The development of square functions in the work of A. Zygmund

I've decided to write this essay about "square functions" for two reasons. First, their development has been so intertwined with the scientific work of A. Zygmund that it seems highly appropriate to do so now on the occasion of his 80th birthday. Also these functions are of fundamental importance in analysis, standing as they do at the crossing of three important roads many of us have travelled by: complex function theory, the Fourier transform (or orthogonality in its various guises), and real-variable methods. In fact, the more recent applications of these ideas, described at the end of this essay, can be seen as confirmation of the significance Zygmund always attached to square functions. This is going to be a partly historical survey, and so I hope you will allow me to take the usual liberties associated with this kind of enterprise: I will break up the exposition into certain "historical periods", five to be precise; and by doing this I will be able to suggest my own views as to what might have been the key influences and ideas that brought about these developments.