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is necessary and sufficient for the existence of a periodic (of period 1) function gCL(,1) such that (2) f(t) = g(t) g(2t). This theorem, without the assertion that geL',1) was apparently established by R. Fortet in his paper [1 ] though his proof is not clear to me. A proof of this theorem in the case a> 1/2, was given by M. Kac [2]. Moreover, it is shown in [2] that there exists a continuous periodic (of period 1) function f satisfying (1) for which there is no function gCL(0,l) such that (2) holds. The method of the proof used here is related to that given by M. Kac. The real difference is in using Haar functions instead of trigonometrical functions. The Lemmas 3 and 4 were already proved for a>1/2in [2] Throughout this note we will consider only measurable and periodic (of period 1) functions. Now, we define the Haar functions as follows:

On the functional equation 𝑓(𝑡)=𝑔(𝑡)-𝑔(2𝑡)