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On Integrals and Summable Trigonometric Series

Published online by Cambridge University Press:  20 November 2018

Cheng-Ming Lee
Cheng-Ming Lee*
Affiliation:
University of Wisconsin-Milwaukee Milwaukee, Wisconsin 53201
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Abstract

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In considering a problem on certain summable (C, k) trigonometric series, R. D. James [13] used a symmetric pk+2- integral defined earlier to recapture the coefficients of the series from the sum function. James' formulas for the coefficients are more complicated than the usual Euler-Fourier form since the pk + 2 - integral is of order k + 2. It is shown that a generalized integral of order one for each non-negative integer k can be suitably defined to reduce James' formulas to the usual form.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 24 , Issue 4 , 01 December 1981 , pp. 433 - 440
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Bullen, P. S., A criterion for n-convexity, Pacific J. Math. 36 (1971), 81-98.Google Scholar
2. Bullen, P. S., and Lee, C.-M., On the integrals of Perron type, Trans. Amer. Math. Soc. 182 (1973), 481-501.Google Scholar
3. Bullen, P. S., The SCnP-integral and the Pn+1-integral, Canad. J. Math. 25 (1973), 1274-1284.Google Scholar
4. Burkill, H., A note on trigonometric series, J. Math. Anal, and Appl. 40 (1972), 39-44.Google Scholar
5. Burkill, J. C., Integrals and trigonometric series, Proc. London Math. Soc. (3). 1 (1951), 46-57.Google Scholar
6. Cross, G., An integral for Cesàro summable series, Canad. Math. Bull. 10 (1967), 85-97.Google Scholar
7., The Pn-integral, Canad. Math. Bull. 18 (1975), 493-497.Google Scholar
8. Cross, G., The representation of (C, k) summable series in Fourier form, Canad. Math. Bull. 21 (1978), 149-158.Google Scholar
9. Cross, G., The SCk+lP-integral and trigonometric series, Proc. Amer. Math. Soc. 69 (1978), 297-302.Google Scholar
10. Den joy, A., Calcul des coefficients d'une série trigonométrique partout convergente, C. R. Acad. Sci. Pari. 172 (1921), 653-655, 833-835, 903-906, 1218-1221; 173 (1921), 127-129.Google Scholar
11. James, R. D., A generalized integral II, Canad. J. Math. 2 (1950), 297-306.Google Scholar
12. James, R. D., Generalized nth primitives, Trans. Amer. Math. Soc. 76 (1954), 149-176.Google Scholar
13. James, R. D., Summable trigonometric series, Pacific J. Math. 6 (1956), 99-110.Google Scholar
14. Marcinkiewicz, J. and Zygmund, A., On the differentiability of functions and the summability of trigonometric series, Fund. Math. 26 (1936), 1-43.Google Scholar
15. Mukhopadhyay, S. N., On the regularity of the Pn-integral and its application to summable trigonometric series, Pacific J. Math. 55 (1974), 233-247.Google Scholar
16. Stein, E. M. and Zygmund, A., On the differentiability of functions, Studia Math. 23 (1964), 247-283.Google Scholar
17. Taylor, S. J., An integral of Perron's type defined with the help of trigonometric series, Quart. J. Math. Oxford (2). 6 (1955), 255-274.Google Scholar
18. Wolf, F., Summable trigonometric series: an extension of uniqueness theorem, Proc. London Math. Soc. (2). 45 (1939), 328-356.Google Scholar
19. Zygmund, A., Trigonometric Series, Cambridge (1959).Google Scholar