On generalization of Fourier and Hartley transforms for some quotient class of sequences

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,e f (Z,^) = 21nj j f (x,y)(a cos Zx + в sin Zx)(p cos py + n sin py) dxdy, (1) RR where (Z,p) E R2. R2 = R x R are the transferm variables and a. в- P- П are arbitrary constants.

A inversion formula of the cited integral be can easily recovered from (1) giving f (X,y') = 2n/j Н^в (Z,p)(a cos Zx + в sin Zx)(p cos py + n sin py) dZ dp. (2) RR

In a special ease, for a = в = 1 P = П = 1 , the Integra 1 transform (1) and the inversion formula. (2) are respectively reduced to tlie double Hartley transform Ad pair (see [10])

Ad (Z’P) = 27r/ j f (x,y)(cos Zx + sin Zx)(cos py + sin py) dxdy          (3)

and

f (х,у) = 2т У j Ad(Z,^Z)(cos Zx + sin Zx)(cos ^y + sin ^y) dZ d^.

RR

Further, with simple computations, the kernel function

(cos Zx + sin Zx)(cos £y + sin ^y) = cas Zx cas ^y

inside the integral signs can be written as

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

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

AddU) = 2П j jf (x.y⁾(cos«x - Cy>⁺sm_Kx + ^y)) dxdy

and

^f (^y^^n У J A ^{d (Z,z)(cos(Zx -} ^y^)+sin(Zx ⁺ £y)) d^Zd^Z, RR

respectively.

By sotting а = 1 ,в = i,P = 1 aiid n = i, wo derive the double Fourier transform F ^d pair.

F d^)^ j j f (x,y)(cos Zx + i sin £x)(cos Z y + i sin ^y) dx dy

RR

and

f (х,У) = 2т У У Fd (Z,^)(cos Zx + i sin £x)(cos Zy + i sin ^y) dZ d^.

RR

By factoring A^A(Z,0 into even and оdd components. A^d(Z,0 = Ed(Z,0 + Od(Z,O- whore

Ed ^{( Z,}^ ⁾ = 2nj jx ^(x,y^)cos(Zx ^- ^ ^dxdy

(ID

and

Od^(Z,^⁾ = 2n У У ^f(x,y^)sin(Zx ^- €y) dxdy

RR

we get

Fd(Z,e) = Ed(Z,0 - iOd(Z,0 and A^a(Z,0 = Re Fd (Z,€) + Im F^d (Z,a      (13)

Denote by L ² the Lebesgue space of iiitegrable functions or"er R² ; then the com'olution product of f ( x,y ) aiid g ( x,y ) iii L² is defined by

(f *2

g )( x, y ) =

RR

f (t, w)g(x — t, y — w) dt dw.

We state and prove the following theorem.

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

Hpe(f *2 g)(Z,^) = J(Z,£)G(Z,a where J(Z,0 ai 1(-I G(Z,O are given ley the integrals

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

G(Z,O = j j 4вп sin(Zz) sin(r£)g(z, r) dzdr.

RR

C Let f (x,y),g(x,y) E L 2 . Then by using the convolution product formula we have

H^pe ( f *2 g ) (Z,£) = y j ( f *2 g ) (x,y)(a cos(xZ) + вsin(xZ))

RR

x (p cos(y^) + n sin(y^)) dx dy

= УУ (У I f (t,w)g(x - t,y - w) dtdw^ RR RR

x (acos(xZ) + вsin(xZ))(pcos(y^) + n sin(y^)) dxdy.

Change of snriables x — t = z and y — w = r imply dx = dz and dy = dr and hence

H O,в ( f *2 g ) ^{( Z,z )} = j j ^{f ( t,w )} j j g^{( z,r )( a cos Z(z +1) + в sin Z(z +} t))

RR

RR

x (p cos Z(r + w) + П sin(r + w)^) dz dr dt dw.

By aid of idie fads cos(a+в) = cos a cos в - sin a sin в aiid sin(a+в) = sin a cos в - sin в cos a and using simple computations we get

HOPe (^f *2 g )^(Z , ^z) = У У ^{f (t, w) a(t, w) dw} dt,

RR where

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

RR

^^^^^^^^.

- cos(tZ )sin(w^) У У g(z,r)(a cos(zZ)+ в sin(zZ)) x (p sin(r^) — n cos(r^)) dzdr

RR

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

RR

+ sin(tZ)sin(w^) У У g(z,r)(a sin(zZ) — в cos(zZ)) x (p sin(r^) — n cos(r^)) dzdr.

RR

Hence, in view of (15). we gel, h$ (f *2 g) «,<) = / / f (t,w)

R R

cos(tz)cos(w£) (H^g«,£)) - cos(tZ)sin(w£) (h^ g(Z,£)) — sin(tZ)cos(w£) (# £— g(Z,€)) + sin(tZ)sin(w£) (H ’-e g(Z,€))

dt dw

= ((H fZ - Hi’»' — HP — ⁺ H S-Z ) g)К-£) / У f (t w) ^cos(t ^z ) cos'wo dtdw-

RR

This can be put into the form h$ (f *2 g) (Z,^) = ^(g)(Z,^) x j(Z,a                    (15)

where Ф = Hp’n — Hp’ A1 — Hp’n R + Hp’ nR. To complete the proof of the theorem, it is а , в а, в а,—в а ,—в sufficiently enough we show that ^(g)(Z,€) = J(Z, €)-

By aid of (15) we derive

»(»)«•« = ( (я^ — HS " — иа — в + HP — в) g) «•« = У У (a cos(zZ)+ в sin(zZ))2n sin(rZ) g(z,r) dzdr RR

— j J (a cos(zZ) — в sin(zZ ))2n sin(r^) g(z, r) dz dr. RR

Hence, it follows that

ф(д)«,<) = j j

4вп sin(zZ) sin(r^) g(z, r) dz dr = J(Z, 0-

B

• 2.    Distributional Hа’g transforms

Denote l>y T ² the space of smoc)tli functions over ^ defined on R² such that

Pk,K(p) = sup \Dk^(x)| < to, x∈K where l,lie supremum traverses all compact subsets K оf R2. Denote l>v T 2 the conjugate space T2 of distributions of compact supports over R2. Then, due I,о Pathak [13], T2 defines a norm and the collection pk,K is separating. Hence it defiiies a Hausdorff topology on T2.

It is easy to notice that the kernel function

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

of (1) is a memlier of T2 and hence, leads I,о the generalized definition

H^pf(Z,0 = h f(x,y),K ( Z,^,x,y )i ,                        (17)

where f is an arlaitrary distriliution in T ² .

Pin

Further simple properties of H^’? can be de rived from (17) as follows:

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

• (i)    Hp? ^1S well-defined:

• (ii)    HOPp ^1S linear:

• (iii)    Hope is one io one:

• (iv)    [ _{is analy}i_{ic and}

dH^        I d        \

-gf f (Z, z) = f (x, y), ^k «, z, x, y))

and дНав        / d \ dzf (Z,Z) = у (x, y), dZK(z,z,x,yy -

C Proof*>f Pan (i) follows from (16). To pros-e Part (ii) Lel a E R arid [f [вg be the Hp? transforms of f and g E T 2, respectively. Then we have

^a* ([ р^{f +} [p g) = h^a*⁽g^(x,y) ^{+ f(x,}y^)),K(C^x^i -

By the concept of addition of distributions we get

^a* ( [р^{f + Н}[ g)^(z,^z) = h^a*^{f (x},y^),K(z,^z,x,y)i⁺ h^a*g^(x,y^{),K (z},^z,x,y)i -

Hence, scalar multiplication in the space T ² implies

“• (^HSf + Hasg) «■ z) = a*^НРвf + a' C? g.

Tins completes the proof of the linearity axiom of Hy|.

To prove that Hy? n one-to-one. we assume H[ff = Hy|g. Then we have hf(x,y),K(z,z,x,y)i = (g(x,y),K(z,z,x,y)i . Hence hf(x,y) - g(x,y),K(Z,z,x,y)i = о in the distributional sense. Therefore, it follows that f(x,y) = g(x,y). This proves Part (iii).

To prove Part (iv) we refer to [13]. Hence the proc if is completed. B

The operation *² can be ex tended to T 0² as

(^f(x, y) *² g⁽x, y), v^(x, y)^ = hf^(x, y), hg^(t, ^w),v^{(t + x,}y ^{+ w)}ii -

We state without proof the following theorem.

Theorem 3. Let f (x,y),g(x, y) E T 0². Then we have

HpP (f (x,y) *2 g(x,y))(z,z) = J(z,z)G(z,z), where

G(z,z) = 4вП hg(t,w), sin(tZ)sin(wz)i, J(z,z) = hf (t,w), cos(tz)cos(wz)i .

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

• 3.    The quotient space 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 commulalive semigroup ( B , * );

• (iii)    An op<'ration © : A x B 4 A such that. for each x E A aiid u 1 ,u ₂ , E B ,

x © (u i * U2) = (x © U1) © U2;

• (iv)    A set A C B N satisfying:

• (a)    If x,y E A. (u _n ) E A. x © un = y © un for all n, then x = y;

• (b)    If (u n ), (a n ) E A, then (u n * a n ) E A (A is the set of all delta sequences).

Consider

A = { (x n ,u n ) : xn E A , (u n ) E A, x _n © u m = x _m © un, V m,n E N} .

If ( x n л), ( y nTO ^E A. xn 0^am = ^ym © v,„ ^V m,n ^E N , then ,re ear ( x „ ,« „) ~ ( y nTO . The relation ~ is an equivaleiice relation in A . The space of equivalence classes in A is denoted by k(A, ( B , * ), © , A). Elemerits of k(A, ( B , * ), © , A) are called Boehmians.

Between A and к (A, (B,*), ©, A) there is a canonical embedding expressed as x © Sn x 4----- as n 4 to.

s _n

The operation © can be ex tended 1.0 к (A, (B,*), ©, A) x A by xn © t = xn © t υn          υn .

In к ( A , ( B , * ) , © , A) , two types of convergence:

• 1)    A sequence (h _n ) E k ( A, ( B , * ), © , A) is said to be 5 convergent. 1.0 h E k ( A, ( B , * ), © , A), denoted by h _n 4 h as n 4 to , if there exists a delta sequence ( u _n) such that ( h _n © u _n) , ( h © u _n) E A , V k, n E N , arid ( h _n © u k) 4 ( h © u k) as n 4 to , i 11 A , for every k E N.

• 2)    A sequence ( h _n) E k ( A, ( B , *) , © , A)is said 1.0 be A convey©nt 1.0 h E k ( A, (B , *) , © , A) , denoted by h _n 4 h as n 4 to , if there exists a ( u _n) E A such that ( h _n - h ) © u _n E A , V n E N , arid ( h n — h ) © u _n 4 Da s n 4 to ii 1 A .

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

Let D ² be the Schwartz space of test finictioiis of bounded supports over R² aiid A² be the subset of D ² of seqiteiices ( 0 _n( x,y )) such that

• (i)    / / ^ n (x,y) dxdy = 1;

R«

• (ii)    J J | 0 _n (x,y) | dxdy 6 M, M is positive real niimlier:

, .RR .....

• (iii)    supp 9 _n (x,y) 4 (D, D) as n 4 to .

( x,y ) ∈ R²

Then A2 is a sei, of della sequences which corresporid I,о the della distribution 5(x,y) II is know from literature that 5(x,y = 0, x = 0, y = 0 and RR RR 5(x,y dxdy = 1 (^ 5(x,y = h(x)h(y)). Il, also verified that j j 6(x - a, y - в) f (x, y) dx dy = f (a, в), RR

αβ

Lei, (h_n(x,y)) E A². Then il is easy I,о see that

(^н а,в Mx^)) ^{( z,}^' ⁾ ^ IP^as n ' ^ .

Lei, B (T 2 , D ² , A ² , * 2) be the Boehmian space having T ² as a group. D ² as a subgroup of T 2. D ² as the sei, of dell, .a sequences and *² being the о peral, ion on T ² then we introduce the following definitions.

Lei, f (t,w) E T 2. 9(t,w) E D ² arid (9n(t,w)) E A ² . We will usitallv choose h (Z, €)• g (Z, €) and e _n(Z,£) to denote

h(Z,^)=4en У У f (t, w)sin(tZ)sin(w£) dtdw,

RR

g ^(Z,z) = У У 9(t, w)cos(tZ)cos(w£) dtdw,

RR en (Z,£) = У У 9n(t, w) cos(tZ)cos(w£) dtdw

RR

provided the integrals exist.

Lei, H1² (Z,^) оr H1² be the space of all H™ transforms of siiiooth functions h (Z, €) such that for some f(t,w) E T ² (18) salisfies. By H₂² (Z, €) оr H₂² denote the set of transforms of g«,£) such that 9(t,w) E D ² and (19) satisfies and, similarly, A3«,£) or A3 denote the set of all sequences e _n (Z,0 such that for some (9_n(t,w)) E A² where (20) holds.

Remark 1. Let (9n(t,w)) E A2. Then we have en (Z,€) =j j 9n(t, w) cos(tZ)cos(w£) dtdw ^ 1asn ^ to.

RR

Tins remark is a straightforward result of (20). Now we are generating I,lie Boehmiaii space B (H1 ² , H2 ² , ^A 2, x² ) .

H ₁²     H ₂ ²

h (Z, £) x² g (Z, €) = h (Z, ^z ) g (Z, ^z ).

We proceed to establish the axioms of the first construction.

Theorem 4. Let h (Z,£) E H2(Z, €) and g (Z,^) E H22 (Z,^)- Then we have h (Z,£) x^{2 д (}“)^E H²’«s  ,             .                      .......

C L e, hK. 0 E H1²K, o. gK,q ^E H2 ( С, _{ ⁾ . тг„, h(u e ⁾ g_K, _{⁾ = H- (f *2 ⁹ « О Ю, cr'erv f ( t,w ) E T ² aiid 9 ( t, w ) E D ² . But since f *² 9 E T ² it followvs I,hat h ( Z, € ) x² g ( Z, € ) E H 1² . Tins completes the t>roof of the theorem. B

Theorem 5. Lei h 1(Z,€), h 2(Z,e) € H i ² (Z,^)- Then for all g (Z, €) € H 2 ² (Z,e) ^m' have ( h i(Z,e) + h 2(Z,e)) x² g (Z,€) = h i(Z,e) x² g (Z,e) + h 2(Z,e) x² g (Z,€)-

C Lel fi(t,w),f2(t, w) € T2 aiid 9(Z,e) € D2 be such that hi(Z,e) = 4вп

У У fi(t,w)sin(tZ)sin(we)

dt dw,

h 2⁽С£)=^{4 в}пУ yf 2⁽t, ^{w )}s^{in( tZ} )

sin(we) dt dw

RR and g-T

RR

9(t, w) cos(tZ) cos(we) dt dw;

then using the definition of x ²

( h i(Z,e) + h 2 (Z,e)) x² g (Z,e) = ( h i(Z, e) + h 2 (Z,e)) g (Z, e)

= h i(z,e) g (z,e) + h 2 (Z,e) g (Z,e) = h i(Z,e) x² g (z,e) + h 2 (Z,e) x² g« ,e)-

B

Theorem 6. Let h _i(Z, e), h ₂(Z, e) € H i2(Z,e). Then for all g (Z,e) € H 22(Z,e) ^we have (^itua ^x2 <,0 = ^a (ы«) ^x2 <■«)

CB

Theorem 7. Let h , (Z,€) ^ h (Z,€) in H i ² (Z,e) aiid g(Z,e) € H 2 ² (Z, e); then h , (Z,e) x² g^ )^H hxx) ^x2^gKX^). ..........

C Ler h n( e,^ ) , h«, ^{e ) €} HMO _M rd _e(«) ^€ H^.o satisfy for some ^fn,^{f €} T ² and 9 € D ². Then c)fcourse f _n ^ f as n ^ to. Therefore.

( h n - h )(Z, e) x² g (Z, e) = ( h n - h )(Z, e) g (Z, e)

= g (Z,e) У Iffn

— f )(t,w)sin(tZ)sin(we) dtdw ^ 0 as

n ^ to .

Hence, (hn ^{— h )}K, ^{e ) x} 2 ^gK, ^{e )} ^ ⁰ „. n ^ ^to . From which we write,

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

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

This completes the proof of the theorem. B

Theorem 8. Let h n (Z, e) ^ h (Z, e) and ( e n (Z, e)) € A3. Then h n (Z, e) x² e n (Z, e) ^ h (Z, e)-

C L=t I,,,(«С h(Z,e) € Hi2(Z.e) ar rd e„ (Z,e) € A3 satisfy for some fn,f € T2 and (9n) € 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) a a n ^ to.

This completes the proof of the Theorem. B

Theorem 9. Let ( e n (Z,e)), ( г , (Z,e)) € A3. Then e n (Z,e) x² r n (Z,e) € A2.

C В у (22) we have en(Z, €) x2 rn(Z, £) = en(Z, £K(Z, €) = j^p (^ *2 £n) •

Hence by the fact that 6_n *2 e_n E A² it foliows that Н^р (^ n *2 ^e n) E A3. Hence the Theorem 9 is proved. B

The Boehmian space B ⁽H², H₂², A3, x ²⁾ is therefore constructed.

A typical element, in B (H2,H22, A3, x2) is of the form [hn]. Addition. multiplication bv a scalar, convolution and differentation in the space B (H2, H22, A3, x2) are defined as hn + dn en      rn

h n x ² r n + d n x ² en ' e n x ² r n

κ

h n e _n

κ h n e _n

κ.

h n e _n

x ²

d n

rn

h n ^{x 2} d n e n x ² r n

D ^α

h n e _n

D ^α h n e _n

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

• 4.    Hpp of generalized Boehmians

From previous analysis given in this article we define the Hpp transform of [fn] as

'гтРаТ f n

^H α,β n

h n e _n

where hn. e_n lias the represeiitation of (18) and (20).

It is clear that [ hn ] E B (H2, H2, A3, x2) .Let [ £ ] = [ gn phen/,, *2 Cm = gm *2 »„. Hα,,ηβ hn x rm = dm

x ² e n ,

where hn. r_m. d_m. e _n Have similar represeiitations as in (18) arid (20) . Tlierefore hn ~ dn. Hence [hn] = [dn]. Therefore. we have H? [ fn ] = H^ [ gn ]. Tlierefore (23) is well-defined.

en       rn                               a,P 9n        a,p tn

Following two theorem are straightforward proofs. We prefer we omit details.

Theorem 10. £§ : B ⁽ T², D ², A², * ^{2 )} ^ B ⁽ H², H₂², A3, x ^{2 )} is linear.

Theorem 11. Ha’p : B (T², D ², A², * ²) ^ B (H(², H2², A2, x ²) is one-one.

Theorem 12. H ap : B ⁽ T² , D² , A² , * ^{2 )} ^ B ⁽ H² , H₂² , A2 , x ^{2 )} is continuous with respect to d convergence.

C Let e n ^ в iii B ⁽ T ² , D ² , A² , *²⁾ as n ^ to . We slit>w that Ha|e n ^ Н0|в in B (H(², H2², A3, x ²) as n ^ to . Let e n ,e E B (T², D ², A², * ²), then we can find f n,k ,f k E T² such that e n = [f nk ] aiid в = [ fk ] aiid fn,k ^ f k as n ^ to , V k E N .

Therefore Н^рв ^[ f nk ^] = ^[h—^] where hn,_k and e k are the the corresponding integral equations of f_n,_k and 9k, see (18) a nd (20). Hence, we have

Hρ,η

f n,k θ k

h n,k e k

h k e k

= в-

B

Theorem 13. H pe : B (T ² , D2, A ² , *2 ) 4 B (H 2 , H 22 , A 3 , x² ) ^1S continuous with respect to A convergence.

C Lel в _п 4 в iii B ⁽T ² , D ² , A ² , *^{2 )} , as n 4 to . Then there is f _n G T ² arid (9k(Z, £)) 6 A ² such that

(вп - в) x2 9 = Ifn-X^k and fn 4 0 as n 4 to. Hence

H$ ((вп

^^^^^^^^r

в) X² 9k ) = H$

fn X² 9k ' θk

h n X² ek ' e k

' h_n 4 0 as n 4 to.

B

• 5.    The inverse problem

  -1

Let ^[hn^] G B ⁽H1², H2², A2, x ²^ Then the inverse transform ^j оf Н^в can be defined by

-1

ρ,η ^H α,β

hn      fn ek       θn in the space B (T2, D2, A2, *2) .

-1

Theorem 14. Н£ в : B ⁽H 1² , H ₂ ² , A 3 , x ^{2 )} 4 B ⁽T2, D2, A ² , *^{2 )} is a well-defined and linear.

C Let [hn] = [dn] G B (H12, H22, A2, x2) . Then it foHows that hn(Z,^) x2 rm(Z,^) = dm(Z,€) x2 en(Z,O- ^here hn(Z,€) =4вп J" Ifn(t, w) sin(tZ) sin(w^) dt dw, en(Z,^) = У У 9n(t, w) cos(tZ)cos(w£) dtdw,

RR dm(z,e) = 4вnj j gm(t, w) sin(tZ)sin(w^) dtdw, and rn(Z,£) = y yen(t,w)cos(tZ)cos(w£) dtdw, en,9n G A2, fn,gn G T2.

RR

The meaning of x2 then leads I,о

h n ((ЛУДСО = d m ((ЛМСО.

Therefore. (22) gives

И™₀ U- * 2 ⁹m) «’ 0 = Н^Ра’в (g_m * 2 6n) (Z, £)•

Since ИОР^в is ^one" to-one, (24) yieIds /_n * 2 9 _m = gm *² e _n . Th us ^ n ~ g n , which then confirms

[fn] = [gn]• This establishes that our transform is well-defined. Vn        ^n

To establish linearity, we assume

^[f ], ^[g n ^] E B (T 2 , D ² , A ² , * 2 ). then

there are «*,«2 E C, field of complex numbers,

- 1

H,n'

Г«1/nl

9n

+

«2 gn ^n

« 1 h n ^{x r}n ⁺ «2 ^dn ^x e n

e n x² r n

- 1

17|{

= ^Иа,в

" «1h n’

e _n

"«1 / n * 2 €n + « 2 gn * 2 9n

9n * 2 ^n

+

«2 d n r n

∗

= «1

h n

en

∗

+ «2

d _n r n

.

B

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