trans. Amer. Math. Soc.,141, 187–194 (1969).

    Google Scholar 

  124. J. L. Walsh, “A closed set of normal orthogonal functions,” Amer. J. Math.,45, 5–24 (1923).

    Google Scholar 

  125. Wang Si-lei, “On the functions represented by Rademacher series,” Chinese Math.,4, No. 4, 703–711 (1963).

    Google Scholar 

  126. C. Watari, “On generalized Walsh-Fourier Series (I),” Proc. Japan. Acad.,33, No. 8, 435–438 (1957).

    Google Scholar 

  127. C. Watari, “On generalized Walsh-Fourier series,” Tohoku Math. J.,10, No. 3, 211–241 (1958).

    Google Scholar 

  128. C. Watari, “Best approximation by Walsh polynomials,” Tohoku Math. J.,15, No. 1, 1–5 (1963).

    Google Scholar 

  129. C. Watari, “Mean convergence of Walsh-Fourier series,” Tohoku Math. J.,16, No. 2, 183–188 (1964).

    Google Scholar 

  130. C. Watari, “Contraction on Walsh-Fourier series,” Proc. Amer. Math. Soc.,15, No. 2, 189–192 (1964).

    Google Scholar 

  131. C. Watari, “On decomposition of Walsh-Fourier series,” Tohoku Math. J.,17, No. 1, 76–86 (1965).

    Google Scholar 

  132. C. Watari, “Multipliers for Walsh-Fourier series,” Tohoku Math. J.,16, No. 3, 239–251 (1964).

    Google Scholar 

  133. P. Weiss, “Zusammenhang von Walsh-Fourierreihen mit Polynomen,” Monat. Math.,71. No. 2, 165–179 (1967).

    Google Scholar 

  134. P. Weiss, “Über der Verwendung von Walshfunktionen in der Codierungstheorie,” Arch. Elektr. Übertragung,21, 255–258 (1967).

    Google Scholar 

  135. P. Weiss, “Die Darstellung von zyklischen Codes mit Hilfe von Walshfunktionen,” Arch. Elektr. Übertragung,22 (1967).

  136. Yano Shigeki, “On approximation by Walsh functions,” Proc. Amer. Math. Soc.,2, No. 6, 962–967 (1952).

    Google Scholar 

  137. Yano Shigeki, “On Walsh-Fourier series,” Tohoku Math. J.,3, No. 2, 223–242 (1951).

    Google Scholar 

  138. Yano Shigeki, “A convergence test for Walsh-Fourier series,” Tohoku Math. J.,6, Nos. 2–3, 226–230 (1954).

    Google Scholar 

  139. Yano Shigeki, “On a lemma of Marcinkiewicz and its applications to Fourier series,” Tohoku Math. J.,11, No. 2, 191–215 (1959).

    Google Scholar 

  140. Yano Shigeki, “Cesaro summability of Walsh-Fourier series,” Tohoku Math. J.,9, No. 3, 267–272 (1957).

    Google Scholar 

Download references

Authors
  1. L. A. Balashov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. I. Rubinshtein
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Itogi Nauki, Seriya Matematika (Matematicheskii Analiz), pp. 147–202, 1970.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Balashov, L.A., Rubinshtein, A.I. Series with respect to the Walsh system and their generalizations. J Math Sci 1, 727–763 (1973). https://doi.org/10.1007/BF01236363

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01236363

Keywords

Search

Navigation