A comparison of the Jordan and Dini tests (Q1101914)

It is well known that the Jordan and Dini tests for everywhere convergence of Fourier series are classically noncomparable, but it is clear that in some sense the Dini test is much stronger. We make this sense precise by use of a natural measure of the complexity of functions with everywhere convergent Fourier series called the Zalcwasser rank. The strength of a test is defined as the strict supremum of the Zalcwasser ranks of the functions for which the test is applicable. It is then shown that the Jordan test has strength 3, while the Dini test has strength \(\omega_ 1\). The latter fact depends on the following key result of \textit{M. Ajtai} and \textit{A. Kechris} [Trans. Am. Math. Soc. 302, 207-221 (1987; Zbl 0633.04004)]: There is no Borel subset of C(T) separating the set D(T) of differentiable functions and the set E(T) of continuous functions with everywhere convergent Fourier series.