Variations on a theme of Hardy's

摘要

Let theta > 1 and let phi : [ 0, 1] -> C be such that the two-sided seriesf phi(x) := Sigma(n is an element of Z) phi(x(theta n))converges for all x. [0,1], ( then necessarily phi(0) = phi(1) = 0). Supposephi(t) = Sigma(k >= 1) a(k)t(k).DefineD phi(s) = Sigma(k >= 1) a(k)/k(s).For different classes of functions phi we show thatf phi(x) = - (1)/(log theta) D '(0) + Sigma m is an element of Z backslash{0} Pi(- (2i pi m)/(log theta)) D phi(-(2i pi m)/(log theta))(_)exp (2i pi m (loglogx-1)/(log theta)).