Lq Estimates for Rearrangements of Fourier Series Citation Greenhall, Charles August (1966) Lq Estimates for Rearrangements of Fourier Series. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/MK4A-AV50. https://resolver.caltech.edu/CaltechTHESIS:10052015-134053571 Abstract This work is concerned with estimating the upper envelopes S* of the absolute values of the partial sums of rearranged trigonometric sums. A.M. Garsia [Annals of Math. 79 (1964), 634-9] gave an estimate for the L 2 norms of the S*, averaged over all rearrangements of the original (finite) sum. This estimate enabled him to prove that the Fourier series of any function in L 2 can be rearranged so that it converges a.e. The main result of this thesis is a similar estimate of the L q norms of the S*, for all even integers q. This holds for finite linear combinations of functions which satisfy a condition which is a generalization of orthonormality in the L 2 case. This estimate for finite sums is extended to Fourier series of L q functions; it is shown that there are functions to which the Men’shov-Paley Theorem does not apply, but whose Fourier series can nevertheless be rearranged so that the S* of the rearranged series is in L q . Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Garsia, Adriano M. Thesis Committee: Unknown, Unknown Defense Date: 17 November 1965 Funders: Funding Agency Grant Number NSF UNSPECIFIED Record Number: CaltechTHESIS:10052015-134053571 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10052015-134053571 DOI: 10.7907/MK4A-AV50 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9199 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 05 Oct 2015 21:17 Last Modified: 28 Feb 2024 17:55 Thesis Files Preview PDF - Final Version See Usage Policy. 7MB Repository Staff Only: item control page

· L ESTIMATES FOR REARRANGEMENTS q OF FOURIER SERIES .,,. ......~- .. . ~ Thesis By ,.,.... Charles August Greenhall In Partia1 Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, Ca1ifornia 1966 (Submitted November 17, 1965) -ii- ACKNCMLEDGMENTS The author expresses his appreciation to his teacher, Pro£essor Adriano M. Garsia, who gave neither too little nor too mu.oh assistance. This work was carried out under a National Science Foundation Regular Graduate FellC?Wship. -iiiCharles August Greenhall Lq Estimates for Rearrangements of Fourier Series ABSTRACT This work is concerned. with estimating the upper envelopes s* of the absolute values of the.. partial sums of rearranged trigonomet.. . ric sums. A. M. Garsia [Annals of Math. ?9 (1964), 634-9] gave an estimate for the L norms of the s*, averaged over all rearrangements 2 of the original (finite) sum. This estimate enabled him to prove that the Fourier series of any :t'unction in 12 can be rearranged so that it converges a.e. The main result of this thesis is a similar estimate of the Lq norms of the s*, for all even integers q. This holds for finite linear combinations of functions which satisfy a condition which is a generalization of orthonormality in the ~ case. This estimate for finite sums is ext.ended to Fourier series of L q :functions; it is shown that there are functions to which the · Men'shov-Paley Theorem does not apply, but whose Fourier series can nevertheless be rearranged so that the S* of the rearranged series -iv- CONTENTS SECTION TITLE PAGE I Presentation of Results 1 II Proof of Th~orem 1 12 llI Discussion of Theorem 1 19 DI Proof of Theorem 2 25 v Proof of Theorem J 30 Appendix 1 4o Appendix 2 42 References 44 -1- I. Presentation of Resu1ts A. M. Garsia [2] has recently proved that eveey L2 Fourier series can be rearranged so that it converges almost everywhere. His proof rests on an estimate for the upper envelopes of the absolute va1ues of the parti~ .sums of a rearranged finite sum T= z;,a,,c/.>v are orthonormal functions on a. measure space, and a.,,. .. , an is a. permutation of { 1, ... , n} are rea1 munbers. I£ <r = (er, l ··· 1<r'n) write (I. I) S*(", 0"1T) = S"(k ,er) = ?rtax lsm~n His estimate is (l, 2) • Here A is a constant not depending on rt or the particular system (i;b,,) used. (We will use the notation i\ 1 A{li,s,. .. ) for positive constants, not necessarily the same at each occurrence, which depend only on the indicated parameters.) n! permutations. of any i in L~ The sum is over all Garsia. uses (1.2) to show that the Fourier series can be partitioned into f'ini te b1ocks of terms, each of which can be rearranged in such a way that the entire rearranged series converges a.a. Theorem 1 of this work is an estimate of the l\ norms of the -2- S11('1f. 1<t) :for all even C\, • It holds for a di:ff'erent class of <l>v than Garsia's result, and the right-hand side of (1.2) is replaced by a The property of the ¢>. that we v need is the following generalization of orthonormality: stronger norm of T than II T ll't • ( 1. 3) Definition. Let IL be · a )?ositive integer. of complex-valued !'unctions on a measure space. is a P-system for 1l i f for all indices ~ ( ~) be a system We will say ( ~) v,, ···, v11 >f IJ ... >flt we have (J.Lt) (Here and elsewhere in this work we assume that all integrals written down exist.) Any orthonormal system is a P-system for 1. The systems (1. 5") b) ¢y(x):= cos(v-l)X (v=l,i.,- ·· ) c) the Rademacher !'unctions Ryl"x):: s9n sin 2v'1Tx Cv=1,.t,···; osxs-J) are P-systems for all 't. either 0 or 2 'Ir • In example (a), the integrals (1.4) are Then write and use tha.t (a) is a P-system to see that (1.4) holds for (b). Ex.ample (c) is taken care 0£ by the :facts that the Rademacher £'unctions are independent random variables and that f R~ ::. 1 or 0 i f °' is even or odd, respectively. H~ever, (sin lfX : \/ =1,a 1···) is not a P-system for a:ny 1' > 1 since -3r2'7T . <1t-1 j . -,,)) = -"Slt1(.21t-l)dx 51>1 - 7r// , 11/f.• I (It - 2 "{ ' "•) 0 <'11' s.0 ('iin .2x) 21t-.2. ' sin x si., ('l//.-S)x. d.x = From now on we will refer to {1.5)(a) and (b), with this particular indexing, as the expo,n ential and cosine systems. Theorem 1. /\ T Let ¢. .·. ¢? - =L,, Iav I ~ /1 rt, be a P-system for .ft • >t I. where a. 1,. •., a" are complex numbers. 1 !~ J[s*cx,a-,T)]"'dx < A" JITl 1 rt for all n and 2 (1,6) 1t a 1 , ... ,an. Curiously enough, no other special property of the 'i'v is needed. L2. Th eorem 1 incl udes Garsia' s Corollary 1. result ( 1 • 2 ) since I IT I JTA1 .i -- J 2 • There is a a> {which may depend on T) such that (1 . 7) The spaces X2 'l- • Theorem 2 will be a statement analogous to (1.6) about trigonometric Fourier series of certain functions in l .(It ( O1 ,('7T) • For a given positive integer Jt consider are those 00 av et'vx f""' .[~ , the functions we will such that L~oc lavl eivx " in L.Zit • This class of func- is the Fourier series of a function f tions will be denoted by X.2-'L • In this definition we may replace -4(e 0 '"> by another complete orthonormal. system(~) , in which case we will write xitt (¢) • Observe first that xl = L2 ' :ror if f E x,, then its Fourier coef'ficients are square-summable, and vice versa. (Thus ItayJeitlX.-v ~ E: Ll't series.) (1.7) im~es. that "La" eivl(, Also II f /I :z. = II f" 11.'.l. is an L2 Fourier We will establish the :following facts about the classes X2tz. EQr. 'Z=2 1 3J· ·· Theorem. a) the class X21t is strictly contained .!!'.! L1" ~ (I. 8) :for all b) f c- X2 't. • For all '1. the functional N(f) = llf//2'£ is a norm on X4tt )~ X.zii is a Banach space under this norm. Part (b) is proved in Appendix 1~ Part (a) is a special case of a well-known result, which we now state. notation o:t Hardy and Littlewood [ 3] • First let us adopt a We will say that the series ~ 00 is a ma..iorant of the series .~ _ eo av e la11I ~A.., 1:11x. if for all v, and we will write [,a.11 ei)lx.-< I A.,,e<v". If these are Fourier series we use the same terminology :for the respective functions. general series In Appendix 1 we use the same notation for more L a </>,, , 11 -5- (1.9) -- )oo 1! LA A,, eiv" /'\J F ~ L.vi . (l. IO) We present a proof of (1.9) which gives a simple example of calculation with P-srstems. First we establish (1.9) for trigonometric polynomials: '» L av,··· a:vlL a"; .. al'tt v1... VIL f, ··'fn =-n fexp[i cv,+··+ v!t-f,- ···-,.un)J dx (1. I I) Here we ha.ve used only that the exponential system is a P-system for 't • Passing to the case of infinite series, we note that FE L.z.ti implies F € L.2, • Hence L la..., la.~ LA~ < oc, and so I all et\ix.,.,, i € Ll,. and er"'-(K 1 F ) Let 0"'1f.. ("' , f) be the Cesaro means o:f f relation 1-< F implies ~ (·, t) -< ~ (·, F) for all >1. • r.2.'TT' l"ll' S 0 lcr"'(x i)J.21ia'x < l .2'1 jO">i(x,F)I dx. 1 0 and F • The Thusby (1.11), (1.12) -6But O",,,(x,F) ~ F(x) strongly in L.z-i and <r>i.(;ic,f)~ f(~) a.e~, f"rom which Fatou•s Lemma. implies that f E l.z't To prove X.lll c l .ZJt. and (1.10) holds. ti.= 2, ~, • • · , strictly; for we choose with positive coef'f'icients (:for example, see (5.11). There are sign changes of these coefficients which put the resulting series in Li for all ~ < o0 [ Statement of' Theorem 2. L; ""1 a" ¢v , where ¢, , ~' ·· · (1.5)(a) or (b). Let 0 1, vol. I, p. 215]. We will consider rearrangements of is the exponential or cosine system = ?t0 < n, < n.t < · · · be a sequence o:f integers such that nk•I /rtk > O£ > 1 (~=I, iJ ... ) for some fixed Dl • We will make a probability space {J1.,J11) from the permutations er= ('t,,O'.z.J ···) o:f {1,2.,···} which permute the indices in each block {v: n~+I ~ V ~ h'+ 1 } (ft:: 0,1,.t, ···) among themselves. The permutations in each block will be independent random variables, with each permutation of the k+J.i block having probability J/(11k•l- nk) a O"E:fi ! . Given f "J L,~a.,,¢..., and set f l1. S*(x,<r,.;) = S* (x.,o-) = s~p J /l41"v q}v (x)}. ~ ff Xv, Theorem 2. ( lW • ~ s*(·,-,f) E L2tz(.O.x (0,2.'ITJ) - ~ . J f [ s*<",o-,f)] '1xdcr s A(«,ri) A o 2 . l j?/ 1~ o 21 • (t.13) -7Corollary 2. • The £unction belongs to (l.l'f) holds for all er in a set of positive measure in 11. Comparison with the Men'shov-Paley Theorem. This result [1, vol. II, p. 189-92] gives a condition which guarantees that S*E LI) without rearrangement. Let 4>= (¢j ,~, .. ) be a uniformly bounded orthonormal. system ( l~I ~ M (""I,~, ... )) • I f a= (a,, a.p ... ) . * "' is a sequence of complex numbers tending to zero let a 1 , a..z. ) ·· · denote the sequence obtained by rearranging tlie sequence Ia, IJ / a1 I,-· · (excluding the av which are zero) in nonascending order. If several lavl are equal then there is a corresponding .repetition in * the av • Then write oo B~ (a) = ( I a~ 't v ~ - ) 2 I /ct • y:q If'\? 2. we always have La~ vi-..z. ~ La:~ v~- • 2 If write Denote by HLi(</>)= HL't (for Hardy and Littl.ewood) the set o:f co f"' I,av'A for a: (4.) * such that B't(a)<oo. • (For ~~2, We will then write Bq,"Q- (f) it can easily be shown that if and hence [avcit s; (a) (PO. is an L,t. Fourier series. -8- We will suppose that ( <k,) is complete . in L2 • ) The Men'shov-Pal.ey Theorem is stated in (1.15)(b). Ir 'b~ 2. ~ HL~ £ Li (1.15)a) JI i /Ii for all ~ ~ -~ (i, M) 8~ (f) (1.16) f € H Li • rri>.t ~ fc:HL't ~ S*(·,:f)EL'b b) llS*lli 5 A(~>M) B~(f). and (1.17) (1.15)(a) is due to Hardy, Littlewood, and Paley [1, vol. II, p. 121], and is a weaker form of (1.15)(b). Applying (1.15)(a) to · f for & in HL2/¢) gives (1.18) . Then Let(~) be a uniformly bounded(ls6\ll~M) orthonormal system• HL.21t (~) ~ X1,,, (¢) and II f II.Zit ~ A(tt) M) B~ (f) for all (1.1q) f €- HL2't. However, the following theorem shows that more than (1.18) is true. Theorem 3. ~ ~., ~, · · · be a uniformly bounded orthonormal P-system :for an '1..2 ~ on a bounded interval.. contained in Xi1t ( <fo). Ih!m HL,i'°' (¢) is strictly Hence Theorem 2 applies to a wider class of trigonometric Fourier series in L.ztz (It ~ 2) than the Men'shov-Paley Theorem (1.-15)(b). -9Of course in Theorem 2 a rearrangement is required to put S1l in L ~ 2 , whereas this is ·not necessary in (1.15)(b). Actually, for the exponential system, the fact that Hli~ is strictly contained in X:a (It ~ l) is a consequence of the f'ollowing well-lmown theorem [ 1, vol. I, p. 215]: Let 0 < n 1 <n.l < 11.3 < · · · be a sequence of integers such that nhl /"1.t) « > / (I<= I, l, · ·) 1! • l (a~e1.n•x+ a_,e-tn*x) 00 • ' ' is the Fourier series of a ~ which belongs to Li for all 't < oo • co :l Thus for 't >I i f we choose positive numbers aft such that E_0oa~ < 00 * but B.2'l (a)= 00 HL.Zit ' the resulting 'a will belong to • . For ex.ample, x,H but not to a~ = ( k I/~ 109 k. )- (k 2.' 1). 1 However, we have discovered a proof of Theorem 3 which does not use the above~mentioned result on gap series and which works for more general orthonormal systems ¢> • Furthermore, it produces examples of functions in X >t ( ¢) - HL.zii (¢) whose coeff'icients are 2 a1l nonzero. · Interpolation of' Theorems 1 and 2 between adjacent values of !"(., • We would like to lmow whether ff .n. l~ [s"'('X)cr,f)]icixdO' < 0 holds for non-even~ • A(O(,~) l~ J If)~ (1.20) " (The notation of Theorem 2 is used here.) -10- We do not believe that this problem admits an easy solution. Even the apparently simpler Majorant Problem of Hardy and Littlewood [J] remains unsolved. They produced an analytic trigonometric polynomial T . such that .IJT/13 > llT/13 , and posed the problem of whether (1.8) can be extended to read (I. 2.1) for non-even 'b • Actually, (1.21) would follow from (1.20). For i f :f is a trigonometric polynomial then lf(x)j ~ s*c~, cr, f) for all Cr' (1.21) would hold for trigonometric polynomials. Applying it to the • Hence I\ Cesaro sums of ;f and ;f and using Fatou's Lemma. would give (1.21) in general. However, there is another kind of connection between (1.20) and I:C J"' l, av ¢v 5 (x,tr, f) = r a ¢.. (x) • m ~ (1.21). let J '~~ m We observe tha.t (1.20) holds i:f and only if l'TT Jfo IS ( >(x,o-,f A. '>It x., rr . iir )j%dxd<r ~ A(«1i)S /fli o (l.ll) for all positive integer-valued functions m (x,<r) (such tha.t 5ni.cx,1r> (")a-) is measurable on .ax [o,.z7TJ) , where AC0(1 ~> does not depend on the choice o:f m(x;a') • Let m.(x,cr) be a fixed :function. The mapping M such that M(f) is the function sm(~,o") (x,cr, f) on .Q.x [o,.2'1T] is a linear transformation from L 1 (H) (H=[o,nrJ) into -11- the measurable :functions on ..(l X H • M can be interpolated between n .for 'b between l'i M is X.Vt to L,ii? ( 12 x H) • . It is tempting to ask whether bounded .from such that Theorem 2 implies that M and Jc+ I • By this we mean that and itr+.2, there may be Banach spaces X2tz 2 Xi 2 X.t.1t+J. 7_ ~ ~ (.Qx H) 2 is bounded .from Xi to ~ Xz and ~ >t 2 Li1t+i (fl x H) and with norm depending on ~ • We wou1d expect that ~ wou1d be L% (Il.x H) , and that Xi wou1d be the set o.f J /\ which have an be equivalent to f We hope that the norm on Xi wou1d in Li( H) • /I f II i • Observe that the Majorant Problem can be looked at in the same way; here we consider the identity trans.formation J' as a bounded linear trans.format.ion .from ask whether J X21 to L2ti (H) with norm 1, and we is bounded .from the proposed space x~ to Li (H) (~ti< i < .ZIC+:Z) with norm!£ 1 /'. • The example of /IT[(3 > llT[/3 shows that with this point o.f view we do not expect that the norm .t\ II f Iii /\ IJ f Iii may not The remaining sections may be read in any order. They consist on X<& will be precisely • In .faot, even be a norm. o.f proofs and discussions of Theorems 1, 2, and 3 of this section. -12- II. froof of Theorem 1 The initial step in this proof is essentially the same as for the case 't = 1 , treated by Garsia. The basic tool is the Bohnenblust-Spitzer Theorem [7] which we state for convenience. Let x, ).. .) "~ be real numbers. of { I 1 • • · , n } er:: (0"1 ,-· ·,<T'n) For a permutation let S(<1) = ma.X /sm~n • I£ O' is the product of the disjoint cycles Cl ) ···, Cp (2.1) There is a 1-1 mapping V of the set of alln! permutations write onto itself such that (.Z •.i) R(V(a-))= 5(cr) for all (]' • Thus we can replace sums over all o' involving S by corresponding sums involving R • Returning to Theorem 1 d.efj.ne the £our :f'unctions 5-+ (x,cr) R t 5-(x,c-) I ( R.e L a~ ¢crv(x.) )T-m Im v=t • -13- Then 5'1t(x ,cr) ~ 5+(x,cr) + 5-(x 1(]") + 5+ (x,(f) + 5 (x,<T'), 1 R R 1 [Slt(x,a'JJi 5 'i~- 1 (s:~ + s;i + s;i + s;i) Ci= .h )} ....!. lCs*(xcr)]Z s '-f'b-' L[.s+i.,..5-i+s+~+s-iJ ' '>t! oR. R. I I • ')t! rt We use (2.1) to rewrite the sum over cf order of summation a little later let cycles C= ( v 1 , .. •>Vt ) • To enable us to reverse Le denote . a sum over fil of indices Vt' ( / ~ V.£ S n.) Xe (a-) , and define 1 if' C is a cycle of er - - 0 otherwise. With these conventions 1 _.!.._ '>!. ! l [ S'*(x,cr)] i ir ~ ~' E { [L Xc(cr) (1?e1'av¢tCx))+Ji+ [Z:Xe(a-)(~eI>~veft(x)rJi 11! <t c vfC e vEc + [ z Xc(cr) (lm L tly¢:,(x)) t] i + [ L Xe (v) (I~ r "v ~(x>)-J ~ } c l1EC . c VfC ~ !Li I ( E Xc(i!) 'h.! '1 C !L lZv~(x) I)'b. vEC Integrating, expanding the qth power, and exchanging orders of summation, we have -14- We now classify the nonzero terms in the right hand side of (2.3). r Xe (rr) ' ' .Xe't (rr/ In order that (C 1 >•· •> Cq,) a' I be nonzero f'or a given . , any two Ct·, Ci must be identical as cycles or have disjoint sets . of' indices, otherwise the C,, .. -, Ci Each such ( C" .. '> Ci ) determines belong to the same permutation. a partition P of {1,. .. ,iJ C.;= C~ and j d i t -i .. into p disjoint classes such that are in the same class, CinCj = ¢ are in dif':f'erent classes. .let s~ indices of the cycle f~ - (15 k ~ p) it i and be the set of' Ci where i. is in the ~+k · class of P. 5fc be the number of indices in _I couldn't all. Let • Write I J [s*(x <T) Ji dx ~ 1i rp I ~ I "D (2. t/) ) lt.) If runs over all. ( C, > • • · , C1 ) which determine the partiti.on E S,, .. '> Sp the disjoint sets and • For given ( S, ,. .. , S,,) the number ·of (C,,···> Ci) which can be chosen is £1 ! ···1p ! / (11• • • f. p ) , since there are f !/i cycles corresponding to a given set • . The number lrr Xe (<r>- ·· Xe (er) , I \ of . is just ( n- J,- ··· -,lr) l since we are p free to permute {1, ...,x.}- U~=I The integrals in (2.5) er containing C1 > • .. >Cq, s, depend only on P and S1, •• ·, SP • . Theref'ore · -15- By H8lder's i~equa11ty for i !'unctions, the last integral is not more than The concavity of the'{,+" root then gives I, (fr JIL 4.s0, Ii SI ... 5p But L 1 v'C; t=t ' the number of (Sp·.. ) s,) is equal to (b ... LP ~I> 1 ti IS'k I= if< (-K.:: I) ...) r) such that :-i-. ·-tp) > and SO I I Ip~ lL...,(. (f,· .. fp) I p Now write •1 ( n (. ... L n-1-·.. -t p ) ,, I f j /[avil i)1 (l.8) n s ... sr • )_ ..!..1 ( [ i i=/ I VEC,,. , -16- where ~ (N,,- .. >N"',M'J ... >M1t) is the number of (S,, ... 7 Sp) such that /S~I =i* (-A.=l, ..·,rJ, and in the cl')( ... )( c't N1, ... , NIL> M1 >'·'> MIL determined by the partition p are al1 and by (s,, .. -,5p). The crucial step in the proof comes now - we use the fact that is a P-system for It sTT ~. • This gives us that It. j: I ~ (X) ¢M.(X)dv- a z OJ this integral being a product of integrals of form (1.4). L fr JIZc.a, 4t I~ 51 ... s,, t= I Vf t Hence -17- (2..JO) where~ is maximized over all 111 .. . > lp that fixed. (N 1 ,. .. ,N1t>M 1>"·· 1 M,J This maximum is easy to obtain. (i= I,-··, 1) • Vii) ... , Vtti > fti >'">f!Z.i E Ci For each ~ ( f<. = I ) .. > p) choose an i and let 11.k be one of the Vti> '" 'fni. ='Yl'k if i { NI) ... ) Nft) MI) .../M/I, } c- is, when Np-= (vt,, ... ,v1'b), M.e= (f1.,, ... ,fi~) (l=l,-··,n)), written out (letting where Vi The requirement all belong to C1x · ··x Ci Np···, Nit, Ml> ... , MJt with c, • is in the f<. th. Co1' implies X"·X in the k Ht. class o:f P Define N class. Then =(v1,. .. ')vi), Ne c,x ... x CaII • Hence s(N,, ..·,M1t) ~ s(N,···,N) , which is simply the number of (5 0 · .. ,Si) . such that n.~ESkl IS~l=fk C'-= ,, ..·,p) disjoint. , and the sk are This number is 11.-P ) 1 . ( i - I · ·· .R. - I n- £ - .. ·I ) 1p) I f l ::::: max t; · . Putting this in (2.10) and referring to (2.8) gives Ip~ [n(n-1) .. ·(n-p+I)] - ll'fi l (.f, ... fp)°& J/T}i "" ..!.. - I l,· ...tp J (i. JI) -18- where the sum is over all (.f p .. '> f p) and such that ,R{: ~ I ( ~ = I,-··> p) f 1+···+£P ~ n . To complete the proof of (1.6), the crude estimate is good enough. (It turns out that J (f I.'· l p ) 'f. df. ... d f p - [r(1/i)JP .!.-1 r (I+ pli) Rf<~ o) L.J* ~ n r-.r ( n pii ~ n 11'b ) P tt ~ 'b ~ oo • ) Now since n ~ p and the left side is a decreasing function of -n. Finally Ip 5. q,f (pP/p!)vi ~ ~1+/ (q,!f tli since p~ q, is proved. f /Tli f 1f1i • This, together with (2.4), gives (1.6), and Theorem 1 -19- III. Discussion of Theorem 1. '11. In this section we will write T.::. L v =1 ct v if>v where the ¢.,v .will be specified as needed. denote the expression as before, Given the (l/?t! )_f _~ S*(:x,0"1 T)]'i>dx A. "1'V we will J't (T), by Theorem 1 says that (3. 1) if ( ~) is a P-system for 1t • can replace f IT candidate is J IT/i1t, l..i-t We would like to know whether we by a "smaller" quantity. The obvious but we do not know whether holds or not for trigonometric polynomials if ~ >2 . Theorem 1 would be "best possible" and (3.2) could not hold if we could prove that (3. 3) but we will see later that (3.3) is false for ~ = l't. >2 if the cty are real it is easy to see that (3.4) For some a' (depending on T .!!:E 't~ I • However ), (3.S) -20- We merely write T= T +- T - > rt= Ia:<fi , T- =L, t:i~ ~ l/T+llcr, ~ ll T- llo,, • ~ 2115*(·,0',T)il't , where O' is any permutation which puts the terms of -r+ first. Thus , where • We may suppose l/T/I~ ~ /IT+lli + ur-11., 5 2 llT+I/" (There are ·at least (-x./l) l of these O" • ) In the complex case, a basically similar argument (see Appendix 2) gives 11 s*c~, <1, T) 112 ~ Alt 11T-112 ( A. ) 'tV for some <:r , provided that ti 't is a P-system for "L • Returning to the problem of whether (J.1) is best possible in some sense, we can obtain some definite results by comparing to expressions sequence Ni (I Gt I ) which depend only on the absolute value lo.I= (ra. 1 1, [a.~I, · · ·) S*Cx ,<7', T) ~ I T(x)j J't (T) • The trivial observation for all er gives (3 . b) which in turn yields (J.7) Theorem 1 is the best result of the form. J't (T) :5 N~ (la.I) within a factor (3. g) Ai . For i f (J.7) holds then Ni(l~I) ~ J\(T) ~ JITI'~» is the smallest N~ (/a.I) i.e., such that (J.8) can hold. -21- The observation (J.6) easily gives us a result of form J~ (T) ~ Ni (lo.I) in the opposite direction~ (3.9) ¢v are orthonormal on Assume that the h< 00 . _, _ so that for q. ~ .2. an interval of length fi.l/t-1/q, JITJ/i ~ /ITll.z = llT/1 = ( Z:~lavl~ }' we have 12 • 1 Then (3.6) gives (3. 10) (J.11) ~ °b =2Jt • If ( ~) is a uniformly bounded ( l ¢ti S M) orthonormal system on an interval of length h then (3.10) is the best ineguali ty of form ( 3. 9), within a factor A (ft> M, 'k) • To prove this it will be sufficient to show (J.12) For any positive integer fl, and nonnegative n there is a T= I, I ~..,<Pv !E:.!:h la. 11 1= (,.11 (v=l1··,n) Ji't (T) ~ An J ( i;-')\. ~v ¢y I~)n,. 1 J . -6; >... , 6->t. such that ( ~. f 3) 'II - ' · For then under the assumptions of (J.11) we have J,z"(T) 5 An~M "(L:la."1 2 In (J.12) the ck,y 1 / for this are arbitrary. T. The method of proof will be to average (3.1) over sign changes of coefficients. real functions and coefficients let In the case of R. 1(w), ···, R~ (w) (0 ~ w -S I) be -22- the first n Rademacher f'unctions. In the complex case let be independent random variables each of which takes on the values exp [ t 2'TT k / (IZ + !) equal probability. The latter Ry J (f< = 0, 1) ··/l- ) with have been contrived so that In the real case we have =1 °' even (v=I ··· n) - O ) °' odd ) These properties ensure that the functions ~(w,x.) £orm a P-system for It only if the </> v '>'i.. (/' • (3. /'i R.) =Rv(w)~(x) (The complex Rv are needed are not themselves a P-system.) We can thus apply =tT(w,x) • This gives Theorem1 to Tl(w,x) = ~ I on { (w,x)} . '>\. A. L,-G.v'l.fv(w,x) . JdCA1Jdx [5'\w,x;cr; 1J)J S An fdw Jdx IVC'4>,x.)J ~ (3.15) 2 2 't In the real case, [1] chap. V, theorem 8.4 gives (3.1 ') . and the same is true in the complex case. ( v =I ... ) n '. , x fixed) then For i f cv = tv cft ('X.) -23- JIL Cv R11 ('V) I.Zif. dw 'V: I o.: 1 _A «r\. - A C C n •· • C C t""n I I h. 1't where oti2 O, ~i~O (i==l, .. ·,'>t.), Oi:'. 1 +···+«:" =f1+···+(3'H. By (3.14c) ="· this expression becomes L ( .~. «l'Ly·I c,1 "°'• ··· IC'tliof. !5. All (L 1Cvl2.yi ot, V= I ) where From (3.15) and (3.16) we obtain · Hence (J.13) holds with T(x} = U(w,") for appropriate w • A consequence of (J.11) is that (3.3) cannot hold for the trigonometric system. kernel.) For ii" i > 1., as co n~ there is no upper bound on • In fact it is easy to show that the last statement holds -24for · ury complete orthonormal P-system for n. · ( i =)Ji > 2) on a bounded interval~ The behavior of J211.. (T) under sign changes of coefficients as revealed by (J.1), (J.6), (J.10), and (J.12) gives us reason to believe that (J.2) may be true for systems satisfying the requirements of (J.11), say. "' fixed and vary the signs For if we keep T of the coefficients of T, the quantities [J.M. (T)]l/ltt A and II T 11 1't both run between the extremes I/ T II24 constants not depending on n ) , and furthermore both quantities and l/T /11 (within reach these two extremes (within constants) for the same sign changes. -25- rv. Proo£ 0£ Theorem 2. Given (n~) and er' , set '11. 50 = 0 > 511. = L av <;f>v ) \I: I s""" (x) =sU.p I Sn (x)j fc l ) * . E,fc.(x,<r) =max / 5 (k 1 ct)- 5 (x (J")I hl<,,. s nk-rl .,, l'lk ' ) (~= 01 t) .. n f<.T-/ A oft == L .y Ia) <f?, • v:nfc.+I Notice that 5)1k (x,<r) = Sn f<(x.) the blocks Cnk) '>1.k+-IJ • for all ft , X, and rr since a' preserves Hence for n~ +I $ 'Vt ~ f'tk+ I .we write S (x,cr) = S'k (x> + ( S (" cr)- SI\ (x 0-)) 7 )1. tc 1 1' 1 "' Hence (4.1) -26- * 6~ (x, er) Theorem 1 applied to the can be stated as 2'11" AI? j ISfc l.i • 4 0 Therefore ( tf. 2.) The theory of .Littlewood and Paley [1, chap. XV] on Fourier series of l p functions for p> I yields Jo s:ft ~ A(rx,'l) s/11' ~rr l'Tl' L Jo /Skl:Z'l. s A(tt,tt) so oo 2 '\ .nr )fl.lit ,i'IT' 'k.::o o which with (4.2) proves (1.13). A slight1y weaker form of Theorem 2 in which the sequence (nk) A may depend on f can be proved by using the following special case of M. Riesz• theorem on partial sums of Fourier series of functions in LP (p.>1) (4.3) Let instead of the deep Littlewood-Paley theory. f E xlJf. • . The partial sums Sn. and I\ S 'l1. of the I\ Fourier series of' J and j satisfy /\ I\ a) II s')\ fl i>t ~ II S11 l12tt ~ II f ll.z1t b) "' 't ~ o a.s n-+ oo. II t- sn 111:''1. !S II "5 - S-n.11 2 I\ Part (a) follows f'rom (1.10) and the. f'act that 5')\-< S'h -< :f • -27. " I\ /"'-.. For part (b) we note similarly that f - Sn= j - Sn I'\ I\ onl.y prove that Sn-+ f M. Riesz' theorem. i\ L i.1t strongly in Given f > 0 5n(T) and 5'>'\.( f - T) A o:f T and f - T We proceed as in • let T(x) , where N is chosen so that o'N (x, f) ; thus we need be the Cesko mean I\ I' ll f - o'N(•Jf )Jl~'t < f, • Letting denote _the. partial sums o:f the Fourier series , we have I\ /\ S = S (T) + 5 (5-T) . ')t /\ A Sn. - i ")t ')1, A ) I\ = Sn (T) - f + Sn (f ~ T) I\ /\ T- f + 5)1. (i-T) i:f n~ N • But " ~-T has nonnegative coe:ffioients. By (a), We now prove our weak form of Theorem 2. write Si'I. (x, a-) = f (x) + (S>\ (x.) - f(x)) + ( S,_Jx,O') - S>i (x,cr)). ~ k -28- This decomposition leads to Integrating over .Cl and using Theorem 1 gives /\.. and the partial sums ,,... Fourier series of f S'1. of the I' tend to -f in increase so fast that Li 't • Hence we can make -29- say. (We may assume that :f is not a.e. zero; al1 we are saying is that this . sum may be made arbitrarily small.) "° :nr Then also 27r )Li Jrr 1s - f 1"'t ~ J 1f'1 2 't ~=I ,.... 0 )ii it o /\. majorizes sit - f since :f - SA " . f.. with A(ct,tt.) rep1aced by Alt • ) A and 6k • Hence (1.13) follows -JO- V. Proof of Theorem 3. g"" L 1 a11 sbv 00 We are to show that there exists a function with positive . coe:f:f1o1ents av such that * B.2rz. (a.)= oo • JI~ i < oo but 21t The pr~o~ . is carried out in two st.ages. First . we obtain an estimate on the L 2't. norm of finite sums with nonnegative coefficients, averaged over al1 rearrangements of the coefficients. Next, such sums will be pieced together to form the function ~ • The arguments are due jointly to the author and Professor Garsia. Rearrangements of coefficients in finite sums. with 6v ~ 0 (v = I) ·,·Jn) Given a sum we will establish an upper bound on the expression I = n'! ~ where It. is a positive integer. tations a' = (0" 1 , • • ·, <T'n) 2tt ')1, JI~1 tcrv 'k, } , (S-.1) As usual, ~<7' sums over all permu- of { I) · · ·>n} • Expanding the 2ti th power gives (5~ 2) We classify the (v,, .. ·7 V11,;fi;"·Jµ1r,) r = LL ~(«,f3) L ti ~ o<J(3 N • . Jcf:/·' ¢?_ (31 . . . ¢?. ¢... f3s «s )'I.I '1.1 '11.5 n~ ) (s. 3) -31- 1 ~5 ;5 m1n(2-'l1 n)) C(: (ot,,· ..;.xs), (3= ((3 1 ,- ··y/3 5 ) 1 °'.1 -> 0J 'A. > 0 y, = 0(. +(3. (1. =I . . . ~) I < '\/ < v < ... < '\/ /"'I. ) t t ) J ) - , , - ,i - 1'3 ) where· 'I. 0<'. 1 +··· f' « 5 = (3 1 + ... + /3s = If., N = (n 1 ,. • · J l1.s) where the ni are distinct, and ~(«J~) is a positive combinatoria1 factor bounded above by an An. • The sum over rr in (5.3) does not depend on the exact identity (S. l.f) M = (>'1 1, ...J rn. 5 ) , the mi where the cf> v form a P-system for lZ distinct. , t;_? 0 , and we will remove the restriction that M andN consist of distinct ind.ices. terms to the right side of (5.4). Now since This adds only nonnegative Thus I 11. Y'. )I. Y. I5 L L {c(ot,f) {n-,s). ~ e,.v' ... [ f,.lls 5 ot, f ')t . v=I v ::.1 • sl-K- 1/'1 ¢ (3, ••·• t cj:/"s ¢._ /1s v=1 v V v=1 v II (S'. S) -32- We will estimate the various U (c<.,p) (which are nonnegative). Let I'k, I 5 M(V = I1 2J • • ·) and let the length of the interval be Ii, • First observe R.-2. .f-1 . M n if £ ~ 2. • (This is a primitive form of (1.16) . ) ('n.-s)! nl Case 1: Also note that < A -' - fl ns • I We simply estimate I~ ~ M and thus Y, ? 2. • A v("',(3) ~ -1!:. . ?ts L fr,/Y. ...L .fr./Y. My,In ... MY.~n ft = A -ei M24 > &Y. . .. ) t-Ys = U .n. '-'v Case 2: Y1 .::: • • • = ){ = 11 1~2., 1 Here (using 5.6) (S. 6) > L.tv and ){+.I 2 2.., 1 (G". 7. 1) in case f.< S. -33Case 3. )'1 ity, =1 and ~ ~ 2 in case s ~ 2. By Schwarz' Inequal- J(;: ~°'· ~(J,) CL <ft~'" ~ti4 .. · L 'k,«s ~(3,) ~ [ Si I:~ /2 JI Z: ~«.i tfi 13~ .. · l ~txs if3i0 It t 1 a. . Ya. MYs ):L.P J 1/2. _ , '/4 M21t-1 s-1/..2. < [ n (M n · · · n .-ri - . - n n ) pj_ecin~ 00 .by modifying the series ' L..,v::.t an integer greater than 1. v -~ ¢:. , where " A= 1-1/.Z'l. , n. being I£ 0 = n0 < >t 1 < n 1 < · · · sequence of integers (to be determined later) let be a permutation of {I> 2J .. ·} block { n~ +I) . .. ) nk-+- I} ( ")1.ft~l pl (x, a') = ~ ) IC Then L 0 8~ = ') ..., L.,, y-1 rt is a = (0" <1'2 , • • • ) 1, which rearranges the terms in each "-= 0' I) 2 a---">. cl?. (x) y among themselves. J ... ) y Write J V: '>lie+ I 00 will be obtained Our function ~ together finite sums. (~ = O, I> ... ) = 00 • such that Choose positive convergence l 00 0 Ere < oc • The factor -34- e~ will multiply the . ..f<. th block . of so fast that L v~A <It • Make ( 'l'l*) increase 00 (s. B) . Let Im n be the expression I .of (5.1) associated with I n. -).,,.1.. L h1+- I V 'f'7v (Here n-m replaces n • in (5.1).). show later that Imn$; A(tt.,M1{) for all m. andn. I there exists Since J/Pk (x, a') j2'd'x over is the average of We will all er , ct such that and hence L c Ek 'frt(x tr) 00 , when written out, is the Fou:r-ier series of a 1 ~ E L).'1,, . a.v = E.ft ~ whose coeff'icients are nK. + I ~ \l S 11~+I c:z~ = ef<. v-A states that • ~A for , with decreasing rearrangement (This is why we took EA ~ • ) Thus (5.8) B~ (a.)= oO. It remains to be proved that Im, n $ A(1z,1 M1 h) . By (5.5) and (5.7.1, 2, J) we need only prove that the expressions . associated with z;')1.. v-~ cf? m+I v are all bounded by -35an y ')"11. -3/.2. Evidently L &Y S L...-l!i+I V 5 A(,rz, 1 M,h) absolute constant greater than 1, if' Y~ 2. A and Ji. ~ Z. , an • Since SS 2,-t, Then to prove the assertion f'or -a;_ we obserye that · < (i~) 1. n R. 2~ 1/21(, - I ( I - x) 1-x 1. 1./1."' - I . .z "' (l-x)i-1 (21?.) l 'Y1M.21t- I - - - - - < (21t) n . ::: (.Zll) tt 1+x +· " +X..vz.-I since f ~ 2. 'L Similarly f'or 1T3 consider -)6- . l/H-1/.2.. - 2n. n < ·2;i n 1-x 1/.Ztz. - 1/.i < 2 ,/"f I. ) since 21'[, > 2... If we form a probability space fl from the er as in Theorem 2, we can see that there are many er which give a . ~ E Ll't . • The becomes By convexity o:f the also. Then 'IL +I. ~ power, we have J41 II ':t'k"''""')II21-""" <- Aer., M J) A:. I' J ... I ,11. -Y?00 Hence for a.e. lT in il., L E1c, l!<}k(·,C')//,i't < 00 and 0 is in L.1.ri • A corollary of the proof of Theorem 3 is that a "smaller" expression :for :finite sums than (5.9) .· Let ¢ cl?. ••• . - /) * B,(tz,(a.) be a uniformly bounded ,2.) orthonormal P-system for an 'l.. ~ 2.. there is a sum a.,, > O ( v = I, .. ·, n ) . (lcitl 5 M (v=I,~, .. -)) on an interval of length ~ <. oo • l\. Then for each n is T =L 1 a 11 cf:, ~ such that (S.10) The relation ( 5.10) holds for and choose a' so that T (X) = «}0 (x. 1<i) if we set n 1 = n Slcf>0 (~1 0')j.Z~x ~ A(tt,M,h) • .Always 80 ~ log YI 1 • Theorem 3 for the trigonometric system. In case that ¢, , ~ l · · · is the exponential or cosine system, we can state Theorem 3 in a different form by using the following theorem of Hardy and Littlewood [4][1, vol. II, P. 129]: (5.11) Let a,~ a"~ a~z 00 L a.v 1>v 1 and only if' .. · decrease to zero. The series is the Fourier series of an ~ E Lp > p> / '1! -38- and in this case the ratio LIf.IP/ L, a~ ,r27T . 2 (excluding -ft.::: 0 a,e,) lies between two positive constants which depend only on p , We innnediately deduce modified forms of ~heorem 3 and (5.9) when ¢.I , ¢_}. , · • • is the exponential or cosine system, '* v z Z there is a function da'*~ L I a,,¢ such that with a,* ~ a~* ~ a * ~ • ·, t o 3 (5.12) For each integer Jt 00 2.'Jr . 'A' /.I"(, a) J ~ I . =oo b) there is a rearrangement o ~uch that (.5.13) J of - (l . L a.v ~ For each It T =.L...-, ~ a.y ~v and n there is a trigonometric polynomial a.JI > 0 (v = I) ···) n) -with ( fo~ n) ~· E LJ.'i. is the Fourier series of a I *J' a,* ..• .2..] 00 ?\, .2.1£" S :2 /Tl IL 5 A/I. O We do not know whether the log n l.arger, a., , a..a, ) •. • s~'lr f1, 0 such that * ~It / l av~ I . ( 5.1 lf) '1::. t ·may be replaced by something -J9- where A).tz. =I i f n. is an integer. But we have never seen a:ny strengthening of this result in the direction of (5.12) or (5.1J). The rearrangement of (a.t) which gives (5.14) was not explicitly constructed; we might ask whether this would be _possible for the case of the cosine system. Perhaps the nondecreasing rearrangement of the coefficients gives the least value of JI T 11 /t 2 • -4o- Appendix 1 CA1.1) 11: (ck,) is a linearly independent P-system for 't 1hml a) A N (T) = II T 11.z~ of the cft • X21t C- L;,It The set b) ~ is a norm on the space of finite linear combinations 'x , L- ...,..,,..,.la,,le"v · the Fourier · is seri es or an space under the norm Proof. such that a) ~ f f Li~ is a Banach fl f 112'Z. • We need only show the triangle inequality. Recall. that (1.9) is true £or trigonometric polynomials with no assumptions on the e".v"' other than that they form a P-system. (1.9) here. If T= rv=/"y~) V= Ly-::1 e-v~ ~ T+U ~ Thus we may use YI. ')t. /'- then A T+U> /\. I T +1T 11-lri < JI T + U /J~t ~ A J (Al.2.) -41- b) Applying (1.9) to two elements 0£ X2't (A1.2), we have that norm on it. Xi~ exactly as in · II'' 112't is a linear space and is a. :f">t € Xi-i (n= IJ.2.)• .. ) To prove completeness, given satis£ying (Al.3) we will £irst identi£y the element 0£ ~ converge. fn - f Since /\. - fm A :f,,_ X 24 will f n - fm and is a majorant 0£ both -~ , (A1.3) and (1.9) imply that are both Cauchy sequences in r, ~ E L.Vt to which ( f 1't.) L ~ 2 • ( ;fJIL ) ("f_,. ) and Thus there are such that II .flt - f lli'l- ~ 0) /\. II i~ - ~ 11.2.ri,~ 0. This also implies convergence 0£ the Fourier coe££icients of these /\ :functions. ~ =f Thus , :f € X2,,, • ~ Jl f"' - f /l~'L~ 0 To prove ~ as n -:r oo II f"' - f 11.i = ll :fn - f Iii. ~ O .,,.-.... , first observe that , and that /""-.. fn - ;fm ~ £or all m., n • Then for each n) - :fn. - -;f -( Jm - f . ~ . ............... !!< II ;f?l-f»l-:f?t.-:f )..,- JI f ?1(.-f Ila,~ 0 as m~ o0 • Hence there ~ is a subsequence ~ i~ n. - :f.,,,, ~ f ..)\. - ;f (:fm. ) (depending on n. ) such that * a.a. as *.~oo . • · 0m 11,i't 110112't ~ ej WI infk ~00 u to 0 as n..!)>oo by (A1.J). It By Fatou•s Lemma, ' which tends -42- Appendix 2 (A2.1) Let (¢.,,) be a complex-valued P-system for It , !!E Then there is a er such that , a.v complex. (A2.2) Proof. M Divide up the unit circle of the complex plane into equal segments E0 >- · · · 7 Em- 1 where , Ek= {€i": (i-f)/m< X5 (k+i)/m} , and ~alls into. c l assify a.Y according t o whic h segment e i a,..g a,,. Thus .write T = ~ + ... +Tm , where Tf< = L (a...,~ ; ec' 4.rg a.,, € Ek ) ~ A • As an approximation to T write n L lavle£ 9 ~ v Y:::I • Then = /...J Y1a .. . a a. .. . a I cos(e +.. +e -t;.- .. ·-B ) J¢.. .. ·<k ~ ··· ¢, V, VJ\ ;4ll "f'-/l YI VII. 1'41 f'11o V, '11.ll f'1 f'A (A2.3) I:f we make 2JL • 'IT/m = 'Tr/ 3 m::. 611, , , then the cosine is 2:. 1/2., The integrals in (A2.'.3) are all nonnegative. Hence Now we may asswne with no loss of generality that the largest of • Then :for all O' . which put the terms of T 1 first. -44References 1. A. Zygmund, Trigonometric Series, Cambridge (1959). 2. A. M. Garsia, Existence of Almost Everywhere Convergent Rearrangements for Fourier Series of L Functions, 2 Annals o:f Math • .79 (1964), 623-9. 3. G. H. Hardy and J. E. Littlewood, A Problem Concerning Majorants of Fourier Series, Quarterly J, of Math. (Oxford Series) 6 (1935), 304-15. 4. H.ardy and Littlewood, Notes on the Theory of Series (XIII): Some New Properties of Fourier Constants, J. London Math. Soc. 6 (1931), 3-9. 5. Hardy and Littlewood, Notes on the Theory of Series (VIII): An Inequality, London Math. Soc, J. 3 (1928), 105-10. 6. R. M, Gabriel, The Rearrangement of Positive Fourier Coefficients, Proc. London Math, Soc. 33 (1932), 32-51. 7. F. Spitzer, A Combinatorial ~mma and Its Application to Probability Theory, Trans, Amer. Math, Soc. 82 (1956), 323-39. .