Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The $ {\rm SC}\sb{k+1}\ {\rm P}$-integral and trigonometric series


Author: G. E. Cross
Journal: Proc. Amer. Math. Soc. 69 (1978), 297-302
MSC: Primary 26A39; Secondary 42A24
DOI: https://doi.org/10.1090/S0002-9939-1978-0476946-3
MathSciNet review: 0476946
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Recently P. S. Bullen and C. M. Lee defined a scale of symmetric Cesàro-Perron integrals, but left open the question of whether their $ S{C_{k + 1}}P$-integral solves the coefficient problem for (C, k) summable series. This paper gives an affirmative answer to that question under natural conditions.


References [Enhancements On Off] (What's this?)

  • [1] J. A. Bergin, A new characterization of Cesàro-Perron integrals using Peano derivates, Trans. Amer. Math. Soc. 228 (1977), 287-305. MR 0435312 (55:8272)
  • [2] P. S. Bullen, Construction of primitives of generalized derivatives with applications to trigonometric series, Canad. J. Math. 13 (1961), 48-58. MR 0121448 (22:12186)
  • [3] P. S. Bullen and C. M. Lee, The $ S{C_n}P$-integral and the $ {P^{n + 1}}$-integral, Canad. J. Math. 25 (1973), 1274-1284. MR 0409737 (53:13489)
  • [4] H. Burkill, A note on trigonometric series, J. Math. Anal. Appl. 40 (1972), 39-44. MR 0313696 (47:2250)
  • [5] J. C. Burkill, The Cesàro-Perron scale of integration, Proc. London. Math. Soc. (2) 39 (1935), 541-552.
  • [6] -, The expression of trigonometrical series in Fourier form, J. London Math. Soc. 11 (1936), 43-48.
  • [7] -, Integrals and trigonometric series, Proc. London Math. Soc. (3) 1 (1951), 46-57. MR 0042533 (13:126e)
  • [8] G. E. Cross, The expression of trigonometrical series in Fourier form, Canad. J. Math. 12 (1960), 694-698. MR 0117496 (22:8275)
  • [9] -, The representation of (C, k) summable series in Fourier form, Canad. Math. Bull. 21 (1978). MR 0492118 (58:11270)
  • [10] -, The $ {P^n}$-integral, Canad. Math. Bull. 18 (1975), 493-497.
  • [11] R. D. James, Generalized nth primitives, Trans. Amer. Math. Soc. 76 (1954), 149-176. MR 0060002 (15:611b)
  • [12] -, Summable trigonometric series, Pacific J. Math. 6 (1956), 99-110. MR 0078474 (17:1198f)
  • [13] S. N. Mukhopadhyay, On the regularity of the $ {P^n}$-integral, Pacific J. Math. 55 (1974), 233-247. MR 0374346 (51:10546)
  • [14] S. Verblunsky, On the theory of trigonometric series. VII, Fund. Math. 23 (1934), 193-235.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A39, 42A24

Retrieve articles in all journals with MSC: 26A39, 42A24


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0476946-3
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society