Trigonometric and Wavelet Transforms for Certain Class of ......Trigonometric and Wavelet Transforms for Certain Class of Generalized Functions By S. K. Q. Al-Omari Al-Balqa Applied

© 2012. S. K. Q. Al-Omari. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Global Journal of Science Frontier Research Mathematics and Decision Sciences Volume 12 Issue 13 Version 1.0 Year 2012 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896

Trigonometric and Wavelet Transforms for Certain Class of Generalized Functions

By S. K. Q. Al-Omari Al-Balqa Applied University, Jordan

Abstract - In this article, we discuss certain class of generalized functions for sine and cosine transforms. we, also obtain a new relationship between Fourier sine transform (Fourier cosine transform) and wavelet transform. Other related theorems are also established in concern.

Keywords and phrases : distribution; tempered ultradistribution; wavelet transform; generalized functions; fourier sine transform.

GJSFR-F Classification : 1991 MSC: Primary 54C40, 14E20; Secondary 46E25, 20C20

TrigonometricandWaveletTransformsforCertainClassofGeneralizedFunctions

Strictly as per the compliance and regulations of

:

Trigonometric and Wavelet Transforms

for

Certain Class of Generalized FunctionsS. K. Q. Al-Omari

Author : Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134,

Jordan.

E-mail

: [email protected]

I.

Introduction

© 2012 Global Journals Inc. (US)

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Notes

A function g (t) is said to be a mother wavelet (or simply, a wavelet)if it satis�es the following conditions:

(i)

Z

Rg (t) dt = 0; (ii)

Z

Rjg (t)j2 dt = 0; (iii)

Z

R

jg (t)j2

tdt <1:

The continuous wavelet transform of f (t) with respect to a motherwavelet g (t) is de�ned by [2; 8]

Wgf (b; a) =

Z

Rf (t)

1pjajg

�t b

a

�dt.

Let S (R) denote the space of rapid descent and S 0 (R) its strongdual of tempered distributions over R, the set of real numbers [5].Due to [6], S+ (R) is the set of all those functions in S (R) whoseFourier transform is supported by the positive frequencies and S0 (R)is the space of all those functions from S (R) for which all momentsvanish. That is, g 2 S0 (R) if g 2 S (R) and

��x�g(�) (x)�� < 1 andZ

Rxng (x) dx = 0: Further, if g is a wavelet in S0 (R) and � =

f(b; a) : b 2 R; a > 0g, then the wavelet transform for f 2 S (R) isde�ned by [6]

Wgf (b; a) =Df (t) ; gb;a (t)

E; (b; a) 2 �; t 2 R:

�

Keywords and phrases : distribution; tempered ultradistribution; wavelet transform; generalized functions; fourier sine transform.

Abstract - In this article, we discuss certain class of generalized functions for sine and cosine transforms. we, alsoobtain a new relationship between Fourier sine transform (Fourier cosine transform) and wavelet transform. Other related theorems are also established in concern.

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rTrigonometric and Wavelet Transforms for Certain Class of Generalized Functions

where,

gb;a (t) = TbDag (t) =1

ag

�t b

a

�; t; b 2 R; a > 0;

Tb and Da stands for the translation and dilation of the wavelet g .The Fourier cosine transform of a complex-valued absolutely inte-

grable function f (t) over ( 1;1) is the function of the variable �given by [9, p.42]

Fcn (f (t)) (�) =Z

Rf (t) cos t�dt:

Fourier sine transform Fsn has similar representation.Fourier sine and cosine transforms being the imaginary and real

parts of the complex Fourier transform, their properties for sine andcosine transform can be obtained from the known properties of theFourier transform. Furthermore, the product of di¤erent combinationsof even and odd functions is an odd function and the product of similarcombinations of even and odd functions is an even function, establishesthat the Fourier transform of an even function is, indeed, a Fouriercosine transform and similarly, the Fourier transform of an odd functionin is a Fourier sine transform.

describe a relationship between the cited trigonometric transforms,Fourier sine and Fourier cosine transforms, and the wavelet transformin the sense of generalized functions. However, the approach in thispaper seems di¤erent and interesting.

II. Wavelet Transform of Ultradifferentiable Functions

By an ultradi¤erentiable function we mean a C1 function whosederivatives satisfy certain growth conditions as the order of the deriv-atives increase [1; 2; 3; 4].Denote by Uo (ni; n;R) and Ue (ni; n;R) the spaces of all odd (resp.

even) C1 functions � (x) on R such that jxiDjx� (x)j < Nninj; j =

1; 2; 3; ::: for some positive constant N .Functions in Uo (ni; n;R) and Ue (ni; n;R) are , indeed, ultradi¤er-

entiable functions and decrease to zero faster than every power of 1=x .Denoting by U�o (ni; n;R) and U�e (ni; n;R) the duals of the correspond-ing spaces. The resulting space is called the odd (resp. even) temperedultradistribution spaces which contain the space S 0 (R) properly [5].In view of de�nitions, Uo (ni; n;R) � S (R) and Ue (ni; n;R) �

S (R) ; we write S 0 (R) � U�o (ni; n;R) and S 0 (R) � U�e (ni; n;R) ; see[9, Theorem 2.1] .Denote by Uo (ni; n;R;+) and Ue (ni; n;R;+) the subsets of Uo (ni; n;R)

and Ue (ni; n;R) whose Fourier transforms are supported by the posi-tive frequencies. Whereas, U�o (ni; n;R;+) and U�e (ni; n;R;+) are therespective duals of Uo (ni; n;R;+) and Ue (ni; n;R;+). In notations

Uo (ni; n;R;+) =�� : � 2 Uo (ni; n;R) ; supp b� � R+

�

[9]Sneddon,

I.N.TheUseofIntegral

Transform

s,Tata

McGraw

-HillPublications

Company

Ltd,

NewDelhi,

1974.

Ref.

�

�

approach in this work is infact a motivation of [8] whichOur


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Notes


and

Ue (ni; n;R;+) =�� : � 2 Ue (ni; n;R) ; supp b� � R+

�:

Denoting by Uo (ni; n;R; �) (Ue (ni; n;R; �)) the set of all g 2 Uo (ni; n;R)(Ue (ni; n;R)) where,��x�g(�) (n)�� <1 and

Z

Rxng (x) dx = 0;

the following propostion holds true:Proposition 1: Let f 2 U�o (ni; n;R) (U�e (ni; n;R)) and g 2

Uo (ni; n;R; �) (Ue (ni; n;R; �)) ; then the wavelet transform of f isgiven by

(1.1) Wgf (b; a) = f (t) ; gb;a (t)

�; (b; a) 2 �; t 2 R;

where � has the usual meaning.

Proof : We have Uo (ni; n;R) � S (R) and Ue (ni; n;R) � S (R).Also, by the assumption that

��x�g� (x)�� < 1 andZ

Rxng (x) dx = 0

we deduce that the wavelet g 2 Uo (ni; n;R; �) and g 2 Ue (ni; n;R; �)which justi�es the proposition.

Theorem 1: Let (b; a) 2 � and g 2 Uo (ni; n;R; �) thenFsn (gb;a (x)) (�) = 0; for � =2 R+:

�

Proof : Let g 2 Uo (ni; n;R; �) then

Fsn (gb;a (x)) (�) =

ZR

gb;a (x) sin �xdx

=

ZR

1

ag

�x� b

a

�sin �xdx

=

ZR

g (y) sin � (ay + b) dy;x� b

a= y

=

ZR

g (y) (sin �ay cos b� + sin b� cos �ay) dy:

In view of the fact that multiplication of di¤erent combinations ofeven and odd functions yield odd functions, we have

Fsn (gb;a (x)) (�) = cos b�ZR

g (y) sin �aydy:

Hence, Fsn (gb;a (x)) (�) = 0; for � =2 R+ .

Theorem 2 : Let (b; a) 2 � and g 2 Ue (ni; n;R; �).ThenFcn (gb;a (x)) (�) = 0; for � =2 R+:

Proof is similar to the proof employed above and, thus, avoided.Theorem 3: Let (b; a) 2 � and g 2 Uo (ni; n;R; �). Then

Fsn (Wg�) =W�gsn�;

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Notes


W�gsn (� � ) (y) =

Z

�

0

@Z

R

� (y) (�)

1

A �gb;a (� + �)d�d�

=

Z

�

0

@Z

R

� (y) (�)

1

A

0

@Z

R

gb;a (x) sin (� + �)xdx

1

A d�d�:

Now, usingsin (� + �)x = sin �x cos �x+ sin �x cos �x

for all � 2 Ue (ni; n;R) ;where �gcn = F 1cn (gs;t (x)) :

Proof being similar to that of Theorem 2 we avoid details.

Theorem 5: Let (b; a) 2 �; g 2 Uo (ni; n;R; �) and �; 2 Uo (ni; n;R)then

where �gsn = F 1sn

�gb;a (y)

�(x) =

Zgb;a (y) sin yx dy, and � is the con-

volution product.

Proof : Let �; 2 Uo (ni; n;R) : Then, we have

W�gsn (� � ) (y) =

Z

�

(� � ) (y) �gb;a (y)dy

=

Z

�

0

@Z

R

� (y) (y �) d�

1

A �gb;a (y)dy

Let y � = � then dy = d� .Then ,

�

Wg

�F �1sn �F �1

sn

�=W�gsn (� � ) ;

�

�

�

Fsn (Wg�) (s; t) =

ZR

0@Z�

� (x) gb;a (x) dx

1A sin ((b; a) : (s; t)) d (b� a)

=

Z�

� (x)

0@ZR

gb;a (x) sin ((b; a) : (s; t)) d (b� a)

1A dx

=

Z�

� (x) �gsn (x)dx

= W�gsn�:

This proves the theorem.Theorem 4: Let (b; a) 2 � and g 2 Ue (ni; n;R; �) .Then

Fcn (Wg�) =W�gcn�;

for all � 2 Uo (ni; n;R) ; where �gsn = F�1sn (gs;t (x))

Proof : We have


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Notes


Equivalently,

Fsn (WgT ) (WgT1) = (W�gT ) � (W�gT1) :

The proof of the theorem ,thus, completes. Following theoremsare natural consequence of the above theorem and proofs being almostsimilar, we omit details.

Theorem 8: Given T1; T 2 U�e (ni; n;R), we have

Fcn ((WgT ) (WgT1)) = (W�gcnT ) � (W�gcnT1) :

Theorem 9: Let T1; T 2 U�o (ni; n;R) .Then

Fsn (WgT ) � (WgT1) = (W�gsnT ) (W�gsnT1) :

Theorem 10: Let T1; T 2 U�e (ni; n;R). Then

Fcn (WgT ) � (WgT1) = (W�gcnT ) (W�gcnT1) .

=

Z

�

gb;a (x)

�ZR

(�) cos �xd�

Z

R� (�) sin �xd�

�dx

=

Z

�

gb;a (x)^c (�) �̂sn (�) dx

= Wg

�^cn�̂sn

�;

where ^cn = Fcn and �̂sn = Fsn : This completes the proof of thetheorem.

Theorem 6: Let (b; a) 2 � and g 2 Ue (ni; n;R; �) : Let �; 2Ue (ni; n;R) then

Wg

��̂cn^cn

�=W�gcn (� � ) ,

where � is the convolution product and �gcn is the inverse Fourier cosinefunction of g .Analysis of the proof of above theorem being similar to that em-

ployed for Theorem 5, details are avoided.Theorem 7: Given T1; T 2 U�o (ni; n;R) ,we have

Fsn ((WgT ) (WgT1)) = (W�gsnT ) � (W�gsnT1) :

Proof : Let � 2 Uo (ni; n;R) : Then in view of Theorem 3, we have

h(WgT ) (WgT1) ;�i = F 1sn (Fsn (WgT1)Fsn (WgT1)) ;�

�= F 1sn ((W�gsnT ) � (W�gsnT1)) ;�

�:

Hence,(WgT ) (WgT1) = F 1

sn ((W�gT ) � (W�gT1)) :

�

�

�

and rules of integration for even and odd functions, we have

W�gsn (� � ) (y) =

Z

�

gb;a (x)

0

@Z

R

(�) cos �x

Z

R

� (�) sin �xd�d�

1

A dx

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for all � 2 Uo (ni; n;R) resp. Ue (ni; n;R) ; is said to have a quasi-asymptotic expansion at 0+ related to (cm (t) ; um) : In notations, wewrite

f �X

ui=cm (t) ; at 0+:

Next, we state without proof the following theorems, namely, theorem11,12 and 13. Detailed proof is analoguous to that of Theorem 3.1 andTheorem 3.2 of [8] and hence avoided.Theorem 11: Let am;m = 1; 2; :::; be a sequence of strictly in-

creasing real numbers and Lm;m = 1; 2; :::; be a sequence of slowlyvarying functions at 0+: Let f 2 U�o (ni; n;R;+) has a quasiasymptoticexpansion at 0+ with respect to (tamLm (t) ; um) ; t > 0: Then

Limt!0+

Wgf (tb; ta)mPi=1

Wgui (tb; ta)

tamLm (t)= 0;m = 1; 2; ::::

�

A function f 2 U�o (ni; n;R;+) and U�e (ni; n;R;+) ; respectively, issaid to have a quasiasymptotic behaviour(f � h) at 0+ related to c (t)if there is a non-zero h 2 U�o (ni; n;R;+) and U�e (ni; n;R;+) such that

limt!0+

�f (tx)

c (t);� (x)

�= hh (x) ;� (x)i , � 2 Uo (ni; n;R) ;Ue (ni; n;R) :

A positive function L (x), which is continuous on (0;1) ; is said tobe slowly varying at 0+ if

limt!0+

L (tx)L (t)

= 1; x 2 (0;1) :

By 1 denote the subset of of all (cm (t) ; um) 2 such that um �vm at 0+related to cm (t) and vm 6= 0 when um 6= 0;m = 1; 2; :::;1 andvm 2 U�o (ni; n;R;+) and U�e (ni; n;R;+). Let (cm (t) ; um) 2 1:An ultradistribution f 2 U�o (ni; n;R;+) (resp. U�e (ni; n;R;+))

satisfying

limt!0+

*�f (:)

mPi=1

ui (:)

�(tx)

cm (t);� (x)

+= 0

�

III. Asymptote Properties of the Wavelet Transform

Let (cm)m=1 be a sequence of continuous positive functions de�ned

on (0; am) ; am > 0 and limt!0+

cm+1 (t)

cm (t)= 0: Let (um)

1m=1be a sequence in

U�o (ni; n;R;+) and U�e (ni;n;R;+), respectively, such that um 6= 0;m =1; 2; :::; p < 1 and um = 0;m > p:The set of pairs (cm (t) ; um) isdenoted by :


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[2] Al-Omari, S.K.Q. Certain Class of Kernels for Roumieu-Type ConvolutionTransform of Ultradistribution of Compact Growth, J. Concr. Appl. Math.7(4)(2009), 310-316.

[3] Al-Omari, S.K.Q. Cauchy and Poison Integrals of Tempered Ultradistributionsof Roumieu and Beurling Types, J. Concr. Appl. Math. 7(1) (2009), 36-47.

[4] Al-Omari, S.K.Q. and Al-Omari, J. M. Operations on Multipliers and CertainSpaces of Tempered Ultradistributions of Roumieu and Beurling types for theHankel-type Transformations, J. Appl. Funct. Anal. (2010) (to Appear) .

[5] Banerji, P.K., Al-Omari,S.K.Q. and Debnath,L. Tempered DistributionalFourier Sine(Cosine)Transform, Integral Transforms Spec. Funct. 17(11)(2006), 759-768.

[6] Holschneider, M. Wavelets,an Analysis Tool, Oxford,Clarendon Press, 1995.[7] Raghuveer, M. and Ajit, S.,Wavelet Transforms,Introduction to Theory and

Application, Addison-Wesley, 1998.[8] S.K.Q. Al-Omari, On the Continuous-time Wavelet Transform of Generalized

Functions, Int. Journal of Math. Analysis, 4(21), 2010, 1027-1034.[9] Sneddon, I.N. The Use of Integral Transforms, Tata McGraw-Hill Publications

Company Ltd, New Delhi, 1974.

Theorem 12: Let am;m = 1; 2; :::; be a sequence of strictly increas-ing real numbers and Lm;m = 1; 2; :::be a sequence of slowly varyingfunctions at 0+: Let f 2 U�e (ni; n;R;+) has a quasiasymptotic expan-sion at 0+ with respect to (tamLm (t) ; um) ; t > 0: Then

Limt!0+

Wgf (tb; ta)mPi=1

Wgui (tb; ta)

tamLm (t)= 0;m = 1; 2; ::::

Theorem 13: Let am;m = 1; 2::::;1 be a sequence of strictly realnumbers and (Lm)1m=1 be a sequence of slowly varying functions at0+: Let f 2 U�b+0 (ni; n;R) has a quasiasymptotic expansion at b

+0 with

respect to (tamLm (t) ; um). Then

Limt!0+

Wgf (b0; ta)mPi=1

Wgui (b0; ta)

tamLm (t)= 0;m = 1; 2; ::::

Author is thankful to Prof. P. K. Banerji, Emeritus Fellow, UGC(India), and Dr. Deshna Loonker, DST (SERC), India, Young

Scientist, Department of Mathematics, JNV University, Jodhpur,India, for revising and formatting the paper.

References Références Referencias

[1] Al-Omari, S.K.Q. , Loonker D. , Banerji P. K. and Kalla, S. L. . FourierSine(Cosine) Transform for Ultradistributions and their Extensions to Temperedand UltraBoehmian Spaces, Integral Transforms Spec. Funct. 19(6) (2008), 453�462.

IV. Acknowledgement

�

�

Documents

Trigonometric and Wavelet Transforms for Certain Class of ......Trigonometric and Wavelet Transforms for Certain Class of Generalized Functions By S. K. Q. Al-Omari Al-Balqa Applied