On the order of Fourier coefficients in the sense of Cesàro means (Q1313678)

The author studies the order of decrease of Fourier coefficients for different classes of functions \((L_ p,C,\text{Lip} \alpha)\) under the agreement that convergence to zero and boundedness are meant in the sense of Cesàro means; that is, if the relevant \((C,\beta)\)-means of the coefficients \(\{a_ n\}\) \((\{b_ n\}):\) \((A_ n^{(\beta)})^{-1} \sum^ n_{k=0} a_ k A_{n-k}^{(\beta-1)}\) are \(o(1)\) (or \(O(1))\) as \(n\) tends to infinity. It is also proved that the presented nine orders are best possible.