e."= cos xtzsiux At 21

Cosx - { (e " + e -*)siux - E. (e

"- e

-"I

I. et ahun

i -,lH¥'re- it

• fltolfat

Hearne." ftp.fttgy/=cowiit9witamg.wii-cswj

ci-ixieiz.us?xi.Eit+xdwiIPiu2udcomp.always apply

Complex V. sp. : Cyr > = a,5tkEt. ..

'Mpl- conj.org?when coup. components

dy, - Yu

¥1,43:) l- y, -13gal

A :-(35) : lu#tEA : ex"iy⑧=C4Ay3td✓

Standard

complex = X,- (Ly , -yzltxzl -y,t3ya )-6 conjugatetuner prod.am#Cxig2a--syix7A

hermitian -Hy ,A×7td

Cont.

FT: Given for I periodic , getIt hitau=¥if±flH cosln.tt#ldt/=sfHI=aE-i7Z.aucoslEIl

bu Siu tbusiul- I

EXf =/ I

HKE *.

o -Tete - It or Fetch

and tar- periodic ( take,⇒⇒ ,

f. is every → just cos - coeff.

au -- II fftfcoslntldteftp./tfHcosluHdtereueVlu--tFffIffHcosCuHdttf!fLHcoscuttdt)=¥f"ioslutldt

°

no, a.- ¥ 'd "T'It -

- E. E -- I

" to ' an -- ¥ tnsiuluttfo"'- F. '

a sinful) = -2/0 in eventhe HEE n odd

n is odd.ie , n=dkIcheck for a-Mif

compare nor htt

Adk -_ 0 answers for sign

aah - f¥, Htt -- +7*-11"

Then sfft-E-I.aucosluti-EI-E.it#iYtIosfl2Ht)Peven t=qu vanish 2k-Heidke

sint I = Life ' e

- i I-

period --4 (eine)! )- H -

Normalized (as in 01131 : U normalized if I - Lulu> = HUH ?↳ important to have geometric meaning to certain quantities , like inner products - length

of projection .

HW5,ex. x4=8l

→ One deal) solution : 11=3.

But g. of degree 4 have 4 solutions (counting multiplicities ) :

Get the others , by equi - partitioning the unit circle :4 sol . Ima eat →

different 4th roots of I

→ divide it• io ditz'

the e.#, eTk

,k -- 0,143

circumferencefade ' Reinto 4 equal a ZIE Have fe Iet"ik_ Ilengths

e

⇒ The other "ioo¥s of 81 are given by 3. (roots of 1)

→ 3 e"

,35*2

,3e"

,3e""'

.

""""

Geometrically : Cauwriteanycaupl . E- late (polar decays- lap . )"

#E " 4*91 wherein* - -

have more root of divide anglethan i'root ", length byui.e .

,Sol

. of yU=Zkn a Z

#

%Re

→ This way , we obtain one of the roots

→ Then,w/ above

, get the others .

X 6=64 ; = 26 eittk

"

First"

root . x -- Le'¥=2e'¥Others :

ye .

et# ' k

k -0,1 . . . .

5

÷÷÷÷qtt¥Totataroots

HWIEx.lk: T : Am I fat AT , At R"

.

Oct. 23

AER"'

means

A- taba) → ITAI -- Efta bat '- lab 'd# can be any

-- HEE ÷H¥±⇒

umm

No matter whatwe plugin these are thegame -

So, every element of rangi looks like : (I ¥ )

what ITAI 's can beattained

.

(Cfr :allelements yet . there is xex with

TlxkyCouture page FAI =/Axl - - f-- collat ( see previous weeks) .

→Back to.

range # = fly ¥1 I x.yizauyttuiuubeofHW6e For a- full , V -- It

,have Luis -

- Uintah.

Hmu product properties :

1) For aimer'

,CHR : su

,view>HIM

,the:L:b

⇐aw,

Love ¥!) be given -= (4kt cuiwaltkkhtcuew, )

rule Tui= (qui claw > Vblilkiaearity

1) Symmetry he , U) = U, Kenya Yuu ku ,

= (ya > ✓Not positive

3)Positivity : thus -- human Is this positive, no matter what a.aaaa ? HELMETED

Because it is not positive, it is not an inner product .

Alternatively , we get sum > =o for it Main is not definite

Definite : Chiu >=ooulyfru=# .

↳in class

,are put this property under " positivity " .

X = Xp Wot. . . t xzwz ND p CZ)= Xotx , ZtXzZIxzZ3 Oct

.24

(4WD to, 2*7, sample site. 4 Eze(¥4 ) - Wo

I co Tre*ti÷÷÷÷k ⇒ ii.Haiti : -ii. x* rn *EI÷÷÷÷÷!÷÷,not normalized Because . ( w

'

w. ! hi ! 'w, )-t.mn;%%6iw4I÷÷

" at :c .t÷¥¥÷÷H÷÷¥÷÷÷÷¥÷:-

I*

4 X

Then plz ) = Cot C, 2- t Cz Et g z 's

Inverse DFT : Y ⇒ x ? Have x -- Fn X

""

K. , en !" two : wi. wins) -fi

,÷: ÷)

Ei