The HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line


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1 The HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni, di Tecnologie Chimiche, Automatica e Modelli Matematici (DITCAM) Università degli Studi di Palermo, Viale delle Scienze, Palermo, Italy mail addresses: Abstract The aim of this paper is to introduce an HenstockKurzweilStieltjes type integration process for real functions on a fractal subset of tha real line. Key words: sset; sderivatives; sintegral Contents 1 Introduction 1 2 Preliminaries The sintegral The shenstockkurzweilstieltjes integral Linearity properties Cauchy Criterion and Cauchy extension SaksHenstock Lemma Relationship between the sintegral and the shenstockkurzweil Stieltjes integral The VitaliCarathéodory Theorem Introduction During last years, some mathematicians have been obliged to define a new concept of derivation (see [1] and [2]) and a new concept of integration (see, [6] and [9]) to solve some physical and engineering problems. In fact even if the geometry of fractal 1
2 sets is a well explored subject (see, [3], [4], [5], [7]), in this sets there are so many irregularities to render inapplicable the standard methods and the technique of the ordinary calculus to give reasonable solutions of practical problems. Just think of the fact that the usual derivative of the classical LebesgueCantor staircase function is zero almost everywhere (see [1] and [6]) and of the fact that the Riemann s integral of functions defined on a fractal sets is undefined. So, in 1991, de Guzmann, Martin and Reyes (see [1]) in order to study the problem of the existence and the uniqueness of the solutions of differential equations of the type: d x(t) dt = f(t, x(t)) in which t or x(t) takes values in a noncontinuous set (i.e. a fractal set) introduced, for functions defined on a fractal set, a new concept of derivative, called the sderivative. Later on, Jiang and Su (see [6]) and more recently Parvate and Gangal (see [9]) introduced, for function defined in a fractal set, a new concept of RiemannStieltjes type integral, called the sintegral and the F s integral, respectively. Both authors defined such integration process in an analogous fashion as the classical Riemann integral but with the Hausdorff measure and the mass function taking over the role of the distance, respectively. Since the Hausdorff measure and the mass function are proportional (see [9], section 4), we can say that Jiang and Su in [6] and Parvate and Gangal in [9] defined indipendently the same integral. Both authors developed in a way analogous to the standard calculus the rest of the integration theory. Therefore properties like linearity and additivity with respect to integration domain are valid. About the Fundamental theorem of calculus, both authors use an extra hypothesis to prove it. In this paper we will compare the hypothesis of the Fundamental theorem of Calculus used by Jiang and Su with the one used by Parvate and Gangal and we will notice that without such additional hypothesis the Fundamental theorem of Calculus lose its classical formulation. Therefore in in the fractal case, it is not possible to prove a theorem like this: If R is a closed fractal set and if F : R is sdifferentiable on with a continuous derivative therefore [a,b] F s dh s = F (b) F (a). Moreover, in this paper we introduce an HenstokKurzweilStieltjes type integration process defined for real functions on a fractal subset of the real line of Hausdorff dimension s with 0 < s 1, since it is well known that numerical algorithms are based very rarely on Lebesgue integrals but are based more often on Riemann sums and in the classical literature the best formulation of the Fundamental theorem of calculus is done by the HenstockKurzweil integral that it is based on Riemann sums too (see [?]. Finally, we will show that the HenstokKurzweilStieltjes integral on the fractal set contains properly the RiemannStieltjes integral defined by Jiang and Su and by Parvate and Gangal. 2
3 2 Preliminaries In all the paper we denote by R the set of all real numbers and by a nonempty closed subset of the interval [a, b] R, with a = min and b = max. Moreover, we denote, by L( ) the Lebesgue measure on R. Now, given s, with 0 s 1, we recall that the sdimensional Hausdorff measure of is defined as: { } H s () = lim inf L(A i ) s : A i, L(A i ) δ. δ 0 H s ( ) is a Borel regular measure (see [7]). Moreover, we recall that, the unique number s for which H t () = 0 if t > s and H t () = + if t < s is called the Hausdorff dimension of. Whenever is H s measurable with 0 < H s () <, therefore is said to be an sset. Without loss of generality we can assume in all the paper, that H s () = 1. Definition 2.1. (see [1], [5], and [8]) Let f : R be a function and let x 0. The sderivative of f, on the right and on the left, at the point x 0 is: f s + (x 0 ) = lim x x + 0 x f(x) f(x 0 ) H s ( [x 0, x]) if H s ( [x 0, x]) > 0 for all x > x 0 f s f(x 0 ) f(x) (x 0 ) = lim x x 0 H s ( [x, if H s ( [x, x 0 ]) > 0 for all x < x 0 x 0 ]) x where these limits exist. It is said that the sderivative of f at x 0 exists if f s + (x 0 ) = f s (x 0 ), we be denote by f s(x 0 ) such common value. Remark. Of course, if f is sderivable at the point x 0 then f is continuous at x 0, according to the topology induced on for the usual topology of R. 2.1 The sintegral Definition 2.2. Let A [a, b] be an interval. The set Ã = A is called an interval of. Definition 2.3. A partition of is any collection P = {(Ãi, x i )} p of pairwise disjoint intervals Ãi and points x i Ãi such that = i Ãi. Definition 2.4. (see [5]) Let f : R be a function. A number σ is said to be the sintegral of f on, if for each ε > 0 there is a δ > 0 such that n f(x i ) H s (Ãi) σ < ε 3
4 for each partition P = {(Ãi, x i )} p of with Hs (Ãi) < δ, for i = 1, 2,..., n. By σ = (R) f(t)dh s (t) we will denote the sintegral and by sr() we will denote the collection of all functions that are sintegrable on. Theorem 2.1. (see [5]) Let f : R be a continuous function at all points of. If F : R is a function H s absolutely continuous on and F s(x) = f(x) at H s  a.e. point x, then (R) f(t) dh s (t) = F (b) F (a). Definition 2.5. (see [5]) Let f : R be a function. We say that f is H s absolutely continuous on, if ε > 0, δ > 0 such that n f(b k ) f(a k ) < ε k=1 whenever n k=1 Hs[ (ak, b k ) ] < δ, a k, b k, k = 1,..., n, and a 1 < b 1 a 2 < b 2... a n < b n. Lemma 2.2. Let f : R be H s absolutely continuous on. contiguous interval of therefore If [α, β] is a f(α) = f(β). Proof. Since [α, β] is a contiguous interval of it follows that H s ([α, β]) = 0. Moreover, since f is H s absolutely continuous on, ε > 0 we have that f(β) f(α) < ε. The assertion follows by the arbitrariness of ε. Definition 2.6. (see, [8]) Let f : R R be a function. A point x R is said a point of change of f if f is not constant over any open interval (c, d) containing x. The set of all points of change of f is called the set of change of f and it is denoted by Sch(f). Theorem 2.3. (see [8]) If F : R R is sdifferentiable on, F s is a continuous function on and Sch(F ) then (R) F s(x)dh s (x) = F (b) F (a). 4
5 xample 2.1. Let [0, 1] be the classical Cantor set. Let F be a function on [0, 1] defined by 0, 0 x F (x) = 3x 1, 3 < x < , 3 x 1. and let f be the restriction of F on the set. Note that: f is not H s absolutely continuous on. Sch(f). f is sdifferentiable at each point of and f s(x) = 0 for all points x. f s is sintegrable on and (R) f s dh s = 0 f(1) f(0) = 1. (i) f is not H s absolutely continuous on. (ii) Sch(f). (iii) f is sdifferentiable at each point of and f s(x) = 0 for all points x. (iv) f s is sintegrable on and (R) f s dh s = 0 f(1) f(0) = 1. 3 The shenstockkurzweilstieltjes integral Definition 3.1. A gauge on is any positive real function δ defined on. Definition 3.2. A partition of is any collection P = {(Ãi, x i )} p disjoint intervals of Ãi and points x i Ãi such that = i Ãi. of a pairwise Definition 3.3. Let P = {(Ãi, x i )} p be a partition of. If δ is a gauge on, then we say that P is a δfine partition of whenever Ãi ]x i δ(x i ), x i + δ(x i )[ for all i = 1, 2,..., p. The following Cousin s lemma, addresses the existence of δfine partitions. Lemma 3.1. If δ is a gauge on, then there exists a δfine partition of. 5
6 Proof. If [α, β] [a, b] and if [α, β] =, we said that [α, β] has a δfine partition of. Let c be the midpoint of [a, b] and let us observe that if P 1 and P 2 are δfine partitions of [a, c] and [c, b], respectively, then P = P 1 P 2 is a δfine partition of. Using this observation, we proceed by contradiction. Let us suppose that does not have a δfine partition then at least one of the intervals [a, c] or [c, b] does not have a δfine partition, let us say [a, c]. Therefore [a, c] is not empty. Let us relabel the interval [a, c] with [a 1, b 1 ] and let us repeat indefinitely this bisection method. So, we obtain a sequence of nested intervals: [a, b] [a 1, b 1 ]... [a n, b n ]... Since the length of the interval [a n, b n ] is (b a)/2 n therefore, for the Nested Intervals Property, there is a unique number ξ [a, b] such that: [a n, b n ] = {ξ}. n=0 It is trivial to notice that the interval [an, b n ] is not empty. Let ξ n [a n, b n ], therefore ξ n ξ < b n a n = (b a)/2 n. So lim n ξ n = ξ. Now since is a closed set, ξ. Since δ(ξ) > 0, we can find k N such that [a k, b k ] [ξ δ(ξ), ξ +δ(ξ)]. Therefore {[a k, b k ], ξ} is a δfine partition of [ak, b k ], contrarily to our assumption. Let P = {(Ãi, x i )} p be a partition of, let f : R be a function and let us consider the following sum S(f, P ) = f(x i )H s (Ãi). Definition 3.4. We say that the function f is shkintegrable on, if there exists I R such that for all ε > 0, there is a gauge δ on with: S(f, P ) I < ε for each δfine partition P = {(Ãi, x i )} p of. The number I is called the shkintegral of f on and we write I = (HK) f dh s. The collection of all functions that are shk integrable on will be denoted by shk(). 6
7 Remark. The number I from Definition 5 is determinate uniquely by the shkintegrable function f. In fact, let us suppose that I and J satisfies Definition 5, let us assume that J I and let us take ε = I J /2. Then we can find two gauges δ 1 and δ 2 on so that f(x i )H s (Ãi) I < ε for each δ 1 fine partition P = {(Ãi, x i )} p of and f(x i )H s (Ãi) J < ε for each δ 2 fine partition P = {(Ãi, x i )} p of. Now let δ = min{δ 1, δ 2 }. Then, for each δfine partition of we have I J I f(x i )H s (Ãi) + f(x i )H s (Ãi) J < 2ε = I J, that is a contradiction. 3.1 Linearity properties Theorem 3.2. (a) If f and g are shkintegrable on, then f + g is also shkintegrable on and (HK) (f + g) dh s = (HK) f dh s + (HK) g dh s. (b) If f is shkintegrable on and k R, then kf is shkintegrable on and (HK) kf dh s = k (HK) f dh s. Proof. (a): Let I and J be the shkintegrals of f and g, respectively. Given ε > 0, let δ and δ be two gauges on such that: f(x i )H s (Ãi) I < ε 2, if the partition P = {(Ãi, x i )} p is δ fine and g(x i )H s (Ãi) J < ε 2 7
8 if the partition P = {(Ãi, x i )} p is δ fine. Now let δ = min{δ, δ } therefore if P is a δfine partition, then it is both δ fine and δ fine. So (f + g)(x i )H s (Ãi) (I + J) f(x i )H s (Ãi) I + g(x i )H s (Ãi) J < ε. for each δfine partition of. The proof follows by arbitrariness of ε. (b): Let I be the shkintegral of f. Then, ε > 0 there is a gauge δ on with: f(x i )H s (Ãi) I < ε for each δfine partition P = {(Ãi, x i )} p of. Therefore, k R we have: kf(x i )H s (Ãi) ki = k f(x i )H s (Ãi) I < k ε. The proof follows by arbitrariness of ε. Theorem 3.3. Let f be shkintegrable on and f(x) 0 for all x, then (HK) f dh s 0. Proof. Let δ be a gauge on such that f(x i )H s (Ãi) (HK) if the partition P = {(Ãi, x i )} p is δfine. Since f(x) 0 for all x, then Therefore ε f(x i )H s (Ãi) 0. f(x i )H s (Ãi) ε < (HK) f dh s < f dh s < ε f(x i )H s (Ãi) + ε, 8
9 by the arbitrariness of ε it follows that (HK) f dh s 0. Corollary 3.4. Let f and g be shkintegrable on. If f g for all x, then (HK) f dh s (HK) g dh s. Proof. Let h := g f. By theorem 3.2 (b) h is shkintegrable and (HK) h dh s = (HK) g dh s (HK) f dh s, and since f g then h(x) 0 for all x and (HK) h dhs 0. Therefore the conclusion is immediate. 3.2 Cauchy Criterion and Cauchy extension Theorem 3.5. A function f : R is shk integrable on if and only if for each ε > 0 there exists a gauge δ on such that S(f, P 1 ) S(f, P 2 ) < ε for each pair δfine partitions P 1 and P 2 of. Proof. ( ) Let ε > 0 be given. Since f shk(), there exists a gauge δ on such that S(f, P ) (HK) f dh s < ε 2 for each δfine partition P of. If P 1 and P 2 are two δfine partitions of, we have that S(f, P 1 ) S(f, P 2 ) S(f, P 1 ) (HK) < ε. ( ) For each n N, let δ n be a gauge on such that f dh s + S(f, P2 ) (HK) f dh s S(f, Q n ) S(f, R n ) < 1 n for each pair δ n fine partitions Q n and R n of. 9
10 Let n (x) = min{δ 1 (x),..., δ n (x)} be a gauge on. For each n N, let P n be a n fine partition on. Clearly, if m > n then P m and P n are n fine partitions on ; hence S(f, P n ) S(f, P m ) < 1 n for m > n. Conseguently, {S(f, P n )} n=1 is a Cauchy sequence of real numbers; therefore this sequence converges to some real numer A = lim n S(f, P n ). Then S(f, P n ) A < 1 n for each n. Let ε > 0 be given and choose N such that 1 N < ε 2. If P is an arbitrary N fine partitions on, then S(f, P ) A S(f, P ) S(f, P N ) + S(f, P N ) A < 1 N + 1 N < ε. Thus f shk() and A = (HK) f dhs. Theorem 3.6. If f shk(), then f shk(ã) for each closed interval Ã. Proof. Let Ã be a closed interval of. Since f shk(), it follows that by Theorem 3.4, for each ε > 0 there exists a gauge δ on such that S(f, P ) S(f, Q) < ε for each pair δfine partitions P and Q of. Let {Ã1, Ã2,..., ÃN } be a finite collection of pairwise nonoverlapping intervals of, such that Ã / {Ã1, Ã2,..., ÃN } and such that N = Ã Ã k. For each k {1,..., N} we fix a δfine partition P k of Ãk. If PÃ and QÃ are δfine partitions of Ã, then it is clear that P Ã N k=1 P k and QÃ N k=1 P k are δfine partitions of. k=1 10
11 Thus S(f, PÃ) S(f, QÃ) N = S(f, P Ã ) + N S(f, P k ) S(f, QÃ) S(f, P k ) k=1 k=1 ( N ) ( N ) = S f, PÃ P k S f, QÃ P k < ε. k=1 k=1 Therefore, by Theorem 3.4, f shk(ã) SaksHenstock Lemma Definition 3.5. A subpartition of is a finite collection P = {(Ãi, x i )} p of a pairwise disjoint intervals Ãi and points x i Ãi for i = 1, 2,..., p. Definition 3.6. Let P = {(Ãi, x i )} p be a subartition of. If δ is a gauge on, then we say that P is a δfine subpartition of whenever Ãi ]x i δ(x i ), x i + δ(x i )[ for all i = 1, 2,..., p. Lemma 3.7 (The SaksHenstock Lemma). Let f shk() and ε > 0. If δ is a gauge on such that S(f, P ) (HK) for each δfine partition P of, then and f dh s < ε { } f(x i )H s (Ãi) (HK) f dh s ε f(x i)h s (Ãi) (HK) f dh s 2ε for each δfine subpartition P 0 = {(Ãi, x i )} p of. Proof. Let ε > 0. The set \ p Ãi is a finite union of disjoint intervals. Let K 1, K 2,..., K m be the closure of these intervals. 11
12 From theorem 3.5 we have that f shk( K j ) for each j = 1, 2,..., m. Hence for each η > 0 and for each j there exists a gauge δ j on K j such that S(f, P j) (HK) f dh s K j < η m for each δ j fine partition P j of K j. Let P = P 0 m P j=1 j. Then P is a δfine partition of and since S(f, P ) = S(f, P 0 ) + m j=1 S(f, P j) we have that (HK) f dh s = (HK) f dh s + m j=1 (HK) f dh s. K j Therefore { m } = S(f, P ) S(f, P j ) { } f(x i )H s (Ãi) (HK) f dh s j=1 S(f, P ) (HK) < ε + m η m = ε + η. f dh s + = S(f, P 0) (HK) f dh s m s} f dh s (HK) f dh j=1 K j m S(f, P j) (HK) f dh s kj { (HK) j=1 Since η > 0 is arbitrary, the first inequality is proved. To prove the second part, let } P 0 {(Ãi, + = x i ) P 0 : f(x i )H s (Ãi) (HK) f dh s 0 and let P0 = P 0 \ P 0 +. Let us note that both P 0 + and P 0 of, so they satisfy the first part of Lemma. are δfine subpartition 12
13 Thus, f(x i)h s (Ãi) (HK) f dh s { } = f(x i )H s (Ãi) (HK) f dh s (Ãi,xi) P + 0 (Ãi,xi) P 0 ε + ε = 2ε. { } f(x i )H s (Ãi) (HK) f dh s Theorem 3.8 (Cauchy extension). Let f : R be a function shk integrable on [c, b] for each c (a, b). If lim c a +(HK) [c,b] f dh s exists, then f is shk integrable on and (HK) f dh s = lim (HK) f dh s. c a + [c,b] Proof. Let ε > 0 and let {c n } n=0 be a strictly decreasing seguence of real numbers of such that c 0 = b and inf n N c n = a. Let A = lim c a +(HK) [c,b] f dh s. Choose N such that (HK) [x,b] for x (a, ( ) c N ] and f(a) H s [a, cn ] < ε 4. f dh s A < ε 4 From Theorem 3.5, since f is shk integrable on [c, b] for each c (a, b), then f shk( [c k, c k 1 ]). Therefore, for ε > 0 and for each positive integer k, we let δ k be a gauge on [c k, c k 1 ] such that S(f, Q k) (HK) for each δ k fine partition Q k of [ck, c k 1 ]. Define a gauge δ on by setting [c k,c k 1 ] f dh s < ε 4(2 k ) 13
14 c N a if x = a min{δ k (c k ), δ k+1 (c k ), 1 2 (c k 1 c k ), 1 2 (c k c k+1 )} δ(x) = min{δ k (c k ), 1 2 (x c k), 1 2 (c k 1 x)} if x = c k and k=1,2,.. if x (c k, c k 1 ) and k=1,2, (b c 1) if x = b and let P = {(Ãi, x i )} p be a δfine partition of. Let Ãi = [y i 1, y i ] for i = 1, 2,..., p and a = y 0 < y 1 <... < y p = b. Since a / k=1 [c k, c k 1 ], our choice of δ implies that x 1 = a, y 1 < c N and y 1 (c r+1, c r ] for some unique positive integer r. We also observe that for each k {1, 2,..., r}, our choice of δ implies that {(Ã, x) P : Ã [c k, c k 1 ]} is a δ k fine partition of Since [ck, c k 1 ]. S(f, P ) = f(a)h s ( [a, y 1 ]) + (Ã,x) P Ã [y 1,c r] f(x)h s (Ã) + ( r k=1 (Ã,x) P Ã [c k,c k 1 ] f(x)h s (Ã) ) and (HK) f dh s = (HK) [y 1,b] [y 1,c r] f dh s + ( r (HK) k=1 [c k,c k 1 ] f dh s ) thus 14
15 S(f, P ) lim c a +(HK) f dh s [c,b] f(a) H s ( [a, y 1 ]) + f(x)h s (Ã) (HK) f dh s (Ã,x) P [y 1,c r] Ã [y 1,c r] ( r ) + f(x)h s (Ã) (HK) f dh s k=1 (Ã,x) P [c k,c k 1 ] Ã [c k,c k 1 ] + (HK) f dh s lim [y 1,b] c a +(HK) f dh s [c,b] < ε. 4 Relationship between the sintegral and the shenstock KurzweilStieltjes integral 4.1 The VitaliCarathéodory Theorem Theorem 4.1 (The VitaliCarathéodory Theorem). Let be an sset, let f be L integrable on and let ε > 0. Then there exist functions u and v on such that u f v, u is upper semicontinuous and bounded above, v is lower semicontinuous and bounded below, and (L) (v u) dh s < ε. (1) Proof. Assume first that f 0 and that f is not identically 0. Since f is the pointwise limit of an increasing sequence of simple functions s n (see [9]), f is the sum of the simple functions t n = s n s n 1 (taking s 0 = 0), and since t n is a linear combination of characteristic functions, we see that there are measurable sets M i (not necessarily disjoint) and constants c i > 0 such that f(x) = c i χ Mi (x) (x ). 15
16 Since (L) f dh s = c i H s (M i ) (2) the series in (3) converges. Since H s is Borel regular (see [7], pag. 57) and is an sset, therefore H s L is a Radon measure. So, there are compacts sets K i and open sets V i such that K i M i V i and Put where N is chosen so that c i H s (V i K i ) < 2 i 1 ε (i = 1, 2, 3,...). (3) N v = c i χ Vi, u = c i χ Ki, c i H s (M i ) < ε 2. (4) N+1 Then v is lower semicontinuous, u is upper semicontinuous, u f v, and v u = N c i (χ Vi χ Ki ) + c i χ Vi c i (χ Vi χ Ki ) + N+1 c i χ Vi so that (4) and (5) imply (2). In the general case, write f = f + f, attach u 1 and v 1 to f +, attach u 2 and v 2 to f, as above, and put u = u 1 u 2, v = v 1 v 2. Since v 2 is upper semicontinuous and since the sum of two upper semicontinuous functions is upper semicontinuous, u and v have the desired properties. We now show that the shk integral is more general than the Lebesgue integral. Theorem 4.2. If is an sset, then L() shk() and (L) f dh s = (HK) f dh s for each f L(). Proof. Let f L() and ε > 0. For the VitaliCarathéodory Theorem, there are upper and lower semicontinuos functions g and h, respectively, such that g f h + and (L) (h g) dhs < ε. Find a gauge δ on so that g(t) f(t) + ε and h(t) f(t) ε for each x, t with x t < δ(x). Now if P = {(Ãi, x i )} p is a δfine partition of, then (L) g dh s (L) f dh s (L) h dh s. (5) N+1 16
17 Since g(t) f(x i ) + ε, then g(t) ε f(x i ), for each t Ãi we have that (L) (g ε) dh s (L) f(x i ) dh s and therefore (L) g dh s ε H s (Ãi) f(x i ) H s (Ãi). Moreover, since h(t) f(x i ) + ε, then f(x i ) h(t) + ε, for each t Ãi and therefore f(x i ) H s (Ãi) (L) h dh s + ε H s (Ãi). The last two results lead to (L) g dh s ε H s (Ãi) f(x i ) H s (Ãi) (L) h dh s + ε H s (Ãi), (6) i = 1,..., p. It follows that: Since S(f, P ) (L) f dh s f(x i)h s (Ãi) (L) f dh s For the (7), we have that f(x i)h s (Ãi) (L) f dh s (L) h dh s + ε H s (Ãi) (L) f dh s Ã i = ε Hs (Ãi) + (L) (h f) dh s. and therefore we have that S(f, P ) (L) (L) h dh s (L) f dh s (L) g dh s and then the theorem is proved. 0 (L) (h f) dh s (L) (h g) dh s f dh s f(x i)h s (Ãi) (L) f dh s [ ] ε H s (Ãi) + (L) (h g) dh s < ε (H s () + 1) 17
18 Remark. The converse of previus theorem is not true. The following example explains this assertion. xample 4.1. Let be the classical Cantor set and let F : R be the function defined as follow F (x) = ( 1)n+1 2 n n if x [ ] 2 3 n, 1 3 n 1, n = 1, 2, 3,... We will prove that F is shkintegrable on, but it is not slintegrable on. (HK) [ 2 3 k,1] F dh s = = = k n=1 k n=1 n=1 For the Cauchy extension, we have: (HK) [ 2 3 n, 1 ( 1) n+1 2 n n F dh s = 3 n 1 ] ([ 2 H s 3 n, 1 3 n 1 n=1 ]) = k ( 1) n+1 2 n 1 k n 2 n = ( 1) n+1 n (HK) F dh s = lim (HK) F dh s = [0,1] k [ 2 3 k,1] ( 1) n+1 = log 2 n F is not slintegrable on. In fact, if F were slintegrable on, therefore F would be slintegrable on. Following the previous sequnce of steps, it follows easily that References (HK) F dh s = [0,1] n=1 n=1 1 n = +. [1] M. De Guzman, M. A. Martin and M. Reyes, On the derivation of Fractal functions. Proc. 1st IFIT Conference on Fractals in the Fundamental and Applied Sciences, Lisboa, Portugal (NorthHolland, 1991), [2] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York (2003). [3] K. Falconer, The geometry of fractal sets. Cambridge University Press, Cambridge, (1985). 18
19 [4] K Falconer, Techniques in fractal geometry. John Wiley and Sons (1997). [5] H. Jiang and W. Su, Some fundamental results of calculus on fractal sets, Commun. Nonlinear Sci. Numer. Simul. 3, No.1, (1998) [6] B. B. Mandelbrot, The fractal geometry of nature. Freeman and company (1977). [7] P. Mattila, Geometry of Sets and Measures in uclidean Spaces. Fractal and rectificability, Cambridge University Press, Cambridge, (1995). [8] A. Parvate and A. D. Gangal, Calculus on fractal subsets of real line  I: formulation. Fractals, Vol. 17, No. 1, (2009) [9] W. Rudin, Real and Complex Analysis, McGrawHill Book Company (1987). 19
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