A number theoretic estimate of Davenport from a functional analytic viewpoint (Q2756115)

In 1961, \textit{A. S. Besicovitch} [J. Lond. Math. Soc. 36, 388-392(1961; Zbl 0103.04101)] constructed a continuous function \(f:[0,1] \to {\mathbb R}\), \(f \not\equiv 0\), symmetric about the point \(1\over 2\), for which every Riemann sum \(\displaystyle {1\over n} \cdot \sum_{a=1}^n f\left( {a\over n}\right)\) is zero. Using a classical result of \textit{H. Davenport} [Quart. J. Math., Oxf. Ser. 8, 313-320 (1937; Zbl 0017.39101)], \textit{P. T. Bateman} and \textit{S. Chowla} [J. Lond. Math. Soc. 38, 372-374 (1963; Zbl 0116.26904)] gave the explicit examples NEWLINE\[NEWLINED(x) = \sum_{n=1}^\infty {{\mu(n)}\over n} \cdot \cos(2\pi n x), \quad F(x) = \sum_{n=1}^\infty {{\lambda(n)}\over n} \cdot \cos(2\pi n x).NEWLINE\]NEWLINE The authors show (Theorem 2) that \(D(x)\) changes sign infinitely often in every neighbourhood of the origin, and that NEWLINE\[NEWLINE D(x) = \Omega_{\pm}\left( x^{1\over 2}\right),\text{ as } x\to 0,NEWLINE\]NEWLINE and so \(D\) is not differentiable at the origin. Write NEWLINE\[NEWLINE \delta_f(n) = {1\over n} \cdot \sum_{a=1}^n f\left( {a\over n}\right) - \int_0^1 f(x) dx. NEWLINE\]NEWLINE Denote by \({\mathcal P}_{[0,1]}\) the quotient space of continuous functions \(f:[0,1]\to {\mathbb R}\) satisfying \(f(x)=f(1-x)\) [for all \(x\)] modulo the relation \(\sim\), defined by \(f\sim g\) iff \(f-g\) is constant. \({\mathcal P}_{BV}[0,1],\) resp. \({\mathcal P}_{A}[0,1]\) are the subspaces of \({\mathcal P}_{[0,1]}\) of functions of bounded variation, resp. of the shape NEWLINE\[NEWLINE f(x) = \sum_{k=1}^\infty a_k \cdot \cos(2\pi k x), \text{ where } \sum_{k=1}^\infty |a_k|< \infty. NEWLINE\]NEWLINE \(c_0\) is the vector space of real sequences convergent to zero. The authors prove: (i) The linear transformation NEWLINE\[NEWLINE \Lambda:{\mathcal P}_{[0,1]} \to c_0. \quad \Lambda(f) = \{ \delta_f(n)\}_{n=1,2,\dots},NEWLINE\]NEWLINE is not injective, its kernel contains all functions equivalent to \(D(x)\). (ii) The transformation \(\Lambda\), restricted to \({\mathcal P}_{BV}[0,1]\) is injective with inverse NEWLINE\[NEWLINE \Lambda^{-1}(\{\delta_f(n)\}) = \sum_{n=1}^\infty \left( \sum_{k=1}^\infty \delta_f(nk) \mu(k) \right) \cos(2\pi n x). NEWLINE\]NEWLINE (iii) The transformation \(\Lambda\), restricted to \({\mathcal P}_{A}[0,1]\) is injective, too; but in this case, for an \(\ell_1\)--dense subset of functions \(f\), the series \(\sum\limits_{k=1}^\infty \delta_f(nk) \mu(k) \cos(2\pi n x)\) diverges for any fixed \(n\). \smallskip The proof of (iii) uses properties of the Möbius function and the Banach--Steinhaus theorem.