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{SECT 0 {PARA 260 "" 0 "" {TEXT -1 27 "The Hausdorf Moment Problem" }}
{PARA 256 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 257 "" 0 "" {TEXT -1
21 "School of Mathematics" }}{PARA 258 "" 0 "" {TEXT -1 12 "Georgia Te
ch" }}{PARA 259 "" 0 "" {TEXT -1 21 "herod@math.gatech.edu" }}{PARA 0
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "There is a classi
cal problem called " }{TEXT 259 27 "the Hausdorf Moment Problem" }
{TEXT -1 93 " which this worksheet will introduce. This classical prob
lem addresses the following request:" }}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{PARA 15 "" 0 "" {TEXT -1 31 "Characterize those sequences \{ " }
{XPPEDIT 18 0 "c[i]" "6#&%\"cG6#%\"iG" }{TEXT -1 71 " \} for which the
re is one and only one continuous function p(x) so that" }}{PARA 0 ""
0 "" {TEXT -1 50 " "
}{XPPEDIT 18 0 "c[n]" "6#&%\"cG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT
18 0 "int(x^n*p(x),x=a..b)" "6#-%$intG6$*&)%\"xG%\"nG\"\"\"-%\"pG6#F(F
*/F(;%\"aG%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 50 " A
reference for this ideas could be found in " }}{PARA 0 "" 0 "" {TEXT
-1 21 "Shohat and Tamarkin, " }{TEXT 256 18 "The Moment Problem" }
{TEXT -1 2 ", " }{TEXT 257 19 "Mathematical Survey" }{TEXT -1 96 "s, A
MS Publications, 1943, 1950 (revised), and 1963. (Library of Congress
Listing: QA295.S55)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 452 " We give an introduction to the notions of thes
e ideas by comparing the cosine function with the polynomial that has \+
the same first seven moments. In what follows, we compute the moments \+
for the cosine function over the interval [0, 4 ] and approximate this
cosine function on that interval with a polynomial of degree N that h
as the same N+1 moments as the cosine function. To give perspective, w
e compare the Taylor polynomial of the same degree." }}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 32 "We
use a polynomial of degree N." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 5 "N:=6;" }}}{PARA 0 "" 0 "" {TEXT -1 51 "The moments are computed o
ver the interval [0, 4 ]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57
"for i from 0 to N do\nc[i]:=int(x^i*cos(x),x=0..4*Pi);\nod:" }}}
{PARA 0 "" 0 "" {TEXT -1 46 "A polynomial p(x) of degree with coeffici
ents " }{XPPEDIT 18 0 "b[n]" "6#&%\"bG6#%\"nG" }{TEXT -1 12 " is defin
ed." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p:=x->sum('b[n]'*x^n,
n=0..N);\np(1);" }}}{PARA 0 "" 0 "" {TEXT -1 136 "The requirement that
this polynomial should have the same N + 1 moments as the cosine func
tion enables us to compute the coefficients. " }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 71 "fsolve(\{seq(c[m]=int(x^m*p(x),x=0..4*Pi),m=0..N
)\},\n\{seq(b[i],i=0..N)\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 10 "assign(%);" }}}{PARA 0 "" 0 "" {TEXT -1 66 "For comparison, we c
ompute the Taylor polynomial for cosine at 2 ." }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 63 "series(cos(x),x=2*Pi,N+1);\nconvert(%,polynom);
\nT:=unapply(%,x);" }}}{PARA 0 "" 0 "" {TEXT -1 105 "Finally, we compa
re the three graphs to see how good the polynomial created from the mo
ments makes a fit." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot(
\{T(x),cos(x),p(x)\},x=0..4*Pi,y=-2..2);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 9 "Exercise.
" }{TEXT -1 104 " Find the polynomial of degree 6 that has the same mo
ments as the sine function on the interval [0, 4 ]." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{MARK "0 0" 27 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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