Metric divergence of trigonometric Fourier series (Q5942593)

The following two theorems are proved. Theorem 1. For every sequence \(\{a_n: n= 1,2,\dots\}\) satisfying the conditions \(a_n\downarrow 0\) and \(\limsup_{n\to\infty} na_n= \infty\), there exists a continuous function \(f\) whose Fourier series diverges at zero and for whose Fourier coefficients \(A_n\) and \(B_n\), \[ |A_n|\leq a_n,\qquad |B_n|\leq a_n.\tag{\(*\)} \] Theorem 2. For every sequence \(\{a_n\}\) satisfying the conditions \(a_n\downarrow 0\) and \(\limsup_{n\to\infty} na^2_n= \infty\), there exists a function \(f\in L^1(0, 2\pi)\) whose Fourier series diverges in the metric of \(L^1(0, 2\pi)\) and for whose Fourier coefficients \(A_n\) and \(B_n\) condition \((*)\) is satisfied.