Integration of functions with values in complex Riesz ... - Dmi Unipg

Integration of functions with values in complex Riesz spaces setting and some application in harmonic analysis A. Boccuto

1

∗

V. A. Skvortsov

†

F. Tulone

‡

Introduction

The problem of recovering the coefficients of orthogonal series, which we consider here, was solved in the classical trigonometric case by Denjoy who introduced in [9] an integration process, totalization T2s , so powerful that the sum of any everywhere convergent series P+∞ called 2πinx a e is integrable and the coefficients can be computed by generalized Fourier forn=−∞ n mulas in which this integral is used. The difficulty of the Denjoy totalization process which involves a transfinite sequence of operations, led other authors to look for an easier solution of the coefficient problem. J. Marcinkiewicz and A. Zygmund, J. C. Burkill, R. D. James and some other authors developed the theory of Perron-type integration reducing the problem to the one of recovering a function from its second order symmetric derivative (see [26] for details). The latest step in this direction was done by D. Preiss and B. Thomson in [17] who produced a Henstock-Kurzweil type integral which integrates approximate symmetric derivatives. Later the same coefficient problem was considered for other orthogonal systems. It is important to note that this problem of recovering the coefficients of orthogonal series with respect to a certain system is meaningful only if the uniqueness theorem is proved for this system, i.e., the coefficients of an orthogonal series with respect to this system are uniquely determined by its sum. The uniqueness can be related to the pointwise convergent series or series which are summable in a certain sense, and the convergence or summability can be supposed everywhere or outside some exceptional set. (For references to the literature on the rich theory of uniqueness of Walsh, Haar and Vilenkin series, including subtle theory of sets of ∗

Department of Mathematics and Computer Sciences, University of Perugia, via Vanvitelli 1, I-06123 Perugia, Italy, e-mail: [email protected], [email protected] (corresponding author) † Department of Mathematics, Moscow State University, RU-119992 Moscow, Russia, and Institute of Mathematics Casimirus the Great University, pl. Weyssenhoffa 11, PL-85079 Bydgoszcz, Poland, e-mail: [email protected] ‡ Department of Mathematics and Computer Sciences, University of Palermo, via Archirafi 34, I-90123 Palermo, Italy, e-mail: [email protected] 2010 A. M. S. Classification: 26A39, 28B15. Supported by NSh-979.2012.1, RFFI-11-01-00321a and Universities of Palermo and Perugia. Key words: Riesz space, order convergence, (D)-convergence, Kurzweil-Henstock integral, Hake theorem.

1

uniqueness, see [1], [8], [19], [30], [31]. The classical trigonometric case is treated for example in [33]) If the uniqueness theorem is true for the system then it is natural to expect that the coefficients can be recovered from the sum by Fourier formulas, as it takes place in the simplest cases, for example in the case of the uniform convergence. Indeed for many known systems (trigonometric, Haar, Walsh, Vilenkin systems and other systems of characters of zero-dimensional groups) it is true that every series with respect to those systems, convergent everywhere to a summable function, is the Fourier series of this function. But the point is that the sum of everywhere convergent orthogonal series can fail to be Lebesgue integrable. So some generalization of the Lebesgue integral is required similar to the above mentioned integrals in the trigonometric case. The first integral solving the coefficient problem for Walsh (and Haar) series was introduced in [20] in a descriptive form. A constructive definition of Denjoy type, based on transfinite induction, was given in [21] and later, independently, in [11]. The coefficient problem for Vilenkin series was examined in [22]. It is an advantage of application of the Henstock-Kurzweil theory (see [10], [12], [16]) that it provides a unifying approach to the discussed above coefficient problem for many orthogonal systems. The choice of the Henstock-Kurzweil type integral with respect to the particular derivation basis is determined by the system and by the type of convergence under consideration. The idea is to reduce the problem of coefficients to the one of recovering a function from its derivative with respect to an appropriate basis. At this point ones again it is important to guarantee the uniqueness of the primitive. For a literature about the Henstock-Kurzweil type integral for Riesz-space-valued functions, see for instance [4] and the bibliography therein. In [6], a formula of integration by parts for a Perron-type integral of any order k in the context of Riesz space-valued functions was proved in connection with related tools, like divided differences, k-convexity and Peano derivatives. In [5] Walsh series with coefficients from a Riesz space were considered and the problem of recovering the coefficients of a convergent Walsh series from its sum by generalized Fourier formulas was solved by an appropriate Henstock-Kurzweil-type integral for Riesz-space-valued functions. In this paper we extend this result to the case of series with respect to characters of any zero-dimensional locally compact abelian group. As characters are in general complex-valued functions we shall have to consider complex Riesz spaces which can be obtained by the complexification of the usual Riesz space. We remind the definition in Section 2. This implies that we need also an extension of the HenstockKurzweil integral theory to our case. We discuss this in Section 3. Considering a derivation basis on zero-dimensional locally compact abelian group (see [25]) we extend the HenstockKurzweil integration theory, with respect to this basis, to the case of functions with values in a complex Riesz space. We also consider and solve in this section the problem of recovering the primitive from a generalized derivative with respect to basis. Finally, in Section 4, we apply this result to the above mentioned problem of recovering coefficients of series with respect to characters in the compact case, and to the one of obtaining an inversion formula for multiplicative integral transforms, in the locally compact case. 2

2

Preliminaries

We recall some basic notions related to Riesz spaces and their complexifications (see [4, 15, 29, 32]). A Riesz space R is said to be Dedekind complete if every nonempty subset A ⊂ R, bounded from above, has supremum in R, which we will denote by sup A. A Riesz space R is Archimedean if for every choice of a, b ∈ R, with na ≤ b for all n ∈ N, we get a ≤ 0. The following result holds: Proposition 2.1. (see [4, Proposition 2.10], [15, p. 125]) Every Dedekind complete Riesz space is Archimedean. A net (pλ )λ∈Λ in R, where (Λ, ≥) 6= ∅ is a directed set, is called (o)-net if it is decreasing (i.e. pλ1 ≤ pλ2 whenever λ1 , λ2 ∈ Λ, λ1 ≥ λ2 ) and inf λ pλ = 0. In particular we get the definition of (o)-sequence when Λ = N. Definition 2.1. We say that a net (xλ )λ order converges (or in short (o)- converges) to x ∈ R if there exists an (o)-net (pλ )λ (with the same directed set Λ) satisfying |xλ − x| ≤ pλ for each λ ≥ λ0 for some λ0 ∈ Λ. We shall write in this case x = (o) limλ xλ . In the case Λ = N we get the definition of (o)-convergent sequence. The (o)-convergence of a Riesz-space-valued series is defined in an obvious way by the (o)-convergence of its partial sum. It is known (see [2]) that an order bounded net (xλ )λ in a Dedekind complete Riesz space is (o)-convergent to x if and only if (o) lim inf λ xλ = (o) lim supλ xλ . For Archimedean Riesz spaces another type of convergence can be considered. Definition 2.2. Let e ∈ R, e ≥ 0, e 6= 0. A net (xλ )λ∈Λ is said to converge e-uniformly to x ∈ R if for every ε > 0 there exists λ0 = λ0 (ε) ∈ Λ so that |xλ − x| ≤ εe for all λ ≥ λ0 . We say that a net xλ (r)-converges (relatively uniform converges as in [15] or converges with respect to a regulator as in [29]) to x ∈ R if there exists a positive e ∈ R (a regulator) such that xλ converges e-uniformly to x ∈ R. In the case Λ = N we get the definition of (r)-convergent sequence. We say that the sequence (xn )n in R is e-uniformly Cauchy (with e ∈ R, e ≥ 0, e 6= 0) if for every ε > 0 there is n1 = n1 (ε) ∈ N such that |xn − xn+p | ≤ ε e whenever n ≥ n1 and p ∈ N (see [15, Definition 39.1]). A Riesz space R is said to be uniformly complete if for every e ∈ R, e ≥ 0, e 6= 0, any e-uniformly Cauchy sequence is e-uniformly convergent. Note that every Dedekind complete Riesz space is uniformly complete (see [15, Lemma 39.2 and Theorem 39.4]), but the converse in general is not true: the space C([0, 1]) gives an example. It is easy to check that in an Archimedean Riesz space e-uniform convergence implies (o)convergence, but the converse is in general not true (see [15, 29]). But it is true under some additional assumption (see Proposition 2.2 below). A Riesz space satisfies property σ (see [15, p. 460]) if, given any sequence (un )n in R with un ≥ 0 ∀ n ∈ N, there exists a sequence (rn )n of positive real numbers, such that the sequence (rn un )n is bounded in R. 3

Definition 2.3. (see [15, pp. 478-479]) A Dedekind complete Riesz space is said to be regular if it satisfies property σ and if for each sequence (rn )n in R, order convergent to zero, there exists a sequence (ln )n of positive real numbers, with limn ln = +∞, such that the sequence (ln rn )n is order convergent to zero. Proposition 2.2. ([13], Theorem 1, p. 350) In a regular Riesz space the (o)-convergence is equivalent to the (r)-convergence. We now recall some basic properties of complex Riesz spaces (see [32]). Given a Riesz space R, we endow the Cartesian product R2 := R × R with a structure of complex Riesz space, by setting (x1 , y1 ) + (x2 , y2 ) := (x1 + x2 , y1 + y2 ), (a + ib)(x1 , y1 ) := (ax1 − by1 , ay1 + bx1 ) for each xj , yj ∈ R, j = 1, 2, and a, b ∈ R. For the sake of convenience we write x + iy instead of (x, y). If z = x + iy, we write x = Rez. We denote so defined complexification of a given Riesz space R by R + iR or in short RC. Identifying x ∈ R and (x, 0) we see that R is embedded in RC as a real linear subspace. To develop the theory on integration of functions with value in complex Riesz space we need a suitable notion of absolute value | · | on R + iR defined in such a way that for every z ∈ R the value |z| coincides with the usual one, given by |z| := sup(z, −z). To this end, set n o |z| = sup Re(z e−iϑ ) : ϑ ∈ [0, 2π] , (2.1) provided that the supremum involved exists in R. Observe that, in this case, we get n o |z| = sup |Re(z e−iϑ )| : ϑ ∈ [0, 2π] (see [32]). Note that if z ∈ R, then, since Re (z e−iϑ ) = z cos ϑ, the supremum in (2.1) exists and it is equal to the classical absolute value (see [32]). It is known (see [14] and [32]) that if the Riesz space R is Archimedean and uniformly complete, then the absolute value defined in (2.1) satisfies all the classical basic properties. Indeed we have the following: Theorem 2.1. (see [32, Theorem 91.2]) Let R be an Archimedean and uniformly complete Riesz space. Then the supremum in (2.1) exists in R for every z = x + iy with x, y ∈ R, and we get |x| ≤ |z|, y ≤ |z|, |z| ≤ |x| + |y|. Moreover, |z1 + z2 | ≤ |z1 | + |z2 |, |z1 | − |z2 | ≤ |z1 − z2 | for every z1 , z2 ∈ C, |z| = 0 if and only if x = 0 and y = 0, and |λ z| = |λ| |z| for each z and λ ∈ C. Observe that since, as said before, Dedekind completeness implies uniform completeness and Archimedeanness, we get that the thesis of Theorem 2.1 holds for a Dedekind complete Riesz space. From now on, we assume that the complex Riesz space RC is generated by a Dedekind complete Riesz space R. We now define order limit in the complex Riesz space RC . 4

Definition 2.4. We say that a net (zλ )λ∈Λ in RC , where (Λ, ≥) 6= ∅ is a directed set, (o)converges to z in RC and we write (o) lim zλ = z, if |zλ − z| is (o)-convergent to zero in R where λ

| · | is the absolute value in RC defined in (2.1). Note that if zλ ∈ R then the above definition coincides with Definition 2.1. Due to Theorem 2.1 the main basic properties of the limit are fulfilled if R is Dedekind complete. Theorem 2.2. Let R be a Dedekind complete Riesz space. Then a net (zλ ) in RC is (o)convergent to z ∈ RC in the sense of Definition 2.4 iff (xλ )λ and (yλ )λ are (o)-convergent to x and y in the sense of Definition 2.1, where zλ = xλ + iyλ , λ ∈ Λ, and z = x + iy. Proof. By Theorem 2.1 we get the inequalities |xλ − x| ≤ |zλ − z| ≤ |xλ − x| + |yλ − y| and |yλ − y| ≤ |zλ − z| ≤ |xλ − x| + |yλ − y|. Now it is enough to apply Definitions 2.4 and 2.1. Using this result we can deduce many properties of (o)-convergent nets in RC from the corresponding properties in the usual Riesz spaces. In particular the Cauchy criterion for (o)convergence in Riesz spaces (see [13, Lemma 2]) can be extended to the (o)-convergence in RC .

3

Derivation basis in a zero-dimensional group and HenstockKurzweil type integral with respect to this bases

Let G be a zero-dimensional compact abelian group. The most known example of such groups are Cantor dyadic group and the group of p-adic integers (see [28]). It is know (see [1]) that a topology in such a group can be given by a chain of subgroups G = G0 ⊃ G1 ⊃ G2 ... ⊃ Gn ⊃ ...

(3.1)

T with {0} = +∞ n=0 Gn . The subgroups Gn are clopen sets with respect to this topology. We denote by Kn any coset of the subgroup Gn and by Kn (x) the coset of the subgroup Gn which contains the element x, i.e., Kn (x) = x + Gn . (3.2) T For each x ∈ G the sequence {Kn (x)} is decreasing and {x} = n Kn (x). We denote the order of the factor group Gn /Gn+1 by pn . Then the order of G0 /G1 is p0 , the order of G0 /G2 is p0 ·p1 and by induction the order of G0 /Gn , n = 1, 2, ..., is mn := p0 ·p1 ·...·pn−1 , with pi ≥ 2 for all i (we agree that m0 := 1). Numbering the cosets constituting the group G0 \ Gn by Knj , with j = 1, 2, ..., mn . We have j n G = ∪m j=1 Kn .

(3.3)

We denote by µ the Haar measures on the group G and we normalize it so that µ(G) = 1. Since µ is translation invariant, then µ(Gn ) = µ(Kn ) = 5

1 mn

(3.4)

for all the cosets Kn , n ≥ 0. Now we define a derivation basis B on G. Take any function ν : G → N0 , with N0 = N ∪ {0} and define a basis set by βν = {(I, x) : x ∈ G, I = Kn (x), n ≥ ν(x)}. So our basis B in G is the family {βν }ν where ν runs over the set of all positive integer-valued functions on G. In the terminology of derivation basis theory any coset Kn , n ∈ N, is called B-interval of rank n. We denote by I the set of all B-intervals. This basis has all the usual properties of a general derivation basis (see [16]). First of all it has the filter base property: • ∅∈ / B; • for every βν1 , βν2 ∈ B there exists βν ∈ B such that βν ⊂ βν1 ∩ βν2 (it is enough to take ν = max{ν1 , ν2 }). Definition 3.1. A βν -partition is a finite collection π of elements of βν , where for S each pair of 0 0 00 00 0 00 elements (I , x ) and (I , x ) in π we have I ∩ I = ∅. If L is a B-interval and (I,x)∈π I = L then π is called βν -partition of L. Our basis B has the partitioning property. It means that the following conditions hold: • for each finite collection I0 , I1 , . . . , In of B-intervals with I1 , . . . , In ⊂ I0 and Ii , i = Sn 1, 2, . . . , being disjoint, the difference I0 \ i=1 Ii can be expressed as a finite union of pairwise disjoint B-intervals; • for each B-interval L and for any βν ∈ B there exists a βν -partition of L. This property of B follows easily from compactness of any B-interval and from the fact that any two B-intervals I 0 and I 00 are either disjoint or one of them is contained in the other one. Note that in the case of our basis B, given a point x ∈ G, any βν -partition contains only one pair (I, x) with this point x. Now we define a Henstock-Kurzweil type integral with respect to basis B for functions taking value in a complex Riesz space RC . Definition 3.2. We say that f : L → RC is Henstock-Kurzweil integrable on a B-interval L with respect to B (in brief, HB -integrable) if there exists Z ∈ RC such that     X   inf sup f (x)µ(I) − Z : π is a βν -partition of L  = 0. ν   (I,x)∈π Z In this case we write (HB )

f = Z. L

We note that in particular this definition is applicable for Riesz space-valued functions. It is easy to see that the element A in the previous definition is uniquely determined. Proposition 3.1. A function f : L → RC is HB -integrable with integral value Z = X + iY , X, Y ∈ R, iff Ref and Imf are HB -integrable with X and Y integral values respectively. 6

It is easy to check that the set of all HB -integrable functions on L is a linear space over the complex numbers. Now we can state the Cauchy criterion for HB -integrability of RC -valued functions. Theorem 3.1. A function f : L → RC is HB -integrable iff      X X f (x)µ(I) − f (x)µ(I) : π1 and π2 are βν -partitions of L  = 0. inf sup ν   (I,x)∈π1 (I,x)∈π2 Proof. For R-valued functions the Cauchy criterion can be proved in the same way as it was done in [5] for another basis. Now, passing to supremum over βν -partitions π1 and π2 in the inequalities X X X X Re f (x)µ(I) − f (x)µ(I) f (x)µ(I) − Re f (x)µ(I) ≤ (I,x)∈π1 (I,x)∈π1 (I,x)∈π2 (I,x)∈π2 and

X X X X Im f (x)µ(I) − Im f (x)µ(I) ≤ f (x)µ(I) − f (x)µ(I) (I,x)∈π1 (I,x)∈π1 (I,x)∈π2 (I,x)∈π2

and applying the Cauchy criterion for the functions Ref and Imf , we obtain that both of these functions are HB -integrable on L, and so by Proposition 3.1 the function f is HB -integrable in L. In the opposite side we use the inequality X X X X f (x)µ(I) − f (x)µ(I) ≤ Re f (x)µ(I) − Re f (x)µ(I) (I,x)∈π1 (I,x)∈π1 (I,x)∈π2 (I,x)∈π2 X X + Im f (x)µ(I) − Im f (x)µ(I) . (I,x)∈π1 (I,x)∈π2 Similarly as in [5, Proposition 3.5], using the Cauchy criterion we get the following property of HB -integral. Proposition 3.2. If f is HB -integrable on B-interval L, then f is also on any B-interval K ⊂ L. The additivity of HB -integral is give by the following easily checked proposition. Proposition 3.3. If a B-interval L can be represented in the form L = ∪pj=1 K j , where K j are pairwise disjoint B-intervals and a function f is HB -integrable on each K j , then f is HB -integrable in L and Z Z p X (HB ) f. (HB ) f = L

j=1

7

Kj

It follows from the last two propositions that for any HB -integrable function f : G → RC the indefinite HB -integral is defined as an additive RC -valued B-interval function on the set I. We shall denote it by Z F (I) = (HB ) f. (3.5) I

We say that a function F : B → RC is (o)-continuous with respect to the basis B at the point x ∈ G if (o) lim F (Kn (x)) = 0. n→∞

The order derivative with respect to B or (o)-B-derivative of an additive set function F : I → RC at a point x is defined as (o)

F (Kn (x)) . n→∞ µ(Kn (x))

(3.6)

DB F (x) := (o) lim

Now we consider the problem of recovering a primitive function from its derivative. We observe by the following example that this problem can fail to be solved if we use the (o)B-derivative even in the case of a usual Riesz space-valued function. Indeed we construct a non-trivial function whose (o)-B-derivative is zero at each point of G. Let σ(I) be the σ-algebra generated by I and L0 (G) be the Riesz space of measurable functions. We note that it is a Dedekind complete Riesz space, see [15, p. 126]. We define the function F : σ(I) → L0 (G) as F (I) = χI . (o) Take any element x ∈ G. By definition of DB -derivative we have (o)

DB F (x) = (o) lim mn χKn (x) . n→∞

We want to show that this (o)-limit is equal to zero. To this end we define a monotone (o)-sequence  mj if s ∈ Kj (x) \ Kj+1 (x), j ≥ n; φn (s) = 0 if s ∈ G \ Kn (x). (o)

By construction we get mn χKn (x) (s) ≤ φn (s) and inf φn = 0. Therefore DB F (x) = 0 for all x, while F is not trivial. Therefore, to guarantee the recovering of the primitive, we introduce another type of derivative. Definition 3.3. A B-interval function F is global differentiable with respect to the basis B on a set E or (g)-B-differentiable on E if there exists a function f : E → R such that    F (I) − f (x) : (I, x) ∈ βν , x ∈ E = 0. lim sup ν µ(I) (g)

In this case, f is called the (g)-B-derivative of F with respect to B and denoted by DB F (x). Now we prove two versions of the theorem on recovering a function by the HB -integral of its (g)-derivative. In the first one we do not assume anything else beside the Dedekind completeness of the Riesz space involved, while in the second one we put the additional assumption of regularity (similar theorems with respect to another basis was considered in [3] and [5]). 8

Theorem 3.2. If a function F : I → RC is (g)-differentiable with respect to B on a B-interval (g) L, then the derivative DB F = f is HB -integrable on L, and Z

(g)

DB F (x) = F (L).

(HB ) L

Proof. By (g)-B-differentiability of F on L, there exists an (o)-net (pν )ν , such that   F (I) sup − f (x) : (I, x) ∈ βν , x ∈ L ≤ pν , |I|

ν ≤ ν0 .

(3.7)

Having chosen a βν -partition π = {(Ii , xi ) : i = 1, . . . , q} of L we get: q q X X [f (xi )µ(Ii ) − F (Ii )] f (xi )µ(Ii ) − F (L) = 0 ≤ i=1 i=1 q X F (Ii ) ≤ µ(Ii ) − f (xi ) µ(Ii ) i=1 ! q X ≤ pν µ(Ii ) = pν µ(L). i=1

We shall need the following proposition: Proposition 3.4. Let R be a Dedekind complete Riesz space, satisfying property σ, Q ⊂ L ∈ I be a countable set, and f : [a, b] → RC be a function, such that f (x) = 0 for all x ∈ L \ Q. Z b Then (HB ) f = 0. a

Proof. For the Riesz space R the proposition can be proved as it was done in [5] for another basis. For the complex Riesz space RC the result can be obtained, by Proposition 3.1, passing to the real and imaginary part of the function f . On the basis of the above proposition we can obtain the second version of recovering primitive theorem, following the lines of the proof of [5, Theorem 4.2]. Theorem 3.3. Let RC be the complexification of a regular Riesz space R, L ∈ I, F : I → RC be such that for some countable set Q ⊂ L the function F is (o)-continuous on Q with respect (g) to B and (g)-B-differentiable on L \ Q with f (x) = DB F (x). Then f is HB -integrable on L, and Z (HB )

f = F (L). L

9

4

Applications to the coefficient problem for the series with respect to characters

Let Γ denote the dual group of G, i.e., the group of characters of the group G. It is known (see [1]) that under the assumption imposed on G the group Γ is a discrete abelian group (with respect to the pointwise multiplication of characters) and it can be represented as a sum of an increasing chain of finite subgroups Γ0 ⊂ Γ−1 ⊂ Γ−2 ⊂ ... ⊂ Γ−n ⊂ ...,

(4.1)

(0) where Γ0 = {γ (0) G. S0}, with γ (x) T0= 1 for all x ∈ (0) Then Γ = i=−∞ Γi and i=−∞ Γi = {γ }, where γ (0) (x) = 1 for all x ∈ G. For each n ∈ N the group Γ−n is the annihilator of Gn , i.e.,

Γ−n = G⊥ n := {γ ∈ Γ : γ(x) = 1 for all x ∈ Gn }. Lemma 4.1. If γ ∈ Γ−n , then γ is constant on each coset Kn of Gn . Proof. Having fixed an element xKn ∈ Kn , we can represent any element x ∈ Kn as x = xKn +y where y ∈ Gn . The properties of characters and the definition of the annihilator imply γ(x) = γ(xKn )γ(y) = γ(xKn ). So the value (x, γ) is constant for all x ∈ Kn . ⊥ The factor groups Γ−n−1 /Γ−n = G⊥ n+1 /Gn and Gn /Gn+1 are isomorphic (see [1]), and so they are of finite order for each n ∈ Z. This implies that the group Γ−n /Γ0 is also finite for any n > 0 and Γ/Γ0 is countable. Now, as we have done above for the group G, for each coset J of Γ0 we choose and fix an element γJ . Then we can represent any element γ ∈ Γ in the form

γ = γJ · {γ},

(4.2)

where {γ} ∈ Γ0 . We agree to put γΓ0 = γ (0) , so that γ = {γ} if γ ∈ Γ0 . The characters γ have also the following property (see [25]). Z Lemma 4.2. If γ ∈ Γ \ Γ−n then γ(x)dµ = 0 for each coset Kn . Kn

We note that the integral in the above lemma is understood in the sense of Lebesgue integral with respect to the measure µ. It follows from this lemma that, if γ1 and γ2 are not equal identically on Kn , then they are orthogonal on Kn , i.e. Z γ1 (x)γ2 (x)dµ = 0. Kn

The characters γ constitute a countable orthogonal system on G with respect to the normalized measure µG , and we can consider the series X aγ γ (4.3) γ∈Γ

10

with respect to this system. We shall consider aγ ∈ RC . We define a convergence of this series at a point x as the (o)-convergence of its partial sums X Sn (x) := aγ γ(x) (4.4) γ∈Γ−n

By Lemma 4.1 Sn is constant on each coset Kn . As a constant function is obviously HB -integrable, then by Proposition 3.3 the function Sn as well as the function aγ γ are HB integrable on G. We denote this value by Sn (Kn ). We associate with the series (4.3) a function F : I → RC defined on each coset Kn by Z Sn (x)µ = Sn (Kn )µ(Kn ). (4.5) F (Kn ) := Kn j We check that F is an additive function on I. Indeed let Kn = ∪j Kn+1 . It follows from Lemma 4.2 that Z (Sn+1 (x) − Sn (x)) = 0. Kn

Then Z F (Kn ) =

Z Sn =

Kn

Z =

Sn+1 Kn

(Sn + (Sn+1 − Sn )) = XZ X j = Sn+1 = F (Kn+1 ). Kn

j Kn+1

j

j

Definition (4.5) implies Sn (x) =

F (Kn (x) . µ(Kn (x))

(4.6)

For partial sums (4.4) of series (4.3) we consider pointwise (o)-convergence as well as (g)convergence on a set. Definition 4.1. Let E be any nonempty set, and D = NE . We say that the sequence of RC -valued functions (Sn (x))n is (g)-convergent to the function S(x) if (o) lim(sup{|Sn (x) − S(x)| : x ∈ E, n ≥ ν(x)}) = 0 ν

The following lemma is an immediate consequence of the equality (4.6). Lemma 4.3. The partial sums Sn (x) are (g)-convergent to a function f on a set E iff the associated function F is (g)-B-differentiable to f on E. The next lemma gives a sufficient condition for the (o)-continuity of the associated function F. Lemma 4.4. Suppose that the coefficients {aγ } with aγ ∈ RC of the series (4.3) satisfy the condition (o) lim {sup |aγ | : γ ∈ Γ−(n+1) \ Γ−n }) = 0 (4.7) n→∞

(this in fact means that the sequence (|aγ |) is (o)-convergent in R to zero with any ordering of γ inside Γ−(n+1) \ Γ−n ) then the associated function F is (o)-continuous at each point x ∈ G. 11

Proof. Recall that the order of G0 /Gn , n = 1, 2, ..., is mn := p0 · p1 · ... · pn−1 . Due to the isomorphism between G0 /Gn and Γ−n /Γ0 , the order of the subgroup Γ−n is also mn . For a fix a point x ∈ G0 , having in mind (3.4) and the equality |γ(x)| = 1, we obtain P |Sn (x)|µ(Kn (x)) ≤ m1n γ∈Γ−n |aγ |. On the right side of the previous inequality we have the arithmetic means of the coefficients {|aγ | : γ ∈ Γ−n }. According to [5, Lemma 5.1] if a sequence is (o)-convergent to zero, then the sequence of its arithmetic means is also (o)-convergent to zero. This implies, by (4.5), the (o)-continuity of the associated function F . Now we prove two theorems on recovering the coefficients of the series (4.3) from its sum, which correspond to the two versions of theorems on recovering the primitive. Theorem 4.1. If R is a Dedekind complete Riesz space and a series (4.3) with aγ ∈ RC is (g)-convergent to a function f on the group G, then f is HB -integrable on G and the series is the Fourier series of f in the sense of the HB -integral. Proof. By Lemma 4.3 the associated function F is (g)-B-differentiable on G with (g)-Bderivative f . So we can apply to the functions F and f Theorem 3.2 to get Z (HB ) f = F (I) (4.8) I

for any B-interval I. Take any γ ∈ Γ and choose n such that γ ∈ Γ−n . Then, by Lemma 4.1, both Sn and γ are constant on each coset Knj (see (3.3)). Denote by γ j the constant value of γ on Knj . The function Sn γ is HB -integrable on G (see remark before definition of associated function (4.5)) and by orthogonality of the system of characters we get (all the integral below are understood in the sense of HB -integral): Z aγ =

Sn γ = G

=

mn X j=1

mn Z X j=1

γ

j

F (Kik )

j Kn

=

Sn γ =

mn X j=1

mn X

γ

j

Z j Kn

j=1

γ

j

Z

Z f=

j Kn

Sn

f γ. G

This completes the proof. Theorem 4.2. If R is a regular Riesz space and a series (4.3), with coefficients aγ ∈ RC satisfying (4.7), is (g)-convergent to a function f on a set G \ E, where E is a countable subset of G, then f is HB -integrable on G and the series is the Fourier series of f in the sense of the HB -integral. Proof. By Lemma 4.3 the associated function F is (g)-B-differentiable on G \ E with (g)B-derivative f . By condition (4.7), according with Lemma 4.4 the associated function F is (o)-continuous everywhere on G, in particular on E. Therefore we can apply the Theorem 3.3 and get (4.8). The rest of the proof is the same as in the proof of the previous theorem. 12

Remark 4.1. In the case of locally compact zero dimensional group G the group of characters Γ is also locally compact and instead of g-convergence of partial sum of the series (4.4) we deal with a g-convergence of integrals Z a(γ)(x, γ)dµΓ (HBΓ ) Γ−n

where the measure and integral are considered on the group Γ and (x, γ) = γ(x) is considered as the value of character x at a point γ ∈ Γ. Here the problem of recovering the coefficients is generalized to the one of obtaining an inversion formula: Z f (x)γ(x)dµG a.e. on Γ. (4.9) a(γ) = (o) lim (HBG ) n→∞

G−n

This problem in the general situation is still open and we announce here only a partial result, stating that formula (4.9) holds true under the additional assumption that indefinite integral of function a(γ) is (o)-B-differentiable almost everywhere.

References [1] G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli and A. I. Rubistein: Multiplicative system of functions and harmonic analysis on zero-dimensional groups. Baku 1981. (In Russian.) [2] Ch. D. Aliprantis & K. C. Border: Infinite Dimensional Analysis,(1994), Springer-Verlag. [3] A. Boccuto: Differential and Integral Calculus in Riesz spaces, Tatra Mountains Math. Publ. 14 (1998), 293-323. [4] A. Boccuto, B. Rieˇcan and M. Vr´ abelov´ a: Kurzweil-Henstock Integral in Riesz Spaces, Bentham Science Publ., Singapore, 2009. [5] A. Boccuto and V. A. Skvortsov: Henstock-Kurzweil type integration of Riesz-space-valued functions and applications to Walsh series, Real Analysis Exchange 29(1), 2003/2004, 419-439 [6] A. Boccuto and V. A. Skvortsov: The Perron Integral of order k in Riesz Spaces, Positivity 14 (2010) (4), 595-612. [7] A. Boccuto, V. A. Skvortsov and F. Tulone: A Hake-type theorem for integrals with respect to abstract derivation bases in the Riesz space setting, Math. Slovaca (2013), in press. [8] B. Golubov, A. Efimov and V. A. Skvortsov: Walsh series and transforms: Theory and Applications. Kluwer Academic Publishers, 1991. [9] A. Denjoy: Le¸cons sur le calcul des coefficients d’une s´erie trigonometrique, Paris, 1941-49. [10] R. Henstock: The General Theory of Integration, Clarendon Press, Oxford, 1991. [11] J. P. Kahane: Une th´eorie de Denjoy des martingales dyadiques, Enseign. Math. (2) 34 (1988), 255–268.

13

[12] P. Y. Lee and R. V´yborn´y: The integral: An easy approach after Kurzweil and Henstock, Cambridge University Press, 2000. [13] P. McGill: Integration in vector lattices, J. Lond. Math. Soc., 11 (1975), 347-360. [14] P. Meyer-Nieberg: Banach Lattices, 1991, Springer-Verlag, Berlin-Heidelberg. [15] W. A. J. Luxemburg and A. C. Zaanen: Riesz Spaces, I, North-Holland Publishing Co., Amsterdam, 1971. [16] K. M. Ostaszewski: Henstock integration in the plane, Memoirs of the Amer. Math. Soc., Providence, 353, 1986. [17] D. Preiss and B. S. Thomson: The approximate symmetric integral, Canadian J. Math. 41 (1989), 508-555. [18] B. Rieˇcan and T. Neubrunn: Integral, Measure and Ordering, Kluwer Acad. Publ./Ister Science, Dordrecht/Bratislava, 1997. [19] F. Schipp, W. R. Wade, P. Simon and J. Pal: Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger Publishing, Ltd, Bristol and New York, 1990. [20] V. A. Skvortsov: Calculation of the coefficients of an everywhere convergent Haar series, Mat. Sb. 75 (1968), 349–360. (English transl.: Math. USSR Sb. 4 (1968), 317–327). [21] V. A. Skvortsov: Constructive version of the definition of the HD-integral, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1982 no. 6, (Engl. transl.: Moscow Univ. Math. Bull, 37 (1982), 41–45). [22] V. A. Skvortsov and F. Tulone: Generalized Henstock integrals in the theory of series with respect to multiplicative system, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2004, no.2, 7-12 (in Russian). [23] V. A. Skvortsov and F. Tulone: Henstock type integral on zero-dimensional groups. J. Math. Anal. Appl. 322 (2006), 621-628. [24] V. A. Skvortsov and F. Tulone: p-adic Henstock integral in the theory of series with respect to characters of zero-dimensional groups. Vestnik Moskov. Gos. Univ. Ser. Mat. Mekh. 1 (2006), 25-29; Engl. transl. Moscow Univ. Math. Bull. 61 (1) (2006), 27-31. [25] V. A. Skvortsov and F. Tulone: Kurzweil-Henstock type integral on zero-dimensional group and some of its applications. Czech. Math. J. 58 (2008), 1167-1183. [26] B. S. Thomson: Symmetric properties of real functions, Monographs and Textbook in Pure and Appl. Math. 183, Marcel Dekker, Inc., 1994. [27] B. S. Thomson: Derivation bases on the real line. Real Anal. Exchange 8 (1982/83), 67-207 and 278-442. [28] V. S. Vladimirov, I. V. Volovich and E. I. Zelenov: P -adic Analysis and Mathematical Physics, World Scientific, 1994. [29] B. Z. Vulikh: Introduction to the theory of partially ordered spaces, Wolters -Noordhoff Sci. Publ., Groningen, 1967.

14

[30] W. R. Wade: Recent developments in the theory of Haar series, Colloq. Math. 52 (1987), 213–238. [31] W. R. Wade: Dyadic harmonic analysis, Contemp. Math 208 (1997), 313–350. [32] A. C. Zaanen: Riesz Spaces, II, North-Holland Publishing Co., Amsterdam, 1983. [33] A. Zygmund: Trigonometric series, Cambridge Univ. Press, London, 1968.

Authors’ addresses: Antonio Boccuto, Dipartimento di Matematica e Informatica, Universit`a degli Studi di Perugia, via Vanvitelli 1, 06123 Perugia, Italy, e-mail: [email protected]; Valentin Skvortsov, Department of Mathematics, Moscow State University, Moscow 119992, Russia, and Institute of Mathematics Casimirus the Great University, pl. Weyssenhoffa 11, 85079 Bydgoszcz, Poland, e-mail: [email protected]; Francesco Tulone, Dipartimento di Matematica ed Applicazioni, Universit`a degli Studi di Palermo, via Archirafi 34, 90123 Palermo, Italy, e-mail: [email protected]

15