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## Fourier Analys&Appl

by: Mrs. Preston Lehner

27

0

1

# Fourier Analys&Appl MATH 354

Mrs. Preston Lehner
UM
GPA 3.87

Hugh Montgomery

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COURSE
PROF.
Hugh Montgomery
TYPE
Class Notes
PAGES
1
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 1 page Class Notes was uploaded by Mrs. Preston Lehner on Thursday October 29, 2015. The Class Notes belongs to MATH 354 at University of Michigan taught by Hugh Montgomery in Fall. Since its upload, it has received 27 views. For similar materials see /class/231479/math-354-university-of-michigan in Mathematics (M) at University of Michigan.

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Date Created: 10/29/15
Math 354 Fall 2007 The Minimum of the Dirichlet Kernel N 1 DNW Z 60106 717N sin2N 17r sin 7m denote the Dirichlet kernel where 66 62quot Clearly the maximum of the Dirichlet kernel is DN0 2N1 The Dirichlet kernel has zeros at n2N1 for n 12 2N Between consecutive zeros DN1 must have at least one local extremum making a total of at least 2N local extrema But is a trigonometric polynomial of degree N so it can have at most 2N roots Thus we conclude that there is exactly one local extremum between each pair of consecutive zeros The unique maximum of DN in fl2N1 12N1 is at 1 0 Put 0 0 and for n 1 2N 7 1 let 1 denote the unique root of ng 0 in the interval n2N 1n 12N Since 1sir17r1 is decreasing for 0 lt 1 S 12 it follows that lDN1n71lgtlDN 12N1l gt WNW for 1 S n S N The Dirichlet kernel lies between the two envelopes i1sin7r1 and is tangent to one or the other of these curves at the points 2n 7 122N 1 n 12 2N 1 in particular at the second of these points we have 32 71 D lt 77 2N 1 N 2N1 114312 C where 2 cl 7 2122065907 37139 One might speculate that 34N 2 is close enough to 1 to ensure that DN11 N 701 2N 1 but we show below that the minimum is smaller 2 DN1 N 7002N 1 where 3 co 2172336282

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