Montgomery, Hugh L.

Numérisation de Documents Anciens Mathématiques

A note on rearrangements of Fourier coefficients

Annales de l’institut Fourier (AIF)

ANNALES DE L’INSTITUT FOURIERHUGH L. MONTGOMERYA note on rearrangements of Fourier coefficientsAnnales de l’institut Fourier, tome 26, no 2 (1976), p. 29-34<http://www.numdam.org/item?id=AIF_1976__26_2_29_0>© Annales de l’institut Fourier, 1976, tous droits réservés.L’accès aux archives de la revue « Annales de l’institut Fourier »(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-tion commerciale ou impression systématique est constitutive d’une in-fraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/Ann. Inst. Fourier, Grenoble26, 2 (1976), 29-34.A NOTE ON REARRANGEMENTSOF FOURIER COEFFICIENTSby Hugh L. MONTGOMERYLet {9^} be a sequence of functions on T = R/Z, withthe property that they are uniformly bounded,(1) ' II 9.11 oo ^ M,and satisfy a Bessels inequality(2) SLC^M^1!^.For the sake of simplicity we suppose that M has the samevalue in (1) and (2); this does not occasion any loss of gene-rality. Suppose that ^ [a/J2 < oo. Thenk(3) f(x) == S a^,(x)kis a member of L^T), since the dual of (2) asserts that(4) f^ 2 ^P.|2 ^ M2 S Kl2.In this note we obtain bounds for j \f\2 in terms of themeasure of the set E and the numbers |a^|. Following Hardyand Littlewood, we let the numbers a^, a^, . . . be thenumbers [a^[, permuted so that flC\. Then we set00(5) f^W = 5 a^ cos ̂ nx.n=0THEOREM 1. — Let {<pfc} be a sequence of functions satis-fying (1) and (2), let f and f* be defined by (3) and (5). Then30 HUGH L. MONTGOMERYfor any measurable set E c T, with measure |E| = 26,we have(6) Jjyp ^ 20MP/_9J^12.00If C e L^T), C{x} ~ S ̂  cos 27cna;, and if C,\, thenn=0C = C*, so (6) implies that/JC|^ < 20/_°JCp,where E c T, |E| =26. Thus, although it is not necessar-r i \ .ily true that C{x) is decreasing on ( ) , — ) » in a certainL /sense it is still the case that C is largest near 0.Using a simple inequality of. A. Baernstein [2], we shallderive from Theorem 1 the following.THEOREM 2. — Let ^ be a convex increasing functionfrom [0, oo) to R. Then, in the above notation,J^dfl2) ^ /_>(20]\P|n2).Taking ^(t) = (g/2, we see from the above that(7) 11/1, ^ 5MIIH. {q ^ 2).Inequalities of this type have a long history. Hardy andLittlewood [3, 4] proved that(8) llfll, ^ c,[in, (q ^ 2)in the case ^u{x) = ^ikx, — oo < k < + oo. Littlewood [6]has shown that Cq is bounded in this case, and F. R. Keogh [5]has shown that Cp —> 1 as q —> oo. In the opposite direction,Littlewood [7] showed that Cq > 1 except when q is aneven integer. Consequently, the constant 20 in Theorems 1and 2 can not be replaced by 1. R. E. A. C. Paley [9] exten-ded (8) to the case of arbitrary uniformly bounded orthonor-mal 9^ (see Zygmund [11, XII § 5] for a simple proof).Theorem 2 does not seem to follow from the special case (7),since in general a convex increasing function ^(^ ls notcomparable to a sum ^ c^, c^ ^ 0, a,. ^ 1.A NOTE ON REARRANGEMENTS OF FOURIER COEFFICIENTS 31If one were to consider, in place of /'*, a functionoo/^) = 5 ̂  ̂  y,^),n=0then one does not in genera] expect the inequality11/1, ^ ^ll/'-ll, (q > 2)to be valid, even when the q^ are given in some naturalorder. (See G. A. Bachelis [I], and H. S. Shapiro [10]). How-ever, in the special case of ordinary Dirichlet series, there aregood reasons to believe that something positive may be said.For example, we can formulate aConjecture. — Let s > 0, and 2 ^ q ^ 4. Then for T ^ 2,N > No(e, g), we haveC. S ̂ ""l dt ^ (T + N^N^S7»==1 Ifor arbitrary coefficients a^ satisfying |aJ ^ 1.The above is known to be true when q == 2, q = 4; thusby Holder's inequality it suffices to consider the case T == N^2.The Conjecture is of special interest in multiplicative numbertheory, since from it one can deduce (see Montgomery [8,Theorem 12.6]) that the interval [x, x + x ^ ) contains aprime number, for all x > Xo{e).We now prove Theorem 1. We have only countably manyfunctions 9^, so without loss of generality we may supposethat 0 ^ k < oo. Let TT be the permutation such that^ == ^(n)|. Put N = [(26)-1], and set^ = {n(n) : 0 ^ n ^ N}.Thus ^ is the set of indices of the N + 1 coefficients oflargest absolute value. Break the sum (3) into two parts,f= S + 2 -A+/2 ,ne% ^%say. On one hand,JjAI2 ^ I IAPocr i ^ 26 ("M S ai)\\ n=0 /32 HUGH L. MONTGOMERYin view of (1). On the other hand, from (4) we see that/j/2i2 ^ r i / 2 i 2 ^ M2 s ^2.l /E l/o n>NFor each x, \f\2 ^ 2\f^\2 + 2|/2|2, so on combining the abovewe find that(9) fj/12 ^ 46 (M S a;Y + 2M2 S ^2.l/E \ n=0 / n>NIt now remains to relate the right hand side above to {_ \f*\2.A J-eLet K{x) = max (0, 1 — \x\Q-1) for \x\ < —• ThenLfin2 ^ r^ Kin2 = 4 i ^(^ + ^) + ̂  - ^))-l/ ' î 2 m.n=0I\T f>/ \ n/sin7rm6\2 /.Now K(m) = 6 ( ———— ) ^ 0, so\ TCJTIV J(10) 1 S o;a;fC(m-n) ^ fe |^|2.^ m,n=0 •/ 'If | AH — n) ^ N then/ . 1 \ 2^T^\ 0 / s]n "<T~ 7t \(11) K{m - n) > 6 (sm^6)2 ^ 6 -̂ - = 4.-6,\ y" /since N < (26)-1. But o; > 0, so(12) 6 / S a:\^ 1 ^ ^ ft(m-n)a;a:.\0^n<^ / ^ O < W , T I < NIf 0 < n — N ^ m ^ / z then a^ ^ a ,̂ so from (11) wefind thatS a;K(m - n) ^ 47t-2e(N + l)a; ^ 27^-2a;,n—N.^m<nsince N + 1 > (26)-1. Hence(13) S <»:2 < 1 TC2 S ^{rn - n)a^.n>N ^ n>Nn—N^m^nA NOTE ON REARRANGEMENTS OF FOURIER COEFFICIENTS 33Combining (9) with (12), (13), we find thatf |/12 ^ 7^6M2 S a;a:6:(m - n).t/ E w. n=0But 27i2 < 20, so by (10) our proof is complete.We note that once (9) is established, the remainder of theproof can be effected in several ways. In proving (8), Hardyand Littlewood [3] established thatCiri^.i^(^+i)^-2.*/u n=0One can modify their proof of this (see also Keogh [5]) toshow thats ^( s ^2 < cf\\r\2.n>6-1 \0<m<n / vTheorem 1 follows easily from the above and (9), apart fromthe values of constants.To prove Theorem 2 we require the following result ofA. Baernstein [2].LEMMA. — For fe L^T), 0 ^ 6 < 1 letf^-{Q) = sup f ( f l ,2 E J Ewhere the supremum is taken over all measurable sets E c 0, 1)such that [E| ==26. For two functions r, s e L^T), the fol-lowing are equivalent:(a) For all 6 e JO, i\ r+(6) ^ 5+(6);L 2 /(6) For any ^ ((), convex and increasing on [0, oo), wehave r^H) ^r^i)-In the language of this lemma, we find from Theorem 1that (^^(O) < (20M|/•*|2)+(e). HenceW\fn\i < ll'F(20M|/•*|2)||l.However, with a little more care we obtain the full strengthof Theorem 2. Let E s [0, 1) be a set with [E| =26. Putr = l/p/E, s == 20M!!|/<*|2X(-9,e). Then by Theorem 1, r+ ^ s+,34 HUGH L. MONTGOMERYso by the Lemma above, ||+(r)i|i ^ ||+0?)||i. If ^(0) == 0,then this asserts thatX^i2) ^ /.^(2oM^T).To obtain this for general ^ we have only to add a constantto both sides of the inequality. This completes the proofof Theorem 2.BIBLIOGRAPHY[I] G. A. BACHELIS, On the upper and lower majorant properties ofLP(G), Quart. J . Math. (Oxford), (2), 24 (1973), 119-128.[2] A. BAERNSTEIN, II, Integral means, univalent functions and circularsymmetrizations, Acta Math., 133 (1974), 139-169.[3] G. H. HARDY and J. E. LITTELWOOD, Notes on the theory of series(XIII): Some new properties of Fourier constants, J. London Math.Soc., 6 (1931), 3-9.[4] G. H. HARDY and J. E. LITTELWOOD, A new proof of a theorem onrearrangements, J . London math. Soc., 23 (1949), 163-168.[5] F. R. KEOGH, Some inequalities of Littlewood and a problem on rear-rangements, J . London Math. Soc., 36 (1961), 362-376.[6] J. E. LITTLEWOOD, On a theorem of Paley, J . London Math. Soc., 29(1954), 387-395.[7] J. E. LITTLEWOOD, On inequalities between / and /*, J . London Math.Soc., 35 (1960), 352-365.[8] H. L. MONTGOMERY, Topics in multiplicative number theory, LectureNotes in Mathematics, Springer-Verlag, Vol. 227, (1971), 187 pp.[9] R. E. A. C. PALEY, Some theorems on orthogonal functions, StudiaMath., 3 (1931), 226-238.[10] H. S. SHAPIRO, Majorant problems for Fourier coefficients, to appear.[II] A. ZYGMUND, Trigonometric series, Second Edition, Cambridge Uni-versity Press, 1968.Manuscrit recu Ie 17 avril 1975Propose par J. P. Kahane.Hugh L. MONTGOMERY,University of MichiganAnn Arbor, Mich. (USA).

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A note on rearrangements of Fourier coefficients

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