some remarks on the extended hartley-hilbert and fourier-hilbert ...€¦ · fourier-hilbert...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 348701 6 pageshttpdxdoiorg1011552013348701

Research ArticleSome Remarks on the Extended Hartley-Hilbert andFourier-Hilbert Transforms of Boehmians

S K Q Al-Omari1 and A KJlJccedilman2

1 Department of Applied Sciences Faculty of Engineering Technology Al-Balqarsquo Applied University Amman 11134 Jordan2Department of Mathematics and Institute of Mathematical Research Universiti Putra Malaysia (UPM)43400 Serdang Selangor Malaysia

Correspondence should be addressed to A Kılıcman akilicmanputraupmedumy

Received 19 January 2013 Accepted 15 March 2013

Academic Editor Mustafa Bayram

Copyright copy 2013 S K Q Al-Omari and A Kılıcman This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We obtain generalizations of Hartley-Hilbert and Fourier-Hilbert transforms on classes of distributions having compact supportFurthermore we also study extension to certain space of Lebesgue integrable Boehmians New characterizing theorems are alsoestablished in an adequate performance

1 Introduction

The classical theory of integral transforms and their applica-tions have been studied for a long time and they are appliedin many fields of mathematics Later after [1] the extensionof classical integral transformations to generalized functionshas comprised an active area of research Several integraltransforms are extended to various spaces of generalizedfunctions distributions [2] ultradistributions Boehmians[3 4] and many more

In recent years many papers are devoted to those integraltransforms which permit a factorization identity (of Fourierconvolution type) such as Fourier transform Mellin trans-form Laplace transform and few others that have a lot ofattraction the reason that the theory of integral transformsgenerally speaking became an object of study of integraltransforms of Boehmian spaces

The Hartley transform is an integral transformation thatmaps a real-valued temporal or spacial function into a real-valued frequency function via the kernel

119896 (120592 119909) = cas (120592119909) (1)

This novel symmetrical formulation of the traditional Fouriertransform attributed to Hartley 1942 leads to a parallelismthat exists between a function of the original variable and thatof its transform In any case signal and systems analysis and

design in the frequency domain using the Hartley transformmay be deserving an increased awareness due to the necessityof the existence of a fast algorithm that can substantiallylessen the computational burden when compared to theclassical complex-valued fast Fourier transform

The Hartley transform of a function 119891(119909) can beexpressed as either [5]

A (120592) =1

radic2120587int

infin

minusinfin

119891 (119909) cas (120592119909) 119889119909 (2)

or

A (119891) = intinfin

minusinfin

119891 (119909) cas (2120587119891119909) 119889119909 (3)

where the angular or radian frequency variable 120592 is related tothe frequency variable 119891 by 120592 = 2120587119891 and

A (119891) = radic2120587A (2120587119891) = radic2120587A (120592) (4)

The integral kernel known as cosine-sine function isdefined as

cas (120592119909) = cos 120592119909 + sin (120592119909) (5)

Inverse Hartley transform may be defined as either

119891 (119909) =1

radic2120587int

infin

minusinfin

A (120592) cas (120592119909) 119889120592 (6)

2 Abstract and Applied Analysis

or

119891 (119909) = int

infin

minusinfin

A (119891) cas (2120587119891119909) 119889119891 (7)

The theory of convolutions of integral transforms has beendeveloped for a long time and is applied in many fields ofmathematics Historically the convolution product [2]

(119891 lowast 119892) (119910) = int

infin

minusinfin

119891 (119909) 119892 (119909 minus 119910) 119889119910 (8)

has a relationship with the Fourier transform with the factor-ization property

F (119891 lowast 119892) (119910) = F (119891) (119910)F (119892) (119910) (9)

Themore complicated convolution theorem of Hartley trans-forms compared to that of Fourier transforms is that

A (119891 lowast 119892) (119910) =1

2G (A119891 timesA119892) (119910) (10)

where

G (119891 lowast 119892) (119910) = 119891 (119910) 119892 (119910) + 119891 (119910) 119892 (minus119910)

+ 119891 (119910) 119892 (119910) minus 119891 (minus119910) 119892 (minus119910)

(11)

Some properties of Hartley transforms can be listed asfollows

(i) Linearity if 119891 and 119892 are real functions then

A (119886119891 + 119887119892) (119910) = 119886A (119891) (119910) + 119887A (119892) (119910) 119886 119887 isinR

(12)

(ii) Scaling if 119891 is a real function then

int

infin

minusinfin

119891 (120572120589) cas (2120587119910120589) 119889120589 = 1119886(A119891) (

119910

119886) (13)

2 Distributional Hartley-Hilbert Transformof Compact Support

TheHilbert transform via the Hartley transform is defined by[6 7]

BA(119910) = minus

1

120587int

infin

0

(A119900

(119909) cos (119909119910) +A119890 (119909) sin (119909119910)) 119889119909

(14)

where

A119900

(119909) =A (119909) minusA (minus119909)

2

A119890

(119909) =A (119909) +A (minus119909)

2

(15)

are the respective odd and even components of (2)We denote C(R) C(R) = C the space of smooth

functions and C1015840(R) C1015840(R) = C1015840 the strong dual of C ofdistributions of compact support overR

Following is the convolution theorem ofBA

Theorem 1 (ConvolutionTheorem) Let 119891 and 119892 isin C then

BA(119891 lowast 119892) (119910) = int

infin

0

(1198961(119909) cos (119910119909) + 119896

2(119909) sin (119910119909)) 119889119909

(16)

where

1198961(119909) = A

119890

119891 (119909)A119900

119892 (119909) +A119900

119891 (119909)A119890

119892 (119909)

1198962(119909) = A

119890

119891 (119909)A119890

119892 (119909) minusA119900

119891 (119909)A119900

119892 (119909)

(17)

Proof To prove this theorem it is sufficient to establish that

1198961(119909) = A

119900

(119891 lowast 119892) (119909) (18)

1198962(119909) = A

119890

(119891 lowast 119892) (119909) (19)

Therefore we have

A119900

(119891 lowast 119892) (119909)

= int

infin

minusinfin

(int

infin

minusinfin

119891 (120574) 119892 (119910 minus 120574) 119889119903) cas (119909119910) 119889119910

= int

infin

minusinfin

119891 (120574)int

infin

minusinfin

119892 (119910 minus 120574) cas (119909119910) 119889119910119889119903

(20)

The substitution 119910 minus 120574 = 119911 and using of (1) together withFubini theorem imply

A119900

(119891 lowast 119892) (119909)

= int

infin

minusinfin

119891 (120574)int

infin

minusinfin

119892 (119911) (cos (119909 (119911 + 120574)) + sin (119909 (119911 + 120574)))

times 119889119911119889119903

(21)

By invoking the formulae

cos (119909 (119911 + 120574)) = cos (119909119911) cos (119909120574) minus sin (119909119911) sin (119909120574)

sin (119909 (119911 + 120574)) = sin (119909119911) cos (119909120574) + cos (119909119911) sin (119909120574) (22)

then (18) follows from simple computation Proof of (19) hasa similar techniqueHence the theorem is completely proved

It is of interest to know that cos(119909119910) and sin(119909119910) aremembers of C and therefore A119900119891A119890119891 isin C1015840 This leads tothe following statement

Definition 2 Let 119891 isin C1015840 then we define the distributionalHartley-Hilbert transform of 119891 as

B

A119891 (119910) = ⟨A

119900

119891 (119909) cos (119909119910)⟩ + ⟨A119890119891 (119909) sin (119909119910)⟩ (23)

The extended transformBA119891 is clearly well defined for

each 119891 isin C1015840

Abstract and Applied Analysis 3

Theorem 3 The distributional Hartley-Hilbert transformBA119891 is linear

Proof Let 119891 119892 isin C1015840 then their components A119890119891A119900119891A119890119892A119900119892 isin C1015840 HenceB

A(119891 + 119892) (119910) = ⟨A

119900

(119891 + 119892) (119909) cos (119909119910)⟩

+ ⟨A119890

(119891 + 119892) (119909) sin (119909119910)⟩ (24)

By factoring and rearranging components we get thatB

A(119891 + 119892) (119910) =

B

A119891 (119910) +

B

A119892 (119910) (25)

FurthermoreB

A(119896119891) (119910) = ⟨119896A

119900

119891 (119909) cos (119909119910)⟩

+ ⟨119896A119890

119891 (119909) sin (119909119910)⟩ (26)

HenceB

A(119896119891) (119910) = 119896

B

A119891 (119910) (27)

This completes the proof of the theorem

Theorem 4 Let 119891 isin C1015840 thenBA119891 is a continuous mapping

onC1015840

Proof Let119891119899 119891 isin C1015840 119899 isinN and119891

119899rarr 119891 as 119899 rarr infinThen

B

A119891119899(119910) = ⟨A

119900

119891119899(119909) cos (119909119910)⟩ + ⟨A119890119891

119899(119909) sin (119909119910)⟩

997888rarr ⟨A119900

119891 (119909) cos (119909119910)⟩ + ⟨A119890119891 (119909) sin (119909119910)⟩

=B

A119891 (119910) as 119899 997888rarr infin

(28)

Hence we have the following theorem

Theorem 5 The mappingBA119891 is one-to-one

Proof Let 119891 119892 isin C1015840 and thatBA119891 =BA119892 then using (23)

we get

⟨A119900

119891 (119909) cos119909119910⟩ + ⟨A119890119891 (119909) sin119909119910⟩

= ⟨A119900

119892 (119909) cos119909119910⟩ + ⟨A119890119892 (119909) sin119909119910⟩ (29)

Basic properties of inner product implies

⟨A119900

119891 (119909) minusA119900

119892 (119909) cos (119909119910)⟩

+ ⟨A119890

119891 (119909) minusA119890

119892 (119909) sin (119909119910)⟩ = 0(30)

Hence

A119900

119891 (119909) = A119900

119892 (119909) A119890

119891 (119909) = A119890

119892 (119909) (31)

ThereforeA119891 (119909) = A

119900

119891 (119909) +A119890

119891 (119909)

= A119900

119892 (119909) +A119890

119892 (119909) = A119892 (119909)(32)

for all 119909 This completes the proof of the theorem

Theorem 6 Let 119891 isin C1015840 then 119891 is analytic and

D119896

119910

B

A119891 (119910) = ⟨A

119900

119891 (119909) D119896

119910cos (119909119910)⟩

+ ⟨A119890

119891 (119909) D119896

119910sin (119909119910)⟩

(33)

Proof of this theorem is analogous to that of the previoustheorem and is thus avoided

Denote by 120575 the dirac delta function then it is easy to seethat

A119890

120575 (119910) = 1 A119890

120575 (119910) = 0 (34)

3 Lebesgue Space of Boehmians forHartley-Hilbert Transforms

The original construction of Boehmians produce a concretespace of generalized functions Since the space of Boehmianswas introduced many spaces of Boehmians were defined Inreferences we list selected papers introducing different spacesof Boehmians One of the main motivations for introducingdifferent spaces of Boehmians was the generalization ofintegral transforms The idea requires a proper choice of aspace of functions for which a given integral transform iswell defined a choice of a class of delta sequences that istransformed by that integral transform to a well-behavedclass of approximate identities and finally a convolutionproduct that behaves well under the transform If theseconditions are met the transform has usually an extensionto the constructed space of Boehmians and the extension hasdesirable properties For general construction of Boehmianssee [8ndash12]

LetD be the space of test functions of bounded supportBy delta sequence we mean a subset of D of sequences (120575

119899)

such that

int

infin

minusinfin

120575119899(119909) 119889119909 = 1

10038171003817100381710038171205751198991003817100381710038171003817 = int

infin

minusinfin

1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 ltM 0 ltM isinR

lim119899rarrinfin

int|119909|gt120576

1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 = 0 for each 120576 gt 0

(35)

where 120576(120575119899)(119909) = 119909 isinR 120575

119899(119909) = 0

The set of all such delta sequences is usually denoted asΔEach element in Δ corresponds to the dirac delta function 120575for large values of 119899

Proposition 7 Let (120575119899) isin Δ then

A119890

120575119899(119910) = int

infin

minusinfin

120575119899(119909) cos (119909119910) 119889119909 997888rarr 1 as 119899 997888rarr infin

A119900

120575119899(119910) = int

infin

minusinfin

120575119899(119909) sin (119909119910) 119889119909 997888rarr 0 as 119899 997888rarr infin

(36)

Let L1(R) L1(R) = L1 be the space of complexvalued Lebesgue integrable functions From Proposition 7 weestablish the following theorem


Theorem 8 Let 119891 isin L1 then BA(119891 lowast 120575

119899)(119910) rarr BA

119891(119910)

as 119899 rarr infin

Proof For 119891 isinL1 (120575119899) isin Δ then using of (14) implies

BA(119891 lowast 120575

119899) (119910)

= int

infin

minusinfin

(A119900

(119891 lowast 120575119899) (119909) cos119909119910 +A119890 (119891 lowast 120575

119899) (119909) sin119909119910) 119889119909

(37)

Since

(119891 lowast 120575119899) (120577) = int

infin

minusinfin

119891 (119905) 120575119899(120577 minus 119905) 119889119905 997888rarr 119891 (120577) (38)

as 119899 rarr infin we see that

A119900

(119891 lowast 120575119899) (119909) = int

infin

minusinfin

(119891 lowast 120575119899) (120577) sin119909120577119889120577

= int

infin

minusinfin

119891 (119905) int

infin

minusinfin

120575119899(120577 minus 119905) sin (119909120577) 119889120577119889119905

997888rarr int

infin

minusinfin

119891 (119905) sin (119909119905) 119889119905

= A119900

(119891) (119909)

(39)

Similarly

A119890

(119891 lowast 120575119899) (119909) 997888rarr A

119890

(119891) (119909) as 119899 997888rarr infin (40)

Therefore invoking the above equations in (37) we getBA(119891 lowast 120575

119899)(119910) rarr BA

119891(119910) as 119899 rarr infin Hence thetheorem

Denote by 120588L1 the space of integrable Boehmians then120588L1 is a convolution algebra when multiplication by scalaraddition and convolution is defined as [9]

119896 [119891119899

120575119899

] = [119896119891119899

120575119899

]

[119891119899

120575119899

] + [119892119899

120574119899

] = [119891119899lowast 120574119899+ 119892119899lowast 120575119899

120575119899lowast 120574119899

]

[119891119899

120575119899

] lowast [119892119899

120574119899

] = [119891119899lowast 119892119899

120575119899lowast 120574119899

]

(41)

Each function 119891 isin L1 is identified with the Boehmian [119891 lowast120575119899120575119899] Also [119891

119899120575119899] lowast 120575119899= 119891119899isin L1 for every 119899 isin N

Since [120575119899120575119899] corresponds to Dirac delta distribution 120575 the

119896th-derivative of each 120588 isin 120588L1 is defined as

D119896

120588 = 120588 lowastD119896

120575 (42)

The integral of a Boehmian 120588 = [119891119899120575119899] isin 120588L1 is defined as

[11]

int

infin

minusinfin

120588 (119909) 119889119909 = int

infin

minusinfin

1198911(119909) 119889119909 (43)

It is of great interest to observe the following example

Example 9 Every infinitely smooth function119891(119909) isinL1 suchthat D119896119891(119909) notin L1 is integrable as Boehmian but not inte-grable as function

The following has importance in the sense of analysis

Theorem 10 Let [119891119899120575119899] isin 120588L1 then the sequence

BA(119891119899) (119910)

= int

infin

minusinfin

(A119900

119891119899(119909) cos (119909119910) +A119890119891

119899(119909) sin (119909119910)) 119889119909

(44)

converges uniformly on each compact subsetK ofR

Proof By aid of theTheorem 8 and the concept of quotient ofsequences we have

BA(119891119899) (119910) =B

A(119891119899lowast120575119896

120575119896

) (119910)

=BA(119891119899lowast 120575119896

120575119896

) (119910)

=BA(119891119896

120575119896

lowast 120575119899) (119910)

997888rarrBA119891119896

120575119896

(119910) as 119899 997888rarr infin

(45)

where convergence ranges over compact subsets of R Thetheorem is completely proved

Let [119891119899120575119899] isin 120588L1 then by virtue ofTheorem 10 we define

the Hartley-Hilbert transform of the Lebesgue Boehmian[119891119899120575119899] as

B

A[119891119899

120575119899

] = lim119899rarrinfin

BA119891119899 (46)

on compact subsets ofRThe next objective is to establish that our definition is well

defined Let [119891119899120575119899] = [119892

119899120574119899] in 120588L1 then

119891119899lowast 120574119898= 119892119898lowast 120575119899 for every 119898 119899 isinN (47)

Hence applying the Hartley-Hilbert transform to both sidesof the above equation and using the concept of quotients ofsequences imply

BA(119891119899lowast 120574119898) =B

A(119892119898lowast 120575119899) =B

A(119892119899lowast 120575119898) (48)

Thus in particular for 119899 = 119898 and considering Theorem 3we get

lim119899rarrinfin

BA119891119899= lim119899rarrinfin

BA119892119899 (49)

Hence

B

A[119891119899

120575119899

] =B

A[119892119899

120574119899

] (50)

Definition (46) is therefore well defined

Theorem 11 The generalized transformBA is linear


Proof Let 1205881= [119891119899120575119899] and 120588

2= [119892119899120574119899] be arbitrary in 120588L1

and 120572 isin C then 1205881+1205882= [(119891119899lowast120574119899+119892119899lowast120575119899)120575119899lowast120574119899] Hence

employing (46) yields

B

A(1205881+ 1205882) = lim119899rarrinfin

(BA(119891119899lowast 120574119899) +B

A(119892119899lowast 120575119899))

(51)

ByTheorem 8 we get

B

A(1205881+ 1205882) = lim119899rarrinfin

BA119891119899+ lim119899rarrinfin

BA119892119899 (52)

Hence

B

A(1205881+ 1205882) =B

A1205881+B

A1205882 (53)

Also for each complex number 120572 we have

B

A(1205721205881) =B

A[120572119891119899

120575119899

]

= 120572 lim119899rarrinfin

BA119891119899

= 120572B

A1205881

(54)


Theorem 12 Let 120588 isin 120588L1 and (120598119899) isin Δ then

B

A(120588 lowast 120598

119899) =B

A120588 =B

A(120598119899lowast 120588) (55)

Proof Let 120588 = [119891119899120575119899] isin 120588L1 then

BA(120588 lowast 120598

119899) =BA[119891119899lowast

120598119899120575119899] = lim

119899rarrinfinBA(119891119899lowast 120598119899)

HenceBA(120588 lowast 120598

119899) = lim

119899rarrinfinBA119891119899=BA120588

Similarly we proceed forBA120588 =BA(120598119899lowast 120588)

This completes the theorem The following theorem isobvious

Theorem 13 IfBA1205881= 0 then 120588

1= 0

Theorem14 TheHartley-Hilbert transformBA is continuouswith respect to the 120575-convergence

Proof Let 120588119899

120575

997888rarr 120588 in 120588L1 as 119899 rarr infin then we show thatBA120588119899

120575

997888rarrBA120588 as 119899 rarr infin Using ([11Theorem26]) we find

119891119899119896 119891119896isinL1 (120575

119896) isin Δ such that [119891

119899119896120575119896] = 120588119899 [119891119896120575119896] = 120588

and 119891119899119896rarr 119891119896as 119899 rarr infin 119896 isinN

Applying the Hartley-Hilbert transform for both sidesimplies BA

119891119899119896rarr BA

119891119896in the space of continuous

functions Therefore considering limits we get

B

A[119891119899119896

120575119896

] 997888rarrB

A[119891119896

120575119896

] (56)


Theorem15 TheHartley-Hilbert transformBA is continuouswith respect to the Δ-convergence

Proof Let 120588119899

Δ

997888rarr 120588 as 119899 rarr infin in 120588L1 then there is 119891119899isin L1

and 120575119899isin Δ such that

(120588119899minus 120588) lowast 120575

119899= [119891119899lowast 120575119899

120575119896

] 119891119899997888rarr 0 as 119899 997888rarr infin

(57)

Thus by the aid of Theorem 3 and the hypothesis of thetheorem we have

B

A((120588119899minus 120588) lowast 120575

119899) =B

A[119891119899lowast 120575119899

120575119896

]

997888rarrBA(119891119899lowast 120575119899) as 119899 997888rarr infin

997888rarrBA119891119899

as 119899 997888rarr infin

997888rarr 0 as 119899 997888rarr infin(58)

ThereforeBA(120588119899minus 120588) rarr 0 as 119899 rarr infin ThusBA

120588119899

Δ

997888rarr

BA120588 as 119899 rarr infinThis completes the proof

Lemma 16 Let [119891119899120575119899] isin 120588L1 and 120575 has the usual meaning

of (34) then

B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899

120575119899

] (59)


B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899lowast 120575

120575119899

]

= lim119899rarrinfin

BA(119891119899lowast 120575)


BA119891119899

(60)

Hence

B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899

120575119899

] (61)

Theorem 17 The Hartley-Hilbert transform BA is one-to-one

Proof LetBA[119891119899120575119899] =BA[119892119899120574119899] then by the aid of (46)

we get

lim119899rarrinfin


BA119892119899 (62)

Hence

BA( lim119899rarrinfin

119891119899) =B

A( lim119899rarrinfin

119892119899) (63)

that isBA119891 =BA

119892 The fact thatBA is one-to-one implies119891 = 119892 Hence we have the following theorem


4 A Comparative Study Fourier-HilbertTransform

In [6 7] the Hilbert transform via the Fourier transform of119891(119909) is defined as

BF(119891) (119910)

=1

120587int

infin

0

(F119894(119891) (119909) cos (119909119910) minusF

119903(119891) (119909) sin (119909119910)) 119889119909

(64)

where

F119903(119891) (119909) = int

infin

0

119891 (119905) cos (119909119905) 119889119905

F119894(119891) (119909) = int

infin

0

119891 (119905) sin (119909119905) 119889119905(65)

are respectively the real and imaginary components of theFourier transform of 119891 which are related by

F (119891) (119909) = F119903(119891) (119909) minus 119894F

119894(119891) (119909) (66)

It is interesting to know that a concrete relationship betweenF119903A119890 and F

119894A119900 is described as F

119903(119909) = A119890(119909) and

F119894(119909) = A119900(119909) [2] Those equations above justify the

following statements of the next theorems

Theorem 18 Let [119891119899120575119899] isin 120588L1 then the sequence of Fourier-

Hilbert transforms of (119891119899) satisfies

BF(119891119899) (119910)

=1

120587int

infin

0

(F119894(119891119899) (119909) cos (119909119910) minusF

119903(119891119899) (119909) sin (119909119910)) 119889119909

(67)

and converges uniformly on each compact subsetK ofR

Thus for [119891119899120575119899] isin 120588L1 the Fourier-Hilbert transform of

[119891119899120575119899] is similarly defined by

B

F[119891119899

120575119899


BF119891119899 (68)

on compact subsets ofRThe following theorems are stated and their proofs are

justified for similar reasons We prefer to we omit the details

Theorem 19 The generalized Fourier-Hilbert transformBF

is linear

Theorem 20 BF(120588 lowast 120598

119899) =BF120588 =BF(120598 lowast 120588) (120598

119899) isin Δ

Theorem 21 IfBF1205881= 0 then 120588

1= 0

Theorem 22 If 120588119899

Δ

997888rarr 120588 as 119899 rarr infin in 120588L1 thenBF120588119899

Δ

997888rarr

BF120588 as 119899 rarr infin in 120588L1 on compact subsets

Theorem 23 The Fourier-Hilbert transformBF is continu-ous with respect to the 120575-convergence

Theorem 24 The Fourier-Hilbert transformBF is continu-ous with respect to the Δ-convergence

Proofs of the above theorems are similar to that given forthe corresponding ones of Hartley-Hilbert transform

References

[1] A H Zemanian Generalized Integral Transformations DoverPublications New York NY USA 2nd edition 1987

[2] R S Pathak Integral Transforms of Generalized Functions andTheir Applications Gordon and Breach Science PublishersNorth Vancouver Canada 1997

[3] S K Q Al-Omari D Loonker P K Banerji and S L KallaldquoFourier sine (cosine) transform for ultradistributions and theirextensions to tempered and ultraBoehmian spacesrdquo IntegralTransforms and Special Functions vol 19 no 5-6 pp 453ndash4622008

[4] S Al-Omari and A Kılıcman ldquoOn the generalized Hartley-Hilbert and fourier-Hilbert transformsrdquo Advances in DifferenceEquations vol 2012 p 232 2012

[5] R P Millane ldquoAnalytic properties of the Hartley transform andtheir applicationsrdquo Proceedings of the IEEE vol 82 no 3 pp413ndash428 1994

[6] N Sundararajan and Y Srinivas ldquoFourier-Hilbert versusHartley-Hilbert transforms with some geophysical applica-tionsrdquo Journal of Applied Geophysics vol 71 no 4 pp 157ndash1612010

[7] N Sundarajan ldquoFourier and hartley transforms amathematicaltwinrdquo Indian Journal of Pure and Applied Mathematics vol 8no 10 pp 1361ndash1365 1997

[8] T K Boehme ldquoThe support of Mikusinski operatorsrdquo Transac-tions of theAmericanMathematical Society vol 176 pp 319ndash3341973

[9] P Mikusinski ldquoFourier transform for integrable BoehmiansrdquoThe Rocky Mountain Journal of Mathematics vol 17 no 3 pp577ndash582 1987

[10] S K Q Al-Omari and A Kılıcman ldquoOn diffraction Fresneltransforms for Boehmiansrdquo Abstract and Applied Analysis vol2011 Article ID 712746 11 pages 2011

[11] P Mikusinski ldquoTempered Boehmians and ultradistributionsrdquoProceedings of the American Mathematical Society vol 123 no3 pp 813ndash817 1995

[12] S K Q Al-Omari and A Kılıcman ldquoNote on Boehmians forclass of optical Fresnel wavelet transformsrdquo Journal of FunctionSpaces and Applications vol 2012 Article ID 405368 14 pages2012

Submit your manuscripts athttpwwwhindawicom

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Volume 2014


Stochastic AnalysisInternational Journal of


or

119891 (119909) = int

infin

minusinfin

A (119891) cas (2120587119891119909) 119889119891 (7)

The theory of convolutions of integral transforms has beendeveloped for a long time and is applied in many fields ofmathematics Historically the convolution product [2]

(119891 lowast 119892) (119910) = int

infin

minusinfin

119891 (119909) 119892 (119909 minus 119910) 119889119910 (8)

has a relationship with the Fourier transform with the factor-ization property

F (119891 lowast 119892) (119910) = F (119891) (119910)F (119892) (119910) (9)

Themore complicated convolution theorem of Hartley trans-forms compared to that of Fourier transforms is that

A (119891 lowast 119892) (119910) =1

2G (A119891 timesA119892) (119910) (10)

where

G (119891 lowast 119892) (119910) = 119891 (119910) 119892 (119910) + 119891 (119910) 119892 (minus119910)

+ 119891 (119910) 119892 (119910) minus 119891 (minus119910) 119892 (minus119910)

(11)

Some properties of Hartley transforms can be listed asfollows

(i) Linearity if 119891 and 119892 are real functions then

A (119886119891 + 119887119892) (119910) = 119886A (119891) (119910) + 119887A (119892) (119910) 119886 119887 isinR

(12)

(ii) Scaling if 119891 is a real function then

int

infin

minusinfin

119891 (120572120589) cas (2120587119910120589) 119889120589 = 1119886(A119891) (

119910

119886) (13)

2 Distributional Hartley-Hilbert Transformof Compact Support

TheHilbert transform via the Hartley transform is defined by[6 7]

BA(119910) = minus

1

120587int

infin

0

(A119900

(119909) cos (119909119910) +A119890 (119909) sin (119909119910)) 119889119909

(14)

where

A119900

(119909) =A (119909) minusA (minus119909)

2

A119890

(119909) =A (119909) +A (minus119909)

2

(15)

are the respective odd and even components of (2)We denote C(R) C(R) = C the space of smooth

functions and C1015840(R) C1015840(R) = C1015840 the strong dual of C ofdistributions of compact support overR

Following is the convolution theorem ofBA

Theorem 1 (ConvolutionTheorem) Let 119891 and 119892 isin C then

BA(119891 lowast 119892) (119910) = int

infin

0

(1198961(119909) cos (119910119909) + 119896

2(119909) sin (119910119909)) 119889119909

(16)

where

1198961(119909) = A

119890

119891 (119909)A119900

119892 (119909) +A119900

119891 (119909)A119890

119892 (119909)

1198962(119909) = A

119890

119891 (119909)A119890

119892 (119909) minusA119900

119891 (119909)A119900

119892 (119909)

(17)

Proof To prove this theorem it is sufficient to establish that

1198961(119909) = A

119900

(119891 lowast 119892) (119909) (18)

1198962(119909) = A

119890

(119891 lowast 119892) (119909) (19)

Therefore we have

A119900

(119891 lowast 119892) (119909)

= int

infin

minusinfin

(int

infin

minusinfin

119891 (120574) 119892 (119910 minus 120574) 119889119903) cas (119909119910) 119889119910

= int

infin

minusinfin

119891 (120574)int

infin

minusinfin

119892 (119910 minus 120574) cas (119909119910) 119889119910119889119903

(20)

The substitution 119910 minus 120574 = 119911 and using of (1) together withFubini theorem imply

A119900

(119891 lowast 119892) (119909)

= int

infin

minusinfin

119891 (120574)int

infin

minusinfin

119892 (119911) (cos (119909 (119911 + 120574)) + sin (119909 (119911 + 120574)))

times 119889119911119889119903

(21)

By invoking the formulae

cos (119909 (119911 + 120574)) = cos (119909119911) cos (119909120574) minus sin (119909119911) sin (119909120574)

sin (119909 (119911 + 120574)) = sin (119909119911) cos (119909120574) + cos (119909119911) sin (119909120574) (22)

then (18) follows from simple computation Proof of (19) hasa similar techniqueHence the theorem is completely proved

It is of interest to know that cos(119909119910) and sin(119909119910) aremembers of C and therefore A119900119891A119890119891 isin C1015840 This leads tothe following statement

Definition 2 Let 119891 isin C1015840 then we define the distributionalHartley-Hilbert transform of 119891 as

B

A119891 (119910) = ⟨A

119900

119891 (119909) cos (119909119910)⟩ + ⟨A119890119891 (119909) sin (119909119910)⟩ (23)

The extended transformBA119891 is clearly well defined for

each 119891 isin C1015840




A(119891 + 119892) (119910) = ⟨A

119900

(119891 + 119892) (119909) cos (119909119910)⟩

+ ⟨A119890

(119891 + 119892) (119909) sin (119909119910)⟩ (24)


A(119891 + 119892) (119910) =

B

A119891 (119910) +

B

A119892 (119910) (25)

FurthermoreB

A(119896119891) (119910) = ⟨119896A

119900

119891 (119909) cos (119909119910)⟩

+ ⟨119896A119890

119891 (119909) sin (119909119910)⟩ (26)

HenceB

A(119896119891) (119910) = 119896

B

A119891 (119910) (27)



onC1015840



B

A119891119899(119910) = ⟨A

119900

119891119899(119909) cos (119909119910)⟩ + ⟨A119890119891

119899(119909) sin (119909119910)⟩

997888rarr ⟨A119900

119891 (119909) cos (119909119910)⟩ + ⟨A119890119891 (119909) sin (119909119910)⟩

=B

A119891 (119910) as 119899 997888rarr infin

(28)




we get

⟨A119900

119891 (119909) cos119909119910⟩ + ⟨A119890119891 (119909) sin119909119910⟩

= ⟨A119900

119892 (119909) cos119909119910⟩ + ⟨A119890119892 (119909) sin119909119910⟩ (29)


⟨A119900

119891 (119909) minusA119900

119892 (119909) cos (119909119910)⟩

+ ⟨A119890

119891 (119909) minusA119890

119892 (119909) sin (119909119910)⟩ = 0(30)

Hence

A119900

119891 (119909) = A119900

119892 (119909) A119890

119891 (119909) = A119890

119892 (119909) (31)

ThereforeA119891 (119909) = A

119900

119891 (119909) +A119890

119891 (119909)

= A119900

119892 (119909) +A119890

119892 (119909) = A119892 (119909)(32)



D119896

119910

B

A119891 (119910) = ⟨A

119900

119891 (119909) D119896

119910cos (119909119910)⟩

+ ⟨A119890

119891 (119909) D119896

119910sin (119909119910)⟩

(33)



A119890

120575 (119910) = 1 A119890

120575 (119910) = 0 (34)




119899)

such that

int

infin

minusinfin

120575119899(119909) 119889119909 = 1

10038171003817100381710038171205751198991003817100381710038171003817 = int

infin

minusinfin

1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 ltM 0 ltM isinR

lim119899rarrinfin

int|119909|gt120576

1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 = 0 for each 120576 gt 0

(35)

where 120576(120575119899)(119909) = 119909 isinR 120575

119899(119909) = 0



A119890

120575119899(119910) = int

infin

minusinfin


A119900

120575119899(119910) = int

infin

minusinfin


(36)




119899)(119910) rarr BA

119891(119910)



BA(119891 lowast 120575

119899) (119910)

= int

infin

minusinfin

(A119900


119899) (119909) sin119909119910) 119889119909

(37)

Since

(119891 lowast 120575119899) (120577) = int

infin

minusinfin

119891 (119905) 120575119899(120577 minus 119905) 119889119905 997888rarr 119891 (120577) (38)


A119900

(119891 lowast 120575119899) (119909) = int

infin

minusinfin

(119891 lowast 120575119899) (120577) sin119909120577119889120577

= int

infin

minusinfin

119891 (119905) int

infin

minusinfin

120575119899(120577 minus 119905) sin (119909120577) 119889120577119889119905

997888rarr int

infin

minusinfin

119891 (119905) sin (119909119905) 119889119905

= A119900

(119891) (119909)

(39)

Similarly

A119890

(119891 lowast 120575119899) (119909) 997888rarr A

119890

(119891) (119909) as 119899 997888rarr infin (40)


119899)(119910) rarr BA



119896 [119891119899

120575119899

] = [119896119891119899

120575119899

]

[119891119899

120575119899

] + [119892119899

120574119899

] = [119891119899lowast 120574119899+ 119892119899lowast 120575119899

120575119899lowast 120574119899

]

[119891119899

120575119899

] lowast [119892119899

120574119899

] = [119891119899lowast 119892119899

120575119899lowast 120574119899

]

(41)





D119896

120588 = 120588 lowastD119896

120575 (42)


[11]

int

infin

minusinfin

120588 (119909) 119889119909 = int

infin

minusinfin

1198911(119909) 119889119909 (43)





BA(119891119899) (119910)

= int

infin

minusinfin

(A119900

119891119899(119909) cos (119909119910) +A119890119891

119899(119909) sin (119909119910)) 119889119909

(44)



BA(119891119899) (119910) =B

A(119891119899lowast120575119896

120575119896

) (119910)

=BA(119891119899lowast 120575119896

120575119896

) (119910)

=BA(119891119896

120575119896

lowast 120575119899) (119910)

997888rarrBA119891119896

120575119896

(119910) as 119899 997888rarr infin

(45)




B

A[119891119899

120575119899


BA119891119899 (46)


defined Let [119891119899120575119899] = [119892

119899120574119899] in 120588L1 then



BA(119891119899lowast 120574119898) =B

A(119892119898lowast 120575119899) =B

A(119892119899lowast 120575119898) (48)


lim119899rarrinfin


BA119892119899 (49)

Hence

B

A[119891119899

120575119899

] =B

A[119892119899

120574119899

] (50)




Proof Let 1205881= [119891119899120575119899] and 120588

2= [119892119899120574119899] be arbitrary in 120588L1



B

A(1205881+ 1205882) = lim119899rarrinfin

(BA(119891119899lowast 120574119899) +B

A(119892119899lowast 120575119899))

(51)

ByTheorem 8 we get

B

A(1205881+ 1205882) = lim119899rarrinfin


BA119892119899 (52)

Hence

B

A(1205881+ 1205882) =B

A1205881+B

A1205882 (53)


B

A(1205721205881) =B

A[120572119891119899

120575119899

]


BA119891119899

= 120572B

A1205881

(54)



B

A(120588 lowast 120598

119899) =B

A120588 =B

A(120598119899lowast 120588) (55)


BA(120588 lowast 120598

119899) =BA[119891119899lowast

120598119899120575119899] = lim



119899) = lim

119899rarrinfinBA119891119899=BA120588




1= 0


Proof Let 120588119899

120575


120575


119891119899119896 119891119896isinL1 (120575


119899119896120575119896] = 120588119899 [119891119896120575119896] = 120588



119891119899119896rarr BA



B

A[119891119899119896

120575119896

] 997888rarrB

A[119891119896

120575119896

] (56)



Proof Let 120588119899

Δ



(120588119899minus 120588) lowast 120575

119899= [119891119899lowast 120575119899

120575119896


(57)


B

A((120588119899minus 120588) lowast 120575

119899) =B

A[119891119899lowast 120575119899

120575119896

]


997888rarrBA119891119899




120588119899

Δ

997888rarr



of (34) then

B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899

120575119899

] (59)


B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899lowast 120575

120575119899

]


BA(119891119899lowast 120575)


BA119891119899

(60)

Hence

B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899

120575119899

] (61)



we get

lim119899rarrinfin


BA119892119899 (62)

Hence


119891119899) =B


119892119899) (63)

that isBA119891 =BA





BF(119891) (119910)

=1

120587int

infin

0

(F119894(119891) (119909) cos (119909119910) minusF

119903(119891) (119909) sin (119909119910)) 119889119909

(64)

where

F119903(119891) (119909) = int

infin

0

119891 (119905) cos (119909119905) 119889119905

F119894(119891) (119909) = int

infin

0

119891 (119905) sin (119909119905) 119889119905(65)


F (119891) (119909) = F119903(119891) (119909) minus 119894F

119894(119891) (119909) (66)



119903(119909) = A119890(119909) and





BF(119891119899) (119910)

=1

120587int

infin

0

(F119894(119891119899) (119909) cos (119909119910) minusF

119903(119891119899) (119909) sin (119909119910)) 119889119909

(67)




B

F[119891119899

120575119899


BF119891119899 (68)




is linear


119899) =BF120588 =BF(120598 lowast 120588) (120598

119899) isin Δ


1= 0

Theorem 22 If 120588119899

Δ


Δ

997888rarr





References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






A(119891 + 119892) (119910) = ⟨A

119900

(119891 + 119892) (119909) cos (119909119910)⟩

+ ⟨A119890

(119891 + 119892) (119909) sin (119909119910)⟩ (24)


A(119891 + 119892) (119910) =

B

A119891 (119910) +

B

A119892 (119910) (25)

FurthermoreB

A(119896119891) (119910) = ⟨119896A

119900

119891 (119909) cos (119909119910)⟩

+ ⟨119896A119890

119891 (119909) sin (119909119910)⟩ (26)

HenceB

A(119896119891) (119910) = 119896

B

A119891 (119910) (27)



onC1015840



B

A119891119899(119910) = ⟨A

119900

119891119899(119909) cos (119909119910)⟩ + ⟨A119890119891

119899(119909) sin (119909119910)⟩

997888rarr ⟨A119900

119891 (119909) cos (119909119910)⟩ + ⟨A119890119891 (119909) sin (119909119910)⟩

=B

A119891 (119910) as 119899 997888rarr infin

(28)




we get

⟨A119900

119891 (119909) cos119909119910⟩ + ⟨A119890119891 (119909) sin119909119910⟩

= ⟨A119900

119892 (119909) cos119909119910⟩ + ⟨A119890119892 (119909) sin119909119910⟩ (29)


⟨A119900

119891 (119909) minusA119900

119892 (119909) cos (119909119910)⟩

+ ⟨A119890

119891 (119909) minusA119890

119892 (119909) sin (119909119910)⟩ = 0(30)

Hence

A119900

119891 (119909) = A119900

119892 (119909) A119890

119891 (119909) = A119890

119892 (119909) (31)

ThereforeA119891 (119909) = A

119900

119891 (119909) +A119890

119891 (119909)

= A119900

119892 (119909) +A119890

119892 (119909) = A119892 (119909)(32)



D119896

119910

B

A119891 (119910) = ⟨A

119900

119891 (119909) D119896

119910cos (119909119910)⟩

+ ⟨A119890

119891 (119909) D119896

119910sin (119909119910)⟩

(33)



A119890

120575 (119910) = 1 A119890

120575 (119910) = 0 (34)




119899)

such that

int

infin

minusinfin

120575119899(119909) 119889119909 = 1

10038171003817100381710038171205751198991003817100381710038171003817 = int

infin

minusinfin

1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 ltM 0 ltM isinR

lim119899rarrinfin

int|119909|gt120576

1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 = 0 for each 120576 gt 0

(35)

where 120576(120575119899)(119909) = 119909 isinR 120575

119899(119909) = 0



A119890

120575119899(119910) = int

infin

minusinfin


A119900

120575119899(119910) = int

infin

minusinfin


(36)




119899)(119910) rarr BA

119891(119910)



BA(119891 lowast 120575

119899) (119910)

= int

infin

minusinfin

(A119900


119899) (119909) sin119909119910) 119889119909

(37)

Since

(119891 lowast 120575119899) (120577) = int

infin

minusinfin

119891 (119905) 120575119899(120577 minus 119905) 119889119905 997888rarr 119891 (120577) (38)


A119900

(119891 lowast 120575119899) (119909) = int

infin

minusinfin

(119891 lowast 120575119899) (120577) sin119909120577119889120577

= int

infin

minusinfin

119891 (119905) int

infin

minusinfin

120575119899(120577 minus 119905) sin (119909120577) 119889120577119889119905

997888rarr int

infin

minusinfin

119891 (119905) sin (119909119905) 119889119905

= A119900

(119891) (119909)

(39)

Similarly

A119890

(119891 lowast 120575119899) (119909) 997888rarr A

119890

(119891) (119909) as 119899 997888rarr infin (40)


119899)(119910) rarr BA



119896 [119891119899

120575119899

] = [119896119891119899

120575119899

]

[119891119899

120575119899

] + [119892119899

120574119899

] = [119891119899lowast 120574119899+ 119892119899lowast 120575119899

120575119899lowast 120574119899

]

[119891119899

120575119899

] lowast [119892119899

120574119899

] = [119891119899lowast 119892119899

120575119899lowast 120574119899

]

(41)





D119896

120588 = 120588 lowastD119896

120575 (42)


[11]

int

infin

minusinfin

120588 (119909) 119889119909 = int

infin

minusinfin

1198911(119909) 119889119909 (43)





BA(119891119899) (119910)

= int

infin

minusinfin

(A119900

119891119899(119909) cos (119909119910) +A119890119891

119899(119909) sin (119909119910)) 119889119909

(44)



BA(119891119899) (119910) =B

A(119891119899lowast120575119896

120575119896

) (119910)

=BA(119891119899lowast 120575119896

120575119896

) (119910)

=BA(119891119896

120575119896

lowast 120575119899) (119910)

997888rarrBA119891119896

120575119896

(119910) as 119899 997888rarr infin

(45)




B

A[119891119899

120575119899


BA119891119899 (46)


defined Let [119891119899120575119899] = [119892

119899120574119899] in 120588L1 then



BA(119891119899lowast 120574119898) =B

A(119892119898lowast 120575119899) =B

A(119892119899lowast 120575119898) (48)


lim119899rarrinfin


BA119892119899 (49)

Hence

B

A[119891119899

120575119899

] =B

A[119892119899

120574119899

] (50)




Proof Let 1205881= [119891119899120575119899] and 120588

2= [119892119899120574119899] be arbitrary in 120588L1



B

A(1205881+ 1205882) = lim119899rarrinfin

(BA(119891119899lowast 120574119899) +B

A(119892119899lowast 120575119899))

(51)

ByTheorem 8 we get

B

A(1205881+ 1205882) = lim119899rarrinfin


BA119892119899 (52)

Hence

B

A(1205881+ 1205882) =B

A1205881+B

A1205882 (53)


B

A(1205721205881) =B

A[120572119891119899

120575119899

]


BA119891119899

= 120572B

A1205881

(54)



B

A(120588 lowast 120598

119899) =B

A120588 =B

A(120598119899lowast 120588) (55)


BA(120588 lowast 120598

119899) =BA[119891119899lowast

120598119899120575119899] = lim



119899) = lim

119899rarrinfinBA119891119899=BA120588




1= 0


Proof Let 120588119899

120575


120575


119891119899119896 119891119896isinL1 (120575


119899119896120575119896] = 120588119899 [119891119896120575119896] = 120588



119891119899119896rarr BA



B

A[119891119899119896

120575119896

] 997888rarrB

A[119891119896

120575119896

] (56)



Proof Let 120588119899

Δ



(120588119899minus 120588) lowast 120575

119899= [119891119899lowast 120575119899

120575119896


(57)


B

A((120588119899minus 120588) lowast 120575

119899) =B

A[119891119899lowast 120575119899

120575119896

]


997888rarrBA119891119899




120588119899

Δ

997888rarr



of (34) then

B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899

120575119899

] (59)


B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899lowast 120575

120575119899

]


BA(119891119899lowast 120575)


BA119891119899

(60)

Hence

B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899

120575119899

] (61)



we get

lim119899rarrinfin


BA119892119899 (62)

Hence


119891119899) =B


119892119899) (63)

that isBA119891 =BA





BF(119891) (119910)

=1

120587int

infin

0

(F119894(119891) (119909) cos (119909119910) minusF

119903(119891) (119909) sin (119909119910)) 119889119909

(64)

where

F119903(119891) (119909) = int

infin

0

119891 (119905) cos (119909119905) 119889119905

F119894(119891) (119909) = int

infin

0

119891 (119905) sin (119909119905) 119889119905(65)


F (119891) (119909) = F119903(119891) (119909) minus 119894F

119894(119891) (119909) (66)



119903(119909) = A119890(119909) and





BF(119891119899) (119910)

=1

120587int

infin

0

(F119894(119891119899) (119909) cos (119909119910) minusF

119903(119891119899) (119909) sin (119909119910)) 119889119909

(67)




B

F[119891119899

120575119899


BF119891119899 (68)




is linear


119899) =BF120588 =BF(120598 lowast 120588) (120598

119899) isin Δ


1= 0

Theorem 22 If 120588119899

Δ


Δ

997888rarr





References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119899)(119910) rarr BA

119891(119910)



BA(119891 lowast 120575

119899) (119910)

= int

infin

minusinfin

(A119900


119899) (119909) sin119909119910) 119889119909

(37)

Since

(119891 lowast 120575119899) (120577) = int

infin

minusinfin

119891 (119905) 120575119899(120577 minus 119905) 119889119905 997888rarr 119891 (120577) (38)


A119900

(119891 lowast 120575119899) (119909) = int

infin

minusinfin

(119891 lowast 120575119899) (120577) sin119909120577119889120577

= int

infin

minusinfin

119891 (119905) int

infin

minusinfin

120575119899(120577 minus 119905) sin (119909120577) 119889120577119889119905

997888rarr int

infin

minusinfin

119891 (119905) sin (119909119905) 119889119905

= A119900

(119891) (119909)

(39)

Similarly

A119890

(119891 lowast 120575119899) (119909) 997888rarr A

119890

(119891) (119909) as 119899 997888rarr infin (40)


119899)(119910) rarr BA



119896 [119891119899

120575119899

] = [119896119891119899

120575119899

]

[119891119899

120575119899

] + [119892119899

120574119899

] = [119891119899lowast 120574119899+ 119892119899lowast 120575119899

120575119899lowast 120574119899

]

[119891119899

120575119899

] lowast [119892119899

120574119899

] = [119891119899lowast 119892119899

120575119899lowast 120574119899

]

(41)





D119896

120588 = 120588 lowastD119896

120575 (42)


[11]

int

infin

minusinfin

120588 (119909) 119889119909 = int

infin

minusinfin

1198911(119909) 119889119909 (43)





BA(119891119899) (119910)

= int

infin

minusinfin

(A119900

119891119899(119909) cos (119909119910) +A119890119891

119899(119909) sin (119909119910)) 119889119909

(44)



BA(119891119899) (119910) =B

A(119891119899lowast120575119896

120575119896

) (119910)

=BA(119891119899lowast 120575119896

120575119896

) (119910)

=BA(119891119896

120575119896

lowast 120575119899) (119910)

997888rarrBA119891119896

120575119896

(119910) as 119899 997888rarr infin

(45)




B

A[119891119899

120575119899


BA119891119899 (46)


defined Let [119891119899120575119899] = [119892

119899120574119899] in 120588L1 then



BA(119891119899lowast 120574119898) =B

A(119892119898lowast 120575119899) =B

A(119892119899lowast 120575119898) (48)


lim119899rarrinfin


BA119892119899 (49)

Hence

B

A[119891119899

120575119899

] =B

A[119892119899

120574119899

] (50)




Proof Let 1205881= [119891119899120575119899] and 120588

2= [119892119899120574119899] be arbitrary in 120588L1



B

A(1205881+ 1205882) = lim119899rarrinfin

(BA(119891119899lowast 120574119899) +B

A(119892119899lowast 120575119899))

(51)

ByTheorem 8 we get

B

A(1205881+ 1205882) = lim119899rarrinfin


BA119892119899 (52)

Hence

B

A(1205881+ 1205882) =B

A1205881+B

A1205882 (53)


B

A(1205721205881) =B

A[120572119891119899

120575119899

]


BA119891119899

= 120572B

A1205881

(54)



B

A(120588 lowast 120598

119899) =B

A120588 =B

A(120598119899lowast 120588) (55)


BA(120588 lowast 120598

119899) =BA[119891119899lowast

120598119899120575119899] = lim



119899) = lim

119899rarrinfinBA119891119899=BA120588




1= 0


Proof Let 120588119899

120575


120575


119891119899119896 119891119896isinL1 (120575


119899119896120575119896] = 120588119899 [119891119896120575119896] = 120588



119891119899119896rarr BA



B

A[119891119899119896

120575119896

] 997888rarrB

A[119891119896

120575119896

] (56)



Proof Let 120588119899

Δ



(120588119899minus 120588) lowast 120575

119899= [119891119899lowast 120575119899

120575119896


(57)


B

A((120588119899minus 120588) lowast 120575

119899) =B

A[119891119899lowast 120575119899

120575119896

]


997888rarrBA119891119899




120588119899

Δ

997888rarr



of (34) then

B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899

120575119899

] (59)


B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899lowast 120575

120575119899

]


BA(119891119899lowast 120575)


BA119891119899

(60)

Hence

B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899

120575119899

] (61)



we get

lim119899rarrinfin


BA119892119899 (62)

Hence


119891119899) =B


119892119899) (63)

that isBA119891 =BA





BF(119891) (119910)

=1

120587int

infin

0

(F119894(119891) (119909) cos (119909119910) minusF

119903(119891) (119909) sin (119909119910)) 119889119909

(64)

where

F119903(119891) (119909) = int

infin

0

119891 (119905) cos (119909119905) 119889119905

F119894(119891) (119909) = int

infin

0

119891 (119905) sin (119909119905) 119889119905(65)


F (119891) (119909) = F119903(119891) (119909) minus 119894F

119894(119891) (119909) (66)



119903(119909) = A119890(119909) and





BF(119891119899) (119910)

=1

120587int

infin

0

(F119894(119891119899) (119909) cos (119909119910) minusF

119903(119891119899) (119909) sin (119909119910)) 119889119909

(67)




B

F[119891119899

120575119899


BF119891119899 (68)




is linear


119899) =BF120588 =BF(120598 lowast 120588) (120598

119899) isin Δ


1= 0

Theorem 22 If 120588119899

Δ


Δ

997888rarr





References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




Proof Let 1205881= [119891119899120575119899] and 120588

2= [119892119899120574119899] be arbitrary in 120588L1



B

A(1205881+ 1205882) = lim119899rarrinfin

(BA(119891119899lowast 120574119899) +B

A(119892119899lowast 120575119899))

(51)

ByTheorem 8 we get

B

A(1205881+ 1205882) = lim119899rarrinfin


BA119892119899 (52)

Hence

B

A(1205881+ 1205882) =B

A1205881+B

A1205882 (53)


B

A(1205721205881) =B

A[120572119891119899

120575119899

]


BA119891119899

= 120572B

A1205881

(54)



B

A(120588 lowast 120598

119899) =B

A120588 =B

A(120598119899lowast 120588) (55)


BA(120588 lowast 120598

119899) =BA[119891119899lowast

120598119899120575119899] = lim



119899) = lim

119899rarrinfinBA119891119899=BA120588




1= 0


Proof Let 120588119899

120575


120575


119891119899119896 119891119896isinL1 (120575


119899119896120575119896] = 120588119899 [119891119896120575119896] = 120588



119891119899119896rarr BA



B

A[119891119899119896

120575119896

] 997888rarrB

A[119891119896

120575119896

] (56)



Proof Let 120588119899

Δ



(120588119899minus 120588) lowast 120575

119899= [119891119899lowast 120575119899

120575119896


(57)


B

A((120588119899minus 120588) lowast 120575

119899) =B

A[119891119899lowast 120575119899

120575119896

]


997888rarrBA119891119899




120588119899

Δ

997888rarr



of (34) then

B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899

120575119899

] (59)


B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899lowast 120575

120575119899

]


BA(119891119899lowast 120575)


BA119891119899

(60)

Hence

B

A([119891119899

120575119899

] lowast 120575) =B

A[119891119899

120575119899

] (61)



we get

lim119899rarrinfin


BA119892119899 (62)

Hence


119891119899) =B


119892119899) (63)

that isBA119891 =BA





BF(119891) (119910)

=1

120587int

infin

0

(F119894(119891) (119909) cos (119909119910) minusF

119903(119891) (119909) sin (119909119910)) 119889119909

(64)

where

F119903(119891) (119909) = int

infin

0

119891 (119905) cos (119909119905) 119889119905

F119894(119891) (119909) = int

infin

0

119891 (119905) sin (119909119905) 119889119905(65)


F (119891) (119909) = F119903(119891) (119909) minus 119894F

119894(119891) (119909) (66)



119903(119909) = A119890(119909) and





BF(119891119899) (119910)

=1

120587int

infin

0

(F119894(119891119899) (119909) cos (119909119910) minusF

119903(119891119899) (119909) sin (119909119910)) 119889119909

(67)




B

F[119891119899

120575119899


BF119891119899 (68)




is linear


119899) =BF120588 =BF(120598 lowast 120588) (120598

119899) isin Δ


1= 0

Theorem 22 If 120588119899

Δ


Δ

997888rarr





References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






BF(119891) (119910)

=1

120587int

infin

0

(F119894(119891) (119909) cos (119909119910) minusF

119903(119891) (119909) sin (119909119910)) 119889119909

(64)

where

F119903(119891) (119909) = int

infin

0

119891 (119905) cos (119909119905) 119889119905

F119894(119891) (119909) = int

infin

0

119891 (119905) sin (119909119905) 119889119905(65)


F (119891) (119909) = F119903(119891) (119909) minus 119894F

119894(119891) (119909) (66)



119903(119909) = A119890(119909) and





BF(119891119899) (119910)

=1

120587int

infin

0

(F119894(119891119899) (119909) cos (119909119910) minusF

119903(119891119899) (119909) sin (119909119910)) 119889119909

(67)




B

F[119891119899

120575119899


BF119891119899 (68)




is linear


119899) =BF120588 =BF(120598 lowast 120588) (120598

119899) isin Δ


1= 0

Theorem 22 If 120588119899

Δ


Δ

997888rarr





References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014










Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



some remarks on the extended hartley-hilbert and fourier-hilbert ...€¦ · fourier-hilbert...

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