Two UC-Sets Whose Union is Not a UC-Set

John J. F. Fournier
1982 Proceedings of the American Mathematical Society  
It is shown that the union of two sets of uniform convergence need not be a set of uniform convergence. We use the standard terminology of harmonic analysis on the unit circle as in [41 We recall some notions discussed in [8] and [9], and in the references cited in these papers. Definition. Given a subset E of the integers, call an integrable function/ on the circle, an E-function if fin) = 0 for all integers « outside E, and denote the space of continuous ^-functions by CE. Call E a set of
more » ... Call E a set of uniform convergence, or a UC-set, if every function in CE has a uniformly convergent Fourier series. The union problem for UC-sets is mentioned as an open problem in [5, p. 86; 9, p. 283]. To solve it, we need a few more facts about UC-sets. It is known that Tí is a UC-set if and only if there is a constant « so that, for each function / in CE, the partial sums SN(f) of the Fourier series of/ satisfy the inequality HSVCDII» K ll/lloo for all nonnegative integers N. Furthermore, when Tí is a UC-set, there is a smallest value of k for which the inequality above holds for all such / and N; this smallest value of k is called the UC-constant of E, and is denoted by k(E). If Tí is a UC-set, then so is every translate of E, but it turns out that the translates of a UC-set do not all have to have the same UC-constant. Definition. Call E a CUC-set, or a set of completely uniform convergence if E is a UC-set with the additional property that the sequence {k(E + «)}"_x is bounded. This notion was introduced, independently by G. Travaglini [9, Lemma 6] and F. Ricci [7, p. 426]. In [8], P. M. Soardi and Travaglini gave some nontrivial examples of CUC-sets, and they showed that if there is a UC-set that is not a CUC-set, then there is a pair of UC-sets whose union is not a UC-set. In the present paper, we exhibit a class of UC-sets that are not CUC-sets, thereby showing that the union of two UC-sets need not be a UC-set. Recall that a set 77 of positive integers is called a Hadamard set if there is a constant r > 1 so that, when 77 is enumerated in increasing order as {k}í-i, then hJ+x > rhj for ally. Also, if E and F are two sets of integers then E -F denotes the set of all integers of the form m -« where m E E and n E F.
doi:10.2307/2043811 fatcat:p2mjrrv3r5hc7eb7kfrpi5aquq