expenses of
TRANSCRIPT
Section 7 Overview of Fourier Series
Fourier Expansions of vectors on 1123
Fourier Expansions of Functions
Odd and Even Functions
A Sine and Cosine series
o
tourer Expenses of Vectors in IRI
Defy The dot product of L u az Us andCY ik Vrs is
U U Y t Uj t Uz Vz
Properties i
Positivity I Z O and ii ie o u E
Symmetry ii I I
Linearity a bit a CI t b
These properties imply
Thi The dot product of ie E c 123 es
ie Illia KEK cosco
where i a
Kii11 a u
VO p
iD
a 11
Remarks i Hill VE I VCU 2uz Us 2
U unit off HIM L ii u
Theirs it to are orthogonal ii 5 0
perpendicular
Exempt i The set I 8 is called an
orthonormal set because
2 a I o i I1123 V i
sq J Ele R e O yL VzTEA T.IE TE E e I
Uy l
ftp.g DILLES t 3 orthogonality Normality
Ux A
Thrun The orthonormal set F I is oi3
basis of IR's so for any vector C112Fourier
a k I t ry I t Vz k fand Vx Vy Vz can be computed by
b Vx F it Vy F I vz E E
Reks L holds because E is a basis
2 holds because I is orthonormal
Proof b We have a i
F E xe7tvyJtVzE1 I
Vx I I t Vy F I t Vz E eD o
Ux Eiain 1 E is 1 TE
Fourier Expenses of functions
Remarki We extend the ideas above from 1123to Tt the space of piecewise continuousfunctions on a b a f b
Three The space TT of o LL piecewise continuous
real functions on a b AL b is o Vector
Space over LR that is iff f f E TT
a b E IR then
ft b g Cx a fax t b fax C IT
Rema rki The dot product in 1123 can beextended to functions
Idea
E i r I DxXo ee X Xz BEX
Discretization ofa b Xo 4,721 3
we approximate fax Ica by
E L fCao f CX f Xz food C 424
L Sexo Sox Scad Fox c 1124
Then the dot product in 1124 is
F fcxojfcxojtfcx.jfcx.jtfcxzgcxzjtfcxg.TLf g fCan 9CXm
By refining the discretization and rescaling the dot productwe get i
Cocot
DIE The inner product of f g on E L L is
f f J f Cx fax DX
Remarksi 11th VET µ J Cx dx
f is unit off 11711 1
Peep i a Positivity f of Zo and f af o f o
Symmetry i f of f f
Linearity o f tbf h acf.be tbCg.h
Re ak No geometric interpretation of f f
Def f f ate orthogonal iff f of O
Thes The set in TTOO
F Uo Iz Una cos zTX Unix Sirsn 1
is an orthogonal set
Proof i show that i
Un Um J cosf.net cosfnITzXJolX 0 fornt inL for n m 102L for him 0
exercise4 0 Meo
ao ao f ta iz dx XI L C 4Uo Uo Lz
use Casco cosCO 21 cos 0 0 t cos 0 0
Un Um J cos Ctm t cos ml ok
u f M h M to y htm fo j Uzo in o
L
Un Um tzfyfm aSirs tmIIIJtfn.m Sifu mL
Lz 01 0 Coto
UuoUm n t m into or into
Guidelines Show that
Um Um J Sirs r Sirs q_x DX O for htmL for him to
use SeisCol Sisco co's toll cos 0 01
Show that
Um Um cosCrgT_xJsin q_XJolXe0foraKm.nuse SeisCA Casco z sirs 0 10 t since
Remark i Uo Uo Lz HUON HE FA
Un Un L Unh VI
Un.vn L 11411 1 7
Therefore an orthonormal set c's analogous to I FIOO
To yay Incas costs Fiscal SimCityn
People use the orthogonal set
Them i The orthogonal set F c's a basis for
the subspace WC of continuous
functions on f L L and for every f C W
the function
fFox doz II an cos n t bn sirs CEIsatisfies fecx f Cx where
a t J fCx DX Z f Uo
un f J f G cos dx I f un
bn y J fcx Seis dx I t.vn
Remar ki Instead of infinite sums we can use
finite sums approximations i
f µCx doz t EN an cosn t bn Seis X