f Ebt Let f is Lip Then Insaf
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ThmI5 Let f be a 21T periodic function integrable on Ebt Suppose that f is Lip zcmtmuaes at X Then Insaf converges to fix as n to Pf later at the end of this section leg of application Recall fix X or T Ty ITT I 21T periodic extension Fourier series z EI HnY sin Cnx It is clear that fix is hip cts at any X C C Tbt affias 2 sin X f Xt C IT On the other hand SF is discts at X IT and we we have seen ft CIF 2n ftp.t seisnfHT7
Transcript of f Ebt Let f is Lip Then Insaf
ThmI5 Let f be a 21T periodic function integrableon Ebt
Suppose that f is Lip zcmtmuaes at X Then Insafconverges to fix as n to
Pf later at the end of this section
leg of application
Recall fix X or T Ty ITT
I 21Tperiodicextension
Fourier series zEI HnY sinCnxIt is clear that fix is hip cts at any XCCTbt
affias 2 sin X f Xt C IT
On the otherhand SF is discts at X IT
and wewe have seen ft CIF 2n ftp.tseisnfHT7
Thin let f be a 21T periodicfunction integrable onEBTSuppose that for Xo C E IT TI
i fcxots fjiyofcxs.fcxot ftfo.tk botharighthandlimit left handhint
di I Lso and 8 so such that
I fix fexots I E L x Xo Osx Xo so
Ifk fixed E Hxo x Oaxo X d
Then Snf flat tfCXo2
as n th
Pf Omitted foolSHE.ioYlXXoj4XoX l LXXo
Uxo 40 fix1XO
egofapplication f x with I
At xo T I is discontinuous
is ft fTx3 T1
ft't ftp.Icxt IT
Cii Fa Os x Xo EL ie Osx TLEEE
wehanIf fit IIf et f IT x 21TGET D
I x 21T C F IX T E LX TD with f 1
Similar for oxo Xa
Hence conditionsofThou1.6 are satisfied
Fourier series Snf ft5tf EITI0 0
Next we turn to uniform convergence and need
Def A function f defined on EaBJ is called tosatisfya Lipschitz condition if I L o such that
I fix fly I E L IXy J V X y Ela b
Notes LD L o is indeep of X y Gea ba kind of uniform hip condition
Z f satisfies a Lipendition f a hipctsat every point onTaub
of If f C C Eab3 Iffy fix f IS StadtIE M ly x I t X y ELaobJ
where M faffIf'tBut fix 1 1 satisfies a hip condition but not d
Thmit Leff be a 21T periodic function satisfying a
Lipschitz condition Then its Fourier series convergesuniformly to f itself Pf Omitted
egofqicatiaefzcxj xcn LITJI.atperiodicextension LINNIT T
satisfies a Lip condition check
4 GItosnx converges uniformly
to 2 on FIT e
Ex put x o and get It EI
Proofofthmtslet f be tip Is at a point to C ET ITT
Steps Snf cxo5 aotfzyakoskxotbp.snbio
Dnc f Hotz dZ
where sina.fnat.atzZz if O
2h 11It If 7 0
is called the Dirichlet teruel i
Pfi nf7cxo5aotpECakoskxo.tbpsinkko
fifty dy tf Istysaskydy askXo
fifty sinkydy sinkXo
f tEi kyuskxotsinkysiukxojffcysdy
I.FI t EI as key Xo fig dy
n
IT It Fiske fEtxddz G xDI 21Tperiodic
Since ask z sina.IE fazt o
Ex Calculate e not it eine usingltz c.f zk zb.tl I
Snf Xo sina.ttzZ flxotZ3dZ
SI Dae fexottidZstepe Properties of Dnb
c S Da dz t
Z Dn Z is even Cts 21Tperiodic on ET IT
Dn 34770 for k n Q n
Neff Disk Dodo 74 too SoolDn lolz to as n 71 0
854 2
If is Easy by integrating S tEIask dz
2 are easy exercise
