On the exactness of a theorem of G. A. Fomin (Q1185383)

\textit{G. A. Fomin} [Mat. Zametki 23, 213-222 (1978; Zbl 0379.42004)] proved the following: If a sequence \(\{a_ n:n=0,1,2,\dots\}\) is such that (1) \(\lim a_ n=0\) \((n\to\infty)\) and (2) for some \(p>1\) we have \[ \sum^ \infty_{m=n}2^ m\left(2^{-m}\sum^{2^{m+1}-1}_{k=2^ m}| a_ k-a_{k+1}|^ p\right)^{1/p}<\infty \] then the series \({1\over 2}a_ 0+\sum^ \infty_{n=1}a_ n\cos nx\) converges, except possibly at \(x\equiv 0\pmod{2\pi}\), to an integrable function and \({1\over 2}a_ 0+\sum^ \infty_{n=1}a_ n\cos nx\) is the Fourier series of \(f(x)\). The author proves the following in order to show how the above result is the best possible in a certain sense. The theorem proved by the author is as follows: If \(p>1\) and \(\{B_ m:m=0,1,\dots\}\) is a sequence of nonnegative numbers such that \(\sum^ \infty_{m=0}2^ mB_ m=\infty\) then there exists a sequence \(\{a_ n\}\) such that \(\lim_{n\to\infty}a_ n=0\) and \[ \left(2^{-m}\sum^{2^{m+1}-1}_{k=2^ m}| a_ k- a_{k+1}|^ p\right)^{1/p}\leq B_ m\;(m=0,1,\dots) \] and the series \({1\over 2}a_ 0+\sum^ \infty_{n=1}a_ n\cos nx\) converges, except possibly at \(x\equiv 0\pmod{2\pi}\), to a nonintegrable function.