Book Review
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1567978
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Book Information:
Author:
Efimov Golubov, and Skvortsov
Title:
Walsh series and transforms
Additional book information:
Kluwer Academic Publishers, Dordrecht, The Netherlands 1991, 367 pp., US$169.00. ISBN 0-7923-1100-0.
[1] G. N. Agaev, N. Ya. Vilenkin, G. M. Dzafarli, and A. I. Rubinshteĭn, Multiplicative systems and harmonic analysis on zero-dimensional groups, Baku "ELM," 1981.
Georges Alexits, Sur la sommabilité des séries orthogonales, Acta Math. Acad. Sci. Hungar. 4 (1953), 181–189 (French, with Russian summary). MR 61193, DOI 10.1007/BF02127579
F. G. Arutjunjan and A. A. Talaljan, Uniqueness of series in Haar and Walsh systems, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1391–1408 (Russian). MR 0172056
Bruce Aubertin, Algebraic elements and sets of uniqueness in the group of integers of a $p$-series field, Canad. Math. Bull. 29 (1986), no. 2, 177–184. MR 844896, DOI 10.4153/CMB-1986-029-7
[5] L. A. Balashov and A. I. Rubinshteĭn, Series with respect to the Walsh system and their generalizations, J. Soviet Math. 1 (1973), 727-763.
L. A. Balashov and V. A. Skvortsov, The Gibbs phenomenon for the Walsh system, Dokl. Akad. Nauk SSSR 268 (1983), no. 5, 1033–1034 (Russian). MR 697817
K. G. Beauchamp, Walsh functions and their applications, Techniques of Physics, No. 3, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1975. MR 0462758
[8] A. D. Bethke, Genetic algorithms as function optimizers, Dissertation, Univ. of Michigan, 1980.
S. V. Bočkarev, The Fourier-Walsh coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 203–208 (Russian). MR 0259474
D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504. MR 208647, DOI 10.1214/aoms/1177699141
P. L. Butzer and H. J. Wagner, Walsh-Fourier series and the concept of a derivative, Applicable Anal. 3 (1973), 29–46. MR 404978, DOI 10.1080/00036817308839055
Ronald R. Coifman and Robert S. Strichartz, The school of Antoni Zygmund, A century of mathematics in America, Part III, Hist. Math., vol. 3, Amer. Math. Soc., Providence, RI, 1989, pp. 343–368. With the collaboration of Gina Graziosi and Julia Hallquist. MR 1025352
Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
Richard B. Crittenden and Victor L. Shapiro, Sets of uniqueness on the group $2^{\omega }$, Ann. of Math. (2) 81 (1965), 550–564. MR 179535, DOI 10.2307/1970401
Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 402468, DOI 10.1007/BF02392270
W. Engels, On the characterization of the dyadic derivative, Acta Math. Hungar. 46 (1985), no. 1-2, 47–56. MR 819052, DOI 10.1007/BF01961004
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372–414. MR 32833, DOI 10.1090/S0002-9947-1949-0032833-2
N. J. Fine, The generalized Walsh functions, Trans. Amer. Math. Soc. 69 (1950), 66–77. MR 42535, DOI 10.1090/S0002-9947-1950-0042535-2
S. Fridli and P. Simon, On the Dirichlet kernels and a Hardy space with respect to the Vilenkin system, Acta Math. Hungar. 45 (1985), no. 1-2, 223–234. MR 779509, DOI 10.1007/BF01955039
John B. Garnett and Peter W. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2, 351–371. MR 658065
Adriano M. Garsia, Martingale inequalities: Seminar notes on recent progress, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. MR 0448538
[23] J. E. Gibbs and M. S. Millard, Walsh functions as solutions of a logical differential equation, National Physical Lab. 1969, Middlesex, England, DES Report No. 1.
D. J. Grubb, Sets of uniqueness in compact, $0$-dimensional metric groups, Trans. Amer. Math. Soc. 301 (1987), no. 1, 239–249. MR 879571, DOI 10.1090/S0002-9947-1987-0879571-5
D. J. Grubb, Summation methods and uniqueness in Vilenkin groups, Proc. Amer. Math. Soc. 102 (1988), no. 3, 556–558. MR 928979, DOI 10.1090/S0002-9939-1988-0928979-4
Richard F. Gundy, Martingale theory and pointwise convergence of certain orthogonal series, Trans. Amer. Math. Soc. 124 (1966), 228–248. MR 204967, DOI 10.1090/S0002-9947-1966-0204967-0
Alfred Haar, Zur Theorie der orthogonalen Funktionensysteme, Math. Ann. 69 (1910), no. 3, 331–371 (German). MR 1511592, DOI 10.1007/BF01456326
Henning F. Harmuth, Sequency theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Foundations and applications; Advances in Electronics and Electron Physics, Suppl. No. 9. MR 0434596
Henning F. Harmuth, Transmission of information by orthogonal functions, 2nd ed., Springer-Verlag, New York-Heidelberg, 1972. MR 0347427
David C. Harris, Compact sets of divergence for continuous functions on a Vilenkin group, Proc. Amer. Math. Soc. 98 (1986), no. 3, 436–440. MR 857936, DOI 10.1090/S0002-9939-1986-0857936-X
Richard A. Hunt, Almost everywhere convergence of Walsh-Fourier series of $L^{2}$ functions, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 655–661. MR 0511000
S. V. Konyagin, Limits of indeterminacy of trigonometric series, Mat. Zametki 44 (1988), no. 6, 770–784, 862 (Russian); English transl., Math. Notes 44 (1988), no. 5-6, 910–920 (1989). MR 983549, DOI 10.1007/BF01158029
N. R. Ladhawala, Absolute summability of Walsh-Fourier series, Pacific J. Math. 65 (1976), no. 1, 103–108. MR 417678
S. V. Levizov, Some properties of a Walsh system, Mat. Zametki 27 (1980), no. 5, 715–720, 829 (Russian). MR 578256
Mohammad Maqusi, Applied Walsh analysis, Heyden & Son, Inc., Philadelphia, PA, 1981. MR 825742
K. H. Moon, An everywhere divergent Fourier-Walsh series of the class $L(\textrm {log}^{+}\textrm {log}^{+}L)^{1-\varepsilon }$, Proc. Amer. Math. Soc. 50 (1975), 309–314. MR 377406, DOI 10.1090/S0002-9939-1975-0377406-8
George W. Morgenthaler, On Walsh-Fourier series, Trans. Amer. Math. Soc. 84 (1957), 472–507. MR 91370, DOI 10.1090/S0002-9947-1957-0091370-4
C. W. Onneweer, On uniform convergence for Walsh-Fourier series, Pacific J. Math. 34 (1970), 117–122. MR 275048
C. W. Onneweer and Daniel Waterman, Fourier series of functions of harmonic bounded fluctuation on groups, J. Analyse Math. 27 (1974), 79–83. MR 481938, DOI 10.1007/BF02788643
[40] R. E. A. C. Paley, A remarkable system of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-279.
J. J. Price, Orthonormal sets with non-negative Dirichlet kernels. II, Trans. Amer. Math. Soc. 100 (1961), 153–161. MR 126124, DOI 10.1090/S0002-9947-1961-0126124-4
Walter Rudin, Fourier analysis on groups, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1038803, DOI 10.1002/9781118165621
F. Schipp, Pointwise convergence of expansions with respect to certain product systems, Anal. Math. 2 (1976), no. 1, 65–76 (English, with Russian summary). MR 415190, DOI 10.1007/BF02079908
F. Schipp, Über die Divergenz der Walsh-Fourierreihen, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 12 (1969), 49–62 (German). MR 256054
F. Schipp, W. R. Wade, and P. Simon, Walsh series, Adam Hilger, Ltd., Bristol, 1990. An introduction to dyadic harmonic analysis; With the collaboration of J. Pál. MR 1117682
[46] A. A. Schneĭder, Uniqueness of expansions with respect to the Walsh system of functions, Mat. Sb. 24 (1949), 279-300.
[47] A. H. Siddiqi, Walsh functions, Aligarh Muslim Univ. Press, Aligarh, India, 1977.
P. Simon, Investigations with respect to the Vilenkin system, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 27 (1984), 87–101 (1985). MR 823096
Per Sjölin, An inequality of Paley and convergence a.e. of Walsh-Fourier series, Ark. Mat. 7 (1969), 551–570. MR 241885, DOI 10.1007/BF02590894
V. A. Skvorcov, Differentiation with respect to nets, and Haar series, Mat. Zametki 4 (1968), 33–40 (Russian). MR 236606
V. A. Skvorcov, Certain generalizations of the uniqueness theorem for series with respect to a Walsh system, Mat. Zametki 13 (1973), 367–372 (Russian). MR 318767
V. A. Skvorcov and W. R. Wade, Generalization of some results concerning Walsh series and the dyadic derivative, Anal. Math. 5 (1979), no. 3, 249–255 (English, with Russian summary). MR 549241, DOI 10.1007/BF01908907
[53] R. S. Stanković and J. E. Gibbs, Bibliography of Gibbs derivatives, Theory and Applications of Gibbs Derivatives, Institute of Mathematics, Belgrade, 1989, pp. XV-XXIV.
M. H. Taibleson, Fourier analysis on local fields, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. MR 0487295
[55] A. A. Talalyan and F. G. Arutunyan, Convergence of series with respect to the Haar system to , Mat. Sb. 66 (1965), 240-247.
N. Vilenkin, On a class of complete orthonormal systems, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 11 (1947), 363–400 (Russian, with English summary). MR 0022560
I. V. Volovich, $p$-adic string, Classical Quantum Gravity 4 (1987), no. 4, L83–L87. MR 895889
William R. Wade, Recent developments in the theory of Walsh series, Internat. J. Math. Math. Sci. 5 (1982), no. 4, 625–673. MR 679409, DOI 10.1155/S0161171282000611
[59] -, Vilenkin-Fourier series and approximation, Colloq. Math. Soc. J. Bolyai (to appear).
William R. Wade and Kaoru Yoneda, Uniqueness and quasimeasures on the group of integers of a $p$-series field, Proc. Amer. Math. Soc. 84 (1982), no. 2, 202–206. MR 637169, DOI 10.1090/S0002-9939-1982-0637169-9
J. L. Walsh, A Closed Set of Normal Orthogonal Functions, Amer. J. Math. 45 (1923), no. 1, 5–24. MR 1506485, DOI 10.2307/2387224
Daniel Waterman, On systems of functions resembling the Walsh system, Michigan Math. J. 29 (1982), no. 1, 83–87. MR 646374, DOI 10.1307/mmj/1029002617
Shigeki Yano, On Walsh-Fourier series, Tohoku Math. J. (2) 3 (1951), 223–242. MR 45236, DOI 10.2748/tmj/1178245527
Shigeki Yano, Cesàro summability of Walsh-Fourier series, Tohoku Math. J. (2) 9 (1957), 267–272. MR 95384, DOI 10.2748/tmj/1178244781
Kaoru Yoneda, Perfect sets of uniqueness on the group $2^{\omega }$, Canadian J. Math. 34 (1982), no. 3, 759–764. MR 663317, DOI 10.4153/CJM-1982-052-1
Wo Sang Young, A note on Walsh-Fourier series, Proc. Amer. Math. Soc. 59 (1976), no. 2, 305–310. MR 410247, DOI 10.1090/S0002-9939-1976-0410247-2
L. A. Zalmanzon, Preobrazovaniya Fur′e, Uolsha, Khaara i ikh primenenie v upravlenii, svyazi i drugikh oblastyakh, “Nauka”, Moscow, 1989 (Russian). With an English summary. MR 1044560
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
- [1]
- G. N. Agaev, N. Ya. Vilenkin, G. M. Dzafarli, and A. I. Rubinshteĭn, Multiplicative systems and harmonic analysis on zero-dimensional groups, Baku "ELM," 1981.
- [2]
- G. Alexits, Sur la sommabilité des séries orthogonales, Acta Math. Acad. Sci. Hungar. 4 (1953), 181-188. MR 0061193 (15:788i)
- [3]
- F. G. Arutunyan and A. A. Talalyan, On uniqueness of Haar and Walsh series, Izv. Akad. Nauk SSSR 28 (1964), 1391-1408. MR 0172056 (30:2282)
- [4]
- B. Aubertin, Algebraic elements and sets of uniqueness in the group integers of a p-series field, Canad. Math. Bull.. 29 (1986), 177-184. MR 844896 (87j:43011)
- [5]
- L. A. Balashov and A. I. Rubinshteĭn, Series with respect to the Walsh system and their generalizations, J. Soviet Math. 1 (1973), 727-763.
- [6]
- L. A. Balashov and V. A. Skvortsov, Gibbs phenomenon for the Walsh system, Dokl. Akad. Nauk SSSR 268 (1983), 1033-1034. MR 697817 (84e:42027)
- [7]
- K. G. Beauchamp, Walsh functions and their applications, Academic Press, New York, 1975. MR 0462758 (57:2731)
- [8]
- A. D. Bethke, Genetic algorithms as function optimizers, Dissertation, Univ. of Michigan, 1980.
- [9]
- S. V. Bochkarev, The Walsh Fourier coefficients, Izv. Akad. SSSR, Ser. Mat. 34 (1970), 203-208. MR 0259474 (41:4112)
- [10]
- D. L. Burkholder, Martingale transforms, Ann. Math. Stat. 37 (1966), 1494-1504. MR 0208647 (34:8456)
- [11]
- P. L. Butzer and H. J. Wagner, Walsh series and the concept of a derivative, Applicable Anal. 3 (1973), 29-46. MR 0404978 (53:8774)
- [12]
- R. R. Coifman and R. S. Strichartz, The school of Antoni Zygmund, A Century of Mathematics in America, part III, Amer. Math. Soc., Providence, RI, 1989, pp. 343-368. MR 1025352 (91h:01091)
- [13]
- R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. MR 0447954 (56:6264)
- [14]
- R. B. Crittenden and V. L. Shapiro, Sets of uniqueness on the group , Ann. of Math. (2) 81 (1965), 550-564. MR 0179535 (31:3783)
- [15]
- P. Enflo, A counterexample to the approximation property in Banach spaces, Acta Math. 139 (1973), 308-317. MR 0402468 (53:6288)
- [16]
- W. Engels, On the characterization of the dyadic derivative, Acta Math. Acad. Sci. Hungar. 46 (1985), 47-56. MR 819052 (87a:42042)
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- C. Fefferman and E. M. Stein, spaces of several variables, Acta Math. 129 (1972), 137-194. MR 0447953 (56:6263)
- [18]
- N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372-414. MR 0032833 (11:352b)
- [19]
- -, The generalized Walsh functions, Trans. Amer. Math. Soc. 69 (1950), 66-77. MR 0042535 (13:126g)
- [20]
- S. Fridli and P. Simon, On the Dirichlet kernels and a Hardy space with respect to the Vilenkin system, Acta Math. Acad. Sci. Hungar. 45 (1985), 223-234. MR 779509 (86m:42032)
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- A. M. Garsia, Martingale inequalities, W. A. Benjamin, Inc., Reading, MA, 1973. MR 0448538 (56:6844)
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Review Information:
Reviewer:
W. R. Wade
Journal:
Bull. Amer. Math. Soc.
26 (1992), 348-359
DOI:
https://doi.org/10.1090/S0273-0979-1992-00276-8