On the classes and
Author:
M. Avdispahić
Journal:
Proc. Amer. Math. Soc. 95 (1985), 230234
MSC:
Primary 26A45; Secondary 42A28
MathSciNet review:
801329
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Abstract 
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Abstract: We prove inclusion relations between Waterman's and Chanturiya's classes and point to some corollaries thereof. The situation which occurs in connection with Zygmund's theorem for Waterman's classes is clarified.
 [1]
N.
K. Bari, Trigonometricheskie ryady, With the editorial
collaboration of P. L. Ul′janov, Gosudarstv. Izdat. Fiz.Mat. Lit.,
Moscow, 1961 (Russian). MR 0126115
(23 #A3411)
 [2]
S. V. Bochkarev, On a problem of Zygmund, Math. USSRIzv. 7 (1973), 629637.
 [3]
Z. A. Chanturiya, The modulus of variation of a function and its application in the theory of Fourier series, Soviet. Math. Dokl. 15 (1974), 6771.
 [4]
Z.
A. Čanturija, The absolute convergence of Fourier
series, Mat. Zametki 18 (1975), no. 2,
185–192 (Russian). MR 0394010
(52 #14816)
 [5]
, On uniform convergence of Fourier series, Math. USSRSb. 29 (1976), 475495.
 [6]
, "On the absolute convergence of classes ," in Fourier analysis and approximation theory, Colloq. Math. Soc. Janos Bolyai 19 (1978), 219240.
 [7]
Z.
A. Chanturia, On the absolute convergence of Fourier series of the
classes 𝐻^{𝜔}∩𝑉[𝑣], Pacific J.
Math. 96 (1981), no. 1, 37–61. MR 634761
(83a:42009)
 [8]
Z.
A. Čanturija, The modulus of variation of a function and
continuity, Sakharth. SSR Mecn. Akad. Moambe 80
(1975), no. 2, 281–283 (Russian, with Georgian and English
summaries). MR
0404549 (53 #8349)
 [9]
Elaine
Cohen, On the Fourier coefficients and continuity of functions of
class 𝒱*_{Φ}, Rocky Mountain J. Math. 9
(1979), no. 2, 227–237. MR 519938
(80m:42006), http://dx.doi.org/10.1216/RMJ197992227
 [10]
Casper
Goffman and Daniel
Waterman, Functions whose Fourier series
converge for every change of variable, Proc.
Amer. Math. Soc. 19
(1968), 80–86. MR 0221193
(36 #4245), http://dx.doi.org/10.1090/S00029939196802211937
 [11]
Casper
Goffman and Daniel
Waterman, The localization principle for double Fourier
series, Studia Math. 69 (1980/81), no. 1,
41–57. MR
604353 (82d:42010)
 [12]
S.
Perlman, Functions of generalized variation, Fund. Math.
105 (1979/80), no. 3, 199–211. MR 580582
(81h:26007)
 [13]
S.
Perlman and D.
Waterman, Some remarks on functions of
Λbounded variation, Proc. Amer. Math.
Soc. 74 (1979), no. 1, 113–118. MR 521883
(80e:26007), http://dx.doi.org/10.1090/S0002993919790521883X
 [14]
Michael
Schramm and Daniel
Waterman, On the magnitude of Fourier
coefficients, Proc. Amer. Math. Soc.
85 (1982), no. 3,
407–410. MR
656113 (83h:42008), http://dx.doi.org/10.1090/S00029939198206561131
 [15]
Rafat
N. Siddiqi, Absolute convergence of Fourier series of a function of
Wiener’s class 𝑉_{𝑝}, Portugal. Math.
38 (1979), no. 12, 141–148 (1982). MR 682363
(84d:42011)
 [16]
Si
Lei Wang, Some properties of the functions of Λbounded
variation, Sci. Sinica Ser. A 25 (1982), no. 2,
149–160. MR
669675 (83k:26006)
 [17]
Daniel
Waterman, On convergence of Fourier series of functions of
generalized bounded variation, Studia Math. 44
(1972), 107–117. Collection of articles honoring the completion by
Antoni Zygmund of 50 years of scientific activity. II. MR 0310525
(46 #9623)
 [18]
Daniel
Waterman, On 𝐿bounded variation, Studia Math.
57 (1976), no. 1, 33–45. MR 0417355
(54 #5408)
 [19]
, On the summability of Fourier series of functions of bounded variation, Studia Math. 55 (1976), 8795.
 [20]
, Fourier series of functions of bounded variation, Proc. Amer. Math. Soc. 74 (1979), 119123.
 [21]
A.
Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge
University Press, New York, 1959. MR 0107776
(21 #6498)
 [1]
 N. K. Bari, Trigonometricheskie ryady, Fizmatgiz, Moscow, 1961 (English transl., A treatise on trigonometric series, Vols. 1 and 2, Macmillan, New York, 1964). MR 0126115 (23:A3411)
 [2]
 S. V. Bochkarev, On a problem of Zygmund, Math. USSRIzv. 7 (1973), 629637.
 [3]
 Z. A. Chanturiya, The modulus of variation of a function and its application in the theory of Fourier series, Soviet. Math. Dokl. 15 (1974), 6771.
 [4]
 , Absolute convergence of Fourier series, Math. Notes 18 (1975), 695700. MR 0394010 (52:14816)
 [5]
 , On uniform convergence of Fourier series, Math. USSRSb. 29 (1976), 475495.
 [6]
 , "On the absolute convergence of classes ," in Fourier analysis and approximation theory, Colloq. Math. Soc. Janos Bolyai 19 (1978), 219240.
 [7]
 , On the absolute convergence of Fourier series of the classes , Pacific J. Math. 96 (1981), 3761; Errata, ibid. 103 (1982), 611. MR 634761 (83a:42009)
 [8]
 , The modulus of variation of a function and continuity, Bull. Acad. Sci. Georgian SSR 80 (1975), 281283. (Russian) MR 0404549 (53:8349)
 [9]
 E. Cohen, On the Fourier coefficients and continuity of functions of class , Rocky Mountain J. Math. 9 (1979), 227237. MR 519938 (80m:42006)
 [10]
 C. Goffman and D. Waterman, Functions whose Fourier series converge for every change of variable, Proc. Amer. Math. Soc. 19 (1968), 8086. MR 0221193 (36:4245)
 [11]
 , The localization principle for double Fourier series, Studia Math. 69 (1980), 4157. MR 604353 (82d:42010)
 [12]
 S. Perlman, Functions of generalized variation, Fund. Math. 105 (1980), 199211. MR 580582 (81h:26007)
 [13]
 S. Perlman and D. Waterman, Some remarks on functions of bounded variation, Proc. Amer. Math. Soc. 74 (1979), 11318. MR 521883 (80e:26007)
 [14]
 M. Schramm and D. Waterman, On the magnitude of Fourier coefficients, Proc. Amer. Math. Soc. 85 (1982), 407410. MR 656113 (83h:42008)
 [15]
 R. N. Siddiqi, Absolute convergence of Fourier series of a function of Wiener's class , Portugal. Math. 38 (1979), 141148. MR 682363 (84d:42011)
 [16]
 S. Wang, Some properties of the functions of bounded variation, Sci. Sinica Ser. A 25 (1982), 149160. MR 669675 (83k:26006)
 [17]
 D. Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44 (1972), 107117; Errata, ibid. 44 (1972), 651. MR 46 9623. MR 0310525 (46:9623)
 [18]
 , On bounded variation, Studia Math. 57 (1976), 3345. MR 0417355 (54:5408)
 [19]
 , On the summability of Fourier series of functions of bounded variation, Studia Math. 55 (1976), 8795.
 [20]
 , Fourier series of functions of bounded variation, Proc. Amer. Math. Soc. 74 (1979), 119123.
 [21]
 A. Zygmund, Trigonometric series, 2nd ed., Vol. I, Cambridge Univ. Press, New York, 1959. MR 0107776 (21:6498)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198508013297
PII:
S 00029939(1985)08013297
Keywords:
Bounded variation,
modulus of variation,
Lipschitz classes,
Zygmund's theorem,
Wiener's theorem
Article copyright:
© Copyright 1985
American Mathematical Society
