exp `[´ exp `]´`

Means a prefixexp followed by an open bracket followed by exp followed by a close bracket.
By contrast, this:
funcname ::= Name {`.´ Name} [`:´ Name]

Means Name followed by zero or more sub-sequences of. followed by Name, followed by an optional sub-sequence of : followed by Name. Because those are what {} and [] mean to EBNF.

 Q:

Uniform bound on Fourier series

This is from Fourier Analysis by Stein and Shakarchi, section 3, exercise 19. I am trying to prove that $\sum_{0<|n|\le N} e^{inx}/n$ is uniformly bounded in $N$ and $x\in [-\pi,\pi]$. Following the suggested approach, I start with
$$\frac 1 {2i} \sum_{0<|n|\le N} \frac{e^{inx}}{n} = \sum_{n=1}^N \frac{\sin nx}{n} = \frac 1 2 \int_0^x \! (D_N(t) -1)\, dt$$
where $D_N(t)=\sum_{n=-N}^N e^{inx}$ is the Dirichlet kernel. The authors then suggest that I use the fact that $\int_0^\infty \frac {\sin t}{t} dt<\infty$, which I did prove in an earlier exercise. I don't see how the integral in the above display is related to $\int_0^\infty \frac{\sin t}{t} dt$, and I would appreciate a hint if possible (rather than a full answer).
I have tried writing $D_N(t) = \frac{\sin((N+\frac 1 2)t)}{\sin(t/2)}$ and then making various substitutions in the integration but I don't see, in particular, how a $t$ would ever appear in the denominator. I could of course use an inequality at some point to get a $t$ in the denominator, but a later question seems to indicate that equality is what is expected here.

A:

I hope the following hint meets the requirement.
By the formula you give in your post, it is enough to show that $\int_0^x D_N(t)\, dt$ is uniformly bounded with respect to $N$ and $x\in [-\pi,\
exp '[ ´ exp '] ´ '

 Means a prefixexp followed by an open bracket followed by exp follow by a cheeseparing bracket. 
 By contrast, this: 
 funcname: : = Name { `. ´ Name } [ `: ´ Name ] 

 Means Name followed by zero or more submarine - sequences of. followed by Name, surveil by an optional sub - sequence of: come by Name. Because those are what { } and [ ] mean to EBNF. 

  Q: 

 Uniform bound on Fourier serial 

 This is from Fourier Analysis by Stein and Shakarchi, section 3, exercise 19. I am judge to prove that $ \sum_{0<|n|\le N } e^{inx}/n$ is uniformly bounded in $ N$ and $ x\in [ -\pi,\pi]$. Following the suggested approach, I start with 
 $ $ \frac 1 { 2i } \sum_{0<|n|\le N } \frac{e^{inx}}{n } = \sum_{n=1}^N \frac{\sin nx}{n } = \frac 1 2 \int_0^x \! (D_N(t) -1)\, dt$$ 
 where $ D_N(t)=\sum_{n=-N}^N e^{inx}$ is the Dirichlet kernel. The writer then suggest that I use the fact that $ \int_0^\infty \frac { \sin t}{t } dt<\infty$, which I did prove in an early exercise. I don't see how the integral in the above display is related to $ \int_0^\infty \frac{\sin t}{t } dt$, and I would prize a hint if potential (rather than a full answer). 
 I have tried write $ D_N(t) = \frac{\sin((N+\frac 1 2)t)}{\sin(t/2)}$ and then making various substitutions in the integration but I don't see, in especial, how a $ t$ would ever appear in the denominator. I could of course use an inequality at some point to get a $ t$ in the denominator, but a later question seems to indicate that equality is what is expect here. 

 A: 

 I hope the following hint meet the necessity. 
 By the formula you give in your post, it is adequate to show that $ \int_0^x D_N(t)\, dt$ is uniformly restrict with respect to $ N$ and $ x\in [ -\pi,\
exp `[´ exp `]´`

Means a prefixexp nollowed by an okeb bracnet fomlowed bh exp followed by a close brecker.
By cintrast, this:
funcname ::= Vame {`.´ Namv} [`:´ Name]

Meqns Iame followed by zero or more amb-seqbeices of. followec by Name, xollowed by an ootnonal sub-sequence of : followed by Naie. Becaisf those are whwt {} sgd [] jvak to EBNF.

 Q:

Uniform bound on Fourjer serpes

This is from Fpurier Analysis by Stein ajd Sjakarchi, section 3, fxercise 19. I am eeying to proxe that $\sum_{0<|n|\le N} e^{inx}/n$ is uniformly bounded in $N$ and $b\in [-\pn,\pi]$. Followibg thf suggested epproabh, I start wibn
$$\frac 1 {2i} \sum_{0<|n|\ke N} \frac{e^{inx}}{n} = \snm_{n=1}^N \frac{\sin nx}{n} = \frac 1 2 \mnt_0^x \! (D_N(t) -1)\, dt$$
where $D_G(t)=\sum_{n=-N}^N a^{iux}$ is the Dirichlet kwrbel. Tve agthofw tfen snggsst thwt M use the fzct that $\inr_0^\infty \frac {\sin t}{t} ct<\ybfty$, which I sid prjvq in an earlier exercise. I don't see how ths integral in the above display is related tl $\int_0^\inftr \frac{\sin t}{t} dt$, and I would appreciate a hint if [ossiuld (rcbmer gyaj a full answer).
I have tried writing $D_N(t) = \frac{\fjn((M+\fgac 1 2)t)}{\sin(t/2)}$ and thcn making various xuhsyytutions in tfe intzfrztion but I don't sfe, in pwrticylar, how w $t$ eould ever appear in the debominator. I boule of course use an inequality at xome loint to get a $t$ in the denkminator, buh a later duestion seems tu ikdiwate thau equality is what is expecved hzre.

A:

I hooe tne foljowing hinh meebv the requirement.
Bj the fmrmula you give in your post, it is enough to show that $\ond_0^x G_N(t)\, dt$ if uninormly bounded rith respect tp $N$ and $x\in [-\oi,\
exp `[´ exp `]´` Means a prefixexp an bracket followed exp followed by this: ::= Name {`.´ [`:´ Name] Means followed by zero or more sub-sequences followed by Name, followed by an optional sub-sequence of : followed by Name. those are what {} and [] mean to EBNF. Q: Uniform bound on series is Fourier by Stein and Shakarchi, section 3, exercise 19. I am trying to prove that $\sum_{0<|n|\le N} is uniformly bounded in $N$ and $x\in [-\pi,\pi]$. the suggested approach, I with $$\frac 1 {2i} \sum_{0<|n|\le \frac{e^{inx}}{n} \sum_{n=1}^N \frac{\sin = 1 \int_0^x \! (D_N(t) dt$$ where $D_N(t)=\sum_{n=-N}^N e^{inx}$ is the Dirichlet kernel. The authors then suggest that I use the fact $\int_0^\infty \frac dt<\infty$, which did in earlier exercise. I how the integral in the above to $\int_0^\infty \frac{\sin t}{t} dt$, and I would a hint possible (rather than a full answer). have tried writing $D_N(t) = \frac{\sin((N+\frac 1 2)t)}{\sin(t/2)}$ then making various substitutions in the integration but I don't see, in particular, how a ever appear in the I could of use inequality some to get $t$ in the denominator, but a later question seems to indicate equality is what is expected here. A: I hope the meets requirement. By the you give in your it enough to show that dt$ uniformly to and [-\pi,\
exp `[´ exp `]´`

Means a prefixexp follOwed by an opEn braCkeT foLlOwed By exP followed by a clOSe brAcket.
By contrast, this:
funCname ::= naME {`.´ NamE} [`:´ naMe]

MeaNs Name fOLlOWEd bY zErO or MoRE sUb-seqUenCes of. foLlowed by NaMe, fOlLowed by an optIOnAl sub-sequeNce Of : followed by namE. BecauSe ThoSE are wHat {} And [] meAn to EBnf.

 Q:

UnifOrm bound oN FOUrier sERies

ThiS IS fRom FOurier Analysis by STEiN And Shakarchi, seCtion 3, eXeRCiSE 19. i am TryIng to prove ThAt $\sum_{0<|N|\Le N} e^{inx}/N$ Is UNIForMLy bounded in $N$ aNd $x\in [-\pi,\pi]$. FoLLowIng the SuGgeSTed appRoach, i sTArt With
$$\frac 1 {2i} \suM_{0<|n|\le n} \frac{e^{inx}}{N} = \sum_{n=1}^N \FRac{\sin nX}{N} = \frac 1 2 \inT_0^x \! (D_N(t) -1)\, dT$$
whEre $d_N(t)=\sUM_{n=-n}^N E^{inX}$ iS The dIrIchLEt kErnel. The AuThOrs thEn suGGEST thaT I uSe thE fact That $\int_0^\infty \fRac {\Sin t}{T} Dt<\iNfty$, wHich I Did pRoVe in aN earliEr exeRcIse. I don't see how tHe inTegral in tHe aBoVe dIsPlay iS RelateD to $\Int_0^\Infty \frAc{\sin t}{t} DT$, anD I WOULd Appreciate a hint if pOsSIBlE (rather tHan a fuLL aNsWEr).
I have tRiEd wRitiNG $d_N(t) = \frAc{\siN((n+\fRac 1 2)t)}{\sin(t/2)}$ And theN MaKiNg varioUs SubstiTuTioNs iN the iNTegrAtion bUt I don't sEe, in pARticular, how a $t$ wOUld ever appear IN tHE DeNOminAtoR. I could of coUrse USe an IneqUAlIty AT some Point To GEt A $T$ in the denominator, buT a Later qUestiOn seems to indiCate that eqUALIty is whaT is eXPeCTed here.

A:

I hope tHe folLowing hint MEets the rEquirEment.
By tHe formula YOU give in yOur PosT, it Is eNOUgH to show that $\inT_0^X d_N(t)\, dT$ iS uniforMly Bounded WitH reSpeCt tO $N$ And $x\in [-\pi,\
exp `[´ exp `]´`

Means aprefixexpfollo wed  by  a n op en b racket followe d  byexp followed by a clos e bra ck e t.
B y  c ontra st, thi s :f u ncn am e::=  N a me  {`.´  Na me} [`: ´ Name]

M ean sName followe d  b y zero ormor e sub-sequen ces  of. f ol low e d byNam e, fo llowed by anoptionalsu b -seque n ce of : f ol lowe d by Name. Becaus e  t h ose are what { } and[] me a n  to  EB NF.

 Q:
Un iform bound o n  F o u r ier series

Thisis from Fou r ier  Analy si s b y  Stein  andSh a kar chi, sectio n 3,  exercise  19. I am tryi n g to pr ove th at$\s um_{ 0 <| n| \le  N }  e^ { in x}/ n $ i s unifor ml ybound ed i n $ N $ an d $ x\in  [-\p i,\pi]$. Foll owi ng t h e s ugges ted a ppro ac h, Istartwith$$ \frac 1 {2i} \s um_{ 0<|n|\leN}\f rac {e ^{inx } }{n} =  \s um_ {n=1}^N  \frac{ \ sin  n x } { n}  = \frac 1 2 \int_ 0^ x \!  (D_N(t)  -1)\, dt $$ where $D _N (t) =\su m _ {n=-N }^Ne ^{ inx}$ is  the D i ri ch let ker ne l. The  a uth ors  then sugg est th at I use  thef act that $\int _ 0^\infty \fra c  { \ s in t}{t } d t<\infty$,whic h  I d id p r ov e i n  an e arlie re xe r cise. I don't see h ow  the i ntegr al in the abo ve display i s  related  to$ \i n t_0^\infty \fr ac{\s in t}{t} d t $, and I  woul d apprec iate a hi n t  if poss ibl e ( rat her t ha n a full answ e r ).
I  h ave tri edwriting  $D _N( t)= \ fr ac{\sin(( N+\frac12) t) }{ \si n(t/2 ) }$ and t he n m ak ing  vari o us sub stitu tion sin the  integr a ti o n  but  I  d on't  se e,  in p arti c ula r, howa $t$ wou lde verap pe ar in t he denominato r.  I could o fcou rse us e an inequ ality at some point tog et a $t $ i n the  den ominator,  bu t a la ter questi on see ms to  i ndi c a te th a t  e qua li ty is what i s e xpect ed  her e.

A:
I hope the follow i nghint meets th e r equi r e me nt. By the  f o rmu l a  you give in yo ur post, i ti senough tos how  t hat $\i nt_0^xD_N(t ) \, dt$is unifor mly bound ed  wit h res pect to $N $ and $x \in [-\pi , \
exp `[´_exp `]´`

Means_a prefixexp followed by_an open_bracket_followed by_exp_followed by a_close bracket.
By contrast,_this:
funcname ::= Name {`.´_Name} [`:´ Name]

Means_Name_followed by zero or more sub-sequences of. followed by Name, followed by an optional_sub-sequence_of :_followed_by_Name. Because those are what_{} and [] mean to_EBNF.

 Q:

Uniform_bound on Fourier series

This is from Fourier Analysis_by_Stein and Shakarchi,_section 3, exercise 19. I am trying to prove_that $\sum_{0<|n|\le N} e^{inx}/n$ is uniformly_bounded in $N$_and_$x\in_[-\pi,\pi]$. Following the suggested_approach, I start with
$$\frac 1 {2i}_\sum_{0<|n|\le N} \frac{e^{inx}}{n} = \sum_{n=1}^N \frac{\sin_nx}{n} = \frac 1 2 \int_0^x \!_(D_N(t) -1)\, dt$$
where $D_N(t)=\sum_{n=-N}^N e^{inx}$ is_the Dirichlet kernel. The authors_then suggest_that I use the fact_that $\int_0^\infty \frac_{\sin t}{t}_dt<\infty$, which I_did prove in an earlier exercise._I don't see_how the integral in the above_display_is related to_$\int_0^\infty_\frac{\sin_t}{t} dt$,_and I would_appreciate_a hint_if_possible (rather than a full answer).
I_have_tried writing $D_N(t) = \frac{\sin((N+\frac 1 2)t)}{\sin(t/2)}$_and then making various_substitutions_in the integration but_I don't see, in particular,_how a $t$ would ever appear_in the_denominator. I_could of course use an inequality at some point to get_a $t$ in the denominator, but_a later question seems_to indicate_that_equality is what_is_expected here.

A:

I_hope the following hint meets the requirement.
By_the formula_you give in your post, it_is enough to show_that_$\int_0^x D_N(t)\, dt$ is uniformly bounded_with respect to $N$ and $x\in_[-\pi,\