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Symmetric integrals and trigonometric series

Seria
Rozprawy Matematyczne tom/nr w serii: 319 wydano: 1992
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Warianty tytułu
Abstrakty
EN

CONTENTS
 1. Introduction............................................................................................5
 2. Some preliminary definitions..................................................................6
 3. Mařík's symmetric difference..................................................................9
 4. Basic definitions...................................................................................11
 5. Properties of the second symmetric variation for real functions...........15
 6. Measure properties..............................................................................19
 7. The integral.........................................................................................23
 8. Additivity..............................................................................................26
 9. Relations to the James P²-integral.......................................................27
10. Relations to the Burkill SCP-integral...................................................29
11. Mařík's integration by parts formula....................................................36
12. Burkill's integration by parts formula...................................................39
13. An application to trigonometric series.................................................43
14. Some further applications...................................................................47
References...............................................................................................48
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 319
Liczba stron
49
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCXIX
Daty
wydano
1992
otrzymano
1991-08-05
poprawiono
1991-11-12
Twórcy
  • Department of Pure Mathematics, University of Waterloo Waterloo, Ontario Canada N2L 3G1
  • Department of Mathematics Simon Fraser University Burnaby, B.C. Canada V5A 1S6, thomson@cs.sfu.ca
Bibliografia
  • [1] N. Bary, A Treatise on Trigonometric Series, Vol. I and II, Pergamon Press, New York 1964.
  • [2] A. M. Bruckner, Differentiation of Real Functions, Lecture Notes in Math. 659, Springer, Berlin 1978.
  • [3] P. S. Bullen, Construction of primitives of generalized derivatives with applications to trigonometric series, Canad. J. Math. 13 (1961), 48-58.
  • [4] P. S. Bullen and C. M. Lee, On the integrals of Perron type, Trans. Amer. Math. Soc. 182 (1973), 481-501.
  • [5] P. S. Bullen and C. M. Lee, The $SC_nP$-integral and the $P^{n+1}$-integral, Canad. J. Math. 25 (1973), 1274-1284.
  • [6] P. S. Bullen and S. N. Mukhopadhyay, Integration by parts formulae for some trigonometric integrals, Proc. London Math. Soc. (3) 29 (1974), 159-173.
  • [7] H. Burkill, A note on trigonometric series, J. Math. Anal. Appl. 40 (1972), 39-44.
  • [8] H. Burkill, Fourier series of SCP integrable functions, ibid. 57 (1977), 587-609.
  • [9] J. C. Burkill, The Cesàro-Perron integral, Proc. London Math. Soc. (2) 34 (1932), 314-322.
  • [10] J. C. Burkill, The expression of trigonometric series in Fourier form, J. London Math. Soc. 11 (1936), 43-48.
  • [11] J. C. Burkill, Integrals and trigonometric series, Proc. London Math. Soc. (3) 1 (1951), 46-57.
  • [12] G. E. Cross, The expression of trigonometric series in Fourier form, Canad. J. Math. 12 (1960), 694-698.
  • [13] G. E. Cross, The relation between two definite integrals, Proc. Amer. Math. Soc. 11 (1960), 578-579.
  • [14] G. E. Cross, The relation between two symmetric integrals, ibid. 14 (1963), 185-190.
  • [15] G. E. Cross, On the generality of the AP-integral, Canad. J. Math. 23 (1971), 557-561.
  • [16] G. E. Cross, The $P^n$-integral, Canad. Math. Bull. 18 (1975), 493-497.
  • [17] G. E. Cross, Additivity of the $P^n$-integral, Canad. J. Math. 30 (1978), 783-796.
  • [18] G. E. Cross, The representation of (C,k) summable series in Fourier form, Canad. Math. Bull. 21 (1978), 149-158.
  • [19] G. E. Cross, The $SC_{k+1}P$-integral and trigonometric series, Proc. Amer. Math. Soc. 69 (1978), 297-302.
  • [20] G. E. Cross, The exceptional sets in the definition of the $P^n$-integral, Canad. Math. Bull. 25 (1982), 385-391.
  • [21] A. Denjoy, Calcul des coefficients d'une série trigonométrique convergente quelconque dont la somme est donnée, C. R. Acad. Sci. Paris 172 (1921), 1218-1221.
  • [22] A. Denjoy, Leçons sur le calcul des coefficients d'une série trigonométrique, Hermann, Paris 1941-49.
  • [23] H. W. Ellis, On the relation between the $P^2$-integral and the Cesàro-Perron scale of integrals, Trans. Roy. Soc. Canada III (3) 46 (1952), 29-32.
  • [24] W. H. Gage and R. D. James, A generalised integral, Proc. Roy. Soc. Canada 40 (1946), 25-36.
  • [25] R. Henstock, Linear Analysis, Butterworths, London 1967.
  • [26] R. D. James, A generalized integral II, Canad. J. Math. 2 (1950), 297-306.
  • [27] R. D. James, Integrals and summable trigonometric series, Bull. Amer. Math. Soc. 61 (1955), 1-15.
  • [28] R. L. Jeffrey, Trigonometric series, Canad. Math. Congress Lecture Series (#2), 1953.
  • [29] Y. Kubota, An integral with basis and its application to trigonometric series, Bull. Fac. Sci. Ibaraki Univ. Ser. A 5 (1973), 1-8.
  • [30] C. M. Lee, On the integrals of Perron type, Ph.D. Thesis, Univ. of Brit. Col., 1972.
  • [31] C. M. Lee, On integrals and summable trigonometric series, Real Anal. Exchange 4 (1978), 66-68.
  • [32] C. M. Lee, A symmetric approximate Perron integral for the coefficient problem of convergent trigonometric series, ibid. 16 (1990), 329-339.
  • [33] P. Y. Lee, Lanzhou Lectures on Integration Theory, World Scientific (Series in Real Analysis), Singapore 1989.
  • [34] J. Marcinkiewicz and A. Zygmund, On the differentiability of functions and the summability of trigonometrical series, Fund. Math. 26 (1936), 1-43.
  • [35] J. Mařík, Generalized integrals and trigonometric series, unpublished manuscript.
  • [36] S. N. Mukhopadhyay, On the regularity of the $P^n$-integral and its application to summable trigonometric series, Pacific J. Math. 55 (1974), 233-247.
  • [37] D. Preiss and B. S. Thomson, The approximate symmetric integral, Canad. J. Math. 41 (1989), 508-555.
  • [38] V. A. Sklyarenko, Certain properties of the $P^2$-primitive, Mat. Zametki 12 (1972), 693-700 (in Russian).
  • [39] V. A. Sklyarenko, Integration by parts in the SCP Burkill integral, Mat. Sb. (N.S.) 112 (154) (1980), 630-646 (in Russian).
  • [40] V. A. Skvortsov, Interrelation between general Denjoy integrals and totalization $(T_2S)_0$, ibid. 52 (94) (1960), 551-578 (in Russian).
  • [41] V. A. Skvortsov, On definitions of $P^2$- and SCP-integrals, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 21 (6) (1966), 12-19 (in Russian).
  • [42] V. A. Skvortsov, The mutual relationship between the AP-integral of Taylor and the $P^2$-integral of James, Mat. Sb. 70 (112) (1966), 380-393 (in Russian).
  • [43] V. A. Skvortsov, The Marcinkiewicz-Zygmund integral and its relation to the Burkill SCP-integral, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 27 (5) (1972), 78-82 (in Russian).
  • [44] S. J. Taylor, An integral of Perron's type defined with the help of trigonometric series, Quart. J. Math. Oxford Ser. (2) 6 (1955), 255-274.
  • [45] S. Verblunsky, On the theory of trigonometric series VII, Fund. Math. 23 (1934), 193-236.
  • [46] A. Zygmund, Trigonometric Series, Cambridge University Press, London 1968.
Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: Primary 26A24, 26A39, 26A45; Secondary 42A24.
Identyfikator YADDA
bwmeta1.element.dl-catalog-86ac72d8-3160-4281-8386-423c865281bc
Identyfikatory
ISBN
83-85116-42-8
ISSN
0012-3862
Kolekcja
DML-PL
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