On the Cesàro convergence of restricted Fourier series

1. If the coefficients of the trigonometrical series Σ (a r , cosrx +b r , sinrx ) (1) satisfy the conditionr -k a r → 0, (r →∞). (2) Where 0 ≤k , the series got by integrating (1)p times term-by-term (p th integrated series) is, by the Riesz-Fischer theorem, necessarily a Fourier series ifp >k + ½, and accordingly, ifp is the least integer satisfying this inequality, the (p + 1)th integrated series will converge uniformly to an integral, the (p + 2)th to the integral of an integral, and so on. One consequence is that we can write down an expression for the partial summation of the series (1), whether this be summed in the ordinary way, or in any Cesàro manner, of index integral or fractional. The expression is, of course, thep th differential coefficient of that giving the corresponding partial summation of the Fourier series.