HENSTOCK-KURZWEIL TYPE INTEGRATION OF RIESZ-SPACE-VALUED FUNCTIONS AND APPLICATIONS TO WALSH SERIES

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1 Rel Anlysis Exchnge Vol. 29(1), 2003/2004, pp A. Boccuto, Deprtment of Mtemtics nd Informtics, Universit di Perugi, vi Vnvitelli 1, Perugi, Itly. emil: V. A. Skvortsov, Instytut Mtemtyki, Akdemi Bydgosk, Bydgoszcz, Polnd nd Deprtment of Mthemtics, Moscow Stte University, Moscow , Russi. emil: HENSTOCK-KURZWEIL TYPE INTEGRATION OF RIESZ-SPACE-VALUED FUNCTIONS AND APPLICATIONS TO WALSH SERIES Abstrct Some versions of Henstock-Kurzweil integrl with respect to different derivtion bses for functions with vlues in Dedekind complete Riesz spces re studied. Fundmentl Theorem of Clculus re proved for these integrls nd n ppliction to Wlsh series is given. 1 Introduction. In this pper we extend some definitions nd results of [4] relted to Henstock- Kurzweil integrtion of Riesz-spce-vlued functions, to the cse of integrtion with respect to derivtion bsis. Considering wide clss of Riesz spces we prove for this type of Henstock-Kurzweil integrls some versions of the Fundmentl Theorem of Integrl Clculus. In prticulr cse of the dydic bsis we pply this theorem to the problem of recovering the coefficients of Wlsh series from their sums by generlized Fourier formuls in which integrls re understood in the bove sense. Key Words: Riesz spces, Henstock-Kurzweil integrtion, derivtion bsis, Fundmentl Theorem of Clculus, intervl functions, Wlsh series. Mthemticl Reviews subject clssifiction: 28B15, 28B05, 28A39, 42C10, 42C25, 46G10 Received by the editors April 23, 2002 Communicted by: Peter Bullen This work is supported by RFFI nd NSh

2 420 A. Boccuto nd V. A. Skvortsov In the rel-vlued cse the Henstock-Kurzweil integrls with respect to bses were considered in [22] (see lso [6] nd [15]). For dydic Henstock- Kurzweil integrl nd for its ppliction to Wlsh series with rel coefficients see [16], [19], [20]. The pper is structured s follows. In Section 2 we recll some fundmentl concepts relted to Riesz spces nd derivtion bses, nd introduce some notions of continuity nd differentibility of Riesz-spce-vlued intervl-functions with respect to bses. In Section 3 we investigte some bsic properties of the Henstock-Kurzweil integrl for Riesz-spce-vlued functions with respect to bstrct bses. In Section 4 we prove some version of the Fundmentl Theorem of Clculus for this type of integrl generlizing some results obtined in [4]. Finlly, in Section 5, we consider Wlsh series with coefficients from Riesz spces nd give n ppliction of the bove mentioned theory to the problem of recovering the coefficients of n order convergent Wlsh series from its sum. 2 Preliminries. Let N, R, R + be the sets of ll nturl, rel nd positive rel numbers respectively, nd let R be Dedekind complete Riesz spce. We dd to R two extr elements, + nd, extending ordering nd opertions in nturl wy, nd denote R = R {+, }. A nonempty set T R is sid to be order bounded from bove if there exists s 1 R such tht s 1 t for ll t T, order bounded from below if there exists s 2 R such tht s 2 t for ll t T, order bounded if it is order bounded both from bove nd from below. By convention, we will sy tht the supremum of ny not bounded from bove nonempty subset of R is + nd the infimum of ny not bounded from below nonempty subset of R is. A net (p β ) β Λ in R, where (Λ, ) is directed set, is clled (o)-net if it is decresing (i.e. p β1 p β2 whenever β 1, β 2 Λ, β 1 β 2 ) nd inf β p β = 0. In prticulr we get the definition of (o)-sequence when Λ = N. Definition 2.1. We sy tht net (r β ) β order converges (or in short (o)- converges ) to r R if there exists n (o)-net (p β ) β (with the sme directed set Λ) stisfying r β r p β for ech β Λ. We shll write in this cse r = (o) lim β r β. In the cse Λ = N we get the definition of (o)-convergent sequence. The (o)-convergence of Riesz-spce-vlued series is defined in n obvious wy by the (o)-convergence of its prtil sum. It is known (see [1]) tht n order bounded net (r β ) β in Dedekind complete Riesz spce is (o)-convergent to r if nd only if (o) lim inf β r β =

3 Integrtion of Riesz-Spce-Vlued Functions 421 (o) lim sup β r β. Recll tht ech Dedekind complete Riesz spce is Archimeden, i.e. if 0 nr p for ll n = 1, 2,... nd some positive elements r nd p of R then r = 0. For Archimeden Riesz spces nother type of convergence cn be considered. Definition 2.2. We sy tht net (r β ) β Λ (r)-converges ( reltively uniform converges s in [13] or converges with respect to regultor s in [24]) to r R if there exists positive element u R ( regultor) such tht, for every ε > 0, there exists β 0 Λ so tht r β r εu for ll β β 0. In the cse Λ = N we get the definition of (r)-convergent sequence. It is esy to check tht in n Archimeden Riesz spce (r)-convergence implies (o)-convergence. The converse is true only under some dditionl ssumption (see Proposition 2.4 below). A Riesz spce stisfies property σ if, given ny sequence (u n ) n in R with u n 0 n N, there exists sequence (λ n ) n of positive rel numbers, such tht the sequence (λ n u n ) n is bounded in R. Definition 2.3. A Dedekind complete Riesz spce is sid to be regulr if it stisfies property σ nd if for ech sequence (r n ) n in R, order convergent to zero, there exists sequence (l n ) n of positive rel numbers, with lim n l n = +, such tht the sequence (l n r n ) n is order convergent to zero. Proposition 2.4. ([14], Theorem 1, p. 350) In regulr Riesz spce the (o)-convergence is equivlent to the (r)-convergence. An intervl is lwys compct nondegenerte subintervl of R. If E R, then E denotes the Lebesgue mesure of E. A collection of intervls is clled nonoverlpping if their interiors re pirwise disjoint. Throughout this pper, [, b] is fixed intervl of R nd I is the fmily of ll subintervls of [, b]. A bsis on [, b] is, by definition, ny subset B of I [, b] such tht (I, x) B implies x I (Note tht this definition of bsis is little bit less generl thn in [22] or in [15], but this level of generlity is enough for our purpose here). Given bsis B, n intervl I is clled B-intervl if (I, x) B, for some x I. We ssume tht [, b] is B-intervl. For set E [, b], E, put B(E) = {(I, x) B : I E}, B[E] = {(I, x) B : x E}. (1) For E [, b] we denote by E the directed set of ll positive relvlued functions defined on E nd endowed with the nturl ordering: given

4 422 A. Boccuto nd V. A. Skvortsov two functions δ 1 nd δ 2, we sy tht δ 1 δ 2 if nd only if δ 1 (x) δ 2 (x) for ll x E. For the set [,b] we shll often write simply. A function δ E is often referred to s gge on E. For given gge δ we denote B δ = {(I, x) B : I (x δ(x), x + δ(x))}. (2) We note tht B δ is lso bsis on [, b], so tht it mkes sense to define the sets B δ (E) nd B δ [E], similrly s in (1). We sy tht bsis B is Vitli bsis if for ny δ nd x [, b] the set B δ [{x}] is nonempty. Let E [, b]. A finite subset Π of B[E] is clled B-decomposition on E if for every distinct elements (I, x ) nd (I, x ) of Π, the corresponding intervls I nd I re nonoverlpping. If I = [, b], for Π B, then (I,x) Π we sy tht Π is B-prtition of [, b]. A B δ -decomposition (or B δ -prtition) is clled δ-fine. Throughout this pper we shll ssume tht ech bsis B considered here is Vitli bsis nd hs the following two prtitioning properties : () for ny B-intervl I nd ny gge δ on I there exists B δ -prtition of I; (b) If I 1 nd n I 2 re B-intervls nd I 1 I 2 then I 2 \ I 1 = I i where I i, i = 3,..., q, re some nonoverlpping B-intervls. One of the simplest exmple of Vitli bsis hving bove prtitioning properties is the so clled full intervl bsis on [, b] which consists of ll elements (I, x) I [, b] such tht x I (Prtitioning property () for this bsis is proved for exmple in [8] nd in [12]). Another exmple [ is the dydic i bsis D consisting of ll pirs (I, x), where x I nd I = 2 k, i + 1 ) 2 k, for k N nd i = 0, 1,..., 2 k 1. This bsis is widely used in hrmonic nlysis (see [19], [20], [7], [6]). We consider this bsis in Section 6. If f : [, b] R nd Π = {(J i, ξ i ) : i = 1,..., q} is prtition of [, b], the q sum f(ξ i ) J i will be denoted by S(f, Π) nd will be clled Riemnn sum ssocited with Π. From now on in this pper, the nottion Π will lwys be reserved to men prtition or decomposition. We now formulte suitble concepts of continuity nd differentibility with respect to bsis, for Riesz-spce-vlued intervl functions (For corresponding i=3

5 Integrtion of Riesz-Spce-Vlued Functions 423 concepts concerning Riesz-spce-vlued point functions nd the full intervl bsis, see [4]). Let R be Dedekind complete Riesz spce, B bsis on [, b] nd τ n R-vlued B-intervl function. We sy tht τ is dditive if τ(i I ) = τ(i ) + τ(i ) whenever I nd I re ny two nonoverlpping B-intervls nd I I is lso B-intervl. A function τ is sid to be (o)-continuous t point x 0 [, b] with respect to the bsis B if inf δ [sup { τ(i) : (I, x 0) B δ [{x 0 }]} ] = 0. (3) Given E [, b], we sy tht the function τ is (o)-continuous on E if it is (o)-continuous t every point x 0 E. We sy tht τ is (u)-differentible on E with respect to the bsis B if there exists function g : E R such tht inf δ E [ sup { τ(i) I } ] g(x) : (I, x) B δ[e] = 0. (4) The function g in (4) will be clled the (u)-derivtive with respect to B (or simply derivtive) of τ on E. It is esy to prove tht the (u)-derivtive is determined uniquely. 3 The Henstock-Kurzweil Integrl with Respect to Bsis. We now introduce Henstock-Kurzweil-type integrl with respect to bsis for Riesz-spce-vlued functions (for the rel cse, see [9], [22], [15], [6]). The definition below is generliztion of definition from [4] (For other versions of definitions of Henstock-Kurzweil integrl for Riesz-spce-vlued functions see [18], [14]). Definition 3.1. Let B be fixed bsis on [, b]. We sy tht f : [, b] R is Henstock-Kurzweil integrble on B-intervl E [, b] with respect to B (in brief, H B -integrble ) if there exists n element Y R such tht inf sup δ [,b] f(x) I Y : Π is B δ-prtition of E = 0. (5) (I,x) Π In this cse we write (H B ) E f = Y.

6 424 A. Boccuto nd V. A. Skvortsov It is esy to see tht the element Y in (5) is uniquely determined nd tht the set of ll H B -integrble functions on E is liner spce. Proposition 3.2. Let R be Dedekind complete Riesz spce, stisfying property σ, Q [, b] be countble set, nd f : [, b] R be function, such tht f(x) = 0 for ll x [, b] \ Q. Then (H B ) b f = 0. Proof. Let Q = {t n : n N} nd f(t n ) = u n for ll n N; without loss of generlity, we cn suppose tht u n 0 n N. As R being Dedekind complete is Archimiden, in order to prove the Proposition, it is enough to show tht there exists n element z R, z 0, z 0, such tht ε > 0 there exists gge δ on [, b] such tht S(f, Π) ε z (6) for ech B δ -prtition Π of [, b] (tht mens tht we re proving in fct (r)- convergence nd therefore lso (o)-convergence to zero of sup{ S(f, Π) : Π B δ }). By property σ, in correspondence with the sequence (u n ) n, there exists sequence (λ n ) n of positive rel numbers nd there exists z R such tht 0 λ n u n z for ll n N. Fix now n rbitrry ε > 0 nd set δ(t n ) = ε 2 n 2 λ n n N, nd δ(x) = 1 if x Q. For every B δ -prtition Π = {(J 1, x 1 ),..., (J q, x q )} of [, b] we hve: 0 S(f, Π) = q f(x i ) J i ( ) ( ) u i J i 2 δ(x i ) u i n=1 x i=t n n=1 x i=t n ( ) 4 ε 2 n 2 λ n u n 4 ε 2 n 2 z = ε z, n=1 proving (6) nd the ssertion. n=1 Now we stte the Cuchy criterion for H B -integrbility. Theorem 3.3. Under the bove nottion, necessry nd sufficient condition for H B -integrbility of f on B-intervl E is tht inf δ E [sup { S(f, Π 1 ) S(f, Π 2 ) : Π 1, Π 2 re B δ -prtitions of E}] = 0.

7 Integrtion of Riesz-Spce-Vlued Functions 425 Proof. The necessry prt is strightforwrd. We now turn to the sufficient prt. For every δ E, denote Ψ(δ) = {Π : Π is B δ -prtition of E}, (δ) = sup {S(f, Π) : Π Ψ(δ)}, b(δ) = inf {S(f, Π) : Π Ψ(δ)}, We hve: r(δ) = sup { S(f, Π 1 ) S(f, Π 2 ) : Π 1, Π 2 Ψ(δ)}. Since by hypothesis S(f, Π 1 ) r(δ) + S(f, Π 2 ) Π 1, Π 2 Ψ(δ). (7) (o) lim δ E r(δ) = inf δ E r(δ) = 0, (8) then there exists gge δ 0 E such tht r(δ) R δ E, δ δ 0. Tking in (7) first the supremum on the left over Π 1 we observe tht (δ) R for δ δ 0 with δ 0 chosen bove. Then we tke infimum over Π 2 to get noting tht b(δ) R for the sme δ. Moreover, we hve: (δ) r(δ) + b(δ) (9) S(f, Π 1 ) S(f, Π 2 ) (δ) b(δ) Π 1, Π 2 Ψ(δ), nd tking the supremum we obtin In view of (9) this gives r(δ) (δ) b(δ). r(δ) = (δ) b(δ). (10) We observe tht the nets ((δ)) δ E nd (b(δ)) δ E re monotone decresing nd incresing respectively, nd thus, thnks lso to (8) nd (10), there exists in R (o)-limit Y (o) lim δ E (δ) = (o) lim δ E b(δ). There exists n (o)-net (z δ ) δ E, such tht z δ inf {S(f, Π) : Π Ψ(δ)} Y sup {S(f, Π) : Π Ψ(δ)} Y z δ

8 426 A. Boccuto nd V. A. Skvortsov for every δ E. From this it follows tht S(f, Π) Y z δ Π Ψ(δ), nd hence 0 sup { S(f, Π) Y : Π Ψ(δ)} z δ for ny δ E. This completes the proof. Proposition 3.4. If [, b], [, c] nd [c, b] re B-intervls nd f is H B -integrble on [, c] nd on [c, b], then f is lso H B -integrble on [, b] nd b c b (H B ) f = (H B ) f + (H B ) f. c Proof. For every δ 1 [,c] nd δ 2 [c,b] we cn define δ [,b] putting min(δ 1 (x), c x) if x < c, δ(x) = min(δ 2 (x), x c) if x > c, min(δ 1 (x), δ 2 (x)) if x = c. Note tht with this δ we hve B δ [[, c)] = B δ ([, c)) nd B δ [(c, b]] = B δ ((c, b]) (see nottions (1) nd (2)). Then ech sum S(f, Π) relted to ny B δ -prtition of [, b] cn be written s S(f, Π) = S(f, Π 1 ) + S(f, Π 2 ), where Π 1 nd Π 2 re two suitble B δ1 - nd B δ2 -prtitions of [, c] nd [c, b] respectively (In cse of need we re splitting here the term f(c)(β α) of the sum S(f, Π) into two terms f(c)(c α) nd f(c)(β c)). By this we get sup Π sup Π 1 ( S(f, Π) (H B ) S(f, Π 1) (H B ) c c b f + (H B ) c f) f + sup Π 2 S(f, Π 2) (H B ) b c f,

9 Integrtion of Riesz-Spce-Vlued Functions 427 nd so 0 inf δ [,b] inf δ 1 [,c] + inf δ 2 [c,b] [ ( sup S(f, Π) (H B ) Π [ sup S(f, Π 1) (H B ) Π 1 [ sup S(f, Π 2) (H B ) Π 2 c c c b f + (H B ) ] f ] f = 0. c ] f) Proposition 3.5. If f is H B -integrble on B-intervl [, b], then f is lso H B -integrble on ny B-intervl I [, b]. Proof. Note tht, ccording to the prtitioning properties () nd (b) of bsis B, ny B δ -prtition of I with δ [,b] cn be extended to B δ -prtition of [, b]. Hving two B δ -prtitions of I, sy Π 1 nd Π 2, we cn extend ech of them by the sme prtition of [, b] \ I getting two B δ -prtitions Π 1 nd Π 2 of [, b]. Then we get S(f, Π 1 ) S(f, Π 2 ) = S(f, Π 1) S(f, Π 2) sup { S(f, Π 1) S(f, Π 2) : (11) Π 1, Π 2 re B δ -prtitions of [, b]} Tking the supremum over ll pirs (Π 1, Π 2 ) of B δ -prtitions of I on the left nd then the infimum over ll δ [,b] on both sides of (11) we get, using the necessry prt of Theorem 3.3 for f on [, b], tht the left side is equl to zero. Therefore inf [{sup{ S(f, Π 1) S(f, Π 2 ) : δ Π 1, Π 2 re B δ prtitions of I}] = 0, nd the ssertion follows by the sufficient prt of Theorem 3.3. It follows from the lst two Propositions tht for ny H B -integrble function f : [, b] R the indefinite H B -integrl is defined s n dditive R-vlued B-intervl function on the fmily of ll B-intervls in [, b]. We shll denote it by F (I) = (H B ) f. (12) I We now prove the following version of the Henstock Lemm (for similr versions existing in the literture, see lso [14], Lemm 12, pp ):

10 428 A. Boccuto nd V. A. Skvortsov Lemm 3.6. If f is (H B )-integrble on [, b] nd F is s in (12), then inf δ sup (I,x) Π f(x) I F (I) : Π is B δ -decomposition of [, b] = 0. (13) Proof. By (5), there exists n (o)-net (p δ ) δ such tht sup (I,x) Π b f(x) I (H B ) f : Π is B δ-prtition of [, b] p δ (14) for every δ. Let δ nd Π = {(J i, ξ i ), i = 1,..., q} be B δ -prtition of [, b]. By Proposition 3.5, f is integrble on J i, i = 1,..., q. Thus for ech i there exists n (o)-net(p δi ) δi Ji such tht sup (I,x) Π f(x) I (H B ) J i f : Π is B δ-prtition of J i p δ i. (15) Now, fix rbitrry L {1,..., q}. Let Π i be B δ -prtition of J i, nd set Π 0 = {(J i, ξ i ) Π : i L} ( i L Π i ). Hving fixed the chosen δ we cn suppose tht δ i (x) δ(x) for ll x J i. Then Π 0 is B δ -prtition of [, b] nd hence by (14) S(f, Π 0) (H B ) b f p δ.

11 Integrtion of Riesz-Spce-Vlued Functions 429 Thus we hve 0 J i f(ξ i ) (H B ) f i L i L J i b = S(f, Π 0) (H B ) f + (H B ) f S(f, Π i ) i L J i i L b S(f, Π 0) (H B ) f + (H B ) f S(f, Π i ) i L J i i L b S(f, Π q 0) (H B ) f + (H B) f S(f, Π i ) J i q p δ + p δi. Considering this inequlity for fixed δ nd for ny δ i s we cn pss to the (o)-limit s the δ i s shrink to zero, to get 0 f(ξ i ) J i (H B ) f i L i L J i p δ. (16) We now observe tht, since R is Dedekind complete Riesz spce, by virtue of the Med-Ogswr-Vulikh representtion theorem (see [?]) there exists compct extremely disconnected topologicl spce Ω, such tht R cn be embedded Riesz isomorphiclly s solid subset of C (Ω) = {f : Ω R : f is continuous, nd the set {ω Ω : f(ω) = + } is nowhere dense in Ω}. From (16), for ll ω Ω nd for ech δ-fine prtition Π = {(J i, ξ i ) : i = 1,..., q} of [, b], we hve (using the sme nottions for elements of R nd for the corresponding elements of C (Ω)) 0 f(ξ i ) J i (H B ) f i L i L J i (ω) p δ(ω) for ll L {1, 2,..., q} (with the convention tht the sum long the empty set of ny quntity is zero). Fix now ω Ω. If p δ (ω) = +, there is nothing to prove. Suppose tht p δ (ω) R. Let L [resp. L ] be the sets of ll indices i {1,..., q} such tht [ f(ξ i ) J i (H B ) J i f ] (ω) 0 [ resp. < 0].

12 430 A. Boccuto nd V. A. Skvortsov We hve: q 0 f(ξ i) J i (H B ) = [ f(ξ i ) J i (H B ) i L 2 p δ (ω) J i J i f (ω) f ] [ ] (ω) i L f(ξ i ) J i (H B ) f (ω) J i for ech ω Ω. (13). Coming bck to the corresponding elements of R we get 4 The Fundmentl Theorem of Clculus for the H B - Integrl. In this section we prove two versions of the Fundmentl Theorem of Clculus for the H B -integrl. The first one refers to ny Dedekind complete Riesz spce while in the second one we put n dditionl restriction on the considered Riesz spce ssuming tht it is regulr (For similr theorems existing in other bstrct settings, see [21] nd [23], nd for the H-integrl in the rel cse see [17], [10] nd [12]). Note tht if (X, H, µ) is mesure spce with µ positive, σ-dditive nd σ-finite, then the spces L 0 (X, H, µ) nd L p (X, H, µ), with 1 p < +, re regulr; furthermore the spce of ll rel sequences, with the usul coordintewise ordering, is regulr (see [13], pp ). So the clss of spces, for which Theorem 4.2 is vlid, is rther wide. Theorem 4.1. If R is Dedekind complete Riesz spce, B is bsis nd τ is B-intervl R-vlued function, (u)-differentible with respect to B on [, b] with derivtive τ, then τ is H B -integrble on [, b], nd b (H B ) τ = τ([, b]). Proof. By (u)-differentibility of τ in [, b] (see (4)), there exists n (o)-net (p δ ) δ, such tht { } τ(i) sup τ (x) I : (I, x) B δ[ [, b] ] p δ δ. (17) Choose δ-fine prtition Π = {(I i, x i ) : i = 1,..., q} of [, b], δ. From

13 Integrtion of Riesz-Spce-Vlued Functions 431 (17) we get: 0 S(τ q, Π) τ([, b]) = [ I i τ (x i ) τ(i i )] q { } I i τ(i i ) τ (x i ) I i ( q ) I i p δ = (b ) p δ. Theorem 4.2. Let R be regulr Riesz spce, B fixed bsis, f : [, b] R nd let τ be R-vlued B-intervl function, such tht for some countble set Q [, b] the function f is the (u)-derivtive of τ on [, b] \ Q with respect to B nd τ is (o)-continuous on Q with respect to B. Then f is H B -integrble in [, b], nd b (H B ) f = τ([, b]). Proof. Let Q {x n : n N}. As R stisfies property σ (thnks to regulrity, see Definition 2.3), then without loss of generlity we cn ssume tht f(x n ) = 0 for ll n, becuse this will not chnge the vlue of the considered integrl (see Proposition 3.2). Let Q c = [, b] \ Q. We shll use Proposition 2.4 now. As f is the (u)- derivtive of τ in Q c, then there exists n element u 0 of R, u 0, such tht ε > 0 gge ζ Q c cn be found such tht, if Π = { (I i, x i ) : i = 1,..., q} is B ζ [Q c ]-decomposition, then I i f(x i ) τ(i i ) I i ε u for ll i = 1,..., q. Moreover, by (o)-continuity of τ in Q (see (3)) nd since (o)- nd (r)- convergence coincide in the considered cse, for ech n N there exists positive nonzero element u n of R such tht η > 0, δ n {xn}: sup{ τ([u, v]) : x n δ n (x n ) u x n v x n + δ n (x n )} η u n. (18) By property σ, there exists sequence (λ n ) n in R + nd positive nonzero element w R, such tht λ n u n w for ll n N. Now, we use (18) with η = ε 2 n λ n nd obtin the corresponding δ n. In such wy we hve obtined

14 432 A. Boccuto nd V. A. Skvortsov for ech fixed ε > 0, gge δ Q Q defined by setting δ Q (x) δ n(x) if x = x n (n N) with sup{ τ([u, v]) : x n δ Q (x n ) u x n v x n + δ Q (x n )} ε 2 n λ n u n. For ech ε > 0 we define now gge δ on [, b] by putting ζ(x) if x Q c, δ(x) = δ Q (x) if x Q. For every B δ -prtition Π of [, b], Π = {(I i, x i ), i = 1,..., q}, we hve [ q ] 0 I i f(x i ) τ([, b]) q = { I i f(x i ) τ(i i )} { I i f(x i ) τ(i i )} x i Q + τ(i i ) x i Q ε u (b ) + ε 2 n λ n u n ε u (b ) + εw. From this the ssertion follows. n=1 5 Applictions to Wlsh Series. In this section we consider Wlsh series with coefficients from Riesz spce nd give n ppliction of Theorem 4.2 to the problem of recovering the coefficients of convergent Wlsh series from its sum by generlized Fourier formuls. The integrl in these formuls will be the Henstock-Kurzweil integrl for Rieszspce-vlued functions defined with respect to the dydic bsis D. We begin with the following: Lemm 5.1. Let ( n ) n be sequence (o)-convergent to zero in Dedekind complete Riesz spce R. Then the sequence (σ j ) j of its rithmeticl mens lso (o)-converges to zero.

15 Integrtion of Riesz-Spce-Vlued Functions 433 Proof. We hve, for ech nturl number n, n r n, where (r n ) n is n (o)-sequence of elements of R. For k N nd k! j < (k + 1)! we get σ j = ( j ) ( j ) 1/j i 1/j i 1/j (k 1)! r i + (k 1)! r 1 /k! + (j (k 1)!)r (k 1)! /j r 1 /k + r (k 1)!. j i=(k 1)!+1 So we hve σ j p j, where (p j ) j is n (o)-sequence defined by p j = r 1 /k + r (k 1)! for k! j < (k+1)!, j 2 (Note tht (r 1 /k) k is n (o)-sequence becuse R is supposed to be Dedekind complete nd therefore Archimeden). To define the Wlsh functions (see [2], [7] for detils) we use dydic expnsions of nturl numbers s well s those of rel numbers of the hlf-open intervl [0, 1). Let n = j=0 ε j2 j with ε j = 0 or 1, nd x = j=0 x j2 j 1 with x j = 0 or 1, with stipultion tht for the dydic rtionls x we use only finite expnsions. With this nottion we put w n (x) = ( 1) P j=0 εjxj, n N, x [0, 1). Note tht for n 2 k the functions w n re constnt on ech intervl k i, where [ i k i = 2 k, i + 1 ) 2 k, k N, i = 0, 1,..., 2 k 1. Let S n = n 1 j=0 be the prtil sums of Wlsh series r i j w j (19) j w j (20) j=0 with coefficients j belonging to Dedekind complete Riesz spce R. Becuse of the bove mentioned property of Wlsh functions, the sums S n re constnt for n < 2 k on ech intervl k i. For these prtil sums we shll consider pointwise (o)-convergence (order convergence) s well s the following (u)-convergence on set.

16 434 A. Boccuto nd V. A. Skvortsov Definition 5.2. Let Λ be ny nonempty set, R be ny (rbitrry) Dedekind complete Riesz spce nd D = N Λ. We sy tht the sequence of R-vlued functions (S n (x)) n, x Λ, (u)-converges to the function S : Λ R (with respect to order convergence) if there exists n (o)-net (p ν ) ν D such tht ν D we hve: sup{ S n (x) S(x) : x Λ, n ν(x)} p ν. Note tht (u)-convergence of sequence on set implies (o)-convergence of this sequence t ech point of the set. The crucil step to the solution of the coefficient problem for the Wlsh series is to observe tht the integrl S k 2 k defines n dditive intervl function ψ on the fmily D of the dydic intervls (As S 2 k is constnt on k i, i k N, i = 0, 1,..., 2 k 1, the integrl here cn be understood in the usul Riemnn sense (see [5])). To prove this it is enough to show tht where ψ( k i ) = ψ( k+1 2i ) + ψ( k+1 2i+1 ), k i = k+1 2i k+1 2i+1, k N, i = 0, 1,..., 2k 1. Note tht, if x k+1 2i, then x + 1 k+1 2k+1 2i+1. The function w n, for 2 k n 2 k+1 1, is constnt on ech k+1 l x k+1 2i then ( w n x + 1 ) 2 k+1 = w n (x). nd if This implies for x k+1 2i S 2 k+1 ( x + 1 ) ( 2 k+1 S 2 k x + 1 ) 2 k+1 = (S 2 k+1(x) S 2 k(x)).

17 Integrtion of Riesz-Spce-Vlued Functions 435 Then + = + = ψ( k+1 2i ) + ψ( k+1 2i+1 ) = S 2 k+1 (x) dx k+1 2i S 2 k+1 (x) dx = (S 2 k (x) + (S 2 k+1 (x) S 2 k (x))) dx k+1 2i+1 k i k+1 2i k i S 2 k (x) dx + ( ( S 2 k+1 k+1 2i k i x k+1 S 2 k (x) dx = ψ ( k i ). (S 2 k+1 (x) S 2 k (x)) dx ) ( S 2 k x + 1 )) 2 k+1 dx As the sum S 2 k is constnt on ech k i, then S 2 k (x) = ψ ( ) k i k i, where x k i. (21) It follows from this formul tht, if the Wlsh series is (o)-convergent t dydic-irrtionl point x, then the function ψ is (o)-differentible with respect to the bsis D t this point nd if the Wlsh series is (u)-convergent on some set of dydic-irrtionl points, then the function ψ is (u)-differentible with respect to D on the sme set. Proposition 5.3. If the coefficients of Wlsh series form sequence (o)- convergent to zero, then the corresponding function ψ is (o)-continuous with respect to the bsis D t ech point of [0, 1]. Proof. Under the sme nottions s bove, we hve: 2 ψ( k i ) = S 2 k k 1 2 k 1 k i j = 1/2 k j. k i The lst expression (o)-converges to zero s k + by Lemm 5.1. We re redy now to prove the following theorem on recovering the coefficients of Wlsh series from its sum. Theorem 5.4. If R is regulr Riesz spce nd Wlsh series (20) is (u)- convergent to function f on set [0, 1) \ E, where E is countble subset of [0, 1), then f is H D -integrble on [0, 1] nd the series (20) is the Fourier series of f in the sense of the H D -integrl. j=0 j=0

18 436 A. Boccuto nd V. A. Skvortsov Proof. Our series being (u)-convergent is lso (o)-convergent on [0, 1) \ E. Note tht the (o)-convergence of Wlsh series t lest t one point implies tht the coefficients of this series (o)-converge to zero. Then the function ψ defined for our series is (o)-continuous with respect to the bsis D everywhere on [0, 1] ccording to Proposition 5.3. Denote by Q the set of dydic-rtionl points. It follows from the definitions of (u)-convergence, (u)- differentibility nd from equlity (21) tht the function ψ is (u)-differentible with (u)-derivtive f in [0, 1) \ (E Q). So we cn pply to functions ψ nd f Theorem 4.2 to get β (H D ) f = ψ([α, β]) α for ny dydic intervl [α, β] = k i. Note tht for n < 2k the coefficients n re the Fourier coefficients of the prtil sum S 2 k. Then, denoting by w ni the vlue of the function w n on k i, we get n = = k 1 i=0 S 2 kw n = w ni ψ ( k i This completes the proof. References 2 k 1 i=0 k i 2 ) k 1 = i=0 S 2 kw n = 2 k 1 j=0 w ni (H D ) k i w ni k i S 2 k f = (H D ) 1 0 fw n. [1] Ch. D. Aliprntis nd K.C. Border, Infinite Dimensionl Anlysis, (1994), Springer-Verlg. [2] K. G. Beuchmp, Wlsh functions nd their pplictions, (1975), Acdemic Press, London. [3] S. J. Bernu, Unique representtion of Archimeden lttice groups nd norml Archimeden lttice rings, Proc. London Mth. Soc., 15 (1965), [4] A. Boccuto, Differentil nd Integrl Clculus in Riesz spces, Ttr Mountins Mth. Publ., 14 (1998), [5] A. Boccuto nd A. R. Smbucini, On the De Giorgi-Lett integrl with respect to mens with vlues in Riesz spces, Rel Anl. Exch., 21 (1) (1995/6),

19 Integrtion of Riesz-Spce-Vlued Functions 437 [6] B. Bongiorno, L. Di Pizz nd V. A. Skvortsov, On Vritionl Mesures Relted to Some Bses, J. Mth. Anl. Appl., 250 (2000), [7] B. Golubov, A. Efimov nd V. A. Skvortsov, Wlsh series nd trnsforms, (1991), Kluwer Acdemic Publishers, Dodrecht-Boston-London. [8] R. A. Gordon, The Integrls of Lebesgue, Denjoy, Perron, nd Henstock, Grdute Studies in Mthemtics, 4, (1994), Amer. Mth. Soc., Providence. [9] R. Henstock, The Generl Theory of Integrtion, (1991), Clrendon Press, Oxford. [10] P. Y. Lee, Lnzhou Lectures on Henstock integrtion, (1989), World Scientific Publishing Co. [11] P. Y. Lee nd R. Výborný, Kurzweil-Henstock Integrtion nd the Strong Lusin Condition, Boll. Un. Mt. Itl., 7-B (1993), [12] P. Y Lee nd R. Výborný, The integrl: An esy pproch fter Kurzweil nd Henstock, (2000), Cmbridge Univ. Press. [13] W. A. J. Luxemburg nd A. C. Znn, Riesz Spces,I, (1971), North- Hollnd Publishing Co. [14] P. MCGILL, Integrtion in vector lttices, J. Lond. Mth. Soc., 11 (1975), [15] K. M. Ostszewski, Henstock integrtion in the plne, Mem. Amer. Mth. Soc., 353, (1986), A. M. S. Providence. [16] A Pcquement, Détermintion d une fonction u moyen de s dérivée sur un réseu binire, C. R. Acd. Sci. Pris, 284, A (1977), [17] W. F. Pfeffer, Lectures on Geometric Integrtion nd the Divergence Theorem, Rend. Ist. Mt. Univ. Trieste, 23 (1991), [18] B. Riečn nd T. Neubrunn, Integrl, Mesure nd Ordering, (1997), Kluwer Acd. Publ. [19] V. A. Skvortsov, Vrition nd vritionl mesures in integrtion theory nd some pplictions, Journl of Mth. Sci., 91 (5) (1998), [20] V. A. Skvortsov, Some properties of dydic primitives, in Lecture Notes Mth., Springer-Verlg 1419 (1990),

20 438 A. Boccuto nd V. A. Skvortsov [21] Á. Száz, The fundmentl Theorem of Clculus in n bstrct setting, Ttr Mountins Mth. Publ., 2 (1993), [22] B. Thomson, Derivtion bses on the rel line,i, II, Rel Anlysis Exchnge, 8 (1-2) ( ), , [23] M. Vrábelová, The fundmentl theorem of Clculus in ordered spces, Act Mth., (Nitr) 3 (1998), [24] B. Z. Vulikh, Introduction to the theory of prtilly ordered spces, (1967), Wolters - Noordhoff Sci. Publ., Groningen.

21 439

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