Abstract
m=i ra=l I Jα where 0<a^2,a^t^b, and {^m} is a sequence in U[ay 6], usually orthonormal. In this paper, Fa(t) is studied for the Haar, Walsh, trigonometric, and general orthonormal sequences. For instance, it is proved that for the Haar system Fa(jt) satisfies a Lipschitz condition of order a/2 in [0,1] and that this result is best possible for any complete orthonormal sequence. An application is also given regarding the absolute convergence of Walsh series. Previously, Bosanquet and Kestelman essentially proved [3, p. 91] THEOREM A. Let {φm} be orthonormal. Then the Fourier coefficients of every absolutely continuous function are absolutely convergent if and only if F x{t) e L°°[a, b]. Also, applying ParsevaΓs equality to the characteristic function of [α, t], we obtain THEOREM B. Let {φm} be orthonormal. Then {φm} is complete in L2[a, b] if and only if F2(t) = t — a, a^t^b.
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