# Calculus of Fractal Curves

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```					Calculus on Fractal Curves in Rn

arXiv:0906.0676v1 [math-ph] 3 Jun 2009

Abhay Parvate Seema Satin A.D.Gangal Department of Physics, University of Pune, Pune 411 007, India abhay@physics.unipune.ernet.in, satin@physics.unipune.ernet.in, adg@physics.unipune.ernet.in
Abstract A new calculus on fractal curves, such as the von Koch curve, is formulated. We deﬁne a Riemann-like integral along a fractal curve F , called F α -integral, where α is the dimension of F . A derivative along the fractal curve called F α -derivative, is also deﬁned. The mass function, a measurelike algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize algorithmic aspect. Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between this calculus and ordinary calculus on the real line. The F α -integral and F α - derivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact, they can thus be evalutated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F , and F α - diﬀerentiability is generalized using Sobolev like constructions. Finally we touch upon an example of a diﬀusion equation on fractal curves, to illustrate the utility of the framework.

1

Introduction

It is now well known that fractals pervade nature [1, 2]. The geometry of fractals is also well studied [1, 3, 4, 5, 6, 7]. Fractal curves often lack the smoothness properties required by ordinary calculus. For example, observed path of a quantum mechanical particle [8] or Brownian and Fractional Brownian trajectories [1, 3] are known to be fractals and are continuous but non-diﬀerentiable. A percolating path, just above the percolating phase transition can be considered as an appoximate realization of a fractal curve [9]. If a long polymer is modeled as a fractal curve, then accumulation of a physical property along the curve would amount to integration on such a curve. This is often carried out using ad hoc procedures. While there are some remarkable approaches to develop tools for such situations [10, 11, 12, 13, 14], much more is desired. This paper aims to formulate a calculus speciﬁcally taylored for fractal curves, in a close analogy with ordinary 1

calculus. In particular, we adopt a Riemann-Stieltjes like approach for deﬁning integrals, because of its simplicity and advantage from algorithmic point of view. Such an approach was concieved in [15] and is fully formulated in [16, 17, 18] for fractal subsets of R . In particular, an integral and a derivative of order α are deﬁned [16] on sets F ⊂ R, where α ∈ (0, 1] is the dimension of F . This calculus, called F α - calculus has many results analogous to ordinary calculus and can be viewed as a generalization of ordinary calculus on R. In fact, in [17, 18] a conjugacy between the F α -calculus and ordinary calculus is discussed. The present paper extends that approach to formulate calculus on fractal curves . The organization of the paper is as follows. In Section 2 we deﬁne a mass function and integral staircase function. The mass function gives the content of a continuous piece of the fractal curve F . The staircase function, more appropriately called the rise function, is obtained from the mass function and describes the rise of the mass of the curve with respect to the parameter. We emphasize the algorithmic nature of the mass function: by presenting an algorithm to calculate it. In section 3 we show that the mass function allows us to deﬁne a new dimension called γ-dimension, which is algorithmic and ﬁner than the box dimension. In section 4 the concepts of limits and continuity are adapted to the concepts of F -limit and F -continuity. Section 5 is devoted to the discussion of integral on fractal curves called F α -integral. The formulation is analogous to the Riemann integration [19]. The notion of F α -diﬀerentiation is introduced in section 6. The fundamental theorems of F α -calculus proved in section 7, state that the F α -integral and F α - derivative are inverses of each other. The conjugacy between F α -calculus on F and ordinary calculus on the real line, discussed in section 8, establishes a relation between the two and gives a simple method to evaluate F α -integrals and F α - derivatives of functions on the fractal F . In section 9, function spaces of F α -integrable and F α -diﬀerentiable functions on the fractal F are explored. In particular Sobolev Spaces are introduced and abstract Sobolev derivatives are constructed. Finally as a simple physical application we brieﬂy touch upon, an example of a diﬀusion equation on fractal curves. Section 10 is the concluding section.

2

The mass function and the staircase

In this paper we consider fractal curves, i.e. images of continuous functions f : R → Rn which are fractals. To be precise: Let [a0 , b0 ] be a closed interval of the real line. Deﬁnition 1 A fractal (curve) F ⊂ Rn is said to be continuously paramatrizable (or just paramatrizable for brevity) if there exists a function w : [a0 , b0 ] → F ⊂ Rn which is continuous, one-to-one and onto F . In this paper F will always denote such a fractal curve. Examples: A simple example of such a parametrization is the function s s w : R → R2 deﬁned by w(t) = (t, Wλ (t)) where Wλ (t) is the well known

2

Weierstrass function [3] given by
∞ s Wλ (t) = k=1 s where λ > 1 and 1 < s < 2. The graph of Wλ (t) is known to be a fractal curve with box-dimension s. Our next example constitutes of one important class of parametrizations of self-similar curves in two dimensions (There are other ways of parametrizing fractal curves ; for example see [20]). Let Ti , i = 0, . . . , n−1 be linear operations which are composed of rotation and scaling. Each Ti can be represented by a 2 × 2 matrix: cos θi − sin θi . T i = si sin θi cos θi

λ(s−2)k sin λk t

Further, they should satisfy the condition:
n−1

Ti (v) = v
i=0

for any vector v, and 0 < si < 1 for i = 0, . . . , n − 1. The fractal is deﬁned by the limit set [4] of the similarity transformations:
j−1

Sj (v) =
i=0

Ti (v0 ) + Tj (v),

j = 0, . . . , n − 1

where v0 is a ﬁxed vector.The limit set will be in the form of a curve because of the way Sj are constructed from Ti . Let ⌊nt⌋ denote the integer part of nt. Now, the function w deﬁned implicitly by
⌊nt⌋−1

w(t) =
i=0

Ti (v0 ) + T⌊nt⌋ (w(nt − ⌊nt⌋)),

0≤t≤1

(1)

parametrizes the above fractal curve. To implement it as an algorithm, we stop the recursion at some appropriate depth. The continuity and invertibility of this parametrization can be numerically veriﬁed, when the curve itself is nonself-intersecting. In particular the von Koch curve is realized by setting all si = 1/3, θ0 = θ3 = 0, θ1 = −θ2 = π/3, and v0 = (1, 0) (the unit vector along x axis). • Hereafter symbols such as a,b,c,etc denote numbers in [a0 , b0 ] and θ, θ′ etc denote points of F . Deﬁnition 2 A subdivision P[a,b] of interval [a, b], a < b is a ﬁnite set of points {a = t0 , t1 , . . . , tn = b}, ti < ti+1 . Any interval of the form [ti , ti+1 ] is called a component of the subdivision P . Moreover, if Q is a subdivision such that P ⊂ Q then Q is called a reﬁnement of P . 3

Deﬁnition 3 For a set F and a subdivision P[a,b] , a < b, [a, b] ⊂ [a0 , b0 ]
n−1

σ α [F, P ] =
i=0

|w(ti+1 ) − w(ti )|α Γ(α + 1)

(2)

where | · | denotes the euclidean norm on Rn , and P[a,b] = {a = t0 , . . . , tn = b}. Next we deﬁne the coarsed grained mass function. Deﬁnition 4 Given δ > 0 and a0 ≤ a ≤ b ≤ b0 , the coarse grained mass α γδ (F, a, b) is given by
α γδ (F, a, b) = {P[a,b] :|P |≤δ}

inf

σ α [F, P ]

(3)

where |P | = max0≤i≤n−1 (ti+1 − ti ) for a subdivision P .
α Some properties of γδ (F, a, b): α α Lemma 5 Let δ1 ≤ δ2 . Then γδ1 (F, a, b) ≥ γδ2 (F, a, b).

The proof is obvious in context with deﬁnition 4. α The following lemma states that γδ (F, a, b) is non decreasing in b and nonincreasing in a.
α α Lemma 6 Let δ > 0 and a0 ≤ a < b < c ≤ b0 . Then γδ (F, a, b) ≤ γδ (F, a, c) α α and γδ (F, b, c) ≤ γδ (F, a, c). α Proof: Let ǫ > 0. Then according to the deﬁnition of γδ (F, a, c) there exits a subdivision P[a,c] = {t0 = a, t1 , . . . , tn = c} such that |P | ≤ δ and σ α [F, P ] < α γδ (F, a, c) + ǫ. Let Q[a,b] = {t ∈ P : t < b} ∪ {b} i.e. Q[a,b] = {y0 , y1 , . . . , ym } where yi = ti if ti < b and ym = b. It follows that |Q[a,b] | ≤ |P[a,c] | ≤ δ since [ym−1 , ym ] ⊂ [tm−1 , tm ]. Therefore, α σ α [F, Q[a,b] ] ≤ σ α [F, P[a,c] ] < γδ (F, a, c) + ǫ. α But γδ (F, a, b) ≤ σ α [F, Q] and ǫ is arbitrary, hence α α γδ (F, a, b) ≤ γδ (F, a, c).

This is the proof of the ﬁrst part, the second part is analogous. •
α Theorem 7 For a0 ≤ a ≤ b ≤ b0 , γδ (F, a, b) is continuous in b and a. α Proof: We prove the continuity of γδ (F, a, b) in b (with δ, α and a ﬁxed). In a similar way we can prove the continuity in a. Due to continuity of w, given ǫ > 0, there exists ∆′ > 0 such that

|c − b| < ∆′ =⇒ |w(c) − w(b)| < (ǫΓ(α + 1))1/α 4

Let ∆ = min(∆′ , δ). For ǫ1 > 0, there exists a subdivision P[a,b] such that |P | ≤ δ and α σ α [F, P ] < γδ (F, a, b) + ǫ1 Now, Q = P ∪ {b + ∆} is a subdivision of [a, b + ∆] such that |Q| ≤ δ.Therefore,
α γδ (F, a, b + ∆)

|w(b + ∆) − w(b)|α Γ(α + 1) ≤ σ α [F, P ] + ǫ α < γδ (F, a, b) + ǫ1 + ǫ. = σ α [F, P ] +
α α α Since ǫ1 is arbitrary, we get γδ (F, a, b + ∆) ≤ γδ (F, a, b) + ǫ. As γδ (F, a, b) is a α α nondecreasing function of b, γδ (F, a, b + t) ≤ γδ (F, a, b) + ǫ for 0 < t < ∆. So, given ǫ > 0, there exists a ∆ > 0 such that α α 0 < c − b < ∆ =⇒ γδ (F, a, c) − γδ (F, a, b) ≤ ǫ α which implies that γδ (F, a, b) is continuous in b from right. The continuity from left follows on the replacement of b by b − ∆ and of b + ∆ by b in the above proof. • The mass function is the limit of the coarse-grained mass as δ → 0 :

≤ σ α [F, Q]

Deﬁnition 8 For a0 ≤ a ≤ b ≤ b0 , the mass function γ α (F, a, b) is given by
α γ α (F, a, b) = lim γδ (F, a, b) δ→0

Remark: Since γ is a monotonic function of δ. The limit exists , but could be ﬁnite or +∞.

Properties of γ α (F, a, b)
Theorem 9 : Let a0 ≤ a < b < c ≤ b0 and γ α (F, a, c) < ∞.Then γ α (F, a, c) = γ α (F, a, b) + γ α (F, b, c). (4)

Proof : Let δ ′ > 0. There exists a δ > 0 such that |t − t′ | < δ =⇒ |w(t) − w(t′ )| < δ ′ since w(t) is continuous. Let P1 be any subdivision of [a, b] and P2 be any subdivision of [b, c], such that |P1 | < δ and |P2 | < δ. Then P1 ∪P2 is a subdivision of [a, c], |P1 ∪ P2 | ≤ δ, and σ α [F, P1 ∪ P2 ] = σ α [F, P1 ] + σ α [F, P2 ]. 5 (5)

Taking the inﬁmum of equation (5) over all P1 and P2 such that |P1 | ≤ δ and |P2 | ≤ δ, and noting that not all subdivisions of [a, c] can be written in the form of P1 ∪ P2 where P1 and P2 are subdivisions of [a, b] and [b, c] respectively, we get α α α γδ (F, a, c) ≤ γδ (F, a, b) + γδ (F, b, c) (6) Let 0 < δ1 ≤ δ. Now for every subdivision P[a,c] , |P | ≤ δ1 , we can construct a subdivision P ′ = P ∪ {b}. Obviously |P ′ | ≤ δ1 , and P ′ = P1 ∪ P2 where P1 is a subdivision of [a, c] and P2 is a subdivision of [b, c]. Let {t0 , t1 , . . . , tn } be points of P . If b ∈ P , then P = P ′ and σ α [F, P ] = α σ [F, P ′ ]. Otherwise, let [tk , tk+1 ] be the interval which contains b. In that case σ α [F, P ′ ]−σ α [F, P ] = Hence, σ α [F, P ′ ] − σ α [F, P ] ≤ This implies that σ α [F, P ] + 3δ ′α Γ(α + 1) ≥ = ≥ σ α [F, P ′ ] σ α [F, P1 ] + σ α [F, P2 ]
α α γδ1 (F, a, b) + γδ1 (F, b, c)

|w(b) − w(tk )|α |w(tk+1 ) − w(b)|α |w(tk+1 ) − w(tk )|α + − . Γ(α + 1) Γ(α + 1) Γ(α + 1) 3δ ′ Γ(α + 1)
α

for all P such that |P | < δ1 . Thus if we take inﬁmum over all subdivisions P such that |P | ≤ δ1 , we get
α γδ1 (F, a, c) +

3δ ′α α α ≥ γδ1 (F, a, b) + γδ1 (F, b, c) Γ(α + 1)

(7)

Equation (7) holds for all δ1 such that 0 < δ1 ≤ δ. Taking limit as δ1 → 0, γ α (F, a, c) + As δ ′ is arbitrary, γ α (F, a, c) ≥ γ α (F, a, b) + γ α (F, b, c) (8) 3δ ′α ≥ γ α (F, a, b) + γ α (F, b, c) Γ(α + 1)

Combining limit of equation (6) and equation (8) we get the required result. • α An immediate consequence of the additivity of γF (F, a, b) is Corollary 10 γ α (F, a, b) is increasing in b and decreasing in a. The next theorem states that γ α (F, a, x) takes all values in the range (0, γ α (F, a, b)) for x ∈ (a, b). 6

Theorem 11 Let a < b and let γ α (F, a, b) = 0 be ﬁnite. Let y be such that 0 < y < γ α (F, a, b). Then there exists c ∈ (a, b),where a0 ≤ a < c < b ≤ b0 such that γ α (F, a, c) = y. Proof: Let z = γ α (F, a, b) − y. α Given a δ > 0, consider the set of all points x of [a, b] such that γδ (F, x, b) ≤ z. This set is an interval of the form [sδ , b] for some sδ , a ≤ sδ < b, because α γδ (F, x, b) is continuous (theorem 7) and decreasing in x corollary 10. Since α γδ (F, x, b) increases as δ decreases (lemma 6), sδ increases as δ decreases. α Similarly the set of all points x of [a, b] such that γδ (F, a, x) ≤ y is an interval of the form [a, tδ ], a < tδ ≤ b, and tδ decreases as δ decreases. Let x ∈ (a, b). Then by theorem(9),
α α γ α (F, a, b) = γ α (F, a, x) + γ α (F, x, b) ≥ γδ (F, a, x) + γδ (F, x, b).

(9)

As y, z < γ α (F, a, b), there exists a δ0 > 0 such that δ < δ0 implies that α γδ (F, a, b) > y, z. In the rest of the proof we only consider δ < δ0 without mentioning. α α Since γδ (F, a, b) > y and γδ (F, a, u) is continuous and increasing in u, there α exists an x ∈ (a, b) such that γδ (F, a, x) = y. This implies that x ∈ [a, tδ ]. Further, from equation (9) it follows that
α α z = γ α (F, a, b) − y = γ α (F, a, b) − γδ (F, a, x) ≥ γδ (F, x, b)

implying that x also belongs to [sδ , b]. This can happen only when sδ ≤ tδ . Thus for each δ there exists an interval [sδ , tδ ] such that
α α x ∈ [sδ , tδ ] =⇒ γδ (F, x, b) ≤ z and γδ (F, a, x) ≤ y.

Let s = sup0<δ<δ0 sδ and let t = inf 0<δ<δ0 tδ . Now sδ increases and tδ decreases as δ goes to zero, but sδ ≤ tδ for any δ. Thus s ≤ t and [s, t] =
0<δ<δ0 α α Consequently x ∈ [s, t] implies γδ (F, x, b) ≤ z and γδ (F, a, x) ≤ y for any δ. Hence x ∈ [s, t] =⇒ γ α (F, x, b) ≤ z and γ α (F, a, x) ≤ y. (10)

[sδ , tδ ].

But as γ α (F, a, x) + γ α (F, x, b) = γ α (F, a, b) = y + z, the inequalities in (10) must be equalities. Thus for a given y,0 < y < γ α (F, a, b), there exists a set [s, t] ⊂ [a, b] such that x ∈ [s, t] =⇒ γ α (F, a, x) = y which completes the proof. Corollary 12 If γ α (F, a, b) is ﬁnite, γ α (F, a, t) is continuous for t ∈ [a, b]. Remark : The implication of this result is that no single point has a nonzero mass, or in other words, the mass function is atomless. Let F ⊂ Rn be paramatrizable. Let λ be a positive real number, v ∈ Rn , and T denote a rotation operator. We denote 7

F + v = {w(t) + v : t ∈ [a0 , b0 ]} λF = {λw(t) : t ∈ [a0 , b0 ]}. and T F = {T w(t) : t ∈ [a0 , b0 ]} Then, Theorem 13 1. Translation : γ α (F + v, a, b) = γ α (F, a, b) 2. Scaling : γ α (λF, a, b) = λα γ α (F, a, b) 3. Rotation : γ α (T F, a, b) = γ α (F, a, b)

Algorithmic Nature of the Mass Function
One of the main diﬀerence between the Hausdorﬀ measure and the mass function is that while the Hausdorﬀ measure is based on sums over a countable covers (composed of arbitrary sets) of the given set F , the mass function is based on ﬁnite subdivisions of the parametrization domain. From an algorithmic point of view, the extent of the set of all possible ﬁnite subdivisions is much smaller than that of all countable (ﬁnite and inﬁnite) covers of a set. This makes the mass function much more amenable to an algorithmic computation. We present an algorithm and its results for the von Koch curve in the Appendix. As in any algorithm which intends to approximate the inﬁmum, this algorithm attempts to ﬁnd a subdivision P such that σ α [F, P ] is close to the inﬁmum. Further, we can consider values of δ only as small as practically possible within the reach of numerical calculations. The goal of the algorithm is thus to ﬁnd a subdivision P as described above, given a ﬁxed δ. However, the set of allowed subdivisions is still large, to explore all of it systematically. Further the constraint |P | ≤ δ does not restrict the number of points in P , rendering the standard deterministic optimization algorithms either inapplicable or too complex to implement. More appropriate is a Monte Carlo method where a subdivision is modiﬁed in a variety of ways randomly but consistently with the constraint |P | ≤ δ, and the change is accepted if the sum σ α [F, P ] decreases due to the modiﬁcation. The algorithm presented in the appendix is based on this startegy.

8

Re-parametrization Invariance of Mass Function
α The deﬁnitions of σ α , γδ , and therefore γ α implicitly involve the particular parametrization w. Here we show that although deﬁned through the parametrization, these deﬁnitions are invariant under the change of parametrization. In order to be able to unambiguously and explicitly refer to the parametrization, we introduce a temporary change in the notation to explicitly indicate dependence on parametrization. Thus given a parametrization w : [a, b] → Rn , we use the following notation here: n−1

σ α [F, P ; w]
α γδ (F, a, b; w)

=
i=0

|w(ti+1 ) − w(ti )|α Γ(α + 1)

= =

|P |≤δ δ→0

inf σ α [F, P ; w]

γ α (F, a, b; w)

α lim γδ (F, a, b; w)

Let w1 and w2 be two parametrizations of the given fractal curve. By our deﬁnition of parametrization, w1 and w2 are continuous and one-to-one. Let the domain of w1 be [a1 , b1 ], and that of w2 be [a2 , b2 ]. We further assume that w1 and w2 have the same orientation, i. e. w1 (a1 ) = w2 (a2 ) and w1 (b1 ) = w2 (b2 ). −1 Thus, z = w2 ◦ w1 : [a1 , b1 ] → [a2 , b2 ] is a continuous, one-to-one and strictly monotonically increasing function. Now, given δ2 > 0 and ǫ > 0, there exists a subdivision P2 of [a2 , b2 ] such that α σ α [F, P2 ; w2 ] < γδ2 (F, a2 , b2 ; w2 ) + ǫ. The set of points P1 = {z −1 (t) : t ∈ P2 } forms a subdivision of [a1 , b1 ]. Then, σ α [F, P1 ; w1 ] = σ α [F, P2 ; w2 ] by appropriate substitution. Therefore,
α σ α [F, P1 ; w1 ] < γδ2 (F, a2 , b2 ; w2 ) + ǫ

which implies that
α α γδ1 (F, a1 , b1 ; w1 ) < γδ2 (F, a2 , b2 ; w2 ) + ǫ

where δ1 = |P1 |. Further, since z is continuous, lim δ1 = 0, as δ2 → 0, implying that γ α (F, a1 , b1 ; w1 ) < γ α (F, a2 , b2 ; w2 ) + ǫ. Since ǫ is arbitrary, and the same argument remains valid starting with z −1 = −1 w1 ◦ w2 , we conclude that γ α (F, a1 , b1 ; w1 ) = γ α (F, a2 , b2 ; w2 ).

9

This establishes the fact that the mass function depends only on the fractal curve (i. e. the image of the parametrization), and is independent of the parametrization itself. Since the mass function underlies the calculus developed in the subsequent sections, the calculus is also independent of the particular parametrization chosen. Now we introduce the integral staircase function for a set F of order α. Deﬁnition 14 Let p0 ∈ [a0 , b0 ] be arbitrary but ﬁxed. The staircase function α SF : [a0 , b0 ] → R of order α for a set F is given by
α SF (t) =

γ α (F, p0 , t) −γ α (F, t, p0 )

t ≥ p0 t < p0

(11)

In the rest of this paper we take p0 = a0 unless stated otherwise. Here this function may, more appropriately, be described as a rise function. However we retain the name staircase function because in analogous calculus on fractal subsets of the real line this role is played by a staircase. Let 1 ≤ α ≤ n. If γ α (F, a0 , b0 ) is ﬁnite, then for all t0 ,t1 ∈ (a0 , b0 ) such that α α t0 < t1 , the following statements hold: 1.SF (t) is increasing in t; 2.SF (t1 ) − α α α SF (t0 ) = γ (F, t0 , t1 ); 3.SF is continuous on (a0 , b0 ). α Throughout the paper we consider only those sets for which SF is strictly increasing and thus invertible. Further, we deﬁne
α J(θ) = SF (w−1 (θ)),

where t ∈ [a0 , b0 ].

α which is the function induced by SF on F , and it is also one-to-one. As an example, ﬁgure 1 shows the staircase function for the von koch curve. The curve was paprametrized as given in [20]. α A graph between the staircase function SF (t) against the Euclidean distance between origin and w(t) for the von- koch curve is shown in ﬁg 2 and 3.

θ∈F

(12)

3

The γ- Dimension

We now consider the sets F for which the mass function γ α (F, a, b) gives the most useful information. Due to the similarity of the deﬁnitions of mass function and the Hausdorﬀ outer measure, the former can be used to deﬁne a fractal dimension as follows. If 1 ≤ α < β ≤ n then by deﬁnition
n−1 i=0 ′

σ β [F, P ] =

(|w(ti+1 ) − w(ti )|)β Γ(β + 1)

Let δ > 0. There exists a δ > 0 such that |t − t′ | ≤ δ =⇒ |w(t) − w(t′ )| ≤ δ ′ (uniform continuity). We assume |P | < δ. Then then
n−1

σ β [F, P ] ≤

δ ′β−α
i=0

|w(ti+1 − w(ti )|α Γ(α + 1) Γ(α + 1) Γ(β + 1) 10

Staircase 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0 0.1 0.20.3 0.40.5 0.60.7 X 0.80.9 10

0.3 0.25 0.2 0.15 Y 0.1 0.05

α Figure 1: SF for von Koch curve. The von Koch curve lies in the XY plane. α The vertical lines are drawn to guide the eye (to show how SF rises)

= Let δ1 < δ.

δ ′β−α σ α [F, P ]

Γ(α + 1) Γ(β + 1) Γ(α + 1) Γ(β + 1)

β α γδ1 (F, a, b) ≤ δ ′β−α γδ1 (F, a, b)

In the limit δ1 → 0 γ β (F, a, b) ≤ δ ′ As δ ′ is arbitrary γ β (F, a, b) = 0 for γ α (F, a, b) < ∞ and β > α It follows that γ α (F, a, b) is inﬁnite upto certain value of α, say α0 , and jumps down to zero for α > α0 . Thus Deﬁnition 15 The γ-dimension of F , denoted by dimγ (F ), is dimγ (F ) = inf{α : γ α (F, a, b) = 0} = sup{α : γ α (F, a, b) = ∞} 11
β−α α

γ (F, a, b)

Γ(α + 1) Γ(β + 1)

1 0.9 0.8 0.7 Euclidean distance 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.05 0.1 0.15 0.2 0.25 Staircase 0.3 0.35 0.4 0.45

α Figure 2: SF (t) for t ∈ [0, 1] vs Euclidean distance between the origin and W(t) for von-koch curve

Now we compare the γ-dimension with the Box dimension. Let dimγ (F ) = α. Then γ α (F, a, b) diverges for any β < α. Thus for any β k > 0, there exists δ0 > 0 such that δ < δ0 =⇒ γδ (F, a, b) > k. ′ Let δ > 0. Then there exists δ > 0 such that |t − t′ | < δ implies |w(t) − w(t′ )| < δ ′ . Let P be any subdivision such that |P | ≤ δ. Let Nδ′ (F ) be the number of terms in the sum σ α [F, P ]. Then for arbitrary but ﬁxed k > 0 and δ < δ0
β k < γδ (F, a, b) ≤

Nδ′ (F )δ ′β Γ(β + 1)

where 1 ≤ β < α ≤ n. Thus, ln(k) ≤ ln Nδ′ (F ) + β ln(δ ′ ) − ln(Γ(β + 1)) Dividing by −ln(δ ′ ) −β ln(δ ′ ) ≤ ln Nδ′ (F ) − ln(k) − ln(Γ(β + 1)) ln Nδ′ (F ) ln(k) − ln(Γ(β + 1)) − − ln(δ ′ ) − ln(δ ′ ) 12

β≤

0

-1

-2 Euclidean distance

-3

-4

-5

-6

-7 -10

-9

-8

-7

-6

-5 Staircase

-4

-3

-2

-1

0

α Figure 3: log-log graph of SF (t) for t ∈ [0, 1] vs Euclidean distance between origin and w(t) for von-koch curve

In the limit as δ ′ → 0 the ﬁrst term is the Box dimension and the denominator in the second term diverges, and we get, β ≤ dimB (F ) = lim ′ ln(Nδ′ (F ) δ →0 − ln(δ ′ )

This is true for any β < α = dimγ (F ) so that dimγ (F ) ≤ dimB (F ). Thus the γ-dimension is ﬁner than the box dimension.

γ-dimension for self-similar curves
Let α denote the γ- dimension of a self similar curve , which is made up of m 1 copies of itself, scaled by a factor of n and rotated and translated appropriately. Then using the translation, scaling and rotation properties of the mass function (theorem 13) one can see that the mass of the whole curve is given by 1 γ α (F, a0 , b0 ) = mγ α ( F, a0 , b0 ) n 13

1 γ α (F, a0 , b0 ) = m( )α γ α (F, a0 , b0 ) n Hence, α = log m/ log n This is same as the Hausdorﬀ dimension of self-similar curves [4] . Hence we can see that for self-similar curves dimγ F = dimH F = dimB F

(13) (14)

where dimH F denotes the Hausdorﬀ dimension and dimB F the box dimension of F .

4

F -Limit and F -Continuity

Now we introduce limits and continuity along a fractal curve. Deﬁnition 16 Let F ⊂ Rn be a fractal curve, and let f : F → R. Let θ ∈ F . A number l is said to be the limit of f throught points of F , or simply F - limit, as θ′ → θ, if given ǫ > 0 there exists δ > 0 such that θ′ ∈ F and |θ′ − θ| < δ =⇒ |f (θ′ ) − l| < ǫ If such a number exists it is denoted by l = F - lim f (θ′ ) ′
θ →θ

Deﬁnition 17 A function f : F → R is said to be F- continuous at θ ∈ F if f (θ) = F - limθ′ →θ f (θ′ ). Deﬁnition 18 f : F → R is said to be uniformly continuous on E ⊂ F if for any ǫ > 0 there exists δ > 0 such that for any θ ∈ F and θ′ ∈ E |θ′ − θ| < δ =⇒ |f (θ′ ) − f (θ)| < ǫ

5

F α -Integration

The class of bounded functions f : F → R is denoted by B(F ). Deﬁnition 19 For t1 ,t2 ∈ [a0 , b0 ],t1 ≤ t2 a section or segment C(t1 , t2 ) of the curve is deﬁned as C(t1 , t2 ) = {w(t′ ) : t′ ∈ [t1 , t2 ]} Deﬁnition 20 Let f : F → R and t1 , t2 ∈ [a0 , b0 ],t1 ≤ t2 . Then, M [f, C(t1 , t2 )] = and m[f, C(t1 , t2 )] =
θ∈C(t1 ,t2 )

sup
θ∈C(t1 ,t2 )

f (θ)

inf

f (θ)

14

α Deﬁnition 21 Let SF (t) be ﬁnite for t ∈ [a, b] ⊂ [a0 , b0 ]. Let P be the subdivision of [a, b] with points {t0 , . . . , tn } .The upper and the lower F α -sum for the function f over the subdivision P are given respectively by n−1

U [f, F, P ] =
i=0 n−1

α

α α M [f, C(ti , ti+1 )][SF (ti+1 ) − SF (ti )],

(15)

Lα [f, F, P ] =
i=0

α α m[f, C(ti , ti+1 )][SF (ti+1 ) − SF (ti )].

(16)

From the deﬁniton it is clear that U α [f, F, P ] ≥ Lα [f, F, P ] (17)

The following lemma asserts that with reﬁnements, the upper F α -sum decreases and the lower F α sum increases, both monotonically (but not strictly monotonically). Lemma 22 Let f ∈ B(F ).If Q is a reﬁnement of a subdivision P , then U α [f, F, Q] ≤ U α [f, F, P ] and Lα [f, F, Q] ≥ Lα [f, F, P ]. Proof: Let P = {t0 , t1 , . . . , tn } and Q = P ∪ {t′ } where t′ ∈ (ti , ti+1 ).Then M [f, F, [ti , t′ ]] ≤ M [f, F, [ti , ti+1 ]] and M [f, F, [t′ , ti+1 ]] ≤ M [f, F, [ti , ti+1 ]] Hence, U α [f, F, P ] ≥ U α [f, F, Q]. This conclusion can be extended for any reﬁnement of P . Analogously Lα [f, F, Q] ≥ Lα [f, F, P ]. • Lemma 23 If P and Q are any two subdivisions of [a, b],then U α [f, F, P ] ≥ Lα [f, F, Q] Proof: As P ∪ Q is a reﬁnement of both P and Q, it follows from the above lemma and equation (17) that U α [f, F, P ] ≥ U α [f, F, P ∪ Q] ≥ Lα [f, F, P ∪ Q] ≥ Lα [f, F, Q] • Now we deﬁne the F α -integral

α Deﬁnition 24 Let F be such that SF is ﬁnite on [a, b]. For f ∈ B(F ), the lower α and upper F -integral of the function f respectively , on the section C(a, b) are

C(a,b)

f (θ)dα θ = sup Lα [f, F, P ] F
P[a,b]

(18)

C(a,b)

f (θ)dα θ = inf U α [f, F, P ] F
P[a,b]

(19)

15

Deﬁnition 25 If f ∈ B(f ), we say that f is F α - integrable on C(a, b) if
C(a,b)

f (θ)dα θ = F

C(a,b)

f (θ)dα θ F

and the common value is called the F α -integral f (θ)dα θ. F

C(a,b)

The next lemma is useful in proving many results: Lemma 26 Let f ∈ B(F ). Then f is F α -integrable on C(a, b) if and only if, for any ǫ > 0, there exists a sudivision P of [a, b] such that U α [f, F, P ] < Lα [f, F, P ] + ǫ. The F α -integral is sectionwise additive: Theorem 27 Let f be an F α -integrable function on C(a, b) and a ≤ c ≤ b. Then , f is F α -integrable on C(a, c) and C(c, b). Further, f (θ)dα θ = F f (θ)dα θ + F f (θ)dα θ F (20)

C(a,b)

C(a,c)

C(c,b)

This can be proved in a manner analogous to Riemann integral. Lemma 28 F α -integration is a linear operation. Lemma 29 Let γ α (F, a, b) be ﬁnite, and f (θ) = 1, function. Then f (θ)dα θ = F θ ∈ F denote the constant

C(a,b)

C(a,b)

α α 1dα θ = SF (b) − SF (a) = J(w(b)) − J(w(a)) F

Proof: Let I = C(a, b), M [f, I] = m[f, I] = 1. Thus U α [f, F, P ] = Lα [f, F, P ] = α α SF (b) − SF (a) for any subdivision P of [a, b]. •

6

F α -Diﬀerentiation

Deﬁnition 30 Let F be a fractal curve. Then the F α -derivative of function f at θ ∈ F is deﬁned as
α (DF f )(θ) = F - lim ′ θ →θ

f (θ′ ) − f (θ) J(θ′ ) − J(θ)

(21)

if the limit exists. 16

α Theorem 31 If (DF f )(θ) exists for all θ ∈ C(a, b), then f is F -continuous on C(a, b).

Lemma 32 F α -derivative is a linear operation. We can immediately calculate the derivative for two elementary functions: Remark : The F α -derivative of a constant function f : F → R, f (θ) = k ∈ R is zero, i. e. α DF (f ) = 0. This result is to be contrasted with the classical fractional derivative (RiemannLiouville, and others) of a constant, which is not zero in general [21, 22, 23, 24]. Lemma 33
α (DF J)(θ) = 1

This lemma together with lemma( 29) can be viewed as the special cases of α the fundamental theorems of calculus (Section 7) involving SF and its derivative,viz. unity. Before stating the analogue of Rolle’s theorem, we state the following lemma: Lemma 34 Let f be a F -continuous function on the segment C(a, b). If the maximum or minimum value for f is attained at w(c) where a < c < b and if α α DF (f (w(c))) exists then DF (f (w(c))) = 0. Proof: We present the proof for maximum value. The proof for minimum value α α is similar. Suppose the contrary is true,DF (f (w(c))) = 0. If DF (f (w(c))) > 0 then f (w(t)) − f (w(c)) f (w(t)) − f (w(c)) F - lim > 0 and so >0 α (t) − S α (c) α α t→c SF SF (t) − SF (c) F for 0 < |t − c| < δ1 where δ1 is a suitable positive number. α α If t ∈ (c, c + δ1 ) then SF (t) − SF (c) > 0 and hence f (w(t)) − f (w(c)) > 0. This contradicts the hypothesis that f attains a maximum at w(c). If α DF (f (w(c))) < 0 then f (w(t)) − f (w(c)) <0 α α SF (t) − SF (c)

α α for 0 < |t − c| < δ2 . If t ∈ (c − δ2 , c) then SF (t) − SF (c) < 0 and hence α f (w(t)) − f (w(c)) > 0, which is again a contradiction. Thus (DF f )(w(c)) = 0. • Now the analogue of Rolle’s theorem is: α Theorem 35 Let f : F → R be a F -continuous function such that (DF f )(θ) is deﬁned on C(a, b) and f (w(a)) = f (w(b)) = 0. Then there is some point α c ∈ (a, b) where (DF f )(w(c)) = 0.

17

Theorem 36 (Law of the mean) Let f : F → R be a F -continuous function α such that (DF f )(w(t)) exists on C( a, b), a < b. Then there exists a point c ∈ [a, b] such that f (w(b)) − f (w(a)) α (DF f )(w(c)) = α α SF (b) − SF (a) Proof: This theorem can be proved by applying theorem 35 to the function h: h(w(t)) = f (w(t)) − f (w(a)) − f (w(b)) − f (w(a)) α α (SF (t) − SF (a)) α α SF (b) − SF (a)

for a ≤ t ≤ b. • We had seen earlier that the F α -derivative of a constant f (θ) = k is zero. Now we see that these are the only functions whose F α -derivatives are zero: Corollary 37 Let f : F → R be a F -continuous function such that α (DF f ) = 0. Then f = k where k on C(a, b). Proof : Suppose, if possible, that the function is not a constant. Then there exist y and z, a ≤ y < z ≤ b, such that f (w(y)) = f (w(z)). This implies either f (w(y)) < f (w(z)) or f (w(y)) > f (w(z)). In both cases there there exists α c ∈ (y, z) such that (DF f )(w(c)) = 0 by theorem (36), which is a contradiction. •

7

Fundamental theorems of F α -calculus

In this section we relate the F α -integration and F α -diﬀerentiation as inverse operations of each other. The ﬁrst fundamental theorem states: Theorem 38 Let f ∈ B(F ) is an F -continuous function on C(a, b),and let g : f → R be deﬁned as g(w(t)) =
C(a,t)

f (θ)dα θ F

for all t ∈ [a, b]. Then

α (DF g)(θ) = f (θ)

Proof: From theorem (27), for t′ ∈ (t, b], we have g(w(t′ )) − g(w(t)) = From deﬁnition (30),
α (DF g)(θ) = F - lim ′ C(t,t′ )

f (θ)dα θ F

θ →θ

g(θ′ ) − g(θ) J(θ′ ) − J(θ)

18

i.e
α (DF g)(w(t)) = F - lim ′ C(t,t′ ) α t →t SF (t′ )

f (θ)dα θ F
α − SF (t)

when θ = w(t), θ′ = w(t′ ).

(22)

Now, m[f, C(t, t′ )]
C(t,t′ )

dα θ ≤ F

C(t,t′ )

f (θ)dα θ ≤ M [f, C(t, t′ )] F

C(t,t′ )

dα θ F

and
C(t,t′ ) α α dα θ = SF (t′ ) − SF (t) by lemma (29) F

so that m[f, C(t, t′ )] ≤
C(t,t′ ) α SF (t′ )

f (θ)dα θ F
α − SF (t)

≤ M [f, C(t, t′ )]

(23)

As f is continuous and w is continuous,
t′ →t+

lim m[f, C(t, t′ )] = ′lim M [f, C(t, t′ )] = f (w(t))
t →t+

(24)

Similarly,
t′ →t−

lim m[f, C(t, t′ )] = ′lim M [f, C(t, t′ )] = f (w(t))
t →t−

(25)

From equations (22),(23),(24) and (25),we get the result. • The second fundamental theorem says that the F α -integral as a function of upper limit is the inverse of F α -derivative except for an additive constant. Theorem 39 Let f : F → R be F α - diﬀerentiable function and h : F → R be α F -continuous, such that h(θ) = (DF f )(θ)). Then h(θ)dα θ = f (w(b)) − f (w(a)) F

C(a,b)

Proof: If g(θ) =
C(a,t)

h(θ)dα θ F

α α then (DF g)(θ) = h(θ) by last theorem. Therefore (DF (g − f ))(θ) = 0 for all θ ∈ C(a, b). Now corollary (37) implies that g − f = k, a constant, or g = f + k. Thus,

f (w(b)) − f (w(a)) = g(w(b)) − g(w(a)) = g(w(b)) = • 19

C(a,b)

h(θ)dα θ F

8

Conjugacy of F α -Calculus and Ordinary Calculus

In this section, we deﬁne a map φ which takes an F α -integrable function f : α α F → R to a Riemann integrable function g : [SF (a0 ), SF (b0 )] → R such that their corresponding integrals have equal values. Thus, the map φ exhibits a conjugacy between the two operations. First let us deﬁne certain classes of functions: 1. B(F ) : class of bounded functions f : F → R. 2. B([c, d]) : class of bounded functions f : [c, d] → R 3. L(F ): set of all functions which are F α -integrable on C(a0 , b0 ).
α α α 4. The image of F under SF is denoted by K, i.e K = [SF (a0 ), SF (b0 )], and B(K) denotes the class of functions bounded on K.

5. L(K) denotes the class of functions in B(K) which are Riemann integrable α α over the interval K = [SF (a0 ), SF (b0 )]. In order to ﬁx the notation, here we brieﬂy review the deﬁnition of Riemann integral. Firstly, if g ∈ B([c, d]) and I ⊂ [c, d] is a closed interval, then we denote M ′ [g, I] = supx∈I g(x) and m′ [g, I] = inf x∈I g(x). Further, the upper and lower sum over a subdivision P[c,d] = {y0 , . . . , yn } are given by U ′ [g, P ] = n−1 n−1 ′ ′ ′ i=0 M [g, [yi , yi+1 ]] and L [g, P ] = i=0 m [g, [yi , yi+1 ]]. If the upper and lower integrals given respectively by inf P U ′ [g, P ] and supP L′ [g, P ] are equal, then g is said to be Riemann integrable, and the Riemann integral
d

g(y)dy
c

is deﬁned to be the common value. Now we deﬁne the above mentioned map φ:
α α Deﬁnition 40 The map φ : B(F ) → B([SF (a0 ), SF (b0 )]) takes f ∈ B(F ) to α α φ[f ] ∈ B([SF (a0 ), SF (b0 )]) such that for each t ∈ [a0 , b0 ], α φ[f ](SF (t)) = f (w(t))

Lemma 41 The map φ : B(F ) → B(K) is one to one and onto. The proof is straightforward. Thus we are assured that the inverse map φ−1 exists. The following theorem brings out the conjugacy between F α - integrals of functions along the fractal curve F and the Riemann integrals of their images under φ.

20

Theorem 42 A function f ∈ B(F ) is F α -integrable over C(a, b) if and only if α α g = φ[f ] is Riemann integrable over [SF (a), SF (b)]. In other words,a function f ∈ B(F ) belongs to L(F ) if and only if g ∈ L(K). Further
C(a,b)

f (θ)dα θ F

α SF (b)

=
α SF (a)

g(u)du

Proof: Let f : F → R be F α -integrable. Then there exists a subdivision P[a,b] = {t0 , t1 , . . . , tn } such that U α [f, F, P ] − Lα [f, F, P ] < ǫ (26)

for any ǫ > 0. α Denote yi = SF (ti ). Then Q = {yi : 0 ≤ i ≤ n} is a subdivision of α α [SF (a), SF (b)] For any component [ti , ti+1 ] M [f, C(ti , ti+1 )] = = = = = Therefore,
n−1

sup
w∈C(ti ,ti+1 )

f (w) f (w(t))
α g(SF (t))

sup
t∈[ti ,ti+1 ]

sup
t∈(ti ,ti+1 )

sup
y∈[yi ,yi+1 ]

g(y)

M ′ [g, [yi , yi+1 ]

U α [f, F, P ] =
i=0 n−1

α α M [f, C(ti , ti+1 )][SF (ti+1 ) − SF (ti )]

=
i=0 n−1

M [f, C(ti , ti+1 )][yi+1 − yi ] M ′ [g, [yi , yi+1 ]][yi+1 − yi ] (27) (28)

=
i=0

= Similarly

U ′ [g, Q] Lα [f, F, P ] = L′ [g, Q]

then using equations (26), (27) and (28) U ′ [g, Q] − L′ [g, Q] < ǫ

21

α α which implies that g is Riemann integrable over [SF (a), SF (b)] and
α SF (b)

g(u)du =
α SF (a)

θ

f (θ)dα θ F

Conversely if g is Riemann Integrable, then for given ǫ > 0 there exists a subα α division Q′ = {v0 , . . . , vm } of [SF (a), SF (b)] such that U ′ [g, Q′ ] − L′ [g, Q′ ] < ǫ. Then the converse can be proved by following the above steps in the reverse order. • Let f1 denote the indeﬁnite F α -integral viz. f1 (w(t)) = C(a,t) f (θ)dα θ and F y let g1 denote the ordinary indeﬁnite Riemann integral viz. g1 (y) = S α (a) g(y ′ )dy ′ . F α If we further denote the indeﬁnite F α -integral operator by IF and the indefinite Riemann integral operator by I, then the result of theorem (42) can be expressed as α IF = φ−1 Iφ as displayed in the commutative diagram of ﬁgure 4. The following theorem brings out the conjugacy between F α -derivative and ordinary derivative. Theorem 43 Let h be a function in B(F ) such that g = φ[h] is ordinarily α α diﬀerentiable on K = range of SF . Then DF h(w(t)) exists for all t ∈ (a0 , b0 ) and dg(v) α α DF h(w(t)) = |v=SF (t) dv Proof: Let v ∈ K. Then by deﬁnition dg g(u) − g(v) = lim dv u→v u−v i.e given ǫ0 > 0, there exists δ0 > 0 such that |u − v| < δ0 =⇒ | dg g(u) − g(v) − | < ǫ0 dv u−v

α Let us recall our assumption that SF is monotonically increasing and one-toα −1 ′ α −1 one. Let t = (SF ) (v), t = (SF ) (u). Then t, t′ ∈ [a0 , b0 ],h(w(t′ )) = g(u) and h(w(t)) = g(v). Thus, α α |SF (t′ ) − SF (t)| < δ0 =⇒ |

h(w(t′ )) − h(w(t)) dg − | < ǫ0 α α dv SF (t′ ) − SF (t)

α α Since (w)−1 and SF are continuous, so is their composition SF ◦ (w)−1 . Therefore, there exists δ1 > 0 such that α α |w(t′ ) − w(t)| < δ1 =⇒ |SF (t′ ) − SF (t)| < δ0

=⇒ |

dg h(w(t′ ) − h(w(t)) − | < ǫ0 α α dv SF (t′ ) − SF (t) 22

α which by deﬁnition of F -limit and DF means α DF h(w(t)) =

•

h(w(t′ )) − h(w(t)) dg α = |v=SF (t) α α w(t )→w(t) SF (t′ ) − SF (t) dv F′ - lim

Theorem 44 Let h ∈ B(F ) be an F α -diﬀerentiable function at all w ∈ F . α Further, let g = φ[h]. Then dg/dv exists at v = SF (t) and dg(v) α α |v=SF (t) = DF h(w(t)) dv
α Proof: As g = φ[h], we have g(SF (t)) = h(w(t)) for all t ∈ [a0 , b0 ] from 8. By deﬁnition and substitution α DF h(w(t))

Thus given ǫ0 > 0 there exists δ0 ′ > 0 such that |w(t′ ) − w(t)| < δ0 ′ =⇒ |

h(w(t′ )) − h(w(t)) α α SF (t′ ) − SF (t) w(t )→w(t) α ′ α g(SF (t )) − g(SF (t)) = F′ - lim α α SF (t′ ) − SF (t) w(t )→w(t) = F′ - lim
α α g(SF (t′ )) − g(SF (t)) α − DF h(w(t))| < ǫ0 α (t′ ) − S α (t) SF F

α α α α Let v = SF (t) and u = SF (t′ ). Then, w◦(SF )−1 (v) = w(t) and w◦(SF )−1 (u) = ′ α −1 w(t ). Further, since w ◦ (SF ) is continuous, there exists δ > 0 such that α |u − v| < δ =⇒ |w(t′ ) − w(t)| < δ0 ′ =⇒ | g(u)−g(v) − DF h(w(t))| < ǫ0 . u−v Which by deﬁnition of ordinary derivative gives

α This conjugacy can also be expressed as DF = φ−1 Dφ as shown in the commutative diagram of ﬁgure 4. Remark : Taylor Series One can write a fractal Taylor series for functions on fractal curve F , by using the results of this section. If g = φ[h] be such that the ordinary Taylor series is given by

•

g(u) − g(v) dg α α |v=SF (t) = lim = DF h(w(t)) u→v dv u−v

g(u) =

(u − y)n dn g(y) n! dy n n=0

∞

α α is valid for u, y ∈ [SF (a), SF (b)], then for θ, θ′ ∈ F it can be seen that

h(θ) =

(J(θ) − J(θ′ ))n α n (DF ) h(θ′ ) n! n=0

∞

provided h ∈ B(F ) is F α - diﬀerentiable any number of times on C(a, b) such α that (DF )n h ∈ B(F ) for any integer n > 0. 23

α IF

f φ−1 6 φ ? g I

- f1 φ−1 6 φ ? φ−1

α DF

f 6 φ ? g D

- f1 φ−1 6 φ ?

- g1

- g1

Figure 4: The relation between F α -integral and Riemann integral, also between F α -derivative and Ordinary derivative

9

Function Spaces in F α -Calculus

We deﬁne various function spaces in this section.

9.1

Spaces of F α -diﬀerentiable functions

Deﬁnition 45 We introduce the following spaces: C k [c, d] : Set of all functions k-times continuously diﬀerentiable on [c, d](in the ordinary sense of diﬀerentiation) C 0 (F ) : Set of all functions which are F -continuous, also denoted by C(F ). C k (F ), k ∈ N: Set of all functions f : F → R such that
α (DF )n f ∈ C 0 (F ) for all n ≤ k

i.e Set of all functions that have F -continuous F α -derivatives upto order k. We deﬁne norm on C k (F ) as follows. ||f || =
θ∈F α sup |[(DF )n f ](θ)|,

0≤n≤k

f ∈ C k (F )

We note that the spaces C k (F ) are complete with respect to this norm. The class of functions C k (F ) is mapped one to one onto C k [c, d],with c = α α SF (a0 ), d = SF (b0 ) by φ (def. 8). Due to this mapping, many results related k to C [c, d] can be translated to analogous results for C k (F ). This implies in particular that C k (F ) is separable since C k [c, d] is separable [25].

9.2

F α -Integrable Functions

Now we discuss spaces of F α - integrable functions and their completion. Consider the set L(F ) of F α - integrable functions. This is obviously a vector space with usual operations of addition and scalar multiplication.

24

It is clear that for f ∈ L(F ), the quantity Np (f ) = ||f ||p = [
C(a,b)

|f (θ)|p dα θ]1/p F

1≤p<∞

is well deﬁned. It satisﬁes the homogeniety property ||λf ||p = |λ| ||f ||p 1 1 = 1. + p p′ Then for a, b ≥ 0, p ∈ (1, ∞), Young’s inequality implies that bp ap + ′ ab ≤ p p
′

λ∈R

Now we follow the convention that p and p′ are related by (29)

(30)

Theorem 46 (Analogue of Holder’s inequality) For f, g ∈ L(F ) and p ∈ (1, ∞),
C(a,b)

|f (θ)g(θ)|dα θ ≤ Np (f )Np′ (g) F

(31)

Proof: If either Np (f ) or Np′ (g) is zero, the result is obvious. Otherwise using (30) with |g(θ)| |f (θ)| and b = a= Np (f ) Np (g) we have
′ ˙ |f (θ)| |g(θ)| 1 |f (θ)|p 1 |g(θ)|p ≤ + ′ p′ p Np (f ) Np′ (g) p Np (f ) p Np′ (g)

(32)

for all θ ∈ F . F α - integrating (32) and using eq(29) we get the required result. • Theorem 47 (Analogue of Minskowski’s inequality) For 1 ≤ p < ∞ and f, g ∈ L(F ) we have Np (f + g) ≤ Np (f ) + Np (g) Proof: The case p = 1 is obvious. For p > 1
p Np (f + g) ≤

C(a,b)

|f (θ)| |f (θ)+ g(θ)|p−1 dα θ + F

C(a,b)

|g(θ)| |f (θ)+ g(θ)|p−1 dα θ F (33)

25

using eq (31) |f (θ)||f (θ) + g(θ)|p−1 dα θ F ≤ = = = Similarly
p−1 |g(θ)||f (θ) + g(θ)|p−1 dα θ ≤ Np (g)Np (f + g) F

C(a,b)

Np (f )Np′ |f + g|p−1 ) Np (f )[ Np (f )[
′

(34)
′

C(a,b)

|f (θ) + g(θ)|(p−1)p dα θ]1/p F |f (θ) + g(θ)|p dα θ](p−1)/p F

C(a,b)

p−1 Np (f )Np (f + g)

(35)

(36)

C(a,b)

Thus from equations (34),(35) and (36)
p−1 p−1 Np (f + g) ≤ Np (f )Np (f + g) + Np (g)Np (f + g)

which implies the result. • This proves the triangle inequality for Np . Therefore Np is a seminorm. Now we identify an appropriate space so that Np acts as a norm on it. Lemma 48 For two functions f, g ∈ L(F ), Np (f − g) = 0 for p > 1, if and only if N1 (f − g) = 0. The proof is straightforward and omitted. Deﬁnition 49 Two functions f, g ∈ L(F ) are Np -equivalent if Np (f − g) = 0. This equivalence relation partitions L(F ) into equivalence classes of functions. Now we deﬁne L′ (F ) to be the space of these equivalent classes. p The space L′ (F ) is a vector space with addition and scalar multiplication p deﬁned appropriately. The function ||.||p = Np acts as a norm on L′ (F ). In view of Lemma (48) for p any p, q ∈ [1, ∞), L′ (F ) = L′ (F ). Therefore we drop the subscript p wherever q p irrelevant and denote the space by L′ (F ). L′ (F ) is not complete, but can be completed using standard procedure of p identifying equivalent Cauchy sequences, as follows: Deﬁnition 50 Two cauchy sequences {fn }, {gn } are Np -equivalent if
n→∞

lim ||fn − gn ||p = 0

This equivalence relation partitions the set of sequences in L′ (F ) into equivap lence classes. The set of the equivalence classes of sequences in L′ (F ) is denoted p by Lp (F ). 26

Thus Lp (F ) is complete by deﬁnition and therefore is a Banach space. Constructions of L′ (K), Lp (K), Np -norm can be made in analogy with L′ (F ), p p Lp (F ),Np -norm respectively using Riemann integral. The following theorem states that the conjugacy operator φ, as deﬁned in section (8), preserves Np -equivalence. Theorem 51 If v1 , v2 ∈ L(F ) are Np -equivalent, then φ[v1 ] and φ[v2 ] are Np equivalent (in L(K)) Proof: We see that (i)φ[v1 −v2 ] = φ[v1 ]−φ[v2 ], (ii)φ[|v|] = |[φ[v]|, (iii)φ[|v|p ] = (φ[|v|])p The proof follows from the above properties and theorm (42). • The relation between L′ (F ) and L′ (K) is expressed by deﬁniton (52) and theorem (53) below: ¯ Deﬁnition 52 The map φ : L′ (F ) → L′ (K) is deﬁned, such that if v ∈ v ∈ ¯ ′ ¯ v ] is the equivalence class u ∈ L′ (K) containing u = φ[v]. L (F ), then φ[¯ ¯ ¯ Theorem 53 The map φ is a linear isometric isomorphism between the spaces ′ ′ L (F ) and L (K). The proof follows from linearity of φ and theorems (42), (41) and (51). Next we prove the separability of the space Lp (F ). Theorem 54 The spaces L′ (F ) and Lp (F ) are separable. p
∞ Proof: Let w ∈ C0 (R) be such that w (x) ≥ 0 for all x ∈ R, supp(w ) = [−1, 1], and 1

w (x)dx = 1.
−1 ∞ where we have used the standard notation C0 (R) for the space of all functions ∞ in C (R) with compact support. Let u ∈ L′ (K). Now we deﬁne [25] a molliﬁer:

1 (Rǫ u)(x) = ǫ or (Rǫ u)(x) =

α SF (b)

w(
α SF (a)

x−y )u(y)dy ǫ

(37)

1 −1

u(x − ǫy)w (y)dy.

(38)

Since u ∈ L′ (K), it also belongs to the corresponding function space based on Lebesgue integral. Then theorem 2.5.3 of [25] states Rǫ u ∈ C ∞ (R) and
ǫ→0+

(39) p≥1 (40)

lim ||Rǫ u − u||p = 0, 27

Hence for every u ∈ L′ (K), there exists a sequence {un } in C ∞ (K) converging to u. This implies that C ∞ (K) is dense in L′ (K). Since C 0 (K) ⊃ C ∞ (K), therefore C 0 (K) is also dense in L′ (K). Since C 0 (K) is separable L′ (K) is separable. Further L′ (K) is isomorphic to L′ (F ) by theorem (53 ), hence the latter is separable. Then Lp (F ), being the completion of L′ (F ) by deﬁnition is p also separable.

9.3

Analogues of Abstract Sobolev spaces

Let J be a ﬁnite set of nonnegative integers {j1 , j2 , . . . , jm } , such that 0 ∈ J and ji ≤ k, 1 ≤ i ≤ m, where k is a ﬁxed integer. Let {Xj , ||.||Xj } = {Xj }j∈J be a family of Banach spaces Xj with norms ||.||Xj . We denote the cartesian product of these by X as follows: X=
j∈J

Xj

The members of X are tuples of the form u = (uj1 , . . . , ujm ) and the set X is a vector space with usual addition and scalar multiplication. A norm can be deﬁned on X such that for u = (uj1 , . . . , ujm ) ∈ X, ||u||X =
j∈J

||uj ||Xj

Thus, X is a Banach space and is separable if and only if each of the Xj , j ∈ J is separable [25]. From now on, we take Xj = Lp (F ) for each j ∈ J, where p ∈ [1, ∞) is ﬁxed. α Also we know that C ∞ ⊂ Lp (F ), and if u ∈ C ∞ (F ), then DF u ∈ C ∞ (F ) ⊂ Lp (F ). For u ∈ C ∞ (F ), let α ||u||J = ||(DF )j u||p .
j∈J

Then ||.||J acts as a norm on C (F ). In general, the space is not complete under this norm. A construction analogous to that of Sobolev spaces makes this space complete, as shown below. We deﬁne a mapping IJ : C ∞ (F ) → X by the relation
α α IJ (u) = ((DF )j1 u, . . . , (DF )jm u)

∞

Further we deﬁne a projection operator Pn : X → Xn , n ∈ J, by Pn (u = (uj1 , . . . , ujm )) = un . The mapping IJ is linear and isometric, i.e. for u ∈ C ∞ (F ), ||IJ (u)||X = ||u||J . 28

We now denote the image IJ (C ∞ (F )) by [Y k,p (F )]. Then IJ is isometric isomorphism between C ∞ (F ) and [Y k,p (F )]. The mapping Pn is continuous linear mapping from X to Xn . Now, denote the closure of [Y k,p (F )] in the topology of X by [W k,p (F )]. Since [W k,p (F )] is closed in X, it is a Banach space. Further since X is separable, so is [W k,p (F )]. Now we deﬁne an abstract Sobolev Space W k,p (F ), based on F α -calculus, to be W k,p (F ) = P0 ([W k,p (F )]) As one can easily see, this is a Banach space, and is separable since Lp (F ) is separable. The abstract (j th ) Sobolev F α -derivative of u ∈ W k,p (F ) is deﬁned as
−1 α (DF )j u = Pj (P0 (u)) W

Example: Diﬀusion on fractal curves :Starting from Chapmann-Kolmogorov equation (involving F α -integral on fractal curves), one can arrive at the fractal diﬀusion equation [26] of the form ∂ α V (θ, t) = ν(DF )2 V (θ, t) ∂t (41)

where C(a,b) V (θ, t)dα θ is the probability of ﬁnding the particle in the section F C(a, b) at time t and ν is the fractal diﬀusion constant. Using the method of conjugacy discussed above, this equation can be shown to admit an exact solution 1 (J(θ))2 V (θ, t) = √ ) exp(− 4νt 2νt

10

Conclusion

In this paper we have developed a calculus on parametrizable fractal curves of dimension α ∈ [1, n]. This involved the identiﬁcation of the important role played by the mass function and the corresponding (rise) staircase function which may be compared with the role played by the independent variable itself in ordinary calculus. The deﬁnitions of F α -integral and F α -derivative are speciﬁcally taylored for fractal curves of dimension α.Further they reduce to Riemann integral and ordinary derivative respectively, when F = R and α = 1. Much of the development of this calculus is carried in analogy with the ordinary calculus. Speciﬁcally, we have adopted Riemann-Stieltjes approach for integration, as it is direct, simple and advantageous from algorithmic point of view. The example of a diﬀusion equation on fractal curves mentioned in section 8 demonstrates the utility of such a framework. This example is discussed in [26] in detail. Other applications may include fractal Langevin equation for Brownian motion and Levy processes on such curves, which will follow in future work. This approach may be further useful in dealing with path integrals 29

and other similar applications. Another direction for extension of the considerations in this paper is the extension to crumpled or fractal surfaces which are continuously paramatrizable by a ﬁnite number of variables.

11

Appendix

The von Koch curve Image of P

Figure 5: The image (under w) of a numerically computed near-optimal subdivision P , for δ = 0.05, superimposed on the von-Koch curve. A Monte Carlo Algorithm We now present a Monte Carlo algorithm to calculate the mass function. Let us ﬁrst summarize its deﬁnition:
n−1

γ α (F, a, b) = lim

δ→0 {P :|P |≤δ}

inf

i=0

|w(ti+1 ) − w(ti )|α Γ(α + 1)

(42)

As in any algorithm which intends to approximate an inﬁmum, this algorithm attempts to ﬁnd a subdivision P such that σ α [F, P ] is close to the inﬁmum. Further, we can consider values of δ only as small as practically possible within the reach of numerical calculations. The goal of the algorithm described below is thus to ﬁnd a subdivision P as described above, given a ﬁxed δ.

30

0.9 0.01 0.02 0.05 0.10

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 20 40 60 80 100

Figure 6: The evolution of Γ(α + 1).σ α [F, P ] over the normalized number of iterations. The evolution is shown only up to N ′ = 100, since the latter part (100 < N ′ ≤ 2000) is almost ﬂat and uninteresting. For the purpose of the algorithm, [a, b] denotes the domain of w. Further, “randomly” means with a uniform probability unless stated otherwise. The symbol P always indicates the “current” subdivision in consideration. We begin with a uniform subdivision P such that |P | = δ/4, and iteratively improve it using the following prescription. 1. Choose two numbers x, y ∈ [a, b] randomly, and relabel them if necessary so that x ≤ y. Then [x, y] ⊂ [a, b]. Let P ′ = {ti : 0 ≤ i ≤ m} denote the set of all points of P ∩ [x, y]. We now modify P ′ in one of the following ways with equal probability, and denote the resultant by P ′′ : (a) With a probability pc = min(1, δ/(y − x)), we shift each point ti (except t0 and tm ) by a random amount between [−δ/2, δ/2], if the resultant subdivision P ′′ still satisﬁes |P ′′ | ≤ δ.

(b) With a probability pd = min(1, δ/(y − x)), we remove each point ti (except t0 and tm ) from P ′ , if the resultant subdivision P ′′ still satisﬁes |P ′′ | ≤ δ. (c) With a probability pi = min(1, δ/(y − x)), we insert a point between each ti and ti+1 which is chosen randomly from [ti , ti+1 ]. (However, to 31

avoid accumulating too much of rounding error, we insert the point only if the distance between any two resultant successive points is greater than δ/10.) 2. Form a new subdivision P1 = (P ∩ [a, x)) ∪ P ′′ ∪ (P ∩ (y, b]), i. e. the subdivision of which the points belonging to [x, y] are changed by the above procedure. If σ α [F, P1 ] < σ α [F, P ], then we consider P1 as the “current” subdivision which will be possibly improved further using above steps. Otherwise we consider P again for the purpose. As the sum σ α [F, P ] approaches the inﬁmum, many of the newly formed subdivisions P ′ are rejected since they sum up higher than P . Thus near the inﬁmum, the sum remains constant for many consecutive iterations, and changes only intermittently. Therefore the usual convergence criterion of terminating iteration when the diﬀerence between successive iterations or every K iterations (K being a suitable large integer) goes below certain small number, is not useful in this case. Instead, after examining the sum over a large number of iterations, we observe that the sum stops making signiﬁcant progress between N ′ = 1000 to N ′ = 2000, where N ′ = N/n is the number of iterations N normalized by the current subdivision size n. Further, we need to go through all these iterations more than once, just to ensure that subdivision is really optimal. Occasionally it may happen that the sum settles a little above the optimal value, gettting ”trapped” in a ”local minimum”. We demonstrate the results of this algorithm as applied on the von Koch curve, parametrized as in equation (1). It turns out that the mass of the entire von Koch curve is a little less than 0.51/Γ(α + 1), α = ln(4)/ ln(3). The image (under w) of the optimal subdivision found by the algorithm is shown in ﬁgure 5, superimposed on the von Koch curve. The evolution of the sum over the normalized number of iterations is shown in ﬁgure 6. The above description assumes that the value of α is the same as the γdimension of the set F , say α0 . We expect δ-independence in the values of σ α [F, P (δ)] where P (δ) denotes the resultant subdivision of the algorithm at α the scale δ, since the value of γδ converges to a ﬁnite nonzero value. This is what we observe from the values of σ α [F, P (δ)] obtained for various values of δ (ﬁgure 6). Now we would like to consider cases when α = α0 . Let 0 < δ1 < δ2 . If α < α0 , then γ α (F, a, b) = ∞. Therefore we expect that R(α) = σ α [F, P (δ1 )] /σ α [F, P (δ2 )] > 1. Similarly since α > α0 implies γ α (F, a, b) = 0, we expect that R(α) < 1. This fact can be used to algorithmically calculate the γ-dimension α0 : We need to ﬁnd the number α0 such that R(α0 ) = 1. We already know that α0 ∈ [1, m], m being the embedding dimension, since F ∈ Rm is a curve. Treating this as the initial bracket of values for α0 , we just need to use some algorithm such as bisection to shrink this bracket to suﬃcient accuracy.

32

Acknowledgements
Seema Satin is thankful to Council of scientiﬁc and Industrial Research (CSIR) India for ﬁnancial assistance.

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Description: This is a technical report on Calculus of Fractal Curves R^n, from Pune University.