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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa
tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 42 "Section \+
2.3: Convergence of Fourier Series" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 0 {PARA 3 "" 0 "" {TEXT 267 30 "Maple Packages for Section 2.3" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "     In \+
the Section 2.1, we discussed three general types of convergence in C(
[0, 1]): normed, pointwise, and uniform. Here, we apply these ideas to
 the special case that we have a function " }{TEXT 262 1 "f" }{TEXT 
-1 31 " and we have the Fourier series" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 21 "                     " }{XPPEDIT 18 
0 "S[n](x) = Sum(<f,phi[p]>*phi[p](x),p = 0 .. n);" "6#/-&%\"SG6#%\"nG
6#%\"xG-%$SumG6$*&-%$<,>G6$%\"fG6#&%$phiG6#%\"pG\"\"\"-&F56#F76#F*F8/F
7;\"\"!F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "phi[1],phi[2],phi[3];" "6%&
%$phiG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 200 "  ...   is the usua
l orthonormal sequence. These results are presented here as Theorems, \+
without proofs. The results may be found in many texts in texts on Fou
rier Series and Boundary Value Problems. " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "There are several terms we will us
e in this module." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 14 "A function is " }{TEXT 256 22 "sectionally continuous" }
{TEXT -1 135 " on an interval [a, b] if it is continuous on that inter
val except for possibly a finite number of jumps and removable discont
inuities." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
14 "A function is " }{TEXT 257 18 "sectionally smooth" }{TEXT -1 31 " \+
on an interval [a, b] if both " }{TEXT 268 1 "f" }{TEXT -1 1 " " }
{TEXT 258 3 "and" }{TEXT -1 1 " " }{TEXT 269 1 "f" }{TEXT -1 53 " ' ar
e sectionally continuous on the interval [a, b]." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 9 "Examples:" }{TEXT -1 280 
" The function f(x) = signum(x) is sectionally continuous on [-1, 1], \+
but the function g(x) = 1/x is not sectionally continuous on that inte
rval. Both are continuous on that interval except at zero. The signum \+
function has a jump at x = 0, but 1/x has a more serious discontinuity
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "plot([signum(x),1/x+1/1
0],x=-1..1,y=-3..3,discont=true,\n   color=[BLACK, RED]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 281 9 "Example: " }{TEXT -1 13 "The function " 
}{XPPEDIT 18 0 "sqrt(abs(x));" "6#-%%sqrtG6#-%$absG6#%\"xG" }{TEXT -1 
115 " is continuous, but not sectionally smooth on [-1, 1]. It is not \+
sectionally smooth because the derivative goes to " }{XPPEDIT 18 0 "in
finity;" "6#%)infinityG" }{TEXT -1 22 " as x approaches zero." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "Diff(sqrt(abs(x)),x)=diff(s
qrt(abs(x)),x);\nplot([rhs(%),sqrt(abs(x))],x=-1..1,y=-3..3,discont=tr
ue,color=[red,black]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Here i
s the first  result for convergence of series. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 9 "THEOREM: " }{TEXT -1 17 "
 If the function " }{TEXT 272 1 "f" }{TEXT -1 49 " is sectionally smoo
th and periodic with period 2" }{TEXT 283 2 " c" }{TEXT -1 46 ", then \+
at each point x the Fourier series for " }{TEXT 271 1 "f" }{TEXT -1 
14 " converges and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 7 "       " }{XPPEDIT 18 0 "a[0]+sum(a[n]*cos(n*pi*x/c)+b[n]*
sin(n*pi*x/c),n = 1 .. infinity);" "6#,&&%\"aG6#\"\"!\"\"\"-%$sumG6$,&
*&&F%6#%\"nGF(-%$cosG6#**F0F(%#piGF(%\"xGF(%\"cG!\"\"F(F(*&&%\"bG6#F0F
(-%$sinG6#**F0F(F5F(F6F(F7F8F(F(/F0;F(%)infinityGF(" }{TEXT -1 24 " = \+
[ f(x+) + f(x-)] / 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT 264 8 "Example:" }{TEXT -1 268 " The function signum(x) is se
ctionally smooth. Therefore the Fourier series for this function conve
rges to 1 for 0 < x < 1, to  -1 for -1 < x < 0, and to 0 for x = -1, 0
, or 1. The Fourier series for the function has period 2. Here is a pl
ot for 5 terms of the series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "c:=1;\nf:=x->signum(x);\na[
0]:=int(f(x),x=-c..c)/int(1^2,x=-c..c);\nfor n from 1 to 5 do\n   a[n]
:=int(f(x)*cos((n*Pi*x)/c),x=-c..c)/\n          int(cos(n*Pi*x/c)^2,x=
-c..c);\n   b[n]:=int(f(x)*sin((n*Pi*x)/c),x=-c..c)/\n          int(si
n(n*Pi*x/c)^2,x=-c..c);\nod;\nn:='n':" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "s:=x->a[0]+sum(a[n]*cos((n*Pi*x)/c)+b[n]*sin((n*Pi*x)
/c),n=1..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([[x,f(
x),x=-c..c],[x,s(x),x=-2*c..2*c]]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 254 "In this previous example, the function was not continuou
s. We will see that a discontinuity always leads to the overshoot that
 can be observed, no matter how many terms of the series are taken. Is
 it not clear that the function would be getting close to " }{TEXT 
285 1 "f" }{TEXT -1 1 "(" }{TEXT 286 1 "x" }{TEXT -1 26 ") if more ter
ms were used?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 97 "In this next Theorem, we see a more powerful result that \+
will insure uniform convergence for all " }{TEXT 270 1 "x" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 8 "T
HEOREM:" }{TEXT -1 15 " If the series " }{XPPEDIT 18 0 "sum(abs(a[n])+
abs(b[n]),n = 1 .. infinity);" "6#-%$sumG6$,&-%$absG6#&%\"aG6#%\"nG\"
\"\"-F(6#&%\"bG6#F-F./F-;F.%)infinityG" }{TEXT -1 40 " converges, then
 the Fourier series for " }{TEXT 284 1 "f" }{TEXT -1 45 " converges un
iformly in the interval [-c, c]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 265 8 "Example:" }{TEXT -1 17 " Take the series \+
" }{XPPEDIT 18 0 "a[n] = 2*(-1+(-1)^n)/(Pi*n^2);" "6#/&%\"aG6#%\"nG*(
\"\"#\"\"\",&F*!\"\"),$F*F,F'F*F**&%#PiGF**$F'F)F*F," }{TEXT -1 9 "  ,
 with " }{XPPEDIT 18 0 "b[n] = 0;" "6#/&%\"bG6#%\"nG\"\"!" }{TEXT -1 
26 ". Since the number series " }{XPPEDIT 18 0 "sum(1/(n^2),n = 1 .. i
nfinity);" "6#-%$sumG6$*&\"\"\"F'*$%\"nG\"\"#!\"\"/F);F'%)infinityG" }
{TEXT -1 148 " converges, the series of absolute values converges, and
 so a Fourier Series with these coefficients converges. We draw graphs
 for this series on [-" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 ", \+
" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 4 " ].\n" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 96 "c:=Pi;\na[0]:=Pi/2;\nfor n from 1 to 5 do\n
    a[n]:=2/Pi*(-1+(-1)^n)/n^2;\n    b[n]:=0;\nod;\nn:='n':" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "s:=x->a[0]+sum(a[n]*cos((n*P
i*x)/c)+b[n]*sin((n*Pi*x)/c),n=1..5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "plot(s(x),x=-2*c..2*c);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 282 8 "Example:" }{TEXT -1 6 " If \{ " }{XPPEDIT 18 0 "phi[1],ph
i[2],phi[3];" "6%&%$phiG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 70 ", \+
... \} is an orthogonal sequence and we construct the Fourier Series \+
" }}{PARA 0 "" 0 "" {TEXT -1 22 "                      " }{XPPEDIT 18 
0 "sum(a[p]*phi[p],p);" "6#-%$sumG6$*&&%\"aG6#%\"pG\"\"\"&%$phiG6#F*F+
F*" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "then this series wil
l converge in norm if and only if " }}{PARA 0 "" 0 "" {TEXT -1 22 "   \+
                   " }{XPPEDIT 18 0 "sum(abs(a[p])^2,p);" "6#-%$sumG6$
*$-%$absG6#&%\"aG6#%\"pG\"\"#F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 228 "converges. We established this earlier with the Fourier \+
Inequality. This result and the following example should be contrasted
 with the previous example. In this example, we have normed convergenc
e, but not uniform convergence. " }}{PARA 0 "" 0 "" {TEXT -1 15 "The e
xample is " }{XPPEDIT 18 0 "sum(sin(n*x)/n,n = 1 .. infinity);" "6#-%$
sumG6$*&-%$sinG6#*&%\"nG\"\"\"%\"xGF,F,F+!\"\"/F+;F,%)infinityG" }
{TEXT -1 145 ". I try to convince you and me that the convergence is n
ot uniform by seeing that the series converges in norm, but not to a c
ontinuous function." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "f2:=x
->sum(sin(n*x)/n,n=1..15):\ng2:=x->sum(sin(n*x)/n,n=1..25):\nh2:=x->su
m(sin(n*x)/n,n=1..35):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "p
lot(\{f2(x),g2(x),h2(x)\},x=-Pi..Pi,y=-2..2,\n   color=[red,blue,green
]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Here is the last Theorem \+
that we state in this section. " }{TEXT 303 96 "It gives a condition f
or uniform convergence based on properties of the function, not the se
ries" }{TEXT -1 120 " as the last theorem did. After all, it is usuall
y the function we know at the outset, and not properties of the series
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 8 "THEO
REM:" }{TEXT -1 4 " If " }{TEXT 287 1 "f" }{TEXT -1 1 "(" }{TEXT 288 
1 "x" }{TEXT -1 113 ") is periodic, continuous, and has a sectionally \+
continuous derivative, then the Fourier Series corresponding to " }
{TEXT 274 1 "f" }{TEXT -1 24 " converges uniformly to " }{TEXT 275 4 "
f(x)" }{TEXT -1 26 " for the entire real line." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "As an example of this The
orem, take the function " }{TEXT 276 1 "f" }{TEXT -1 1 "(" }{TEXT 290 
1 "x" }{TEXT -1 5 ") = |" }{TEXT 277 1 "x" }{TEXT -1 19 "| on the inte
rval [" }{XPPEDIT 18 0 "-pi;" "6#,$%#piG!\"\"" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 235 " ]. It is continuous on t
hat interval and its periodic extension is also continuous. While its \+
derivative is not continuous, it is sectionally continuous. Further, t
he coefficients are the ones used in the previous example. Check that.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "On th
e other hand, the function " }{TEXT 278 1 "f" }{TEXT -1 1 "(" }{TEXT 
291 1 "x" }{TEXT -1 4 ") = " }{TEXT 279 1 "x" }{TEXT -1 78 " on the in
terval [-1, 1] is continuous there. While the periodic extension is " 
}{TEXT 261 11 "sectionally" }{TEXT -1 17 " continuous, the " }{TEXT 
260 36 "periodic extension is not continuous" }{TEXT -1 137 ". Thus, y
ou cannot expect the Fourier Series to converge uniformly. At these di
scontinuities, you should expect the series to converge to" }}{PARA 0 
"" 0 "" {TEXT -1 12 "          ( " }{TEXT 289 1 "f" }{TEXT -1 1 "(" }
{TEXT 292 1 "c" }{TEXT -1 5 "-) + " }{TEXT 293 1 "f" }{TEXT -1 2 "(-" 
}{TEXT 294 1 "c" }{TEXT -1 7 "+) )/2," }}{PARA 0 "" 0 "" {TEXT -1 140 
"the average of the jump from the left and the jump from the right at \+
the points of discontinuities. Recall the first Theorem in this Sectio
n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 
58 "Test for Failure of a Fourier Series to Converge Uniformly" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 170 "A common
ly known result in undergraduate mathematics is that if a sequence of \+
continuous functions converges uniformly, the the limit must be contin
uous. A Fourier Series" }}{PARA 0 "" 0 "" {TEXT -1 9 "         " }
{TEXT 295 1 "S" }{TEXT -1 1 "(" }{TEXT 296 1 "N" }{TEXT -1 4 ") = " }
{XPPEDIT 18 0 "sum(A[p]*sin(p*x),p = 1 .. N);" "6#-%$sumG6$*&&%\"AG6#%
\"pG\"\"\"-%$sinG6#*&F*F+%\"xGF+F+/F*;F+%\"NG" }{TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 47 "is a sequence of continuous functions. Indeed, \+
" }{TEXT 297 1 "S" }{TEXT -1 1 "(" }{TEXT 298 1 "N" }{TEXT -1 84 ") is
 simply a sum of trigonometric function (and, hence, infinitely differ
entiable)." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 355 "Thus, if the series converges uniformly, the limit must be con
tinuous. This result is invoked most often if the periodic extension o
f a function is not continuous. The test for failure to have uniform c
onvergence goes like this: If the periodic extension of a function is \+
not continuous, then the Fourier Series for the function cannot conver
ge uniformly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 33 "Here is an illustration: we take " }{TEXT 301 1 "f" }
{TEXT -1 1 "(" }{TEXT 300 1 "x" }{TEXT -1 4 ") = " }{TEXT 299 1 "x" }
{TEXT -1 159 " over [-1,1]. Think about the periodic extension, with p
eriod 2. This periodic extension is NOT continuous. Thus, the Fourier \+
series cannot converge uniformly." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 254 "c:=1;\nf:=x->x;\na[0]:=int(f(x),x=-c..c)/int(1^2,x=-
c..c);\nfor n from 1 to 5 do\na[n]:=int(f(x)*cos((n*Pi*x)/c),x=-c..c)/
\n          int(cos(n*Pi*x/c)^2,x=-c..c);\n   b[n]:=int(f(x)*sin((n*Pi
*x)/c),x=-c..c)/\n          int(sin(n*Pi*x/c)^2,x=-c..c);\nod;\nn:='n'
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "s:=x->a[0]+sum(a[n]*co
s((n*Pi*x)/c)+b[n]*sin((n*Pi*x)/c),n=1..5);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 46 "plot([[x,f(x),x=-c..c],[x,s(x),x=-2*c..2*c]]);" }}
}{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 -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 485 "We have presented two type results in this Section. One \+
type result was information about  convergence based on the character \+
of the coefficients. The other result was based on the character of th
e function to which the series was to converge. The most frequent appl
ication is the last result: if the function is periodic, piecewise smo
oth, and continuous, the Fourier Trigonometric Series converges unifor
mly. If the function is not continuous, the series will not converge u
niformly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
51 "EMAIL: herod@math.gatech.edu   or    jherod@tds.net" }}{PARA 0 "" 
0 "" {TEXT -1 38 "URL: http://www.math.gatech.edu/~herod" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 36 "Copyright \251  \+
2003  by James V. Herod" }}{PARA 257 "" 0 "" {TEXT -1 19 "All rights r
eserved" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 1 1 2 33 1 1 }