Let \({\mathbb{T}}\) denote the reals taken mod \(2\pi\) ; we consider functions f on \({\mathbb{T}}^ d\), where d is a given positive integer. The complex Fourier coefficients of f are denoted by \(\hat f(\)n), \(n=(n_ 1,n_ 2,...,n_ d)\in {\mathbb{Z}}^ d.\) let t be given (1\(\leq t\leq \infty)\). For some values of the parameters \(\alpha\),\(\beta\) there exists a constant K such that \((\sum_{n\in E}| \hat f(n)|^ 2)^{1/2}\leq K| E|^{\alpha}\| f| x^{\beta}| \|_ t,\) where E is any finite subset of \(Z^ d\). In this paper, necessary and sufficient conditions on \(\alpha\),\(\beta\) for the existence of such a constant are obtained. The problem considered may be regarded as the analogue for Fourier series of a problem on Fourier integral recently considered by J. F. Price [J. Math. Phys. 27, 1711-1714 (1983; Zbl 0513.60100)].