Sumudu transform of Weierstrass function

Abstract

Weierstrass function, Weierstrass­ Mandelbrot function and its variants of fractal functions are Sumudu transformed to show that non regular curves can be smoothed.

Full text

Sumudu transform of Weierstrass function
Fethi Bin Muhammad Belgacem1,?, Carlo Cattani2and Rathinavel Silambarasan3
1Department of Mathematics, Faculty of Basic Education, PAAET, Shaamyia, Kuwait. Email :
fbmbelgacem@gmail.com
2Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy.
Email : ccattani@unisa.it
3M. Tech IT-Networking, SITE, V. I. T. University, Vellore, Tamilnadu, India. Email : sil-
ambu_vel@yahoo.co.in
?Corresponding Author.
Abstract
Weierstrass function, Weierstrass-Mandelbrot function and its variants of fractal functions
are Sumudu transformed to show that non regular curves can be smoothened. Obtained results
are verified through Maple graphs to show the simulation between non regular, fractal curves
to regular, smooth curves for different partial sums.
1 Introduction
Theorem in calculus states "Function differentiable at a point is continuous at a point" while the
counterpart of the statement need not be true. So differentiable at a point guarantees conitnuity but
the continuity at a point does not guarantee differentiable. In mathematics later functions are called
"Continuous everywhere differentiable nowhere", in image processing those functions are called as
fractal curves. Karl Weierstrass gave an example for such nowhere differentiable function in [23,24].
Weierstrass functions were extended in the study of wavelets in [3] to give Weierstrass-Mandelbrot
function, whose behaviour and development in [19]. In the context of nowhere differentiable
functions followed by Weierstrass function several functions where studied in [2, 10–18, 20, 21].
Apart from the mentioned functions, several other functions of nowhere differentiable are given
in [22].
Sumudu transform by having dimension, scale and unit preserving nature is closely related to
Laplace transform whose dual is established in [4] with detailed properties and table in [5]. Sumudu
transform is further exposed with number theory in [6] while the reverse process of inverse Sumudu
transform is applied and solved by forward Sumudu to Bessel function in [7]. Further application
of Sumudu transform is given in [1]. The new definition of Sumudu transform for trigonometric
functions were derived and by invesion the trigonometric functions were expressed as infinite series
in terms of Bessel function in [9] where the coefficients of the series are obtained by integrating
1

the function and evaluated at origin [8]. Defined in the set A={f(x)|∃M,τ1,τ2>0,|f(x)|<
Me |x|
τj,if x∈(−1)j×[0,∞)}, the Sumudu transform of f(x)given,
S[f(x)] = Z∞
0
e−xf(ux)dx =1/uZ∞
0
e−x/uf(x)dx ;u∈(−τ1,τ2).(1)
Sumudu transform is applied to Weierstrass function and its related fractal curves whose translation
is studied through Maple plots which can be best understood by graph rather mere equation.
2 Weierstrass, Weierstrass-Mandelbrot function and its Sumudu
Karl Weierstrass in [23] defined the function which is everywhere continuous and nowhere differ-
entiable an example for fractal curve.
W(x) =
∞
∑
k=0
akcos(bkπx);a∈(a,b),ab >1+3π/2,b>1is odd.(2)
Weierstrass-Mandelbrot function (fractal curve) [3,19] is defined by
WM(x) =
∞
∑
k=−∞
(1−eibkx)eiφk
b(2−d)k;d∈(1,2),b>1,φk=arbitrary phase.(3)
(a) W(x)for Partial sum=4 (b) Sumudu of W(x)for
Partial sum=4
(c) W(x)for Partial
sum=10
(d) Sumudu of W(x)for
Partial sum=10
Figure 1: Weierstrass function W(x)and its Sumudu transform (equation (5)) with a=
0.5,b=3for first 4 and 10 partial sum respectively from left to right in [−3,3]
Depends on the arbitrary phase, fractal curve equation (3) studied as deterministic functions
and stochastic functions in [3]. Making the arbitrary phase to zero and considering the real and
imaginary part of equation (3) leads to cosine and sine fractal functions respectively [3].
C(x) = Re(WM(x)) =
∞
∑
k=−∞
1−cos(bkx)
b(2−d)k.and S(x) = Im(WM(x)) =
∞
∑
k=−∞
1−sin(bkx)
b(2−d)k.(4)
2

(a) C(x)for Partial sum=6 (b) Sumudu of C(x)for
Partial sum=6
(c) C(x)for Partial sum=10 (d) Sumudu of C(x)for Par-
tial sum=10
Figure 2: Cosine-fractal function C(x)and its Sumudu transform (first equation of equation
(6)) with b=1.5,d=1.5for first 6 and 10 partial sum respectively from left to right in [−5,5]
Theorem 1. Sumudu transform of Weierstrass function is given by
S[W(x)] =
∞
∑
k=0
ak
1+b2kπ2u2;a∈(a,b),ab >1+3π/2,b>1is odd.(5)
Proof. By Sumudu transform of cosine functionfrom [4,5] and simplifying leads to the theorem. 
Corollary 1. Sumudu transform of cosine and sine fractal functions are given respectively by
S[C(x)] =
∞
∑
k=−∞
bdku2
(1+b2ku2).and S[S(x)] =
∞
∑
k=−∞−b(d−2)k−b(d−1)ku+bdku2
(1+b2ku2).(6)
G. H. Hardy proved the Weierstrass function equation (2) is nowhere differentiable for all values
of b>1and also extended to sine function (theorem 1.31 in [16]). Graph of Weierstrass function
equation (2) and its Sumudu transform equation (5) is shown in figure 1. Graph of Cosine and Sine
fractal functions equation (4) with their Sumudu transforms equation (6) are shown in respective
figures 2 and 3. For all the graphs drawn are for the domain defined to both xthe original image
and uthe transformed image.
Remark 1. For the two different partial sums from figure 1 the wiggle becomes more fracture when
kbecomes large, at the same time for higher values of kthe fractal curve cannot be distinguished
with k−1value. In the Sumudu transform plot it can be seen the curve is regular without any
wiggle similarly for larger values of kthe transfomed image becomes more smoother.
Remark 2. As the fracture becomes more congested which can be seen in figure 2 Sumudu transform
maps to more clear image and enlarged resolution. While from figure 3 the range increases even
when the difference of two partial sums are less. That is when the partial sum 5 is increased to
7, the range increases twice the range of partial sum 5. But this is not the case for C(x)and its
Sumudu in figure 2.
3

(a) S(x)for Partial sum=5 (b) Sumudu of S(x)for Par-
tial sum=5
(c) S(x)for Partial sum=7 (d) Sumudu of S(x)for Par-
tial sum=7
Figure 3: Sine-fractal function S(x)and its Sumudu transform (secone equation of equation
(6)) with b=1.5,d=1.5for first 5 and 7 partial sum respectively from left to right in [−10,10]
(a) WD1(x)for Partial
sum=4
(b) Sumudu of WD1(x)for
Partial sum=4
(c) WD1(x)for Partial
sum=7
(d) Sumudu of WD1(x)for
Partial sum=7
Figure 4: Cosine-fractal function WD1(x)and its Sumudu transform (first equation of equation
(8)) due to Dini with a=7for first 4 and 7 partial sum respectively from left to right in [−1,1]
3 Weierstrass function variants
Dini in [12–15] and Knopp in [18] defined the continuous everywhere and differentiable nowhere
functions which satisfies the conditions given in [22] (Remark 1, pp 25-26 in [22]).
WD1(x) =
∞
∑
k=1
akcos(∏k
i=1(2i−1)πx)
∏k
i=1(2i−1)and WD2(x) =
∞
∑
k=1
aksin(∏k
i=1(4i+1)πx)
∏k
i=1(4i+1);|a|>1+3π/2.
(7)
Now the Sumudu transform of Weierstrass function due to Dini is given by also shown in figure
4 and 5 respectively.
S[WD1(x)] =
∞
∑
k=1
ak
∏k
i=1(2i−1)1+ (∏k
i=1(2i−1)πu)2and S[WD2(x)] =
∞
∑
k=1
akπu
1+∏k
i=1(4i+1)πu2.
(8)
Weierstrass function was studied for arbitrary odd powers by Hertz in [17] to define the every-
4

(a) WD2(x)for Partial
sum=3
(b) Sumudu of WD2(x)for
Partial sum=3
(c) WD2(x)for Partial
sum=5
(d) Sumudu of WD2(x)for
Partial sum=5
Figure 5: Sine-fractal function WD2(x)and its Sumudu transform (second equation of equation
(8)) due to Dini with a=7for first 3 and 5 partial sum respectively from left to right in [−1,1]
(a) WH(x)for Partial
sum=6
(b) Sumudu of WH(x)for
Partial sum=6
(c) WH(x)for Partial
sum=9
(d) Sumudu of WH(x)for
Partial sum=9
Figure 6: Fractal function WH(x)and its Sumudu transform (equation (10)) due to Hertz with
a=3,b=3,p=3for first 6 and 9 partial sum respectively from left to right in [−0.5,0.5]
where continuous and differentiable nowhere by,
WH(x) =
∞
∑
k=1
akcosp(bkπx);a>1,p∈Nis odd ,ab >1+2pπ/3.(9)
whose Sumudu transform calculates to with p=3,
S[WH(x)] =
∞
∑
k=1
ak(1+7b2kπ2u2)
(1+9b2kπ2u2)(1+b2kπ2u2).(10)
Maple plot for Hertz cosine function and its Sumudu transform is shown in figure 6 for two
different partial sum. Porter in [20] studied the Weierstrass function to define Wi:[a,b]→Rby,
W1(x) =
∞
∑
k=0
ak(x)sin(bkπx)and W2(x) =
∞
∑
k=0
ak(x)cos(bkπx).(11)
where akand bkare respective sequence of differentiable functions and integers with the require-
ments given in [20] (pp 179, [20]) also given in [22]. Setting S[ak(x)] = ak(u)Sumudu transform
5

(a) WP1(x)for Partial sum
=4
(b) Sumudu of WP1(x)for
Partial sum =4
(c) WP1(x)for Partial sum
=7
(d) Sumudu of WP1(x)for
Partial sum =7
Figure 7: Fractal Sine function WP1(x)and its Sumudu transform (first equation of equation
(15) due to Porter with a=7for first 4 and 7 partial sum respectively from left to right in
[−3,3]
(a) WP2(x)for Partial sum
=4
(b) Sumudu of WP2(x)for
Partial sum =4
(c) WP2(x)for Partial sum
=7
(d) Sumudu of WP2(x)for
Partial sum =7
Figure 8: Fractal Cosine function WP2(x)and its Sumudu transform (second equation of
equation (15)) due to Porter with a=7for first 4 and 7 partial sum respectively from left to
right in [−3,3]
of equation (11) leads to,
S[W1(x)] =
∞
∑
k=0
ak(x)bkπu
1+ (bkπu)2and S[W2(x)] =
∞
∑
k=0
ak(x)
1+ (bkπu)2.(12)
The following functions where the example for Porter function that are nowhere differentiable
function [20,22].
WP1(x) =
∞
∑
k=0
aksin(k!πx)
k!and WP2(x) =
∞
∑
k=0
akcos(k!πx)
k!;|a|>1+3π/2.(13)
WP3(x) =
∞
∑
k=0
sin(k!akπx)
akand WP4(x) =
∞
∑
k=0
cos(k!akπx)
ak;|a|>1.(14)
6

(a) WP3(x)for Partial sum
=7
(b) Sumudu of WP3(x)for
Partial sum =7
(c) WP3(x)for Partial sum
=9
(d) Sumudu of WP3(x)for
Partial sum =9
Figure 9: Fractal Sine function WP3(x)and its Sumudu transform (first equation of equation
(16)) due to Porter with a=3for first 7 and 9 partial sum respectively from left to right in
[−3,3]
(a) WP4(x)for Partial sum
=7
(b) Sumudu of WP4(x)for
Partial sum =7
(c) WP4(x)for Partial sum
=9
(d) Sumudu of WP4(x)for
Partial sum =9
Figure 10: Fractal Cosine function WP4(x)and its Sumudu transform (second equation of
equation (16)) due to Porter with a=3for first 7 and 9 partial sum respectively from left to
right in [−3,3]
Hence the Sumudu transform for Porter functions are,
S[WP1(x)] =
∞
∑
k=0
akπu
1+ (k!πu)2and S[WP2(x)] =
∞
∑
k=0
ak
k!(1+ (k!πu)2).(15)
S[WP3(x)] =
∞
∑
k=0
akπu
1+ (k!akπu)2and S[WP4(x)] =
∞
∑
k=0
1
ak(k!(1+ (k!πu)2)).(16)
The graph plot for Porter functions WP1through WP4and its Sumudu transform are shown respec-
tively in figure 7 through figure 10.
Apart from the above functions, following are the extension of the Weierstrass functions studied
by various authors. Cellérier function from [10], (3.2, [22]) and its Sumudu transform shown in
figure 11. Riemann function [21], (3.3, [22]) and its Sumudu transform shown in figure 12. Darboux
function [11], (3.5, [22]) and its Sumudu transform shown in figure 13.
7

(a) Partial sum =7(b) Partial sum =7(c) Partial sum =10 (d) Partial sum =10
Figure 11: Fractal function and its Sumudu transform due to Cellerier with a=2for first 7
and 10 partial sum respectively from left to right in [−3,3]
(a) Partial sum =5(b) Partial sum =5(c) Partial sum =7(d) Partial sum =7
Figure 12: Fractal function and its Sumudu transform due to Riemann for first 5 and 7 partial
sum respectively from left to right in [−3,3]
Remark 3. Weierstrass functions and its variants are composition of trigonometric functions hence
those functions can be expressed as infinite series proposed in [8, 9] (proposition 4, pp 929, [8]).
For instance equation (2) is expressed as,
W(x) =
∞
∑
n=0
∞
∑
k=0
(−1)nak
(bkπ)2(n+1)Z∞
0
d2n+1
dx2n+1J0(2√vxdv).(17)
Now Sumudu transform of Weierstrass function is given (using equation (7), [9]) by double sum-
mation.
S[W(x)] =
∞
∑
n=0
∞
∑
k=0
(−1)nak
(bkπu)2(n+1).(18)
Remark 4. All the fractal functions considered in this work falls under remark 3. Hence the
Sumudu transform of trigonometric fractal functions can be obtained by simply integrating the
fractal functions making uthe Sumudu transform variable as factor of function in place of x.
8

(a) Partial sum =3(b) Partial sum =3(c) Partial sum =3(d) Partial sum =3
Figure 13: Fractal function and its Sumudu transform due to Darboux for first 3 and 5 partial
sum respectively from left to right in [−3,3]
4 Conclusion
Following were the results of this work, Sumudu transform of non-regular, fractal curves results
in smooth and regular functions. While the anti-derivative of Weierstrass function and its related
fractal curves are just smooth with less wiggle, Sumudu transform gives the functions with zero
wiggle. It is belived that all everywhere continuous and nowhere differentiable functions which are
not trigonometric functions such as Blancmange curve, Bolzano curve, etc can too be made smooth
by Sumudu transform which will be considered in the future communication.
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