Mathematics
and Computer Science Speaker Series
California State University, Stanislaus
Date: Friday, March 26, 2010
Time: 3:30 - 4:30 p.m
Room: P-101
Speaker: Dennis Nemzer
Title: Trigonometric Series Vanishing on (a,b)
Abstract: Given a trigonometric series, the Fourier
coefficients can be useful in determining properties about the
series. For example, the Fourier coefficients can be used to
determine to which class of functions or generalized functions the
trigonometric series belongs. That is, does the series represent
a square integrable function, an infinitely differentiable function, or
a Schwartz distribution?
An interesting question is: What conditions must the Fourier
coefficients satisfy so that the trigonometric series vanishes on a
given interval? We will discuss a theorem which gives necessary
and sufficient conditions for a trigonometric series to vanish on an
interval. Some examples which illustrate this theorem will be
given. Also, we will discuss how our theorem may be useful when
applied to a differential equation whose solution describes the
displacement of a mass suspended from a spring.
Most of this talk is accessible to anyone having basic familiarity with
Fourier series and complex variables. However, students with some
basic knowledge of at least one of Fourier series, complex variables,
or real analysis may benefit from attending.