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The Complexity of Everywhere Divergent Fourier Series

Published online by Cambridge University Press:  20 November 2018

T. I. Ramsamujh
T. I. Ramsamujh*
Affiliation:
Department of Mathematics Florida International University Miami, FL 33199, USA
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Abstract

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A natural rank function is defined on the set DS of everywhere divergent sequences of continuous functions on the unit circle T. The rank function provides a natural measure of the complexity of the sequences in DS, and is obtained by associating a well-founded tree with each such sequence. The set DF of everywhere divergent Fourier series, and the set DT of everywhere divergent trigonometric series with coefficients that tend to zero, can be viewed as natural subsets of DS. It is shown that the rank function is a coanalytic norm which is unbounded in ω1 on DF. From this it follows that DF, DT and DS are not Borel subsets of the Polish space SC(T) of all sequences of continuous functions on T. Finally an alternative definition of the rank function is formulated by using nested sequences of closed sets.

Keywords

04A1542A20everywhere divergentFourier seriesOrdinal rank
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 2 , 01 April 1991 , pp. 413 - 424
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Ajtai, M. and Kechris, A.S., The set of continuous functions with everywhere convergent Fourier series, Trans. Amer. Math Soc. 302(1987), 207–221.Google Scholar
2. Bary, N.K., A treatise on trigonometric series. Vol. 1, Macmillan, New York, 1964.Google Scholar
3. Belov, A.S., Study of some trigonometric series, Mathematical Notes 13(1973), 291–298.Google Scholar
4. Chen, Y.M., A remarkable divergent Fourier series, Proc. Japan Acad. 28(1962), 239–244.Google Scholar
5. Edwards, R.E., Fourier series: a modern introduction. Vol. 1, Springer-Verlag, New York, 1979.Google Scholar
6. Fatou, R., Séries trigonométriques et séries de Taylor, Acta Math. 30(1906), 335–100.Google Scholar
7. Hardy, G.H. and Littlewood, J.E., Some problems of diophantine approximations: A remarkable trigonometric series, Proc. Nat. Acad. Sci. 2(1912), 583–586.Google Scholar
8. Herzog, F., A note on power series which diverge everywhere on the unit circle, Michigan Math J. 2(1953- 54), 175–177.Google Scholar
9. Kechris, A.S., Sets of everywhere singularfunctions. Lecture Notes in Math. 1141 Springer-Verlag, Berlin, 1985,233–244.Google Scholar
10. Kechris, A.S. and Woodin, W.H., Ranks of differentiable functions, Mathematika 33(1986), 252–278.Google Scholar
11. Kolmogorov, A.N., Une série de Fourier-Lebesgue divergent presque partout, Fund. Math. 4(1923), 324- 329.Google Scholar
12. Kolmogorov, A.N., Une série de Fourier-Lebesgue divergent partout, C. R. de l'Acad. des Sci. Paris 183(1926), 1327-1329.Google Scholar
13. Korner, T., Everywhere divergent Fourier series, Colloq. Math. 45(1981), 103–118.Google Scholar
14. Lusin, N.N., Über eine Potenzreihe, Rendiconti Circolo Matem. di Palermo 30(1911), 386–390.Google Scholar
15. Marcinkiewicz, J., Sur les séries de Fourier, Fund. Math 27(1936), 38–69.Google Scholar
16. Moschovakis, Y.N., Descriptive set theory. North Holland, New York, 1980.Google Scholar
17. Ramsamujh, T.I., Some topics in descriptive set theory and analysis. PhD. Thesis, Calif. Inst, of Tech., Pasadena, 1986.Google Scholar
18. Ramsamujh, T.I., The complexity of nowhere differentiable continuous functions, Canadian J. of Math. 41(1989), 83–105.Google Scholar
19. Stechkin, S.B., O trigonometricheskikh Ryadakh Raskhodyashchikhsya v Kazhdoi toche, Izvestiya Akad. Nauk SSSR 21(1957), 711–728.Google Scholar
20. Steinhaus, H., Sur une série trigonométrique divergente, C. R. de la Soc. Sci. Varsovie 5(1912), 223–227.Google Scholar
21. Steinhaus, H., A divergent trigonometrical series, J. of Lond. Math. Soc, 4(1929), 86-88Google Scholar
22. Steinhaus, H., An example of a thoroughly divergent orthogonal development, Proc. Lond. Math. Soc. (2)20(1921), 123–126.Google Scholar
23. Zygmund, A., Trigonometric series, Vol. 1, Cambridge Univ. Press, Cambridge, 1968.Google Scholar