On Polynomials with Low Peak Signal to Power ( L ∞ to L 2 Norm) Ratios and Theorems of Kashin and Spencer

In many engineering problems such as optical communications, radar and sonar problems, the electronic synthesis of speech, etc., as well as mathematical applications, a problem that arises is that of finding a waveform (trigonometric polynomial) with a specified spectrum, such that its crest factor is minimum, where the crest factor is the ratio of the peak signal energy ( L ∞ norm) to the power ( L 2 norm) of the waveform. The mathematical formulation of the problem is as follows: Given 1 ≤ m 1 < m 2 < … m k ≤ n and arbitrary complex numbers a j j = 1,…, k , we want to find signs ε j = ±1 such that the growth of the crest factoris minimized as k and n are allowed to become arbitrarily large. Recently B. Kashin has solved special cases of this general problem using some deep geometrical results. We solve the general problem just stated and in particular obtain Kashin's results as a special case. The analogous problem where the ε j are replaced by α j and where the α j are complex numbers with | α j | = 1 is also solved. The results obtained are best possible in the sense that the asymptotic growth of the crest factors of the polynomials found is optimal. The preceding results are obtained by a solution of the following geometrical problem of independent interest: Given vectors υ 1 ,…, υ k in complex n ‐dimensional space, where k ≤ n , we want to find signs ε j = ±1 such that the growth of ‖ ε 1 υ 1 + … + ε k υ k ‖ ∞ is minimized as k and n go to infinity. As a special case of this geometrical result we obtain some combinatorial discrepancy results of J. Spencer.