Let \(\phi_ m(f,x,h)=\Delta^ m_ hf(x)+\Delta^ m_{-h}f(x)\) be the symmetrical differences, where \(\Delta^ m_ hf(x)\) denotes the \(m^{th}\) difference of f(x) with step h with respect to x. The main result is: Theorem Let \(f\in L^ p(R)\), \(1<p\leq 2\) such that \(\| \phi_ mf\|_ p=O(h^{\alpha}),\) \(0<\alpha \leq 1\), \(h\to 0\). Then the Fourier transform \(\hat f\) of f belongs to \(L^{\beta}(R)\) if \(p(p+\alpha p-1)^{-1}<\beta \leq p(p-1)^{-1}.\) The result is generalized in various manners: To combinations of different order differences, to the multidimensional case, to the discrete case (Fourier- coefficients) and to the Fourier transform on compact Abelian metric zero dimensional groups.