ON THE COEFFICIENTS OF MULTIPLE SERIES WITH RESPECT TO VILENKIN SYSTEM. 1. Introduction

Transcription

1 t m Mathematcal Publcatons DOI: /tmmp Tatra Mt. Math. Publ. 68 (2017), ON THE COEFFICIENTS OF MULTIPLE SERIES WITH RESPECT TO VILENKIN SYSTEM Valentn A. Skvortsov Francesco Tulone ABSTRACT. We gve a suffcent condton for coeffcents of double seres n,m a n,mχ n,m wth respect to Vlenkn system to be convergent to zero when n + m. Ths result can be appled to the problem of recoverng coeffcents of a Vlenkn seres from ts sum. 1. Introducton It s easy to gve an example showng that the rectangular convergence almost everywhere of a double Walsh seres a n,m w n (x)w m (y) (1.1) n=0 m=0 does not mply n general that the coeffcents a n,m tend to zero when n+m. It suffces to take a non-trval Walsh seres n=0 a nw n (x) convergent to zero almost everywhere on [0, 1] (see [2] or [4]) and to put a n,m = a n for all m. However, a smple suffcent condton to guarantee that the coeffcents a n,m of (1.1) tend to zero when n + m was gven n [5]. In ths paper, we generalze the above theorem to the case of the Vlenkn system (see [1]). Although we are provng the result for the two-dmensonal case, the same method can be used to obtan a smlar result for any dmenson. We present also some applcaton of ths result to the problem of recoverng coeffcents of a Vlenkn seres from ts sum (more detals on ths applcaton wll be gven n [10]). c 2017 Mathematcal Insttute, Slovak Academy of Scences M a t h e m a t c s Subject Classfcaton: 42C10, 26A39. K e y w o r d s: Vlenkn system, multple Vlenkn seres, rectangular convergence, Perron ntegral, Saks contnuty, recoverng the coeffcents. Supported by RFBR (grant no ). 81

2 VALENTIN A. SKVORTSOV FRANCESCO TULONE 2. Prelmnares Let P = {p j } j=0 (2.1) be a fxed sequence of natural numbers wth p j 2 for j =0, 1, 2,...Set m 0 =1 and m k = p 0 p 1... p k 1. For each x [0, 1) we consder ts P-adc expanson x j x =, 0 x j p j 1. (2.2) m j+1 j=0 We denote by Q P the set of all P-adc ratonal numbers,.e., ponts of the form t m k,0 t m k. The elements of the set [0, 1) \ Q P are called P-adc rratonal numbers n [0, 1]. We note that each x Q P has two expansons, a fnte one and an nfnte one. We agree to consder only the fnte expanson for x Q P. Then, to each x [0, 1) there corresponds only one sequence of ntegers {x j } j=0,0 x j p j 1, gven by (2.2). We also consder P-adc expansons of ntegers n 0: s n = α j m j 1 wth 0 α j p j 1. (2.3) j=0 Now, for each n 0wthP-adc expanson (2.3), we defne an nth functon χ n of the so-called multplcatve Vlenkn system (see [1] and [2]) by puttng s α j x j χ n (x) :=exp 2π, (2.4) p j where numbers x j are gven by (2.2). Note that when p j = 2 for all j, weobtan the Walsh system. We consder half-open P-adc ntervals (or P-ntervals) I (k) r := [ r m k, r +1 m k j=0 ), 0 r m k 1, (2.5) where the number k =0, 1, 2,... s called the rank of the nterval. By I (k) we denote an arbtrary nterval of rank k and by I x (k) the ntervals of rank k the closures of whch I (k) contan the pont x. So,fx (0, 1), the notaton I x (k) s related to two ntervals (2.5), and, for x = 0 or 1, to one. We also assgn the rank k toanumbern and to a functon χ n f m k 1 n<m k (χ 0 has rank 0). Note that functons χ n of rank k are constant on the 82

3 ON THE COEFFICIENTS OF MULTIPLE SERIES WITH RESPECT TO VILENKIN SYSTEM nterval of the same rank and I (k 1) χ n dx =0. (2.6) We now defne the double Vlenkn system on the square K =[0, 1) [0, 1) by χ n,m (x, y) :=χ n (x) χ m (y). If the rank of the numbers n and m are equal to k and l, respectvely, we say that the par (n, m) hasrank(k, l), and so has the functon χ n,m (x, y) and an nterval I (k,l) := I (k) I (l). The notaton I (k,l) (x,y) s referred to the nterval I(k) x I y (l) of rank (k, l) the closure of whch I (k,l) contans the pont (x, y). Note that we shall use notaton I (k,l) (x,y) also for ponts (x, y) K \ K. WedenotebyI the set of all two-dmensal P-ntervals I (k,l). Functons χ n,m (x, y)ofrank(k, l) are constant on two-dmensonal P-ntervals of the same rank. A double seres n the Vlenkn system s defned by a n,m χ n,m (x, y) := n,m=0 n=0 m=0 a n,m χ n (x) χ m (y), (2.7) where a n,m are real or complex numbers. The rectangular partal sum S r,s of seres (2.7) at a pont (x, y) s r 1 s 1 S r,s (x, y) := a n,m χ n,m (x, y). n=0 m=0 Note that f r m k and s m l, then the above partal sums are constant on each ntervals of rank (k, l). We say that the seres (2.7) converges rectangularly to asums(x, y) atapont(x, y), and we wrte lm r,s S r,s (x, y) =S(x, y) f S r,s (x, y) S(x, y) as mn{r, s}. 3. Man result We gve a suffcent condton for the coeffcents a n,m of the seres (2.7) to tend to zero when n + m. In what follows, we suppose that the sequence {p } s bounded by a number P 2. Theorem 3.1. Suppose that a double seres (2.7) wth respect to Vlenkn system s rectangularly convergent everywhere on a cross { } { } a (0, 1) (0, 1) b, where a and b are P-adc rratonal ponts, except on a countable set E. Then, lm a n,s =0. (3.1) n+s 83

4 VALENTIN A. SKVORTSOV FRANCESCO TULONE P r o o f. Assume that (3.1) does not hold. Then, for some C > 0andsome sequence of ndexes {n,s },wthn + s when, the nequalty a n,s >C (3.2) holds for all. It follows from the hypothess of the theorem that there exst ponts of convergence of the seres. Ths mples that lm n,s a n,s χ n,s (x, y) =0 at those ponts. Havng n mnd that χ n,s (x, y) =1,weget lm n,m a n,m =0. (3.3) From ths fact t follows that n (3.2) we can consder one of the ndex to be bounded. Therefore, wthout loss of generalty, we can assume that for a constant s and for all n, n 1 <n 2 < <n <, the nequalty a n,s >C (3.4) holds. Let the terms of the sequence {n } n (3.4) be of ranks k, respectvely, and the number s be of rank l. We can assume that k 1 <k 2 < <k < For each, we put t l (y) = m l 1 q=0 a n,qχ q (y). Note that t l s constant on each P-nterval of rank l, and accordng to (2.6), l+j (y)dy = l (y)dy (3.5) I (l) t for any j 1. From condton (3.4) we get t l (y) >C (3.6) on at least one nterval I (l) of rank l. Indeed, f (3.6) does not hold,.e., for all y we have m l 1 t l (y) = a n,qχ q (y) C, q=0 then consderng a n,q as Fourer coeffcents of t l, we obtan n partcular for q =s, 1 1 a n,s = t l(y)χ s (y)dy t l(y) χ s (y) dy max t l(y) C y 0 0 gettng a contradcton wth (3.4). Takng a subsequence of {k } f necessary, we can suppose that the nterval I (l) s the same for each and we fx t for the further dscusson. 84 I (l) t

5 ON THE COEFFICIENTS OF MULTIPLE SERIES WITH RESPECT TO VILENKIN SYSTEM For a fxed, consder the sum p l(x, y) =S n +1,m l (x, y) S n,m l (x, y) m l 1 = χ n (x) a n,qχ q (y) q=0 = χ n (x) t l(y). (3.7) From (3.6), t follows that there exsts a double nterval I (k,l) = I (k ) a I (l) of rank (k,l) whch ntersects a (0, 1) and on whch p l (x, y) >C.Denotethe value of p l on ths nterval by p l (I(k,l) ) and the value of t l on the nterval I(l) by t l (I(l) ). Thus, t l (I(l) ) = p l (I(k,l) ) >C. The ntervals I (k,l) are embedded nto each other as ncreases and the nterval I (l) s ther common projecton onto the y-axs. Now, we consder the sequence {t l+j } j. We note that there exsts a sequence of nested P-ntervals {Îl+j } j such that (Î(l+j+1) ) (Î(l+j) t l+j+1 ) t l+j >C, j=0, 1,... (3.8) Indeed, for j = 0 we put Î(l) = I l and suppose Î(l+j) has been chosen. We have m l+j+1 1 t l+j+1 (y) =t l+j (y) + a n,qχ q (y). q=m l+j From (2.6), we get m l+j+1 1 a n,qχ q (y)dy =0. q=m l+j Î (l+j) Ths easly mples that at least for one of the ntervals I (l+j+1) Î(l+j) whch we denote by Î(l+j+1), the nequalty (3.8) holds. We formulate the next step of the proof as a separate lemma. Lemma 3.1. Under the assumpton of Theorem 3.1 and under the above notaton, suppose that for the nterval I (l),forthesequence{n },andforthe correspondng sequence of ranks {k }, whch are used n the defnton of the sequence {t l },thenequalty t l (I(l) ) >C>0 holds for all =1, 2,... Then, for any B, 0 <B<C there exst two nonntersectng P-adc ntervals Ĩ(l 1) and Î (l 1), whch are subntervals of I (l),wthrankl 1 >l, and a subsequence {n m } m of {n } such that t m l1 (Ĩ(l 1 ) ) >B and t m l1 (Î(l 1 ) ) >B for all m. 85

6 VALENTIN A. SKVORTSOV FRANCESCO TULONE Proof. Take ε = C B Pm l and accordng to (3.3), choose j 0 1and 0 such that a n,q <εfor all > 0 and q m l+j0. Then, for any such and for any y we get t l+j0 +1 (y) t l+j 0 (y) m l+j = a n,qχ q (y) q=m l+j0 <ε(m l+j0 +1 m l+j0 ) = ε(m l+j0 p l+j0 m l+j0 ) εp m l+j0 = (C B)m l+j 0 m l. (3.9) Now, suppose that for each j, 1 j j 0 + 1, and for all ntervals I (l+j) I (l) except one Î(l+j),chosenasn(3.8),wehave t l+j (I (l+j) ) B. Applyng the last nequalty wth j = j 0 and havng (3.5) n mnd, we obtan the followng estmaton t (Î(l+j 0 ) ) 1 l+j 0 = m l+j0 tî(l+j 0 )t l+j 0 (y)dy tl+j 0 (y)dy t l+j 0 (y) dy I (l) I (l)\î(l+j 0 ) l (y)dy t l+j 0 (y) dy I (l) t > C m l B I (l) \Î (l) ( 1 m l 1 m l+j0 So, t (Î(l+j0) l+j0 ) m > l+j0 (C B) m l +B. Snce by assumpton we have t l+j0 +1 (y) B for all y Î(l+j0) \ Î(l+j0+1), for the same y we get t l+j0 +1 (y) t l+j 0 (y) t l+j0 (y) t l+j0 +1 (y) > m l+j 0 (C B) + B B m l = m l+j 0 (C B). m l 86 ).

7 ON THE COEFFICIENTS OF MULTIPLE SERIES WITH RESPECT TO VILENKIN SYSTEM Ths contradcts (3.9). So, there exsts an nterval I (l+j) I (l), I (l+j) Î(l+j) wth j = j 0 or j = j 0 +1 for whch t l+j (I(l+j) ) >B. Ths nterval together wth the nterval Î(l+j) gves a par Ĩ(l1) and Î(l1) we are lookng for, wth l 1 = l + j. As the rank of chosen ntervals s bounded by l + j 0 + 1, we can take a subsequence {n m } of {n } such that the par Ĩ(l1) and Î(l1) s the same for all m > 0. Ths proves the lemma. Proceedng wth the proof of the theorem, we take a sequence C = C 0 > C 1 >C 2 > > C 2 and applyng Lemma 3.1 successvely frst for C = C 0 and B = C 1, then for C = C 1 and B = C 2 and so on, we double n each step the ntervals obtaned n the prevous step. In the qth nductve step we get 2 q dsjont ntervals I (lq) of the rank l q >l q 1 > >l 1 >land some sequence { q } so that n q s a subsequence of {n } gven by (3.4) for whch t q l q (I (lq) ) >Cq. Now, we take an nterval I (k q ) for whch a I (k q ),wherek q s the rank of n q. Then, for any q we have p q l q ( I (k q,l q) ) >Cq > C 2. (3.10) So, we get contnuum of sequences of nested ntervals {I (k q,lq) }. Hence, at least one of these sequences has a common pont (a, y 0 ) / E wth y 0 / Q P and at ths pont, by assumpton of the theorem, the seres s convergent and so, for the sequence p j defned by (3.7) we should have lm q p q l q (a, y 0 ) 0. Ths s n contradcton to (3.10). Therefore, the assumpton (3.2) s false and the theorem s proved. 4. Applcaton to the problem of recoverng the coeffcents Theorem 3.1 s essental for obtanng results on recoverng the coeffcents of convergent double Vlenkn seres by generalzed Fourer formulas. A standard method for soluton to the problem of recoverng the coeffcents n the case of many classcal orthogonal seres s based on reducng ths problem to the one of recoverng a prmtve from ts dervatve where the dervatve s defned wth respect to a properly chosen dervaton bass (see [6]). In the case of Vlenkn seres, a sutable bass s the bass formed by P-ntervals. Gven a set functon F : I R, we defne the upper and the lower P-dervatve at a pont (x, y) as D P F (x, y):= lm sup mn(k,l) respectvely. F (I (k,l) x,y ) I (k,l) x,y and F (I x,y (k,l) ) D P F (x, y):= lm nf mn(k,l) I x,y (k,l), (4.1) 87

8 VALENTIN A. SKVORTSOV FRANCESCO TULONE It s clear that D P F (x, y) D P F (x, y). If D P F (x, y) =D P F (x, y), ths common value s called P-dervatve D P F (x, y). For a complex-valued set functon F =ReF + ImF, we defne P-dervatve at a pont (x, y) as D P F (x, y) :=D P ReF (x, y)+d P ImF (x, y). The dffcultes whch should be overcome n solvng the problem of recoverng a prmtve from ts dervatve n the multdmensonal case are related to the fact that the prmtve, we want to recover, s dfferentable not everywhere but outsde some exceptonal set whch s not countable n a dmenson greater than one. We have to mpose some contnuty assumptons on the prmtve at ponts of exceptonal sets to guarantee ts unqueness. It turns out that the usual contnuty wth respect to the bass s not enough for ths purpose and we ntroduce a stronger noton of contnuty, whch we call local Saks contnuty wth respect to the bass. Defnton 4.1. We say that a P-nterval functon F s locally P-contnuous n the sense of Saks, or brefly PS-contnuous, at a pont (x, y) K f lm F ( I (k,l) ) x,y =0. (4.2) k+l In our case, the role of the exceptonal set wll be played by the set Z of ponts of the unt nterval K havng at least one P-ratonal coordnate,.e., Z := ( [0, 1) Q P ) ( QP [0, 1) ). (4.3) Now, we ntroduce a Perron type ntegral whch solves the problem of recoverng a prmtve from ts P-dervatve whch s defned only outsde the set Z. It s a generalzaton of the dyadc Perron type ntegral (P B S-ntegral) ntroduced n [9]. Propertes of ths ntegral n more detals wll be studed n [10]. Defnton 4.2. Let f be a real pont functon defned at least on K \ Z. An addtve PS-contnuous on K P-nterval functon M : I R (resp., m) s called a PS-major (resp., PS-mnor) functon of f f the lower (resp., the upper) P-dervatve satsfes the nequalty for all (x, y) K \ Z. D P M(x, y) f(x, y) (resp.d P m(x, y) f(x, y)) (4.4) Proof of the followng lemma s the same as the proof of the correspondng lemma n [9] for the dyadc case. Lemma 4.1. Let M and m be a PS-major and a PS-mnor functon for a pont- -functon f on K. Then, for each P-nterval I we have M(I) m(i). 88

9 ON THE COEFFICIENTS OF MULTIPLE SERIES WITH RESPECT TO VILENKIN SYSTEM The last lemma mples that for any functon f we have { } { } nf M(K) sup m(k) M m where nf and sup are taken over all PS-major and PS-mnor functon of f, respectvely. Ths justfes the followng defnton. Defnton 4.3. A pont-functon f : K R, defned at least on K \ Z, ssad to be PS-ntegrable on K f there exsts at least one PS-major functon and at least one PS-mnor functon of f and { } m(k) <, < nf M { M(K) } =sup m where nf and sup are taken as above. The common value s called PS-ntegral of f on K and s denoted by (PS) K f. In the same way, we can defne PS-ntegral on any P-nterval I. Drectly from the defntons we get the followng result whch shows that the PS-ntegral solves the problem of recoverng the prmtve from ts P-dervatve n the form we need. Theorem 4.1. If an addtve PS-contnuous P-nterval functon F s P-dfferentable wth D P F (x, y) =f(x, y) everywhere on K \ Z, then the functon f s PS-ntegrable on K and F s ts ndefnte PS-ntegral. In a natural way, the defnton of the PS-ntegral s extended to the case of complex-valued functons. Defnton 4.4. A pont-functon f : K C, defned at least on K \ Z, ssad to be PS-ntegrable on K f functons Ref and Imf are ntegrable. In ths case, the value of PS-ntegral s defned as (PS) K f := (PS) K Ref +(PS) K Imf. To connect the problem of recoverng a prmtve wth the problem of recoverng coeffcents of the seres (2.7), we ntroduce the followng P-nterval functon. For each P-nterval of rank (k, l) we defne ψ(i (k,l) ):= S mk,m l (x, y)dxdy. I (k,l) It s easy to check that ψ s an addtve functon on algebra generated by I. In P-adc analyss, the functon ψ s referred to as the quas-measure generated by the seres (see [4]). Snce the sum S mk,m l s constant on each I (k,l),weget S mk,m l (x, y) = 1 S I (k,l) mk,m l (x, y)dxdy = I (k,l) for any pont (x, y) nti (k,l). ψ(i (k,l) ) I (k,l) (4.5) 89

10 VALENTIN A. SKVORTSOV FRANCESCO TULONE In fact, any addtve P-nterval functon ψ defnes a Vlenkn seres for whch t s the quas-measure generated by t and (4.5) holds. So, we have one-to-one correspondence between the famly of addtve P-nterval functons and the famly of Vlenkn seres. The equalty (4.5) obvously gves a relaton between P-dfferentablty of ψ at (x, y) and convergence of the seres. In partcular, at least at the ponts (x, y) K \ Z, weget lm S m k,m l (x, y) =D P ψ(x, y), (4.6) k,l and therefore, the convergence of the seres (2.7) at such ponts to a sum f(x, y) mples P-dfferentablty of the functon ψ at (x, y) wthf(x, y) beng the value of P-dervatve. The next lemma demonstrates the mportance of Theorem 3.1 for an applcaton of the PS-ntegral to the problem of recoverng the coeffcents as t gves the requred contnuty propertes of the quas-measure. Lemma 4.2. If the coeffcents of seres (2.7) satsfy condton (3.1), thenat each pont (x, y) K the quas-measure ψ s PS-contnuous,.e., (4.2) holds everywhere on K. P r o o f. Havng n mnd that for all <m k and j < m l functons χ,j are constant on each nterval I (k,l) (x,y) of rank (k, l), we have for a fxed (x, y) (recall that (x, y) can belong to closure of I (k,l) ψ(i (k,l) (x,y) ) (x,y) ) S mk,m l (t, s) dt ds I (k,l) (x,y) m k 1,m l 1,j=0 a,j χ,j (t, s) dt ds I (k,l) (x,y) mk 1,m l 1,j=0 a,j. m k m l But the rght-hand sde expresson s the arthmetc mean of the modulus of the coeffcents a,j and t tends to zero together wth coeffcents when +j. Note that, n fact, we have proved unform PS-contnuty of ψ. Wealsonote that the above statement s not true even under assumpton of convergence everywhere on K of Vlenkn seres f the convergence s consdered wth respect to regular rectangles, for example, wth respect to cubes (see [3]). 90

11 ON THE COEFFICIENTS OF MULTIPLE SERIES WITH RESPECT TO VILENKIN SYSTEM The followng statement s essental for establshng that a gven Vlenkn seres s the Fourer seres n the sense of some general ntegral (see, for example, [6]; a proof n the one-dmensonal verson can be found n [2, Th ]). Proposton 4.1. Let some ntegraton process A be gven whch produces an ntegral addtve on I. Assume a seres of the form (2.7) s gven. Let a P-nterval functon ψ be the quas-measure generated by ths seres and (4.5) holds. Then ths seres s the Fourer seres of an A-ntegrable functon f f and only f ψ(i) =(A) f for any P-nterval I. I In vew of (4.6) and the above proposton, n order to solve the coeffcent problem t suffces to show that the quas-measure ψ generated by the Vlenkn seres s the ndefnte PS-ntegral of ts P-dervatve whch exsts at least on K \ Z. Fnally, usng all the above results, we get Theorem 4.2. If a two-dmensonal seres (2.7) s rectangular convergent to asumf everywhere n K \ Z, thenf s PS-ntegrable on K and the coeffcents of the seres are PS-Fourer-Vlenkn coeffcents of f. Proof. Take any (a, b) K \ Z. Then ntersecton of the cross { a [0, 1] } { } [0, 1] b wth Z s countable, and by Theorem 3.1, condton (3.1) holds. Then, by Lemma 4.2, the quas-measure ψ generated by the seres (2.7) s PS-contnuous everywhere on K. Moreover, the equalty (4.6) mples lm S m k,m l (x) =D P ψ(x, y) =f(x, y) k,l everywhere on K \ Z. Then, applcaton of Proposton 4.1 and Theorem 4.1 to the real and magnary parts of functons ψ and f completes the proof. REFERENCES [1] AGAEV, G. N. VILENKIN, N. YA. DZHAFARLI, G. M. RUBINSTEIN, A. I.: Multplcatve System of Functons and Harmonc Analyss on Zero-Dmensonal Groups. Èlm, Baku, (In Russan) [2] GOLUBOV, B. EFIMOV, A. SKVORTSOV, V.: Walsh Seres and Transforms: Theory and Applcatons. Nauka, Moscow 1987; Translated: Mathematcs and ts Applcatons (Sovet Seres), Vol. 64, Kluwer Academc Publshers Group, Dordrecht, [3] PLOTNIKOV, M. G.: Recovery of the coeffcents of multple Haar and Walsh seres, Real Anal. Exchange 33 (2007), [4] SCHIPP, F. WADE, W. R. SIMON, P. PAL, J.: Walsh Seres: An Introducton to Dyadc Harmonc Analyss. Adam Hlger Publ., Ltd., Brstol, [5] SKVORTSOV, V. A.: On coeffcents of convergent multple Haar and Walsh seres, Vestnk Moskov. Unv., Ser. I Mat. Mekh. 6 (1973), (In Russan); Englsh transl.: Moscow Unv. Math. Bull. 28 (1974),

12 VALENTIN A. SKVORTSOV FRANCESCO TULONE [6] Henstock-Kurzwel type ntegrals n P-adc harmonc analyss, Acta Math. Acad. Paedagog. Nyház. (N.S.) 20 (2004), [7] SKVORTSOV, V. A. TULONE, F.: Perron type ntegral on compact zero-dmensonal Abelan groups, Vestnk Moskov. Unv., Ser. I Mat. Mekh. 63 (2008), (In Russan); Englsh transl.: Moscow Unv. Math. Bull. 63 (2008), [8] Kurzwel-Henstock type ntegral on zero-dmensonal group and some of ts applcatons, Czechoslovak Math. J. 58 (2008), [9] Multdmensonal dyadc Kurzwel-Henstock- and Perrop-type ntegral n the theory of Haar and Walsh seres, J. Math. Anal. Appl. 421 (2015), [10] TULONE, F. SKVORTSOV, V. A.: Multdmensonal P-adc ntegral n some problems of harmonc analyss, Mnmax Theory Appl. 2 (2017), Receved November 11, 2016 Valentn A. Skvortsov Department of Mathematcs Moscow State Unversty Moscow RUSSIA E-mal: vaskvor2000@yahoo.com Francesco Tulone Department of Mathematcs and Computer Scences Palermo Unversty Palermo ITALY E-mal: francesco.tulone@math.unpa.t 92