A Quantitative Version of the Young Test for the Convergence of Conjugate Series Original Research Article

The classical Young test says that if f is a 2π-periodic function of bounded variation on [−π, π], then the conjugate series to the Fourier series of f converges at x if and only if the conjugate function f exists at x. Our main goal is to give estimates of the rate of this convergence in terms of the oscillation of Ψx(t) ≔ f(x + t) −f(x − t) over appropriate subintervals. In particular, we obtain a conjugate version of the well-known Dini-Lipschitz test. As a byproduct, we obtain the rate of convergence in L1-norm.