Fractional Derivatives of Weierstrass-Type Functions

M. Z¨ahle ∗ H. Ziezold †

Abstract

For a special type of fractional diﬀerentiation formulas for the Weierstrass and the Weierstrass-Mandelbrot functions are shown. For integral-valued parameters λ a well-computable form of the derivatives in the mean of these functions is proved and used to compute some values by Monte-Carlo integration.

Keywords: Fractional derivatives; Weierstrass Functions.

∗Mathematical Institute, University of Jena, D-07743 Jena, Germany †Department of Mathematics, University of Kassel, D-34109 Kassel, Germany

1 1 Introduction

Let h : R 7→ R be a Hölder continuous function of order β ≤ 1 with Hölderconstant H(h) and period 1, and let 0 < γ < β. For fixed integral parameter λ = 2, 3,... define the Weierstrass-type function

∞ γ X −kγ k W (x) = Wh (x) := λ h(λ x) . (1) k=1 For h(0) = 0 the corresponding Weierstrass-Mandelbrot function is given by

∞ γ X −kγ k M(x) = Mh (x) := λ h(λ x) . (2) k=−∞ It has the scaling property M(x) = λ−γM(λx) . (3) If h(u) = sin(2πu) and λ = 2 one obtains the classical Weierstrass function with exponent γ. In the case h(u) = 1 − |2u − 1|, u ∈ [0, 1], and λ = 2, W is called the Takagi function. These are traditional examples of continuous nowhere diﬀerentiable functions.

Figure 1. Weierstrass function W and Weierstrass-Mandelbrot function M for λ = 2 and γ = 0.5.

γ It is easy to see that the Hölderexponent of Wh agrees with γ. A well-known fact is γ also that the so-called box counting dimension of the graph of Wh equals 2 − γ (see e. g. [1]). Hu and Lau [3] proved that this number coincides with the fractal dimension of an appropriate measure. The problem whether this is also true for the Hausdorff measure is still open. The best lower estimate for the Hausdorff dimension known from the literature is 2 − γ − c/ ln λ for some constant c (Mauldin and Williams [4]). In the present paper we will continue the study of fractional derivatives of these functions which is closely related to dimension problems. In [6] the first of the authors

2 proved that the fractional degree of diﬀerentiability agrees with γ if M 6≡ 0 and is not less than β if M ≡ 0. In particular, there exist all lower order Weyl-Marchaud derivatives of W and M. For M 6≡ 0 the corresponding gradual fractional derivatives in the mean are shown to be constant at Lebesgue almost all points. The aim of the present paper is to calculate the fractional derivatives explicitly and to develop a corresponding computation procedure. The latter results in numerical integration of rapidly oscillating functions. It turns out that for such functions integration by Monte-Carlo methods works well whereas classical numerical procedures are not practicable here. We thank Christian Glaßer for valuable technical support in assembler program- ming.

2 Basic notions and related results

In [6] relationships between the Weyl-Marchaud derivatives used in the theory of function spaces and some notions of fractal geometry are worked out. Recall that a version of the Weyl-Marchaud derivative of order 0 < α < 1 of a real or complex-valued measurable function f on R at x is given by

Z ∞ α α f(x) − f(x − y) Dl f(x) = dy (left-sided) (4) Γ(1 − α) 0 y1+α α Z ∞ α (−1) α f(x) − f(x + y) Dr f(x) = dy (right-sided) (5) Γ(1 − α) 0 y1+α provided that these integrals exist in the sense of absolute convergence. We additionally introduced (upper) (absolute) fractional derivatives of order α in the mean with respect to the logarithmic measure: If 0 < α < 1,

Z ∞ α 1 |f(x) − f(x − y)| |dl |f(x) = lim sup 1+α dy (6) δ→0 | ln δ| δ y Z ∞ α 1 |f(x + y) − f(x)| |dr |f(x) = lim sup 1+α dy (7) δ→0 | ln δ| δ y Z ∞ α 1 |f(x) − f(x − y)| |dl |f(x) = lim dy (8) δ→0 | ln δ| δ y1+α Z ∞ α 1 |f(x + y) − f(x)| |dr |f(x) = lim dy (9) δ→0 | ln δ| δ y1+α Z ∞ α 1 f(x) − f(x − y) dl f(x) = lim dy (10) δ→0 | ln δ| δ y1+α Z ∞ α 1 f(x + y) − f(x) dr f(x) = lim dy (11) δ→0 | ln δ| δ y1+α where (8) to (11) are determined if the limits exist. For α ≥ 1 one applies these deﬁnitions replacing f by its ordinary derivative of order [α], provided that it exists, and then α by α − [α].

3 The degree of diﬀerentiability of f at x is deﬁned by:

γ = γ(x) = sup{α : |dα|f(x) = 0} = inf{α : |dα|f(x) = ∞} (12) in the left-sided or right-sided version (cf. [6], Proposition 2). In [6], Theorem 2, we proved that the Weyl-Marchaud derivatives exist for all α < γ, R ∞ −1−α provided that 1 |f(x ± y) − f(x)|y dy < ∞, and do not exist if α > γ. γ Finally, the (absolute) gradual derivative in the mean of f at x is given by (|dl,r|f(x)) γ dl,rf(x) (left- or right-sided version) if the corresponding expression is determined. γ Our aim is to calculate these derivatives for the Weierstrass-type functions Wh introduced above. From now on we will assume h(0) = 0. First note that in the γ case Mh 6≡ 0 the exponent γ agrees with the left-sided and the right-sided degree of γ γ differentiability of Wh and of Mh at all points (cf. [6], Theorem 5(i) and Proposition γ γ 8). If Mh ≡ 0 then the degree of differentiability of Wh is not less than the Hölder exponent β of h at all points (cf. [6], Proposition 8). Moreover, in the first case γ γ γ γ γ γ γ γ the gradual derivatives dl Wh (x), dr Wh (x), |dl |Wh (x), and |dr |Wh (x) are constants approximated by

−nγ 1 Z 1 Z λ W (z) − W (z − y) γ −n(β−γ) dl W (x) − dy dz ≤ constλ , (13) ln λ 0 λ−(n+1)γ y1+γ n = 1, 2,..., at Lebesgue a. a. x (cf. [6], Theorem 5(ii)). (For the right-sided and absolute versions replace W (z) − W (z − y) by the corresponding expressions.)

3 Calculation of the fractional derivatives

We ﬁrst consider the Weyl-Marchaud derivatives. Note that Dαh is a H¨older continuous function of order β − α with period 1. From this we infer immediately

α γ γ−α D Wh (x) = WDαh (x) (14)

(left- and right-sided versions) for all α < γ and x ∈ R (with the same parameter λ). Note that in the classical case, h(u) = sin(2πu), one obtains

α α Dl,rh(u) = (2π) sin(2πu ± πα/2) . A table of the Weyl-Marchaud derivatives for some basic functions may be found in Samko, Kilbas, and Marichev [5], Chapter II, § 9. We now turn to the gradual derivatives. The following formulas will be used as an auxiliary tool.

Proposition 1 At Lebesgue a. a. x we have

Z 1 Z 1 γ γ γ γ 1 ±(Wh (z) − Wh (z ∓ y)) (i) dl,rWh (x) = lim dy dz n→∞ n ln λ 0 λ−n y1+γ

4 Z 1 Z 1 γ γ γ γ 1 |Wh (z) − Wh (z ∓ y)| (ii) |dl,r|Wh (x) = lim dy dz n→∞ n ln λ 0 λ−n y1+γ (iii) On the right or left hand sides of (i) and (ii) W may be replaced by M.

Proof. In the notation

Z λ−k γ γ 1 |Wh (x) − Wh (x ∓ y)| gk(x) := dy ln λ λ−(k+1) y1+γ

γ γ the derivative |dl,r|Wh (x) may be rewritten as

n−1 1 X lim gk(x) . n→∞ n k=0 Furthermore, for any n and m

n+m−1 n−1 m−1 1 X 1 X 1 X gk(x) = gk(x) + gn+k(x) . n + m k=0 n + m k=0 n + m k=0

k In [6], Theorem 5, we have proved that gn+k(x) = gn(λ x)+Θn,k(x), where |Θn,k(x)| ≤ 1 −(β−γ)n Θn = ln λ H(h)e . Therefore

m−1 γ γ 1 X k |d |W (x) − lim gn(λ x) ≤ Θn . l,r h m→∞ m k=0 The limit on the left hand side exists at almost all x by the following arguments: γ k k Because of the periodicity of Wh we have gn(λ x) = gn(A x) for A : [0, 1] 7→ [0, 1] mapping x onto the fractional part of λx. The Lebesgue measure is A-invariant so that Birkhoﬀ’s ergodic theorem yields

m−1 Z 1 1 X k lim gn(λ x) = gn(z) dz m→∞ m k=0 0 at Lebesgue a. a. x (cf. Falconer [2]). Consequently,

Z 1 n−1 Z 1 γ γ 1 X |d |W (x) = lim gn(z) dz = lim gk(z) dz l,r h n→∞ n→∞ 0 n k=0 0 which proves (ii). Similar arguments (without the absolute value signs) lead to (i). The replacement γ γ of the continuous limits as δ → 0 in the definition of dl,rWh by the discrete limits as n → ∞ is justified by the existence of the absolute derivatives. For (iii) we estimate the difference on the right hand sides by

1 n−1 Z 1Z λ−k y−(1+γ) lim sup X |M γ(z) − M γ(z ∓ y) − W γ(z) + W γ(z ∓ y)| dy dz −(k+1) h h h h n→∞ n k=0 0 λ ln λ

5 1 Z 1 Z 1 1 0 ≤ lim sup X λ−lγy−(1+γ)|h(λlz) − h(λl(z ∓ y)| dy dz −n n→∞ n 0 λ ln λ l=−∞ 1 1 Z 1 0 ≤ lim sup X λ−lγH(h)λlβyβy−(1+γ)dy −n ln λ n→∞ n λ l=−∞ 1 1 0 1 = H(h) X λl(β−γ) lim sup (1 − λ−n(β−γ)) = 0 . ln λ β − γ l=−∞ n→∞ n Similar arguments (see also [6] Proposition 8) lead to

γ γ γ γ γ γ γ γ dl M (x) = dl W (x), dr M (x) = dr W (x) and γ γ γ γ γ γ γ γ |dl |W (x) = |dl |M (x), |dr |W (x) = |dr |M (x) proving (iii). 2 Remark. A by-product of the last proof is the estimation

−n γ γ 1 Z 1 Z λ |W (z) − W (z ∓ y)| γ γ h h |dl,r|Wh (x) − dy dz ≤ Θn . ln λ 0 λ−(n+1) y1+γ However, for large n it is numerically diﬃcult to compute the inner integral. Therefore we suggest another approximation of the absolute derivatives combining the periodicity γ γ of Wh with the scale invariance of Mh . (For brevity we will omit from now on the lower index h in the notation.)

Theorem 1 At Lebesgue a. a. x the following holds:

(a) γ γ γ γ γ γ γ γ dl W (x) = dr W (x) = dl M (x) = dr M (x) = 0 ;

(b) γ γ γ γ γ γ γ γ |dl |W (x) = |dr |W (x) = |dl |M (x) = |dr |M (x);

Z 1Z 1 M γ (λn1 z + y) − M γ (λn1 z) (c) γ γ 1 n1,n2 n1,n2 γ γ |d |W (x) − dy dz ≤ε1,n +ε2,n −1 1+γ 1 2 ln λ 0 λ y

γ Pn2 −kγ k where Mn1,n2 (x) = k=−n1 λ h(λ x) is the truncated Weierstrass-Mandelbrot function and H(h)λ−(n1+1)(β−γ) 2khk εγ = , εγ = λ−n2γ , 1,n1 (β − γ) ln λ 2,n2 γ ln λ n1, n2 = 1, 2,....

Proof. Since W γ has period 1 we have Z 1 Z 1 W γ(z) − W γ(z ∓ y) dy dz 0 λ−n y1+γ Z 1 1 Z 1 Z 1∓y = W γ(z) dz − W γ(z)dz dy = 0 . λ−n y1+γ 0 ∓y

6 Therefore Proposition 1 (i) implies (a). The equality of the derivatives of the Weierstrass- and the Weierstrass-Mandelbrot functions is given in Proposition 1 (iii). Furthermore, in view of the periodicity of W γ, Z 1 Z 1 |W γ(z) − W γ(z + y)| Z 1 1 Z y+1 dy dz = |W γ(z − y) − W γ(z)|dz dy 0 λ−n y1+γ λ−n y1+γ y Z 1 Z 1 |W γ(z − y) − W γ(z)| = dy dz , 0 λ−n y1+γ so that Proposition 1 (ii) yields

γ γ n1 γ Next we replace M by Mn1,n2 , which has period λ , and n by n1, we set δn1,n2 (y, z) = γ γ Mn1,n2 (z + y) − Mn1,n2 (z), and we continue with

m−1 Z 1 Z λk 1 X −(1+γ) −k γ γ lim y λ |δn ,n (y, z)| dz dy + Rn ,n m→∞ m ln λ λ−1 0 1 2 1 2 k=n1 1 m−1 Z 1 Z λn1 X −(1+γ) −k k−n1 γ γ = lim y λ λ |δn ,n (y, z)| dz dy + Rn ,n m→∞ m ln λ λ−1 0 1 2 1 2 k=n1 1 Z 1 Z λn1 −(1+γ) −n1 γ γ = y λ |δn ,n (y, z)| dz dy + Rn ,n . ln λ λ−1 0 1 2 1 2 γ For the remainder term Rn1,n2 we get because of

|(M(z + y) − M(z)) − (Mn1,n2 (z + y) − Mn1,n2 (z))|

−∞ ∞ X −kγ k k X −kγ k k = λ (h(λ (z + y)) − h(λ z)) + λ (h(λ (z + y)) − h(λ z))

k=−n1−1 k=n2+1 −∞ ∞ ≤ X λ−kγH(h)λkβ|y|β + X λ−kγ2khk

k=−n1−1 k=n2+1 H(h)λ−(n1+1)(β−γ) 2khkλ−(n2+1)γ ≤ |y|β + 1 − λ−(β−γ) 1 − λ−γ

7 the estimation

H(h) λ−(n1+1)(β−γ) Z 1 2khk λ−n2γ Z 1 Rγ ≤ · y−(1+γ)yβ dy + · y−(1+γ) dy n1,n2 ln λ 1 − λ−(β−γ) λ−1 ln λ λγ − 1 λ−1 H(h)λ−(n1+1)(β−γ) 2khk = + λ−n2γ (β − γ) ln λ γ ln λ γ γ = ε1,n1 + ε2,n2 . This proves (c). 2

As a by-product of the proof the case n1 = −1 and n2 = 0 provides the result H(h) 2khk |dγ|W γ(x) ≤ + . (15) (β − γ) ln λ γ ln λ

4 Numerical computations

Theorem 1 (c) enables us to compute the values of the absolute fractional derivatives in the mean with arbitrary exactness. In order to approximate |dγ|W γ(x) for a. a. x up to 2ε, we solve the inequalities

γ γ ε1,n1 ≤ ε for n1 and ε2,n2 ≤ ε for n2. This yields ln H(h) − ln(ε(β − γ) ln λ) n ≥ − 1 1 (β − γ) ln λ and ln(2 k h k) − ln(εγ ln λ) n ≥ . 2 γ ln λ In the classical case, h(x) = sin 2πx, we get e. g. for λ = 2, γ = 0.5, ε = 0.001 as minimal values for n1 and n2 : n1 = 39, n2 = 31. Because of the large values of λn2 the integrand of

1 1 1 Z Z |M γ (λn1 z + y) − M γ (λn1 z)| Iγ = n1n2 n1n2 dydz n1n2 ln λ y1+γ 0 λ−1 is rapidly oscillating in the interval [0, 1]. So classical numerical integration by smooth- ing the integrand does not give reliable results to say nothing of the estimation of the error. Therefore we use a Monte-Carlo method. For this purpose we substitute in the above integral y = λ−t and obtain

1 1 Z Z Iγ = λtγ M γ (λn1 z + λ−t) − M γ (λn1 z) dt dz . n1n2 n1n2 n1n2 0 0

γ Denote the integrand by ∆n1n2 (t, z).

8 Let (t, z) be chosen at random from the uniform distribution on the unit square [0, 1]2. Then γ γ In1n2 = E(∆n1n2 ) , γ where E(.) denotes expectations. From N independent realisations of ∆n1,n2 we get the meanx ¯N and the empirical standard deviation sN as estimators for the expectation and the standard deviation of ∆γ . Thus the mean is an estimator for Iγ and √sN n1n2 n1,n2 N measures the error in the conﬁdence interval sense; e. g. with probability 0.99 we get x¯N and sN such that sN γ sN x¯N − 2.6√ ≤ I ≤ x¯N + 2.6√ N n1n2 N and therefore sN γ γ sN x¯N − 2.6√ − 2ε ≤ |d |W (x) ≤ x¯N + 2.6√ + 2ε . N N For our values in the ﬁgures 2 to 4 we have chosen ε ≤ 0.001 and N such that √sN ≤ N 0.005 giving with probability 0.99

γ γ |d |W − x¯N ≤ 0.015 a. e.

Furthermore, to get the relevant parts of λk+n1 z and λk−t for the evaluations of the periodic function h, i. e. the parts modulo 1, we have to use enough digits of these expressions. In order that the number of decimal digits of these relevant parts is at least 10 the number of used decimal digits has to be at least

n1+n2 10 + lg(λ ) = 10 + (n1 + n2) lg λ ln H(h) − ln(ε(β − γ) ln λ) ln(2 k h k) − ln(εγ ln λ) ! ln λ ≥ 10 + + + 1 (β − γ) ln λ γ · ln λ ln 10 1 H(h) 1 2 k h k 1 1 = 10 + · lg + lg + lg λ − + lg(ln λ) . β − γ ε(β − γ) γ εγ β − γ γ For ﬁxed γ we may thus use 1 H(h) 1 2 k h k 10 + lg + lg + lg λ digits. β − γ ε(β − γ) γ εγ In Figure 2 the (approximated) absolute fractional derivatives in the mean for the γ Weierstrass functions Wh with h(u) = sin(2πu) for varying λ and γ = 0.3, 0.5, 0.7 as well as the upper bound (15) for γ = 0.7 are presented. The λ-values between 10k and 10k+1 are 2 · 10k and 4 · 10k, k = 1, 2,..., 11. The asymptotics as λ → ∞ may be determined in the general case: γ γ −(β−γ) By using (ε1,0 + ε2,1) ln λ ≤ const · λ we get from (15) : H(h) 2khk a¯ := + ≥ lim inf |dγ|W γ(x)ln λ (β − γ) γ λ→∞ Z 1 Z 1 = lim inf( y−1−γ|h(z + y) − h(z) + λ−γh(λz + λy) − λ−γh(λz)| dy dz λ→∞ 0 λ−1 γ + R0,1(x) ln λ) Z 1 Z 1 ≥ y−1−γ|h(z + y) − h(z)| dy dz =: a > 0 . 0 0

9 Thus the absolute fractional derivatives in the mean of the Weierstrass-type functions converge as aλ/ ln λ to 0 with a ≤ lim infλ→∞ aλ ≤ lim supλ→∞ aλ ≤ a¯.

Figure 2. Absolute fractional derivatives in the mean for the Weierstrass func- γ tions Wh with h(u) = sin(2πu) for γ = 0.3 (lower point row), γ = 0.5 (middle point row), γ = 0.7 (upper point row) and the upper bound (15) for γ = 0.7 .

The (approximated) absolute fractional derivatives in the mean for classical Weier- strass functions with varying parameter γ are shown in Figure 3 for λ = 2 and in Figure 4 for λ = 1012.

Figure 3. Absolute frac- Figure 4. Absolute fractional deriva- tional derivatives in the tives in the mean for the mean for the Weierstrass Weierstrass functions for functions for λ = 2 and λ = 1012 and varying γ. varying γ.

10 References

[1] K. J. Falconer, Fractal Geometry, Wiley, 1990.

[2] K. J. Falconer, Wavelet transforms and order-two densities of fractals, J. Statist. Physics 67 (1992), 781-793.

[3] T.-Y. Hu and K.-S. Lau, Fractal dimension and singularities of Weierstrass type functions, Trans. Amer. Math. Soc. 335 (1993), 649-665.

[4] R. D. Mauldin and S. C. Williams, On the Hausdorﬀ dimension of some graphs, Trans. Amer. Math. Soc. 298 (1986), 793-803.

[5] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives. Theory and applications, Gordon and Breach, 1993.

[6] M. Zähle, Fractional differentiation in the self-affine case. V – The local degree of differentiability, to appear in Math. Nachrichten.

Martina Z¨ahle Herbert Ziezold Mathematical Institute Department of Mathematics University of Jena University of Kassel D-07740 Jena D-34109 Kassel Germany Germany e-mail: [email protected] e-mail: [email protected]