The Complexity of Everywhere Divergent Fourier Series

Ramsamujh, T. I.

Canadian journal of mathematics, Tome 43 (1991) no. 2, pp. 413-424

Résumé

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.

Détail

DOI : 10.4153/CJM-1991-024-6

Mots-clés : 04A15, 42A20, everywhere divergent, Fourier series, Ordinal rank

Bibliographie

[1] 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] 2. Bary, N.K., A treatise on trigonometric series. Vol. 1, Macmillan, New York, 1964. Google Scholar

[3] 3. Belov, A.S., Study of some trigonometric series, Mathematical Notes 13(1973), 291–298. Google Scholar

[4] 4. Chen, Y.M., A remarkable divergent Fourier series, Proc. Japan Acad. 28(1962), 239–244. Google Scholar

[5] 5. Edwards, R.E., Fourier series: a modern introduction. Vol. 1, Springer-Verlag, New York, 1979. Google Scholar

[6] 6. Fatou, R., Séries trigonométriques et séries de Taylor, Acta Math. 30(1906), 335–100. Google Scholar

[7] 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] 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] 9. Kechris, A.S., Sets of everywhere singularfunctions. Lecture Notes in Math. 1141 Springer-Verlag, Berlin, 1985,233–244. Google Scholar

[10] 10. Kechris, A.S. and Woodin, W.H., Ranks of differentiable functions, Mathematika 33(1986), 252–278. Google Scholar

[11] 11. Kolmogorov, A.N., Une série de Fourier-Lebesgue divergent presque partout, Fund. Math. 4(1923), 324- 329. Google Scholar

[12] 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] 13. Korner, T., Everywhere divergent Fourier series, Colloq. Math. 45(1981), 103–118. Google Scholar

[14] 14. Lusin, N.N., Über eine Potenzreihe, Rendiconti Circolo Matem. di Palermo 30(1911), 386–390. Google Scholar

[15] 15. Marcinkiewicz, J., Sur les séries de Fourier, Fund. Math 27(1936), 38–69. Google Scholar

[16] 16. Moschovakis, Y.N., Descriptive set theory. North Holland, New York, 1980. Google Scholar

[17] 17. Ramsamujh, T.I., Some topics in descriptive set theory and analysis. PhD. Thesis, Calif. Inst, of Tech., Pasadena, 1986. Google Scholar

[18] 18. Ramsamujh, T.I., The complexity of nowhere differentiable continuous functions, Canadian J. of Math. 41(1989), 83–105. Google Scholar

[19] 19. Stechkin, S.B., O trigonometricheskikh Ryadakh Raskhodyashchikhsya v Kazhdoi toche, Izvestiya Akad. Nauk SSSR 21(1957), 711–728. Google Scholar

[20] 20. Steinhaus, H., Sur une série trigonométrique divergente, C. R. de la Soc. Sci. Varsovie 5(1912), 223–227. Google Scholar

[21] 21. Steinhaus, H., A divergent trigonometrical series, J. of Lond. Math. Soc, 4(1929), 86-88 Google Scholar

[22] 22. Steinhaus, H., An example of a thoroughly divergent orthogonal development, Proc. Lond. Math. Soc. (2)20(1921), 123–126. Google Scholar

[23] 23. Zygmund, A., Trigonometric series, Vol. 1, Cambridge Univ. Press, Cambridge, 1968. Google Scholar