ZEROS OF FINITE WAVELET SUMS

On the zeros of finite sums of exponential functions

We establish an upper bound for the number of zeros of finite sums of exponential functions defined in the set of real numbers, and then discuss some applications of this result to linear algebra. Also, we record a few questions for classroom conversations on the modifications of our theorem.

On zeros of Kloosterman sums

Zeros of Sections of Exponential Sums

We derive the large n asymptotics of zeros of sections of a generic exponential sum. We divide all the zeros of the nth section of the exponential sum into “genuine zeros,” which approach, as n → ∞, the zeros of the exponential sum, and “spurious zeros,” which go to infinity as n → ∞. We show that the spurious zeros, after scaling down by the factor of n, approach a “rosette,” a finite collecti...

Zeros of Exponential Sums and Integrals

It is of frequent occurrence in problems of both pure and applied mathematics that certain values sought may be specified and must be determined as the roots of a tran-scendental equation. In particular, the equation may be of the class in which the unknown is involved only through the medium of exponential or trigonometric functions, with coefficients which are power functions or essentially s...

The Complex Zeros of Random Sums

Mark Kac gave an explicit formula for the expectation of the number, νn(Ω), of zeros of a random polynomial, Pn(z) = n ∑ j=0 ηjz j , in any measurable subset Ω of the reals. Here, η0, . . . , ηn are independent standard normal random variables. In fact, for each n > 1, he obtained an explicit intensity function gn for which Eνn(Ω) = ∫ Ω gn(x)dx. Inspired by that result, Larry Shepp and I found ...