Tutorial I '

Selected

problems of Assignment I

Leon Li

12/9/2018

Contact Information

Name : Li Yan Lung ,Leon

Office : Academic Building I,

Room 505

( ¥kE#*F¥ - ¥ )

Email : ylli@math.cuhk.edu.hk

Office hour : Mon,

Tue,

Thurs 10:00a.m. -

1:00p.m.

Q1 ) ( Ex.

1 QI )

Let F- { finite Fourier series )

= {

*= aot ¥

,

Ian cos nxt basin nx ) }

P = { trigonometric polynomials }

= { plcosx , sin X ) =

?⇐oAik Cosi x sin

"x }

Show that I

=pomen

Sol : Recall Euler formula : e"

= cos x ti sin x

i . tnzo,

e" "

= cos nxt I sin nx

On the other hand, by binomial theorem

,

einx = ( cos × + i sin x )"

= FEW cosh- "

x lisinxsk

= Indios x. sin x ) t I rnlwsx, sin X )

,where In

,rnEP

.

.

.

Cos hx = In Coos x,

sin x )EP{

sin nx = rn Loos x , sin x )

EP:

.

tf

TEF

,

Tex) - aot Ee

,

@n Int burn )

EP,

Hence

FEP.

Conversely : Fn EIN,

cos"

x = (e

"

ye. "

Y

= In ( Eo (Ya ) e" " " "

e- " "

)( where Ck EIR )

= Tze G. ei" - " " ×

=EICK cos Cn - 2k ) x E F-

fnlx )

Similarly , sin"

ex , = (e

"- e

- *

yn2 i

= fEoc- is" (9) e'

'

" " "

e- " "

)

ftp.dd.%9cniiksiiiiesinin.zksxle#YI7sioted-gn#I

. Vp E P , p=

¥,

aikarix sink x

=

Eka ; " fix ) 9k Cx ) E F

( by product - to -

sum formula )

Hence P E F

Combining above, we have F- =P

.

Q2 ) I Ex.

1 Q2 )

Let f : I -

A , Ti ] → IR be 21T - periodic integrable ,

show that

Ii ) f is integrable over anyfinite closed interval

Cii ) f 2. JE IR closed interval of length 21T,

{ fix )dx =

§ this dx

Pf : Li ) f is integrable over I - a. TD

Hr ( f : 21T -

periodic )

§ is integrable over [ kn . 1) IT , Until IT ] ,th EI

*

§ is integrable over E- Kmt 1)IT , Kmt 1) IT ] ,that 7L

Then tf K EIR finite closed interval,

then existsN EIN see .

KE [ -

GN-11 )

,UNH ) ]

,

: .

f- is integrable over K .

Iii ) It suffices to show that { fix , dx = fcxsdx :

Write I = [ a,

at 2h ],

I a E IR.

Then 7- NE I sit . HIT E I.

:c

{ scxsdx = fat'

flxldx t Si!"

six , dx

= S

'

fly, dy t Santa's d× ( bondageof

, ,

variabley

-

- x

the integral

= SY!" fly

,dy= S faddy I by change of variable x=y

- Chiba )

.

'

. { faddy = Siatkddx =

§ tcxsdx

Q

3)( Ex

.1

,

Q 7)

Let f :[ .

q , Tt ] → 112 be ZI - periodic differentiable function

such that J'

: ET , I ] → IR is integrable

an = at [ha fix ) cos nx dx

let { bn = ÷, gjnga , s

,nn×d× be the Fourier wessicients

of $,

Showthat lanl,

lbnl → 0 as n → a without using

Riemann - Lebesgue lemma.

Pf : Note that [",

# cos hxdx

= [ six ) cos nx ]Ya - [j # th sin nx ) dx

= O + nbn = nb

: '

.

'

f'

is bounded,

HH.

E M,

I MEIR )

;. Ibnl = In | Ea fan wsnxdx / E tnfijmdx → 0

as h → A

Similarly for an :sin . nxdx = - nan

:.lant-LIEafcxssinnxdxlf.tnMdx → O

as h → as