MATH 172: CONVERGENCE OF THE FOURIER SERIES

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1 MATH 72: CONVEGENCE OF THE FOUIE SEIES ANDÁS VASY We now discuss convegence of the Fouie seies on compact intevas I. Convegence depends on the notion of convegence we use, such as (i L 2 : u j u in L 2 if u j u L 2 0 as j. (ii unifom, o C 0 : u j u unifomy if u j u C 0 sup x I u j (x u(x 0. (iii unifom with a deivatives, o C : u j u in C if fo a non-negative integes k, sup x I k u j (x k u(x 0. (iv pointwise: u j u pointwise if fo each x I, u j (x u(x, i.e. fo each x I, u j (x u(x 0. Note that pointwise convegence is too weak fo most puposes, so e.g. just because u j u pointwise, it does not foow that I u j(x dx u(x dx. This woud foow, howeve, if one assumes unifom convegence, o indeed L 2 (o L convegence, since u j (x dx u(x dx (u j (x u(x dx u j u, u j u L 2 L 2. I I I Note aso that unifom convegence impies L 2 convegence since ( /2 u j u L 2 u j u 2 dx I ( /2 ( /2 u j u 2 C dx dx u 0 j u C 0, I so if u j u unifomy, it aso conveges in L 2. Unifom convegence aso impies pointwise convegence diecty fom the definition. On the othe hand, convegence in C impies unifom convegence diecty fom the definition. On the faiue of convegence side: the unifom imit of a sequence of continuous functions is continuous, so in view of the continuity of the compex exponentias, sines and cosines, the vaious Fouie seies cannot convege unifomy uness the imit is continuous. On the othe hand, even if the imit is continuous, the convegence may not be unifom: undestanding conditions, unde which it is, is one of ou fist tasks. Thee ae two issues egading convegence: whethe the seies in question conveges at a (in whateve sense we ae inteested in, and second whethe it conveges to the desied imit, in this case the function whose Fouie seies we ae consideing. The fist pat is easie to answe: we have aeady seen that even the geneaized Fouie seies conveges in L 2 (but not necessaiy to the function!. Now conside unifom convegence. eca that the typica way one shows convegence of a seies is that to show that each tem in absoute vaue is M n, whee M n is a non-negative constant, such that n M n conveges. Simiay, one shows that a seies n u n(x conveges unifomy by showing that thee ae non-negative constants M n such that sup x I u n (x M n and such that n M n conveges. I

2 2 ANDÁS VASY The Weiestass M-test is the statement that unde these assumptions the seies n u n(x conveges unifomy. In paticua, if one shows that fo n 0 this boundedness hods with M n C/ n s whee C > 0 and s >, the Weiestass M-test shows that the seies conveges unifomy. (Since a finite numbe of tems do not affect convegence, one can aways ignoe a finite numbe of tems if it is convenient. So suppose that φ is a function on [, ] and its 2-peiodic extension, denoted by φ ext, is C k, k intege. The fu Fouie seies is ( C n e inπx/, C n φ(x e inπx/ dx. 2 n This gives us the bound (2 C n 2 φ(x e inπx/ dx 2 φ(x dx sup φ(x, x [,] so the coefficients ae bounded, but it is not cea if they have any decay, hence the unifom convegence of the Fouie seies is uncea. Now we integate by pats, putting additiona deivatives on φ. Thus, fo n 0, C n 2 ( inπ ( x φ(x e inπx/ dx, since the bounday tems vanish in view of the fact that φ ext is C k. Note that the ight hand side ooks ike the oigina expession fo C n, except that φ has been epaced by x φ, and that the facto inπ appeaed in font. Thus, epeating this agument k times, we deduce that C n ( k ( 2 inπ xφ(x k e inπx/ dx. In paticua, using the anaogue of the estimate (2 to estimate the intega, we deduce that ( k C n sup ( n π xφ(x k C ( k x [,] n k, C sup ( π xφ(x. k x [,] In paticua, fo k 2, we deduce by the Weiestass M-test that the Fouie seies conveges unifomy. It is possibe to impove this concusion as foows. Note that ( k C n C k,n, n 0, inπ whee C k,n is the Fouie coefficient of k xφ. Thus, as ong as k xφ is continuous, o indeed L 2, we have fom Besse s inequaity, with X n (x e inπx/, that 2 C k,n 2 n Z n Z C k,n 2 X n 2 k xφ 2 L 2. Now, in ode to appy the Weiestass M-test we need that n Z C n conveges. We now use the Cauchy-Schwaz inequaity fo sequences, with inne poduct on the sequences given by {a n }, {b n } 2 n Z a n b n.

3 { ( } k Beow by n π we mean the sequence whose n 0 enty is 0. We thus obtain using k > /2 that n Z, n 0 C n n Z, n 0 { ( } k { C k,n } 2 n π xφ k L 2 2 ( { k ( } k C k,n { C k,n }, n π n π n Z, n 0 2 ( n Z ( 2k n π /2 /2 C k,n 2, n Z, n 0 ( 2k n π so it indeed conveges. We thus deduce by the Weiestass M-test that the Fouie seies conveges unifomy if meey k. Now, the tem-by-tem m-times diffeentiated Fouie seies is n C n ( inπ m e inπx/, C n 2 so fo n 0 its coefficients satisfy the estimate ( m inπ ( C n n π C ( π k sup ( xφ(x. k x [,] φ(x e inπx/ dx, k m sup ( xφ(x k x [,] C n k m, We thus deduce that the tem by tem m times diffeentiated Fouie seies conveges unifomy fo m k 2, and thus (by the standad anaysis theoem the unifom imit of the Fouie seies is actuay k 2-times diffeentiabe, with deivative given by the tem-by-tem diffeentiated seies. Modifying this agument as above, using Besse s inequaity, we in fact get that the Fouie seies is actuay k -times diffeentiabe, with deivative given by the tem-by-tem diffeentiated seies. In the exteme case, when the 2-peiodic extension of φ is C, this tes us that in fact the Fouie seies conveges in C. One way of summaizing ou esuts thus fa is that the map T : φ C n e inπx/, C n φ(x e inπx/ dx, 2 is continuous as a map and aso as as we as n T : L 2 ([, ] L 2 ([, ], T : C pe([, ] C 0 pe([, ], T : C pe([, ] C pe([, ]. whee the subscipt pe states that the 2-peiodic extensions of these functions must have the stated eguaity popeties. Note the oss of deivatives as a map /2 3

4 4 ANDÁS VASY on C, o moe geneay C k. We woud ike to say that this is the identity map in each case; by the density of Cpe([, ] in these spaces it woud suffice to show this in the ast case. A moe systematic way of achieving the same concusion egading convegence is the foowing. The functions X n (x e inπx/ ae eigenfunctions of A i d dx with peiodic bounday conditions and with eigenvaue λ n nπ/. Now, in genea, suppose that X n ae othogona eigenfunctions of a symmetic opeato A : D V on an inne poduct space V with eigenvaue λ n (which ae thus ea, and suppose that φ, Aφ,..., A k φ D. Then fo n such that λ n 0, the geneaized Fouie coefficients C n φ, X n X n 2 satisfy C n λ φ, λ n X n n X n 2 epeating k-times, we deduce that Thus, λ φ, AX n n X n 2 C n λ k A k φ, X n n X n 2. C n λ n k A k φ L 2 X n L 2. In the specia case X n (x e inπx/, we get the estimate C n ( k k φ L 2 n π 2 C n k, C 2 λ Aφ, X n n X n 2. ( k k φ L 2, π which is amost the same estimate we had befoehand, with a sighty diffeent constant, and the nom of the deivative we use is weake, not the unifom (o sup nom, but the L 2 -nom. But now note that this agument woks even fo instance fo the cosine and sine Fouie seies, using that both sine and cosine ae bounded by in absoute vaue, i.e. we do not have to wite down these cases sepaatey. We have thus shown that unde appopiate assumptions, depending on the notion of convegence, the vaious kinds of Fouie seies a convege. Note that we may need stonge assumptions than the kind of convegence we woud ike, fo instance we needed to know something about deivatives of φ to concude unifom convegence. Howeve, we sti need to discuss what the Fouie seies convege to! Note that when the Fouie sine seies, B n sin (nπx/, B n 2 n 0 φ(x sin(nπx/ dx, conveges, as it does say in L 2 when φ L 2, o unifomy if φ is C 2 and satisfies the bounday conditions φ(0 0 φ(, then it conveges to an odd 2-peiodic function since each tem sin(nπx/ is such. Simiay, the Fouie cosine seies, when it conveges, conveges to an even 2-peiodic function. Moeove, note that

5 5 if a function Φ is odd on [, ], its fu Fouie seies, A 0 + A n cos (nπx/ + B n sin (nπx/, satisfies A 0 2 B n n Φ(x dx, A n Φ(x sin(nπx/ dx, n, Φ(x cos(nπx/ dx, A 0 0, A n 0, B n 2 Φ(x sin(nπx/ dx, 0 i.e. if Φ is the odd 2-peiodic extension of φ then B n B n, B n being the Fouie sine coefficient of φ on [0, ]. In view of an anaogous agument fo the Fouie cosine seies, convegence issues fo both the Fouie sine and cosine seies on [0, ] can be educed to those fo the fu Fouie seies on [, ], so we ony conside the atte. Moeove, the change of vaiabes of θ nπx peseves a the notions of convegence, so it suffices to conside the Fouie seies on [, π]. (The genea case woks diecty, by the same aguments, but we have to wite ess afte this escaing. One way of poving the convegence of the Fouie seies is the foowing modification of Hömande s poof of the Fouie invesion fomua. We wite C (S Cpe([, π], and s(z fo sequences {c n } n Z such that n k c n is bounded fo a k 0 (s stands fo Schwatz sequences, with notion of convegence that a sequence {{c j,n } n Z } j conveges to {c n} n Z in s(z as j if fo a k 0 intege, sup n k c j,n c n 0 n Z as j. Thus, the map is continuous, and it satisfies and Simiay, with we have whie F : C (S s(z, (Fφ n F dφ dθ infφ, φ(x e inθ dθ, (F(e iθ φ n φ(x e i(n θ dθ (Fφ n. F : s(z C (S, F {c n }(θ F {c n } n F {inc n } d dθ F {c n }, c n e inθ n n c n e inθ, c n e i(n+θ e iθ F {c n }.

6 6 ANDÁS VASY Thus, the map T F F : C (S C (S satisfies We have a emma: T d dθ d dθ T, T eiθ e iθ T. Lemma 0.. Suppose T : C (S C (S is inea, and commutes with e iθ and d dθ. Then T is a scaa mutipe of the identity map, i.e. thee exists c C such that T f cf fo a f C (S. Poof. Let ω. We show fist that if φ(ω 0 and φ C (S then (T φ(ω 0. Indeed, if we et φ (θ φ(θ/(e iθ e iω then by Tayo s theoem (o L Hopita s ue φ is C, and it is -peiodic as both the denominato and the numeato ae. Thus, φ (e iθ e iω φ, and hence T φ (e iθ e iω (T φ, whee we used that T is inea and commutes with mutipication by e iθ. Substituting in θ ω yieds (T φ(ω 0 indeed. Thus, fix ω, and et g C (S be the function g. Let c(ω (T g(ω; thus c C (S. We caim that fo f C (S, (T f(ω c(ωf(ω. Indeed, et φ(θ f(θ f(ωg(θ, so φ(ω f(ω f(ωg(ω 0. Thus, 0 (T φ(ω (T f(ω f(ω(t g(ω (T f(ω c(ωf(ω, poving ou caim. We have not used that T commutes with θ so fa. But c(ω( θ f(ω T ( θ f(ω θ (T f θω θ (c(θf(θ θω ( θ c(ωf(ω + c(ω( θ f(ω. Compaing the two sides, and taking f such that f neve vanishes, yieds ( θ c(ω 0 fo a ω. Thus, c is a constant, poving the emma. The constant c can be evauated by appying T to a singe function, e.g. to f, which yieds (Ff n 0 if n 0, (Ff 0, hence T f f, so T is indeed the identity map. Now, if φ L 2 ([, π], we take φ n whose -peiodic extensions ae C, which convege to φ in L 2. Since T φ n φ n, and T : L 2 ([, π] L 2 ([, π] is continuous, we deduce that T φ im n T φ n im n φ n φ, i.e. the Fouie seies conveges in L 2 to φ. If φ C (S, then we have seen that the Fouie seies of φ conveges unifomy to some function ψ, and as C (S L 2 (S, it conveges in L 2 to φ. But unifom convegence impies L 2 convegence, so in fact the Fouie seies conveges in L 2 to ψ, so ψ φ, and thus we concude that the Fouie seies of φ conveges unifomy to φ. As we coud educe a the Fouie seies we have consideed to the fu Fouie seies on [, π], we deduce the foowing esut:

7 7 Theoem 0.2. The foowing othogona sets of functions ae compete in the espective vecto spaces V, and thus the coesponding Fouie seies of any φ V conveges in L 2 to φ. (i In V L 2 ([0, ], with the standad inne poduct, ( nπx f n (x sin, n, n Z. (ii In V L 2 ([0, ], with the standad inne poduct, ( nπx f 0 (x, f n (x cos, n, n Z. (iii In V L 2 ([, ], with the standad inne poduct, f n (x e inπx/, n Z. (iv In V L 2 ([, ], with the standad inne poduct, ( nπx f 0 (x, f n (x cos g n (x sin ( nπx, n, n Z. Moeove, if the appopiate extension, namey (i odd, 2-peiodic, (ii even, 2-peiodic, (iii 2-peiodic, (iv 2-peiodic,, n, n Z, of φ is C, then the coesponding Fouie seies conveges to φ unifomy, and if the appopiate extension is C, then the convegence is in C. We discuss a moe taditiona poof, using the Diichet kene, in the Appendix. We aso connect these esuts to soving the Diichet pobem fo the Lapacian on the disk of adius, namey D B 2 {x 2 : x < }, u 0, x D, u D h, with h a given function on D. The genea sepaated soution was (3 u(, θ A 0 + n (A n cos(nθ + B n sin(nθ, n and using othogonaity, we have deduced the coefficients must be A 0 A n π n B n π n h(θ dθ, h(θ cos(nθ dθ, h(θ sin(nθ dθ.

8 8 ANDÁS VASY Now, if h is C, then we have seen that its Fouie seies conveges unifomy to h, i.e. u(, θ h(θ. Aso, if h is C 2, then A n, B n C/ n n 2 fo n. This suffices to concude that fo <, n (A n cos(nθ + B n sin(nθ 2C ( n. n 2 By the Weiestass M-test, the seies (3 conveges unifomy on the cosed disk, i.e. fo [0, ], and in addition the tem-by-tem m-times diffeentiated seies sti conveges unifomy in [0, ρ] fo any ρ < since the factos of n m this diffeentiation gives ae countebaanced by ( ρ n, and n nm α n conveges wheneve m and α (0,, so u is a C function of and θ fo [0,. A simpe modification of ou agument fo unifom convegence of the Fouie seies of C functions woud in fact extend this concusion to h meey C. As v n cos(nθ and v n sin(nθ sove v 0 fo (0,, u itsef satisfies u 0 fo (0, the oigin may a pioi be a pobem, since poa coodinates ae singua thee. That this is not the case foows fom n e inθ (e iθ n (x + iy n, so taking ea and imaginay pats shows that n cos(nθ and n sin(nθ ae simpy poynomias in D, and in paticua the tem-by-tem diffeentiated seies, diffeentiated with espect to x o y, actuay sti conveges in x 2 + y 2 <, unifomy in x 2 + y 2 ρ, ρ <, so in fact u is C at the oigin as we, and soves u 0 fo each summand does so. Thus, fo at east h C 2 (and as a simpe agument shows, h C in fact, we have soved the PDE: we found u C 2 (D C 0 (D soving ou pobem. It tuns out that one can wite the soution in a moe expicit fom, which enabes one to obtain a bette esut. Namey, fo ρ, whee ρ <, in view of the absoute convegence of the seies (with penty of oom eft, so one coud even diffeentiate abitaiy many times and have absoute and unifom convegence, u(, θ + h(ω dω ( n π ( n Now, denoting (4 h(ω + 2 K(, θ, ω + 2 h(ω (cos(nω cos(nθ + sin(nω sin(nθ dω ( n (cos(nω cos(nθ + sin(nω sin(nθ dω. n times the tem in paantheses by K, n0 ( n (cos(nω cos(nθ + sin(nω sin(nθ n ( n cos(n(θ ω n ( n (e in(θ ω + e in(θ ω n ( n ( n e in(θ ω + e in(θ ω. n

9 9 Both sums ae those of geometic seies, so the summation yieds ( K(, θ, ω ( + e i(θ ω e i(θ ω ( e i(θ ω ( ( e i(θ ω 2 ( e i(θ ω ( 2 ( cos(θ ω ( ( cos(θ ω + 2 ( 2 ( ( cos(θ ω + 2, whee the second equaity was binging the two tems to common denominato. We thus deduce that fo <, u(, θ 2 2 h(ω 2 2 cos(θ ω + 2 dω. ewiting in tems of Eucidean coodinates, if y ω, x θ, then the numeato is 2 x 2, whie the denominato is x y 2, so atogethe we have (5 u(x h(y 2 x 2 x y 2 ds(y. This is caed the Poisson fomua. In paticua, u(0 h(y ds(y, D D i.e. since the cicumfeence of D is, the vaue of u in the cente is the aveage vaue of h, which is caed the mean vaue popety of soutions of u 0. Now, the Poisson fomua woud give an independent (i.e. new, not eying on Lemma 0. way of poving that the Fouie seies of h sums to h. Namey, one can show diecty, using (5, that fo h C the Fouie seies, which we aeady know conveges unifomy to some function, conveges to h. Indeed, in view of the unifom convegence of the seies (3 on the cosed disk, at it conveges to im u(, θ im K P (, ψ h(ωk P (, θ ω dω, cos ψ + 2. We caim that this imit is h(θ. Using the seies definition of K, (4, and noting that so K(, θ, ω K P (, θ ω, K P (, ψ ( + 2 ( n cos(nψ one sees that by the othogonaity of cosines to the constant function, fo each <, n K P (ψ dψ, 2

10 0 ANDÁS VASY and K P is -peiodic in ψ, so u(, θ h(θ K P (, θ ω (h(ω h(θ dω K P (, ω (h(θ ω h(θ dω, whee the ast equaity invoved a change of vaiabes, but in view of the peiodicity, the domain of integation did not need to change (i.e. the intega ove [, π] is the same as that ove [θ π, θ + π]. Now, the advantage of the Poisson kene, K P, ove the Diichet kene discussed in the Appendix is that fo < one has K P 0, diecty fom its definition. Moeove, K P (, ψ 0 as unifomy outside [ δ, δ] fo any δ > 0. Thus, δ K P (, ω (h(θ ω h(θ dω δ K P (, ω (h(θ ω h(θ dω + K P (, ω (h(θ ω h(θ dω. [,π]\[ δ,δ] Now even if h is meey continuous, given any ɛ > 0, choosing δ sufficienty sma, h(θ ω h(θ < ɛ fo a ω with ω δ, and thus fo any (0,, the absoute vaue of the fist tem is δ ɛ K P (, ω dω ɛ K P (, ω dω ɛ. δ On the othe hand, by the unifom convegence of K P (, ω to 0 outside ( δ, δ, the second tem aso goes to 0 as, in paticua is < ɛ in absoute vaue if is sufficienty age. Thus, even if just h is continuous, u(, θ h(θ unifomy as. In paticua, if h is C, the patia sums of the Fouie seies convege to h unifomy, giving the pomised poof. We fomaize the popeties of K P fo a genea pupose esut. Poposition 0.3. Suppose that fo each N N, K N is a -peiodic measuabe function on, and (i K N L ([, π], and thee is M > 0 such that K N L ([,π] M fo a N. (ii [,π] K N, (iii Fo a δ > 0, K N χ [,π]\( δ,δ 0 in L as N. Then fo h C 0 pe( C 0 (S, h N (θ K N (θ ωh(ω dω (K N h(θ conveges to h unifomy. A famiy K N satisfying (i-(iii is caed a famiy of good kenes. Note that (ii impies the bound of (i (with M if K N 0 pointwise. Note aso that one can epace N N by anothe paamete set, such as N (0, ], and et N 0, etc., without any changes to the esut.

11 Poof. As befoe, (6 h N (θ h(θ δ δ K N (θ ω (h(ω h(θ dω K N (ω (h(θ ω h(θ dω K N (ω (h(θ ω h(θ dω + K N (ω (h(θ ω h(θ dω. [,π]\[ δ,δ] Now as h is continuous on [, ], it is unifomy continuous thee, so given any ɛ > 0, choosing δ sufficienty sma, h(θ ω h(θ < ɛ 2M fo a ω with ω δ. Thus fo any N, the absoute vaue of the fist tem of (6 is ɛ δ K N (ω dω ɛ 2M δ 2M K N L ([,π] ɛ 2. On the othe hand, the absoute vaue of the second tem is 2 K N L ([,π]\( δ,δ sup h, so by (iii, it goes to 0 as N, so in paticua is < ɛ/2 if N is sufficienty age, say N > N 0. Summing up, we see that N > N 0 impies that sup h h N < ɛ, poving the unifom convegence. Note that the standad definition of convegence of a seies, via patia sums, is a kind of a eguaization. If a tems of a seies u n ae bounded, a diffeent eguaization is to conside n u n fo <, when the seies conveges by the Weiestass M-test, and et. This is a bette behaved eguaization than the standad definition, and is caed Abe summabiity. Thus, fo h meey continuous, the Fouie seies is Abe summabe to h, but it need not convege unifomy to h (thee ae actua counteexampes. Appendix A. The Diichet kene We now conside a moe cassica poof of the convegence of the Fouie seies. So suppose now that φ is a function on [, π] whose -peiodic extension, Φ, is C. Then the Fouie seies of φ is given by (, with π. We wok out patia sums of the seies, S N (θ N n N C n e inθ, C n φ(ω e inω dω.

12 2 ANDÁS VASY Thus, S N (θ N n N π φ(ω e in(θ ω dω N n N e in(θ ω φ(ω dω K N (θ ωφ(ω dω, K N (ψ N n N e inψ. Now, the sum fo K N is a geometic seies, so using the summation fomua fo a geometic seies, (7 K N (ψ N n N e inψ ei(2n+ψ e inψ e iψ e i(n+/2ψ e i(n+/2ψ e iψ/2 e iψ/2 sin ((N + /2ψ. sin (ψ/2 Note that the ight hand side has the fom 0/0 at ψ 2kπ, k Z, but by L Hopita s ue it is continuous, with vaue K N (0 (2N +, and it is indeed C at these points. Moeove, K N has the additiona popety, which is cea fom the seies fomua since e inψ is othogona to in L 2 ([, π] fo n 0, K N (ψ dψ K N,, fo a N, and, as a the summands in the sum ae -peiodic, K N (ψ + K N (ψ. One cas K N the Diichet kene. Now, we need to anayze the diffeence S N (θ φ(θ K N (θ ωφ(ω dω φ(θ K N (θ ω(φ(ω φ(θ dω. K N (θ ω dω It is convenient to ewite the intega by a change of vaiabes. In view of the -peiodicity of the integand, we do not need to change the imits of the intega: S N (θ φ(θ K N (ψ(φ(θ ψ Φ(θ dψ. Note that this intega, much ike the one befoe, is vey simia to a convoution on ; the diffeence is that we meey integate ove an inteva of ength. Since the integand is -peiodic, this is best thought of as the convoution of K N and the function u θ (ψ Φ(ψ Φ(θ on the cice, S. Now, oughy, the K N ae appoximations to δ 0. Howeve, this is in a weake sense than e.g. χ N (x Nχ(x/N, whee χ 0 vanishes outside [, ], o indeed the Poisson kene K P above, is such, since K N (ψ dψ is not bounded by a fixed constant C (independent of N: it is the osciatoy natue of K N that makes the anaysis moe invoved.

13 3 Witing out K N (ψ as in (7, an othogonaity agument can be used to deduce the decay of S N (θ φ(θ as N. Namey, suppose that Φ is C. Then by Tayo s theoem, the function is continuous. Thus, Now, the functions v θ (ψ Φ(θ ψ Φ(θ sin(ψ/2 S N (θ φ(θ sin ((N + /2ψ v θ (ψ dψ. Z n (ψ sin ((n + /2ψ, n 0, n Z, ae othogona to each othe on [, π] (and indeed on [0, π], since they ae eigenfunctions of the symmetic opeato d2 dθ with Neumann bounday condition at 2 and (as cos((n + /2π 0 with distinct eigenvaues (n + /2 2. Moeove, Thus, Z n (ψ 2 dψ π. S N (θ φ(θ 2 v θ, Z N π Z N 2 Z N L 2. L 2 But we have shown that fo any othogona set {x n } n0 in an inne poduct space V, and any v V with geneaized Fouie coefficients c n, c n 2 x n 2 conveges, and c n v,xn x n 2. Thus, S N (θ φ(θ 2 4π c N 2 Z N 2 L 2, c N v θ, Z N Z N 2 L 2. But as N c N 2 Z N 2 L 2 conveges, its tems go to 0, i.e. c N 2 Z N 2 L 2 0 as N, so we deduce that S N (θ φ(θ 0 as N, poving the desied convegence. Since we aeady know that fo functions φ whose -peiodic extensions ae C the Fouie seies conveges unifomy (and thus pointwise to some function, and we just showed pointwise convegence of the seies to φ, we deduce that fo such φ the Fouie seies conveges unifomy to φ. Appendix B. Fejé s kene We now discuss anothe kene that is bette behaved than the Diichet kene. This is the Fejé kene, given by aveaging the Diichet kene K N ove N.

14 4 ANDÁS VASY Namey, et K M+ (ψ M + M K N (ψ N0 M N0 e i(n+/2ψ e i(n+/2ψ e iψ/2 e iψ/2 ( iψ/2 ei(m+ψ iψ/2 e i(m+ψ e (M + e iψ/2 e iψ/2 e iψ e e iψ ( e i(m+ψ e i(m+ψ (M + e iψ/2 e iψ/2 e iψ/2 eiψ/2 e iψ/2 e iψ/2 ( e i(m+ψ + e i(m+ψ (M + (e iψ/2 e iψ/2 2 (M + 4 sin 2 (2 2(cos(M + ψ (ψ/2 (M + sin 2 ((M + 2 ψ sin 2. (ψ/2 Since K M is an aveage of the K N, each of which has intega, K M (ψ dψ, just ike the Diichet kene. Howeve, it has the significant advantage thatk M 0 pointwise, and this K M (ψ dψ as we, just ike fo the Poisson kene. Futhemoe, fo any δ > 0, KM 0 unifomy on [, π] \ ( δ, δ, just as fo the Poisson kene, since 0 sin 2 ((M + 2 ψ, sin2 (ψ/2 is bounded away fom 0 thee, and M. Theefoe Poposition 0.3 gives, fo h Cpe( 0 C 0 (S, the unifom convegence of s M (θ K M (θ ωh(ω dω to h as M. Note that at the eve of the patia sums S N of the Fouie seies, s M M N0 S N, so what we have shown is that athough the patia sums do not M necessaiy convege unifomy fo a meey continuous h, the aveaged patia sums do. As fo Abe summation, this is a bette eguaization of the sum of an infinite seies than the standad definition; this is caed Cesào summabiity.