On generalization of Fourier and Hartley transforms for some quotient class of sequences Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2016, Том 18, Выпуск 4, С. 3-14

ON GENERALIZATION OF FOURIER AND HARTLEY TRANSFORMS FOR SOME QUOTIENT CLASS OF SEQUENCES

S. K. Q. Al-Omari

In this paper we consider a class of distributions and generate two spaces of Boehmians for certain class of integral operators. We derive a convolution theorem and generate two spaces of Boehmians. The integral operator under concern is well-defined, linear and one-to-one in the class of Boehmians. An inverse problem is also discussed in some details.

Mathematics Subject Classification (2000): 44A15, 46F12.

Key words: ff,P'/3 transform, Hartley transform, Fourier transform, quotient space.

Integral transforms have been introduced and found their applications in applied mathematics and diverse fields of science. The Hartley transform is an integral transformation that maps a real-valued temporal or spacial function into a real-valued frequency function via the kernel cas(-) = cos(-) + sin(-). Advantages of Hartley transforms comes over that of Fourier transforms since they avoid the use of complex arithmetic which results in faster algorithms. Hartley transforms can further be analytically continued into the complex plane, and for real functions they are Hermitian symmetry or reflection in the real axis. In this article we consider an integral transform related to Hartley and Fourier transforms defined for functions of two variables as

Ha]if((>0 = y- I I f(x,y)(acos(x +f3sm(x)(pcos£y + r]sm£y)dxdy, (1)

where ((,£) G R2, R2 = R x R are the transform variables and a, p, n are arbitrary-constants.

A inversion formula of the cited integral be can easily recovered from (1) giving

In a special case, for a = @ = 1, p = n = 1, the integral transform (1) and the inversion formula (2) Me respectively reduced to the double Hart ley transform Ad pair (s ee [10])

1. Introduction

№.»> = ¿7/

нРа]з (Z, f )(a cos (x + в sin (x)(p cos £y + n sin f y) d( d£. (2)

R R

R

R

© 2016 Al-Omari S. K. Q.

and

f(x,y) = ^ J J Ad((, £)(cos(x + sin C®)(cos£y + sin(4)

R R

Further, with simple computations, the kernel function

(cos (x + sin Zx)(cos + sin ) = cas Zx cas (5)

inside the integral signs can be written as

cas Zx cas vy = cos(Zx — £y) + sin(Zx + ^y). (6)

Hence, the integral Equations (3^d (4) can also be rearranged in terms of (6) as

= ^ J J f(x,y)(cos(Cx-^y)+sin((x + ^y))dxdy (7)

RR

and

f{x'y) = h J J Ad(('0(cos(Cx-^y)+sm(Cx + ^y))d(d^ (8)

RR

respectively.

By setting a = 1, ^ = i, p = 1 and n = i, we derive the double Fourier transform Fd pair, = J f (x, y)(cos (x + i sin£x)(cos (y+ i sin£y)dxdy (9)

RR

and

f(x>y) = ~^f J F<i ((, (x + i sin £x) (cos (y + i sin £y)d(d£. (10)

RR

By factoring Ad(Z,e) into even and odd components, Ad(Z,e) = Ed(Z,e) + Od(Z,e)> where

Ed((= J f(x,y)cos((x-£y)dxdy (11)

RR

and

°d(C= J f(x,y)sin((x-£y)dxdy (12)

RR

we get

Fd(Z, e) = Ed(Z, e) — iOd(Z, e) and Ad(Z, e) = Re Fd(Z, e) + Im Fd(Z, e). (13)

Denote by L2 the Lebesgue space of integrable functions over R2; then the convolution product of f (x, y^d g(x, y) in L2 is defined by

(f *2 g)(x,y) = J J f (t,w)g(x — t,y — w) dtdw.

RR

We state and prove the following theorem.

Theorem 1 (Convolution Theorem). Let f (x,y),g(x,y) G L2. Then we have

K'}(f *2 g)(C,0 = J(C,e)G(C,e),

where J(Z,£) and G(Z,£) are given by the integrals

J((,£) = J J f (t, w) cos(tZ) cos(w£) dtdw

and

G(Z ,0 = J J 4£n sin(Zz) sin(r£)g(z, r) dzdr.

R R

< Let f (x,y),g(x,y) G L2. Then by using the convolution product formula we have

HPa} (f *2 g) (Z,0 = / J (f *2 g) (x,y)(acos(xZ) + £sin(xZ))

RR

x(p cos(y£) + n sin(y£)) dxdy

/ f (t, w)g(x — t, y — w) dt dw

x(acos(xZ) + £sin(xZ))(pcos(y£) + nsin(y£)) dxdy. Change of variables x — t = ^d y — w = r imply dx = dz and dy = dr and hence

Ha} (f *2 g) (Z,0 = / J f(t,w)| J g(z,r)(a cos Z(z +1) + £ sin Z(z +1))

R R R R

x(p cos Z (r + w) + n sin(r + w)£) dz dr dtdw.

By aid of the facts cos(a+£) = cos a cos £—sin a sin £ and sin(a+£) = sin a cos £—sin £ cos a and using simple computations we get

Ha} (f *2 g) (Z, 0 = / J f (t, w) a(t, w) dw dt, (14)

RR

where

a(t, w) = cos(tZ) cos(wv) J J g(z,r)(a cos(zZ) + £ sin(zZ)) x (p cos(r£) + n sin(r£)) dzdr

R R

— cos(tZ )sin(w£)^ J g(z,r)(a cos(zZ) + £ sin(zZ)) x (p sin(r£) — n cos(r£)) dzdr

R R

— sin(tZ)cos(w£)^ J g(z,r)(a sin(zZ) — £ cos(zZ)) (p cos(r£) + n sin(r£)) dzdr

R R

+ sin(tZ) sin(w£) J J g(z,r)(a sin(zZ) — £ cos(zZ)) x (p sin(r£) — n cos(r£)) dzdr.

Hence, in view of (15), we get

( cos(i()cosK) (V'sg(C,£)) - ^

H^p (/ *2 g) (Z,0 = f j / (t,w)

dtdw

cos(i() sin(w^) (Hpa-ng(-sin(i() cos(w^) [Hp-pg(Z,Oj + y sin(i() sin(w^) (H£:pg(Z,0) )

Hpa} - Hpap - Hp-p + H^:) g) (z, 0 / J / (t, w) cos(i() cos(we) dt dw.

This can be put into the form

Hpavp (/ *2 g) (Z,0 = *(g)(Z,0 x J(Z,0, (15)

where ^ = H'p - Ha— - H^—p + H'—p. To complete the proof of the theorem, it is sufficiently enough we show that ^(g)(Z>0 = J (CO-(15)

*(g)(Z,0 = ((h™ - - Hj—p + H£:p) g) (Z,0

= J J (a cos(zZ) + £ sin(zZ ))2n sin(r£) g(z,r) dzdr

R R

(a cos(zZ) - £sin(zZ))2n sin(r£) g(z, r) dz dr.

Hence, it follows that

^(g)(Z,0 = y J 4£n sin(zZ )sin(rf) g(z, r) dzdr = J (Z,0-

RR

Hence the theorem is completely proved. >

2. Distributional H^'p transforms

Denote by T2 the space of smooth functi ons over ^defined on R2 such that

PkK= sup |Dk^(x)| < to,

xeK

/

where the supremum traverses all compact subsets K of R2. Denote by T 2 the conjugate space T2 of distributions of compact sup ports over R2. Then, due to Pa thak [13], T2 defines a norm and the collection pk,K is separating. Hence it defines a Hausddrfit topology on T2. It is easy to notice that the kernel function

K (Z,Z,x,y) = (a cos(Zx) + £ sin(Zx))(p cos(£y) + n sin(fy)) (16)

(1) T2

H^p/(Z,0 = (/(x,y),K(Z,£,x,y)) , (17)

!

/ T 2.

Further simple properties of H'/ can derived from (17) as follows:

/

Theorem 2. Let f (x,y) e T 2, then we have

(i) H/ 3S well-defined;

(ii) H'/ is linear;

(iii) H'/Z 3S one one'

(iv) H'/ 3S anafytic and

dH / d \

C, a = (/Or, y), ^(C, 6 y))

and

dH / d \

C, a = (/Or, y), vYj ■

< Proof of Part (i) follows from (16). To prove Part (ii) Let a e R and H'/f H'/g be the HP'/ transforms of ^d g e T 2, respectively. Then we have

a* (H3f + H^sg) = (a*(g(x, y) + f (x, y)), K(Z, e, x, y)) . By the concept of addition of distributions we get

a* (Hf + Hg) (Z,e) = (a*f (x, y), K(Z,e,x, y)) + (a*g(x, y),K(Z,e,x, y)) .

/

T2

This completes the proof of the linearity axiom of H'/-

To prove that H'/ one-to-one, we assume H'/f = H/g. Then we have (f (x, y), K(Z,e,x, y)) = (g(x, y), K(Z,e,x, y)) . Hence

(f (x, y) — g(x, y),K (Z, e, x, y)) = 0

in the distributional sense. Therefore, it follows that f (x,y) = g(x, y). This proves Part (iii). To prove Part (iv) to refer to [13]. Hence the proof is completed. > The operation *2 can be extended to T'2 as

<f (x, y) *2 g(x, y), <p(x, y)) = (f (x, y), (g(t, w), <p(t + x,y + w))).

We state without proof the following theorem. Theorem 3. Let f (x,y),g(x,y) e T'2. Then we have

H (f (x, y) *2 g(x, y)) (z, e) = j (Z, e)G(Z, e),

where

g(z,e) = (g(t,w),sin(tz)sin(we)), j(z,e) = (f(t,w),cos(tz) cos(we)).

For similar proof see Theorem 1. Hence we delete the details.

3. The quotient SpflCG of Boehmians

The idea of construction of Boehmians was initiated by the concept of regular operators. Construction of Boehmians is similar to that of field of quotients and in some cases, it gives just the field of quotients. The construction of Boehmians consists of the following elements:

(i) A set A;

(ii) A commutative semigroup (B, *) ;

(iii) An operation 0 : A x B ^ A such that for each x e A and ui, u2, e B,

x 0 (ui * U2) = (x 0 Ui) 0 U2;

(iv) A set A C BN satisfying:

(a) If x, y e A, (un) e A x 0 un = y 0 un for all n, then x = y;

(b) If (un), (an) e A, then (un * an) e A (A is the set of all delta sequences). Consider

A = {(x„,u„) : x„ e A, (un) e A, x„ 0 um = xm 0 un, Vm,n e N} .

If (xn, Un), (yn, a„) e A, xn 0 ^m — ym 0 unj Vm, n e N, then we say (x„,u„) ~ (y«,CT„). The relation ~ is an equivalence relation in A. The space of equivalence classes in A is denoted by k(A, (B, *), 0, A). Elements of k(A, (B,*), 0, A) are called Boehmians.

Between ^^d k (A, (B,*), 0, A) there is a canonical embedding expressed as

x 0 Sn

a; —>- as n —» 00.

sn

The operation 0 can be extended to k (A, (B,*), 0, A) x A by

xra , _ XTJ 0 t

W t — •

un un

In k (A, (B, *) , 0, A), two types of convergence:

1) A sequence (hn) e k(A, (B, *), 0, A) is said to be ¿convergent to h e k(A, (B,*), 0, A),

denoted by hn A h as n ^ to, if there exists a delta sequence (un) such that (hn 0 un), (h 0 un) e A, V k, n e N^d (hn 0 uk) ^ (h 0 uk) as n ^ to, in A, for every k e N.

2) A sequence (hn) e k(A, (B,*), 0, A)is said to be A convergent to h e k(A, (B,*), 0, A),

denoted by hn A h as n ^ to, if there exists a (un) e A such that (hn — h) 0 un e A, V n e N, and (hn — h) 0 un ^ 0 as n ^ to in A.

For further details we refer to [1-9] and [11-14].

Let D2 be the Schwartz space of test functions of bounded supports over R^d A2 be the subset of D2 of sequences (0n(x,y)) such that

(i) / I ^n(x,y) dxdy = 1;

R R

(ii) J J |#n(x,y)| dxdy ^ M, M is positive real number;

RR

(iii) supp 0n(x, y) ^ (0, 0) as n ^ to.

(x,y)eR2

Then A2 is a set of delta sequences which correspond to the delta distribution 5(x,y). It is know from literature that S(x, y) = 0 x = 0, y = ^d JR JR S(x, y) dxdy = 1 S(x, y) = S(x)S(y)). It also verified that

/ S(x - a, y - p) f (x, y) dx dy = f (a, p),

where a mid в are constants.

Let (Sn(x, y)) G A2. Then it is easy to see that

(Н™5п(х,у)) (C.O^^asn^oo.

Let B (T2, D2, A2, *2) be the Boehmian space having T2 as a group, D2 as a subgroup of T2, D2 as the set of delta sequences and *2 being the operation on T2 then we introduce the following definitions.

Let f (t,w) G T2, 9(t,w) G D^d (9n(t,w)) G A2. We will usually choose h(CO> fl(CO and en(CO to denote

h(Z,i)=4enj Jf (t,w)sin(t()sin(w£) dtdw, (18)

R R

.(C,fl = // W^a^OM«« dtdw, (19)

RR

en(C0 = ^ J On(t,w)eos(t()eos(w£) dtdw (20)

RR

provided the integrals exist.

Let H|2(CO or H|2 be the фасе of all H^'p transforms of smooth functions f)(CO such that for some f (t,w) G T2 (18) satisfies. By H2(CO or H2 denote the set of transforms of g(CO such that 9(t,w) G D^d (19) satisfies and, similarly, A2(CO or A3 denote the set of all sequences en(Z, О such that for some (9n(t,w)) G A2 where (20) holds.

Remark 1. Let (9n(t,w)) G A2. Then we have

en ((,0 = J J 9n(t,w)eos(t( )eos(w£) dtdw ^ las n ^ <x. (21)

RR

This remark is a straightforward result of (20). Now we are generating the Boehmian space B(Hf,H22, A2, X2).

To this aim, we define an operation between Hp and H2 as

h(CO x2s(CO = h(COs(CO. (22)

We proceed to establish the axioms of the first construction.

Theorem 4. Let h(CO G Hf((,0 and g(CO G H22(CO. Then we have h(CO x2

fl(CO G H2(CO.

< Let h(CO G Hf(CO, fl(CO G H22 (CO. Then h(COs(CO = Hpa} (f *2 9) (CO for every f (t, w) G T2 Mid 9(t, w) G D2 .But since f *2 9 G T2 it follows t hat h(C О x2 й(С О G H2. This completes the proof of the theorem. >

Theorem 5. Let fji(C,€), h2(Z,e) e Hi2(Z,e). Then for all g(Z,e) e H22(Z,e) we have

(hi(Z,e) + h2(C,e)) x2g(Z,e) = hi(Z,e) x2g(Z,e) + h2(CO x2g(Z,e)-

< Let /i(t,w),/2(t,w) e T2 and 0(C,O e D2 be such that

hi(C,^)=4^^ y"/i(i,w)sin(iC)sin(w^) dtdw,

RR RR

g(Z.0 = // W,w)cos(t( )cos(wi) did.;

and

x2

(hi(c,e) + h2(co) x2 g(Z,e) = (hi(c,e) + h2(c,e)) g(Z,e) = hi(Z,e)g(Z,e) + h2(C,e)g(C,e) = hi(C,e) x2g(Z,e) + h2(Z,e) x2g(Z,e).

This completes the proof. >

Theorem 6. Le%(Z,e), h2(Z,e) e H2(Z,e). Then for all g(Z,e) e H22((,£) we have (a*hi(Z,0) x2 g(Z,e) = (hi(Z,e) x2 g(Z,0)•

< Proof of this theorem is analogous to the previous proof. Details are omitted. > Theorem 7. Let hn(Z,e) ^ h(Z,e) ™ H2(Z,e) and g(Z,e) e H2(Z,e); hn(Z,e) x2

g(z,e) ^ h(z,e) x2 g(z,e).

< Let hn(Z,e), h(Z,e) e Hi2(Z,e^d g(Z,e) e H2(Z,e) satisfy for some /n,/ e T2 and d e D2. Then ofcourse /n ^ / as n ^ to. Therefore,

(hn — h)(Z,e) x2 g(Z,e) = (hn — h)(Z,e)g(Z,e)

= fl(Z,e)/ J(/n — /)(t,w)sin(tZ)sin(we) dtdvj ^ 0 as n ^ to.

RR

Hence, (hn — h)(Z,e) x2 g(Z, e) ^ 0 as n ^ to. From which we write,

(hn — h)(Z,e)g(Z,e) = hn(Z,e)g(Z,e) — h(Z,e)g(Z,e) ^ 0 as n^ to.

Thus

hn(Z,e) x2 g(Z,e) ^ h(Z,e) x2 g(Z,e) as n ^to.

This completes the proof of the theorem. >

Theorem 8. Let hn(Z,e) ^ h(Z,e) ^d (en(Z,e)) e A2. Tto hn(Z,e) x2en(Z,e) ^ h(Z,e)-

< Let hn(Z,e), h(Z,e) e Hf(Z,e) and en(Z,e) e A§ satisfy for some /n,/ e T2 and (0n) e A2. Then employing Remark 1 gives

hn(Z,e) x2 en(Z,e) = hn(Z,e)en(Z,e) ^ hn(Z,e) ^ h(Z,e) as n ^ to.

This completes the proof of the Theorem. >

Theorem 9. Let MZ,0), (*n(Z,e)) e A2. Tlien en(Z,e) x2 tn(Z,e) e A2.

< By (22) we have

<n(C,0 x2 rra(C,0 = e^UMCO = fa} {On *2 £n) .

Hence by the fact that On *2 en G A2 it follows that H^ (On *2 en) G A3- Hence the Theorem 9 is proved. >

The Boehmian space B (Hp,H32, A3, x2) is therefore constructed. A typical element in B (Hp, Hp, A3, x2) is of the form . Addition, multiplication by a scalar, convolution and differentation in the space B (Hp, H2, A3, x2) are defined as

hn + dn

- - =

en rn

hn X + dn X en X 2 rn

K

In Ki)n

_ en _ en

k being complex number.

hn 2 dn

- X2 - =

en rn

,2

en x2 rn

A and ¿-convergence are defined as usual for Boehmian spaces.

and Da

tyn

en

d ahn

4. H^'ß of generalized Boehmians

From previous analysis given in this article we define the H^} transform of [j2-] as

H P'n

V

. en _

(23)

where f)n, en has the representation of (18) and (20).

It is clear that G B (Hp, H22, A3, x2) . Let [£] = [f^], then fn *2 em = gm *2 9n. Applying H^ transform and using the convolution theorem yield

hn x rm — dm x en

where f]n, xm, dm, tn have similar representations as in (18) and (20). Therefore ^ ~

Hence = [I11]. Therefore, we have H^} [I2-] = H^} [I2-]. Therefore (23) is well-defined. Following two theorem are straightforward proofs. We prefer we omit details.

Theorem 10. tf^ : B D2, A2, *2) B (Hp, H22, A3, x2) is linear.

Theorem 11. H^ : B (^2, D2, A2, *2) B (Hp, H22, A3, x2) is one-one.

Theorem 12. Hp} : B D2, A2, *2) ->■ B (Hp, H22, A3, x2) is continuous with respect to 5 convergence.

< Let pn ->■ /3 in B (^2,D2,A2,*2) as n ^ co. We show that ->■

in B (Hp,H22, A3, x2) as n —y too.Let pn,p G B (T2,D2, A2,*2), then we can find fn,k,fk G such that /3n = and /3 = and fn>k ->■ fk as n ->■ 00, VA; G

e

n

Therefore H^ [^f-] = where f)^ and are the the corresponding integral equations

of /n,k and 9k, see (18) and (20). Hence, we have

H P'n

Ha,ß

fn,k hn,k —

Ok ek ek

= ß.

The proof is completed. >

Theorem 13. : B (T2, D2, A2, *2) a B (Hf,H22, A2, x2) is continuous with

respect to A convergence.

< Let £n A £ in B (T2, D2, A2, *2) ^n A to. Then there is /n £ T 2 and (9k ((, £)) £

fn X2 9k"

A2 such that

(ßn - ß) x2 9n =

and fn — 0 as n —^ to. Hence

öfc

Hence the theorem is completely proved. >

7« x2 6>fc" "f)n X2 tk

L 6k \ ek

— hn — 0 as n — to.

5. The inverse problem

-1

by-

Let £ B (j^2, Jf22, A I, x2). Then the inverse transform H^'l of Hpal can be defined

-1

Tt™

Ha,ß

"fIn fn

ek ßn_

in the space B (T2, D2, A2, *2) .

-1

Theorem 14. Hp'ß : B (Hf,H22, A2, x2) — B (T2, D2, A2, *2) is a well-deßned and

linear.

< Let = G B (j^2, Jf22, A2, x2) . Then it follows that x2 tm(C,f) =

MCO x2 en(C,^), where

bn((,0 = 4/3^ J J /n(t,w)sm(t()sin(wf) dtdw,

R R

en(Z>0 = J J 9n(t, w) cos(tZ) cos(w£) dtdw,

RR

dm(Z ,0=4^ J J 9m(t, w)sin(t( )sin(wf) dtdw,

and

rn((,0=J J €n(t,w)cos(t()oos(w£) dtdw, €n,0n £ A2, fn, gn £ T2

The meaning of x2 then leads to

Therefore, (22) gives

hn (Z,0tm(C0= dm (C,0en(C0-

H"ß fn *2 em) (Z, i) = H™ (gm *2 e^) (Z, i).

(24)

Since H^ is one-to-one, (24) yields fn *2 6m = gm *2 en. Thus j2- ~ j2-, which then confirms [j2-] = [I2-]. This establishes that our transform is well-defined.

To establish linearity, we assume there are G C, field of complex numbers,

[fe]> IS] then

-1

H a,ß

a* fn

+

a2 gn

-1

-

= H a,ß

a* fn *2 en + gn *2 en

en *2 en

a*it)n x2 t„ + a^dn x2 tn + ~a*2Dn * = a* -f)n 1 * + a2 "fn"

en X rn en rn en rn

This completes the proof of the theorem. >

The author would like to express many thanks to the anonymous referee for his/her corrections and comments on this manuscript.

e

n

n

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Received, December 16, 2015.

Al-Omari Sheideh Khalaf, Prof. Al-Balqa Applied University, Faculty of Engineering Technology, Department of Physics and Applied Sciences Amman, 11134, JORDAN

University of Dammam, Faculty of Science (Girls), Department of Mathematics Dammam 31113, SAUDI ARABIA E-mail: s.k.q.alomariOfet.edu. jo

ОБ ОБОБЩЕНИИ ПРЕОБРАЗОВАНИЙ ФУРЬЕ И ХАРТЛИ ДЛЯ ОДНОГО ФАКТОР-КЛАССА ПОСЛЕДОВАТЕЛЬНОСТЕЙ

Эль-Омари Ш. X.

В работе рассматривается некоторый класс распределений и строятся два пространства Боэхмианов для одного класса интегральных операторов. Устанавливается конволюционная теорема относительно пространств Боэхмианов. Возникающий при этом интегральный оператор корректно определен, линеен и однозначно задается соответствующим Боэхмианом. В работе также подробно рассматривается некоторая обратная задача.

Ключевые слова: интегральное преобразование, преобразование Хартли, преобразование Фурье, фактор пространство.