Generalizations of Cesàro continuous functions and integrals of Perron type

Lee, Cheng Ming

doi:10.1090/S0002-9947-1981-0617545-4

Generalizations of Cesàro continuous functions and integrals of Perron type
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Trans. Amer. Math. Soc. 266 (1981), 461-481 Request permission

Abstract:

The linear space of all the Cesàro continuous functions of any order is extended by introducing pointwisely Cesàro continuous functions and exact generalized Peano derivatives. Then six generalized integrals of Perron type are defined and studied. They are based on three recent monotonicity theorems and each depends on an abstract upper semilinear space of certain functions. Some of the integrals are more general than all the integrals in the Cesàro-Perron scale provided that the abstract semilinear space is taken to be the linear space of all the pointwisely Cesàro continuous functions or all the exact generalized Peano derivatives. That such a concrete general integral is possible follows from the fact proved here that each exact generalized Peano derivative is in Baire class one and has the Darboux property. Relations between the pointwisely Cesàro continuous functions or the exact generalized Peano derivatives and functions defined by means of the values of certain Schwartz’s distributions at "points" are also established.

References

Bruce S. Babcock, On properties of the approximate Peano derivatives, Trans. Amer. Math. Soc. 212 (1975), 279–294. MR 414803, DOI 10.1090/S0002-9947-1975-0414803-0
J. A. Bergin, A new characterization of Cesàro-Perron integrals using Peano derivates, Trans. Amer. Math. Soc. 228 (1977), 287–305. MR 435312, DOI 10.1090/S0002-9947-1977-0435312-0
L. S. Bosanquet, A property of Cesàro-Perron integrals, Proc. Edinburgh Math. Soc. (2) 6 (1940), 160–165. MR 2909, DOI 10.1017/S0013091500024688
A. M. Bruckner, An affirmative answer to a problem of Zahorski, and some consequences, Michigan Math. J. 13 (1966), 15–26. MR 188375, DOI 10.1307/mmj/1028999475
Andrew M. Bruckner, Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Springer, Berlin, 1978. MR 507448, DOI 10.1007/BFb0069821
Andrew Bruckner, Current trends in differentiation theory, Real Anal. Exchange 5 (1979/80), no. 1, 9–60. MR 557963, DOI 10.2307/44151488
P. S. Bullen, The $P^{n}$-integral, J. Austral. Math. Soc. 14 (1972), 219–236. MR 0320244, DOI 10.1017/S1446788700010065
P. S. Bullen and C. M. Lee, On the integrals of Perron type, Trans. Amer. Math. Soc. 182 (1973), 481–501. MR 338291, DOI 10.1090/S0002-9947-1973-0338291-6
P. S. Bullen and S. N. Mukhopadhyay, Peano derivatives and general integrals, Pacific J. Math. 47 (1973), 43–58. MR 327990, DOI 10.2140/pjm.1973.47.43
J. C. Burkill, The approximately continuous-Perron integral, Math. Z. 34 (1932), no. 1, 270–278. MR 1545252, DOI 10.1007/BF01180588

The Cesàro-Perron integral

34

The Cesàro-Perron scale of integration

39

George Cross, The expression of trigonometrical series in Fourier form, Canadian J. Math. 12 (1960), 694–698. MR 117496, DOI 10.4153/CJM-1960-062-8

Sur l’intégration des coefficients différentials d’ordre supérieur

25

Michael J. Evans, $L\,_{p}$ derivatives and approximate Peano derivatives, Trans. Amer. Math. Soc. 165 (1972), 381–388. MR 293030, DOI 10.1090/S0002-9947-1972-0293030-1
H. W. Ellis, Mean-continuous integrals, Canad. J. Math. 1 (1949), 113–124. MR 28932, DOI 10.4153/cjm-1949-012-6
H. W. Ellis, On the compatibility of the approximate Perron and the Cesàro-Perron integrals, Proc. Amer. Math. Soc. 2 (1951), 396–397. MR 43867, DOI 10.1090/S0002-9939-1951-0043867-0
H. W. Ellis, Darboux properties and applications to non-absolutely convergent integrals, Canad. J. Math. 3 (1951), 471–485. MR 43872, DOI 10.4153/cjm-1951-048-x
Ralph Henstock, The use of convergence factors in Ward integration, Proc. London Math. Soc. (3) 10 (1960), 107–121. MR 121459, DOI 10.1112/plms/s3-10.1.107
R. D. James, Generalized $n$th primitives, Trans. Amer. Math. Soc. 76 (1954), 149–176. MR 60002, DOI 10.1090/S0002-9947-1954-0060002-0
R. L. Jeffery, Non-absolutely convergent integrals, Proc. Second Canadian Math. Congress, Vancouver, 1949, University of Toronto Press, Toronto, 1951, pp. 93–145. MR 0044614
R. L. Jeffery and H. W. Ellis, Cesàro totalization, Trans. Roy. Soc. Canada Sect. III 36 (1942), 19–44 (French). MR 7529
R. L. Jeffery and D. S. Miller, Convergence factors for generalized integrals, Duke Math. J. 12 (1945), 127–142. MR 11710, DOI 10.1215/S0012-7094-45-01211-7
Y. Kubota, On a characterization of the $\textrm {CP}$-integral, J. London Math. Soc. 43 (1968), 607–611. MR 228648, DOI 10.1112/jlms/s1-43.1.607
Yôto Kubota, The mean continuous Perron integral, Proc. Japan Acad. 40 (1964), 171–175. MR 165064
Yôto Kubota, An integral of the Denjoy type, Proc. Japan Acad. 40 (1964), 713–717. MR 178113
Cheng Ming Lee, On the approximate Peano derivatives, J. London Math. Soc. (2) 12 (1975/76), no. 4, 475–478. MR 399378, DOI 10.1112/jlms/s2-12.4.475
Cheng Ming Lee, An approximate extension of Cesàro-Perron integrals, Bull. Inst. Math. Acad. Sinica 4 (1976), no. 1, 73–82. MR 412358
Cheng Ming Lee, An analogue of the theorem of Hake-Alexandroff-Looman, Fund. Math. 100 (1978), no. 1, 69–74. MR 486362, DOI 10.4064/fm-100-1-69-74

Note on the oscillatory behavior of certain derivatives

4

S. Lojasiewicz, Sur la valeur et la limite d’une distribution en un point, Studia Math. 16 (1957), 1–36 (French). MR 87905, DOI 10.4064/sm-16-1-1-36

On the differentiability of functions and summability of trigonometric series

26

An integral of Perron type

J. Mikusiński and R. Sikorski, The elementary theory of distributions. I, Rozprawy Mat. 12 (1957), 54. MR 94702
H. William Oliver, The exact Peano derivative, Trans. Amer. Math. Soc. 76 (1954), 444–456. MR 62207, DOI 10.1090/S0002-9947-1954-0062207-1
Richard J. O’Malley and Clifford E. Weil, The oscillatory behavior of certain derivatives, Trans. Amer. Math. Soc. 234 (1977), no. 2, 467–481. MR 453940, DOI 10.1090/S0002-9947-1977-0453940-3

Ueber die gegenseitigen Beziehungen verschieder approximativ stetiger Denjoy-Perron-Integrall

22

Theory of integral

W. L. C. Sargent, A descriptive definition of Cesàro-Perron integrals, Proc. London Math. Soc. (2) 47 (1941), 212–247. MR 5897, DOI 10.1112/plms/s2-47.1.212
W. L. C. Sargent, On generalized derivatives and Cesàro-Denjoy integrals, Proc. London Math. Soc. (2) 52 (1951), 365–376. MR 41199, DOI 10.1112/plms/s2-52.5.365
W. L. C. Sargent, Some properties of $C_\lambda$-continuous functions, J. London Math. Soc. 26 (1951), 116–121. MR 41198, DOI 10.1112/jlms/s1-26.2.116
L. Schwartz, Théorie des distributions. Tome I, Publ. Inst. Math. Univ. Strasbourg, vol. 9, Hermann & Cie, Paris, 1950 (French). MR 0035918
S. Verblunsky, On the Peano derivatives, Proc. London Math. Soc. (3) 22 (1971), 313–324. MR 285678, DOI 10.1112/plms/s3-22.2.313
S. Verblunsky, On a descriptive definition of Cesàro-Perron integrals, J. London Math. Soc. (2) 3 (1971), 326–333. MR 286954, DOI 10.1112/jlms/s2-3.2.326
Clifford E. Weil, On properties of derivatives, Trans. Amer. Math. Soc. 114 (1965), 363–376. MR 176007, DOI 10.1090/S0002-9947-1965-0176007-2
Clifford E. Weil, Monotonicity, convexity and symmetric derivates, Trans. Amer. Math. Soc. 221 (1976), no. 1, 225–237. MR 401994, DOI 10.1090/S0002-9947-1976-0401994-1
Zygmunt Zahorski, Über die Menge der Punkte in welchen die Ableitung unendlich ist, Tôhoku Math. J. 48 (1941), 321–330 (German). MR 27825
Z. Zahorski, Sur la première dérivée, Trans. Amer. Math. Soc. 69 (1950), 1–54 (French). MR 37338, DOI 10.1090/S0002-9947-1950-0037338-9