SUBSYSTEMS OF THE WALSH ORTHOGONAL SYSTEM WHOSE MULTIPLICATIVE COMPLETIONS ARE QUASIBASES FOR L p [0, 1], 1 p<+

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 4, Pages S (2) Article electronically published on July 25, 22 SUBSYSTEMS OF THE WALSH ORTHOGONAL SYSTEM WHOSE MULTIPLICATIVE COMPLETIONS ARE QUASIBASES FOR L p [, 1], 1 p<+ M. G. GRIGORIAN AND ROBERT E. ZINK (Communicated by Andreas Seeger) Abstract. If one discards some of the elements from the Walsh family, an ancient example of a system that serves as a Schauder basis for each of the L p -spaces, with 1 < p < +, the residual system will not be a Schauder basis for any of those spaces. Nevertheless, Price has shown that each member of a large class of such subsystems is complete on subsets of [, 1] that have measure arbitrarily close to 1. In the present work, it is shown that subsystems of this kind can be multiplicatively completed in such a way that the resulting systems are quasibases for each space L p [, 1], 1 p<+, fromwhichthe earlier completeness result follows as a corollary. 1. In one of the earliest monographs treating the subject, Paley [9] showed that the Walsh system is a Schauder basis for each space L p [, 1], 1 <p<+. If some of the Walsh functions are deleted from the system, the residual family will no longer be a Schauder basis for any of these spaces, of course, but, as Price [1] has shown, the subfamily may be complete on a set of large measure. In the present work, it is shown that subsystems of the type considered in [1] can be multiplicatively completed in such a way that the new systems are, in fact, quasibases for every space L p [, 1], with 1 p<+. It thus follows that for each such Walsh subfamily, for each such p, for every ɛ>, there is a measurable subset, E ɛ,of[, 1], such that the subfamily is closed in L p (E ɛ ), E ɛ > 1 ɛ. 2. Let B be a Banach space, let B be the associated conjugate space, let Φ=ϕ n : n N} be a subset of B. Then Φ is a quasibasis for B, ifthereisacorresponding subset of B, ψn : n N}, such that for every f B, n=1 ψ n (f)ϕ n converges to f in the norm of B. This notion, introduced by Gelbaum [3] in 1958, is a weaker concept than that of a Schauder basis, since the sequence of coefficient functionals need not be unique [12, p. 278]. The Walsh system, an extension of the Rademacher system, may be obtained in the following manner. Let r be the periodic function, of least period 1, defined on [, 1) by r = χ [, 1 2 ) χ [ 1 2,1). Received by the editors August 22, 21, in revised form, November 2, Mathematics Subject Classification. Primary 42C c 22 American Mathematical Society

2 1138 M. G. GRIGORIAN AND ROBERT E. ZINK The Rademacher system, R = r n : n =, 1,...}, is defined by the conditions r n (x) =r(2 n x), x R,n=, 1,...,, in the ordering employed by Paley, the n th element of the Walsh system, W = W n : n =, 1,...} is given by W n = r n k k, k= where k= n k2 k is the unique binary expansion of n, witheachn k either or 1. One shows that the Walsh subsystems introduced by Price can be multiplicatively completed so as to become quasibases, by constructing appropriate sequences of coefficient functionals. For this purpose one employs the Haar system h n : n N}. In the stard notation, one has h 1 =id [,1], ) h (j) k =2 (χ k/2 ( ) 2j 2 2j 1 χ ( 2k+1, 2j 1 ) 2 k+1 2 k+1, 2j, k =, 1,...; j =1,...,2 k ; 2 k+1, for n =2 k + j, h n = h (j) k. 3. In his AMS Colloquium Lectures, Levinson [8] proved the following completeness theorem for families of exponential functions. Let S = n j } be an increasing sequence of natural numbers, let Λ(n) bethe number of elements of S that are less than n, let D(S) = limsup ξ 1 limsup n Λ(n) Λ(ξn). (n ξn) If D(S) = 1, then, for every ɛ>, there is a measurable set E ɛ [, 1] such that E ɛ > 1 ɛ e 2πinj(.) : j N } is complete in L 2 (E ɛ ). The corresponding result for the Walsh functions involves a different set function which yields a weaker notion of density; viz., for S N, let ρ(s) = limsup k limsup n Λ(n + k) Λ(n) k One always has ρ(s) D(S), equality need not hold. Theorem A (Price). Let S = n j } be an increasing sequence of natural numbers, let W S = W nj : j N}. Ifρ(S) =1, then, for every ɛ>, thereexists a measurable set E ɛ [, 1] such that E ɛ > 1 ɛ W S is complete in L 2 (E ɛ ). At the time of this writing, it was not known whether it would be possible to replace D(S) byρ(s) in Levinson s theorem. The equivalence of the notions of completeness on a set of large measure multiplicative completability was demonstrated in [11]. Theorem B. Let E be a measurable set of finite, positive measure, let Φ= ϕ n : n=1, 2,...} be a subset of L 2 (E). The following conditions are equivalent: (BP) There exists a bounded measurable function, m, such that mϕ n : n =1, 2,...} is complete in L 2 (E); (M) Φ is complete in measure on E; (T) For every positive ɛ, thereexistse ɛ E such that E ɛ > E ɛ, Φ is complete in L 2 (E ɛ )..

3 SUBSYSTEMS OF THE WALSH ORTHOGONAL SYSTEM 1139 The notion of multiplicative completion is due to Boas Pollard [1]; the equivalence (M) (T ) is due to Talalyan [13], [14] who also showed that if Φ has these properties, then so also does every family obtained from Φ by deleting a finite number of its members. Subsequently, Goffman Waterman [4] gave a new proof of the latter result of Talalyan observed that it is always possible to make certain infinite deletions from a system that satisfies (T )soastoleavea residual system that also enjoys this property. Although it was not shown there, adaptations of the arguments given in [11] can be used to establish a companion theorem to Theorem B in which the rôle of L 2 is played by any space L p,with1 p<+. It follows that any system that is multiplicatively completable in L p (E) isclosedinl p (E ɛ ), for some sets E ɛ with E ɛ > E ɛ, for every ɛ>. Moreover, Braun [2] has extended the work of Boas Pollard in the following manner. Theorem C. Let E be a measurable set of finite, positive measure, let Φ= ϕ n : n = 1, 2,...} be a Schauder basis for some space L p (E), with 1 p < +. Then, to every natural number N there corresponds a bounded measurable function, M, such that every element f of L p (E) can be represented by a series k=n+1 a kmϕ k that converges to f in the L p -norm. In other terminology, Theorem C asserts that the systems Mϕ k : k = N +1,...} are systems of representation for L p (E). Neither does the theorem claim nor are the arguments used to establish it sufficient to show that these systems are quasibases for L p (E); nevertheless, subsequent analysis of the problem has shown this to be the case [7]. 4. Theorem. Let S = n j } be an increasing sequence of natural numbers such that ρ(s) =1,letW S = W nj : j N} be the corresponding subsystem of the Walsh system. Then, there exists a bounded, measurable function, M, such that MW nj : j N} is a quasibasis for each space L p [, 1], 1 p<+. The demonstration of the theorem depends upon a proposition of Menshov- Talalyan type (see, for example, [14]), to which the following lemmata lead. Lemma 1. Let S = n j } be an increasing sequence of natural numbers. The following are equivalent assertions: (α) ρ(s) =1; (β) There exists an increasing sequence of natural numbers, M k } k=1, such that lim j (M 2j M 2j 1 )=+,, for every j, each integer in [M 2j 1,M 2j ] is an element of S. Proof. In his demonstration of the completeness in measure of a Walsh subsystem W S,withρ(S) = 1, Price showed that, for every natural number p, thereisan increasing sequence of natural numbers, j i } i=1, such that S contains all of the integers in [j i 2 p, (j i +1)2 p ], i; thus,(α) (β). The other implication is even simpler. Lemma 2. Let S = n k } k=1 be an increasing sequence of natural numbers, with ρ(s) =1,letM k } k=1 be a corresponding sequence, of the type guaranteed by Lemma 1; let be an interval of the form [ i 1 i 2, m 2 ],forsomem N m

4 114 M. G. GRIGORIAN AND ROBERT E. ZINK 1 i 2 m ;letk be a natural number greater than 1; let the real numbers ɛ, p, γ satisfy the conditions <ɛ<1, p 1, γ. Then, there exist a measurable set E, a Walsh polynomial such that K Q = c nk W nk Span W S k=k Q(x) = γ, if x E;, if x/ ; E > (1 ɛ) ; [ ] 1/p 1 m max c nk W nk p dt : k m K 4 γ ɛ 1/2 A 2 1/2, if p =1, 4 γ ɛ 1/q A k=k p 1/p, if p>1, where A p is a constant depending only upon p, 1 p + 1 q =1. Proof. Let ν = 1 + [log 2 (1/ɛ)], where [x] denotes the greatest integer less than or equal to x. Because is a dyadic interval, there is a natural number µ 1 such that µ 1 S µ1 (γχ )= a j W j = γχ, j= where the a j are the Walsh-Fourier coefficients S n f denotes the n th partial sum of the Walsh-Fourier series. Suppose that the greatest power of 2 that appears in the binary expansion of µ 1 is 2 l. If m is any positive multiple of 2 l+1, then, for every j [,µ 1 ], one has W m+j = W m W j. From this observation from the definition of the sequence M k } k=1, there follows the existence of natural numbers k 1 m 1 >µ 1 such that, for all j [,µ 1 ], both Setting W m1+j = W m1 W j m 1 + j [M 2k1 1,M 2k1 ]. b (1) t = ( ) = x : W m1 (x) = 1}, (+) = x : W m1 (x) =+1}, a j, if t = m 1 + j, for some j [,µ 1 ];, otherwise;

5 SUBSYSTEMS OF THE WALSH ORTHOGONAL SYSTEM 1141 one has, since W m1 Q 1 =: M 2k1 t=m 2k1 1 b (1) t W t = µ 1 is orthogonal to a j W j, j= µ 1 j= a j W m1+j µ 1 = a j W j W m1 j= = γχ (+) γχ ( ), ( ) = (+) = 1 2. Let 1 = ( ). Then 1 is a finite union of congruent, dyadic intervals; thus, there exists a natural number µ 2 such that µ 2 S µ2 (2γχ 1 )= a (2) j W j =2γχ 1, j= where the a (2) j are the Walsh-Fourier coefficients. Choose natural numbers k 2 m 2 such that k 2 >k 1, m 2 > 2maxm 1,µ 2 },, j [,µ 2 ], both W m2+j = W m2 W j m 2 + j [M 2k2 1,M 2k2 ]. Setting one has Q 2 =: b (2) t = M 2k2 t=m 2k2 1 a (2) j, if t = m 2 + j, j [,µ 2 ];, otherwise; ( ) 1 = x 1 : W m2 (x) = 1} ; (+) 1 = x 1 : W m2 (x) =+1} ; b (2) t W t = µ 2 j= µ 2 = j= a (2) j W m2+j a (2) j W j W m2 =2γχ 1 W m2 =2γ ( χ (+) 1 χ ( ) 1 ),

6 1142 M. G. GRIGORIAN AND ROBERT E. ZINK so that γ, on (+) (+) Q 1 + Q 2 = 3γ, on ( ) 1,, otherwise,, because m 2 > 2m 1, (+) 1 = ( ) 1 = = 1 4. Proceeding inductively, suppose that, for all i [1,ν 1], one has found natural numbers m i k i, Walsh polynomials Q i,sets i, such that, i [1,ν 1], m i [M 2ki 1,M 2ki ], m i+1 > 2m i k i+1 >k i, (+) i 1 = x i 1 : W mi (x) =+1} ( ) i 1 = x i 1 : W mi (x) = 1}, Q i = M 2ki t=m 2ki 1 1, b (i) t W t =2 i 1 γχ i 1 W mi, i = ( ) i 1, (+) i = ( ) i = 1 2 i = 1 2 i+1, for i<ν 1. Let ν 1 = ( ) ν 2. As in the earlier work, there is a Walsh polynomial µ ν P ν = a (ν) j W j, j= such that P ν =2 ν 1 γχ ν 1. Choose natural numbers k ν >k ν 1 m ν > 2maxm ν 1,µ ν } so that both Let let Then b (ν) t = W mν +j = W mν W j m ν + j [M 2kν 1,M 2kν ], j [,µ ν ]. a (ν) j, if t = m ν + j, j [,µ ν ];, otherwise; Q ν = M 2kν t=m 2kν 1 b (ν) t W t. Q ν = P ν W mν =2 ν 1 γχ (+) χ ( ) ν 1 }, ν 1

7 SUBSYSTEMS OF THE WALSH ORTHOGONAL SYSTEM 1143 where (+) ν 1 ( ( ) ν 1 )=x ν 1 : W mν (x) =+1( 1)}. Finally, let b (i) t, if M 2ki 1 t M 2ki, 1 i ν; c t =, otherwise ; in the ordering of W S,letn k = M 2k1 1 n K = M 2kν ;let ν K Q = Q i = c nk W nk, i=1 k=k let E = ( ) ν 1. Then Q = γχ 2 ν γχ ( ), ν 1 so that Q(x) = Thus, for p>1, 1 p + 1 q =1, ( 1 ) 1/p Q p = Q(t) p dt γ, if x E;, if x/ ; E =(1 2 ν ) <ɛ. γ 1/p +2 ν γ ( ) ν 1 1/p = γ 1/p (1 + 2 ν/q ) γ 1/p ( /q ɛ 1/q ) < γ 1/p 2 1+1/q ɛ 1/q < 4 γ 1/p ɛ 1/q. Since Q is a Walsh polynomial, it is its own Walsh-Fourier series; that is, c t = 1 Q(x)W t (x)dx, thus, from Paley s theorem, S n Q p A p Q p, n N p>1, where A p is a constant depending only upon p. Hence } m max c nk W nk p : k m K 4A p ɛ 1/q γ 1/p, p >1, k=k } } m m max c nk W nk 1 : k m K max c nk W nk 2 : k m K k=k k=k t; 4A 2 ɛ 1/2 γ 1/2.

8 1144 M. G. GRIGORIAN AND ROBERT E. ZINK Lemma 3. Let f be a (dyadic) step function on [, 1], f = 2 n i=1 γ i χ i, where ( ) i 1 i = 2 n, i 2 n,i=1,...,2 n, let W nk : k N} be a subsystem of the Walsh system that satisfies the conditions of the Theorem. To each p 1, eachɛ (, 1), eachk N, there corresponds a Walsh polynomial K Q = c nk W nk k=k +1 a measurable set E [, 1] such that E > 1 ɛ, max m k=k +1 for every measurable set e E. Q(x) =f(x), x E, } c nk W nk L p (e) : K <m K ɛ + f L p (e), Proof. Without loss of generality, one may assume that, for all i [1, 2 n ], 4 γ i A 2 ɛ 1/2 i 1/2 + A p ɛ 1/q i 1/p } <ɛ, for p>1, 1 p + 1 q =1, since, if necessary, one could refine the dyadic partition upon which f is defined. Successive applications of Lemma 2 yield measurable sets E i i Walsh polynomials such that for all i =1,...,2 n, Q i = Q i (x) = K i k=k i 1+1 c (i) n k W nk, γ i, if x E i ;, if x/ i ; max j E i > (1 ɛ) i ; c nk W nk p : K i 1 +1 j K i k=k i γ i A p ɛ 1/q i 1/p, for p>1; 4 γ i A 2 ɛ 1/2 i 1/2, for p =1.

9 SUBSYSTEMS OF THE WALSH ORTHOGONAL SYSTEM 1145 Let E = 2 n i=1 E i,letk = K 2 n,let Q = 2 n i=1 K i k=k i 1+1 c (i) n k W nk = K k=k +1 c nk W nk, where c nk = c (i) n k, for K i 1 +1 k K i,i=1,...,2 n. Then E > 1 ɛ Q(x) =f(x), x E. To each j [K +1,K] there corresponds a unique i j such that 1 i j 2 n K ij 1 +1 j K ij ; thus, i j j 1 j c nk W nk = Q i + c nk W nk. k=k +1 i=1 k=k ij 1+1 Since Q coincides with f on E, one has for each measurable set e E, ( 1/p 1/p j i j 1 c nk W nk dx) p Q i p dx e k=k +1 e + i=1 1 j k=k ij 1+1 c nk W nk p dx ( 1/p f dx) p + ɛ, p 1. e To complete the proof of the theorem, one employs the methods devised by Braun. Choose p>1, let h n : n =1, 2,...} be the Haar system. For each n, let f n = h n / h n p,letg n : n =1, 2,...} be the family of coefficient functionals associated with f n : n =1, 2,...}. By virtue of Lemma 3, one may, following Braun, construct a measurable function M, with M(t) 1, for all t in [, 1], a double sequence of W S -polynomials P kj } P kj = n k (j) i=n k 1 (j)+1 k=1,j=k, a i W ni, with N = n (1) <n 1 (1) =n (2) <n 1 (2) <n 2 (2) = n (3) <... n (j) <n 1 (j) < <n j (j) =n (j +1)<..., a sequence E n } n=1 of measurable subsets of [, 1], such that (i) [, 1] \ E n <δ n, δ n } n=1 ; (ii) f k M l j=k P kj p 2 l, k, l k; 1/p

10 1146 M. G. GRIGORIAN AND ROBERT E. ZINK (iii) sup M s i=n k 1 (l)+1 a iw ni p : s n k (l)} 2 l+2,ifl>k; (iv) sup M s i=n l 1 (l)+1 a iw ni L p ([,1]\E l ) : s n l (l)} 2 l+2 ;, for every measurable set e E l, (v) sup M s i=n l 1 (l)+1 a iw ni Lp (e) : s n l (l)} 2 l+1 + f l Lp (e). One associates with Φ = MW ni : i>n} the system Ψ = ψ i : i>n}, where ψ i = a i g k, if n k 1 (l) <i n k (l), for some l k. Then, with the coefficient functionals defined on L r [, 1] by setting b i ( ) = 1 ( )ψ i dt, for i = N +1,N +2,..., one finds that Φ is a quasibasis for each space L r [, 1], with 1 r p. The abbreviated demonstration below essentially duplicates the argument employed in [7, 437ff]. Let r [1,p], let f be an arbitrary element of L r [, 1], let S n (f) bethenth partial sum of the series n b i (f)mw ni, i=n+1 let c k f k be the expansion of f in the p-normalized Haar system. k=1 The estimation of f S n (f) r depends upon two critical properties of the (α) ck k f r, k N, sequence c k } k=1 : since c k = c k (f) = 1 fg k dt f r g k r, 1 r + 1 r =1, g k r = h k p h k r = h k 2 k 1 p + 1 r, where k is the support of h k, = k 1 p + 1 r 1 k 1 ( 1 p + 1 r ) k; (β) since c k = c k (f) = lim k c k / g k r =, 1 fg k dt = 1 k 1 f c j f j r g k r, k 1 (f c j f j )g k dt since f n : n =1, 2,...} isaschauderbasisforl r [, 1] (as well as for L p [, 1]).

11 SUBSYSTEMS OF THE WALSH ORTHOGONAL SYSTEM 1147 If n = n k (l), for some natural numbers k l (with l k), then, for k<l, f S n (f) r = f k i=j l c j MP ji f c j f j r + + j=k+1 j=k+1 i=j k c j f j c j f j MP ji r, i=j f c j f j r +2 l+1 c j,, if k = l, then a similar estimate yields From (α), one has so that f S n (f) r f l c j f j r +2 l l c j (l(l +1)/2) f r c j MP ji r l MP ji r i=j l c j. n 1 f S n (f) r f c j f j r +2 l l(l +1), for n = n k (l). Finally, if n k 1 (l) <n<n k (l) k<l,then f S n (f) r f c j f j r +2 l l(l +1) f r + c k f c j f j r +2 l [l(l +1) f r +4k]. On the other h, if k = l, then f S n (f) r f c j f j r +2 l l(l +1) f r + c l Setting one has σ ln = n i=n l 1 (l)+1 a i MW ni, σ ln r r = σ ln r L r ([,1]\E l ) + σ ln r L r (E l ) n i=n k 1 (l)+1 n i=n l 1 (l)+1 2 ( l+2)r + σ ln r L r (E l l ) + σ ln r L r (E l \ l ), a i MW ni p a i MW ni r.

12 1148 M. G. GRIGORIAN AND ROBERT E. ZINK by virtue of condition (iv),, from condition (v), follow σ ln r L r (E l \ t) σ ln r L p (E l \ l ) (2 l+1 + f l L p (E l \ l )) r =2 ( l+1)r σ ln r L r (E l l ) σ ln r L p (E l l ) l 1 r p Thus, from these estimates (β), with lim l c l n i=n l 1 (l)+1 2 r l 1 r p (2 ( l+1)r + f l r p )=2r l 1 r p (2 ( l+1)r +1). a i MW ni r (2 ( l+2)r+1 +2 r+1 l 1 r p ) 1 r cl θ(l) =, it follows that =(2 ( l+2)r+1 +2 r+1 l 1 r p ) 1 r l 1 r 1 p θ(l), lim n S n (f) f r =. Since p>1 is otherwise arbitrary, Φ is a quasibasis for every space L p [, 1], with p [1, + ). As has been mentioned, in the discussion following Theorem B, it follows that W S is complete (or total) in measure, on [, 1], thus also, for each ɛ>, W S is closed in L p (E ɛ ) for some measurable set E ɛ [, 1], with E ɛ > 1 ɛ. 5. The theorem is, in a sense, complementary to earlier results of Grigorian [5], [6] who has established the following proposition. Let S be an increasing sequence of natural numbers, with ρ(s) = 1, let W S be the corresponding family of Walsh functions. For every ɛ>, there exists a measurable set E ɛ [, 1] such that E ɛ > 1 ɛ W S is a system of representation, in L 1 (E ɛ ), in the sense of convergence a.e., as well as in the sense of convergence in the L 1 -norm. As these results the theorem of this article seem to indicate, the subsystems W S,withρ(S) = 1, are incredibly rich. On the other h, Walsh families of this kind may be quite sparse indeed, for there are sets S N, of asymptotic density, such that ρ(s) =1. References 1. R. P. Boas Harry Pollard, The multiplicative completion of sets of functions, Bull. Amer. Math. Soc. 54 (1948), MR 1:189b 2. Ben-Ami Braun, On the multiplicative completion of certain basic sequences in L p, 1 <p<, Trans. Amer. Math. Soc. 176 (1973), MR 47: Bernard R. Gelbaum, Notes on Banach spaces bases, An. Acad. Brasil3 (1958), MR 2: Casper Goffman Daniel Waterman, Basic sequences in the space of measurable functions, Proc. Amer. Math. Soc. 11 (196), MR 22: M. Zh. Grigoryan, Convergence almost everywhere of Walsh-Fourier series of integrable functions, Izv. Akad. Nauk. Arm. SSR Ser. Math. 18 (4) (1983), (Russian). MR 85a: M. G. Grigorian, Convergence of Walsh-Fourier series in the L 1 metric almost everywhere; English translation in. Soviet Math. Izv. VUZ 34 (11) (199), 9 2.

13 SUBSYSTEMS OF THE WALSH ORTHOGONAL SYSTEM K. S. Kazarian Robert E. Zink, Some ramifications of a theorem of Boas Pollard concerning the completion of a set of functions in L 2,Trans.Amer.Math.Soc.349 (1997), MR 99d: N. Levinson, Gap Density Theorems, Amer. Math. Soc. Colloq. Publ. 26, Amer. Math. Soc., Providence, 194. MR 2:18d 9. R. E. A. C. Paley, A remarkable set of orthogonal functions, Proc. London Math. Soc. 34 (1932), J. J. Price, A density theorem for Walsh functions, Proc. Amer. Math. Soc. 18 (1967), MR 35: J. J. Price Robert E. Zink, On sets of functions that can be multiplicatively completed, Ann. Math. 82 (1965), MR 31: Ivan Singer, Bases in Banach Spaces II, Springer Verlag, Berlin, Heidelberg, New York, MR 82k: A. A. Talalyan, On the convergence almost everywhere of subsequences of partial sums of general orthogonal series, Izv. Akad. Nauk. Arm. SSR Izv. Fiz. Mat. Tehn Nauki 1 (1957), MR 19:742b 14., The representation of measurable functions by series; English translation in Russian Math. Surveys 15 (196), Department of Mathematics, Erevan State University, Alex Manoogian Str., Yerevan, Armenia address: gmarting@ysu.am Department of Mathematics, Purdue University, West Lafayette, Indiana address: zink@math.purdue.edu