Some polynomial extremal problems which emerged in the twentieth century

Most of the “extremal problems” of Harmonic (or Fourier) Analysis which emerged before the year 2000 were actually born in the twentieth century, and their emergences were scattered throughout that century, including the two world war periods. A great many of these problems pertain to polynomials, trigonometric polynomials and (finite) exponential sums. Writing a reasonably complete monograph on this huge subject (even if we choose to restrict it to polynomials only) would be a monumental task, although the literature does indeed contain some valuable monographs on various aspects of the subject. The present text just touches upon a number of extremal problems on polynomials and trigonometric polynomials, with the hope of expanding this same text in the near future to a much larger version, and ultimately to a “reasonably complete” monograph (but only with the help of other mathematicians.) The theory of polynomials on the unit circle is, of course, part of classical Fourier Analysis, studied with the tools of real and complex analysis. But it also leads to studying polynomials on the (cyclic) finite subgroups of the unit circle, and this is part of Fourier Analysis on finite groups. In many ways this leads to cyclotomy, which is part of Number Theory and Algebra. Also, some combinatorial designs (cyclic difference sets) show up in connection with this study. Thus the analysis of polynomials and trigonometric polynomials, even in one single variable, is at the crossroad of many important areas of contemporary mathematics. It is also much connected with some areas of engineering, such as signal processing. 202 Bahman Saffari / Some polynomial extremal problems which emerged in. . .