5 zak transform for boehmians

Page 1

International Journal of Applied Mathematics & Statistical Sciences (IJAMSS) ISSN 2319-3972 Vol. 2, Issue 3, July 2013, 33-40 Š IASET

ZAK TRANSFORM FOR BOEHMIANS VASANT GAIKWAD1 & M. S. CHAUDHARY2 1

Department of Mathematics, Shri Chhatrapati Shivaji College, Omerga, Maharashtra, India 2

Department of Mathematics, Shivaji University, Kolhapur, Maharashtra, India

ABSTRACT It is known that the classical Zak Transform is a linear unitary transformation from L2(R) onto L2 (Q) whose image can be completely characterized. In this paper, we shall construct a Boehmian space B1 containing L2(R) and another Boehmian space B2 containing L2 (Q) and define Zak transform as a continuous linear map of B1 onto B2. We shall also prove that this extended definition is consistent with the classical definition and that there are Boehmains which are not L2 – functions but for which we can define the generalized Zak transform.

KEYWORDS: Boehmians, Convolution, Lebesgue Measurable Functions, Sequence, and Zak Transform Mathematics Subject Classification: 46F30; 42A38; 65R10; 46T30; 46F99.

INTRODUCTION Motivated by the concept of Boehme’s regular operator [4] the theory of Boehmian spaces is developed in the literature [2], [5], [7], [8], and [11]. Further various integral transforms are also studied in the context of Boehmians with their properties [1], [6], [9], and [10]. In this paper we shall extend the concept of Zak transform in the context of Boehmians and study its properties. We shall recall the classical theory of Zak transform in section 2 and we construct suitable Boehmian spaces for our definition of Zak transform in section 3. In section 4 we define the Zak transform and study its properties. Let R, ₾ denote the usual real line, the complex plane respectively and Q = [0, 1) x [0, 1). Let L2(R) and L2 (Q) denote the set of all Lebesgue measurable functions f on R with set of all Lebesgue measurable functions F on Q with the double integral

1 1 |F 0 0

∞ |f −∞

(t, w)|2 dt dw < ∞ respectively, where dt,

dw denotes the Lebesque measures on [o, 1). | f |22 =

∞ |f −∞

đ?‘Ľ |2 dx and | F |22 =

1 1 |F 0 0

(t, w)|2 dt dw.

PRELIMINARIES We first recall the theory of Zak transform from L2 (R) onto L2 (Q). The Zak transform đ?‘?đ?‘Ž [f] of a function f is defined by đ?›ˇ (f) = đ?‘?đ?‘Ž [f] (t, w) = (đ?‘?đ?‘Ž f) (t, w) ≜ đ?‘Ž

∞ đ?‘˜=−∞ f

(at + ak) đ?‘’ −2đ?œ‹ikw ,

where a > 0, t and w are real, f ∈ L2(R). It follows that đ?›ˇ (f) ∈ L2 (Q) and đ?›ˇ is a linear unitary transformation from L2(R) onto L2 (Q). Moreover an inversion formula is also given by

đ?‘Ľ |2 dx < ∞ and the

34

Vasant Gaikwad & M. S. Chaudhary

f (t) =

1 0

đ?‘? f t, w dw, - ∞ < t < ∞,

2đ?œ‹

1 −2đ?œ‹ixt đ?‘’ 0

1

f (2đ?œ‹x) =

1 −2đ?œ‹iwt đ?‘’ 0

1

f (-2đ?œ‹đ?‘¤) =

2đ?œ‹

(Z f) (t, w) dt and

(Z f) (t, x) dt,

where f is Fourier transform of f [12].

BOHEMIAN SPACE We recall the L2 – Boehmian space đ??ľđ?‘…2 , see [1]. As in the context of Boehmians in [3] we take the complex vector space as L2(R) and the commutative semi-group as D(R) with usual convolution defined by ∞ Ď• −∞

(Ď• ∗ Ńą ) (x) =

đ?‘Ľ − t Ńą t dt, x ∈ R,

and the operation from L2(R) x D(R) into L2(R) also as the same convolution functions on R defined above. We take ∆ as the set of all sequences (Ď•n ) whose elements from D(R) satisfying ∆1

∞ Ď• (x) −∞ n

∆2

∞ |Ď•n (x)| −∞

∀n∈N

dx = 1 dx ≤ M

∀ n ∈ N for some M > 0

∆3 S (Ď•n ) → 0 as n →∞, where S (Ď•n ) = sup {|x|: x ∈ R, Ď•n (x) ≠0}. Now we construct a new Boehmian space as follows: Take the vector space as L2 (Q) and the commutative semigroup as D(R). Define ⊗: L2 (Q) x D(R) → L2 (Q) by (F⊗ Ď•) (t, w) =

∞ F −∞

t, w − đ?‘Ľ Ď• x dx.

Lemma 3.1: If F ∈ L2 (Q) and Ď• ∈ D(R) then F ⊗ Ď• ∈ L2 (Q). Proof: If F ∈ L2 (Q) ⇨ F: Q →ℂ such that

1 1 |F 0 0

(t, w)|2 dt dw < ∞,

Where Q = [0, 1) x [0, 1). We have to prove that F ⊗ Ď• ∈ L2 (Q) for Ď• ∈ D(R). Consider

1 1 | 0 0

F ⊗ ϕ (t, w)|2 dt dw = ≤

1 1 ∞ | −∞ F 0 0 1 1 ∞ ( 0 0 −∞

t, w − đ?‘Ľ . Ď• đ?‘Ľ dđ?‘Ľ |2 dt dw

F t, w − đ?‘Ľ

Ď• đ?‘Ľ dđ?‘Ľ )2dt dw

Since R has finite measure with respect to the measure | Ď•(x) |dx, now we can apply Jensen’s inequality and get the last integral dominated by 1 1 ∞ |F 0 0 −∞

=

∞ |Ď• −∞

=

∞ −∞

t, w − đ?‘Ľ |2 | Ď•(x)| dx dt dw

đ?‘Ľ | dx

Ď• đ?‘Ľ

1 1 |F(t, w 0 0

− đ?‘Ľ)|2 dt dw, (By Fubini’s Theorem)

dx . | F |2 = M.| F |2 < ∞,

(since F ∈ L2 (Q) and ∆2)

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Zak Transform for Boehmians

đ?‘‡đ?‘•đ?‘–đ?‘ đ?‘–đ?‘šđ?‘?đ?‘™đ?‘–đ?‘’đ?‘ that, F ⊗ Ď• ∈ L2 (Q). Lemma 3.2: If Ď•1 , Ď•2 ∈ D then Ď•1 ∗ Ď•2 = Ď•2 ∗ Ď•1 ∈ D. For this Boehmian space also we take the ‘Delta Sequences’ as ∆. Lemma 3.3: If F ∈ L2 (Q), (Ď•n ) ∈ ∆, then F ⊗ Ď•n →F as n →∞. Proof: Let ∈ > 0 be given. Using the fact that Cc (Q) is dense in L2 (Q) we can choose H ∈ Cc (Q) such that | F − H| 2 < ∈

(1)

Now | F ⊗ Ď•n − F| 2 ≤ | F ⊗ Ď•n − H ⊗ Ď•n | 2 +| H ⊗ Ď•n − H| 2 + | H − F|

(2)

2

Consider | F ⊗ Ď•n − H ⊗ Ď•n |22 = =

1 1 | 0 0

F ⊗ Ď•n − H ⊗ Ď•n (t, w)|2 dt dw

1 1 ∞ | −∞(F 0 0

≤

1 1 ∞ ( 0 0 −∞

≤

1 1 ∞ 0 0 −∞

t, w − đ?‘Ľ − H t, w − đ?‘Ľ ) Ď•n đ?‘Ľ dđ?‘Ľ|2 dt dw

F t, w − đ?‘Ľ − H t, w − đ?‘Ľ . |Ď•n đ?‘Ľ | dđ?‘Ľ)2 dt dw F t, w − đ?‘Ľ − H t, w − đ?‘Ľ

2

|Ď•n đ?‘Ľ |dx dt dw

(By Jensen’s inequality) ≤

∞ |Ď•n −∞

1 1 | 0 0

đ?‘Ľ | dđ?‘Ľ .

F t, w − đ?‘Ľ − H(t, w − đ?‘Ľ |2 dt dw (By Fubini’s Theorem)

≤

∞ |Ď•n −∞

đ?‘Ľ | dđ?‘Ľ . | F − H| 2 ≤ M | F − H|

2

< M ∈2 → 0 as n →∞.

Therefore, | F ⊗ Ď•n − H ⊗ Ď•n |2 → 0 as n → ∞

(3)

Next | H ⊗ Ď•n − H |22 = =

1 1 | 0 0

H ⊗ Ď•n − H (t, w)|2 dt dw

1 1 ∞ | −∞(H 0 0

≤

1 1 ∞ ( 0 0 −∞

≤

1 1 ∞ 0 0 −∞

t, w − đ?‘Ľ − H t, w ) Ď•n đ?‘Ľ dđ?‘Ľ|2 dt dw,

(By ∆1)

H t, w − đ?‘Ľ − H t, w |Ď•n (x) |dx) 2 dt dw H t, w − đ?‘Ľ − H t, w

2

|ϕn (x) |dx dt dw (By Jensen’s inequality)

≤

∞ |Ď•n −∞

đ?‘Ľ | dđ?‘Ľ .

1 1 | 0 0

H t, w − đ?‘Ľ − H(t, w |2 dt dw (By Fubini’s Theorem)

≤

∞ |Ď•n −∞

đ?‘Ľ | dđ?‘Ľ .

đ?‘˜

| H t, w − đ?‘Ľ − H(t, w) |2 dt dw ∀ n ≼ N

Where K compact subset of Q, with support of H ∠K ∠Q and N1 ∈ N is such that S (Ď•đ?‘› ) < 1 ∀ n ≼ N1.

(4),

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Vasant Gaikwad & M. S. Chaudhary

Since H is uniformly continuous on compact set K. Hence for given ∈ > 0 there exists đ?›ż > 0 such that |H (x, y) – H (u, v)| < ∈ whenever |(x, y) – (u, v)| < đ?›ż. Now choose N2 ∈ N such that S (Ď•n ) < δ Therefore R.H.S. of (4) < M ∈2

∀ n ≼ N2.

dt dw ≤ C ∈2, for some 0 < C < ∞ → 0 as n → ∞.

đ?‘˜

Therefore, | H ⊗ Ď•n − H |→ 0 as n → ∞

(5)

Using (1), (3) and (5) in (2), we get, | F ⊗ Ď•n − F |→ 0 as n → ∞ ⇒ F ⊗ Ď•n →F in L2 (Q) as n →∞. Lemma 3.4: If F1, F2 ∈ L2 (Q) and (Ď•n ) ∈ ∆, such that F1⊗ Ď•n = F2⊗ Ď•n ∀ n ∈ N then F1 = F2 in L2 (Q). Proof: As F1⊗ Ď•n = F2⊗ Ď•n ∀ n ∈ N Letting n →∞, we get F1 = F2. Lemma 3.5: If đ??šđ?‘› → F as n → ∞ in L2 (Q) and Ď• ∈ D(R) then đ??šđ?‘› ⊗ Ď• → F⊗ Ď• as n → ∞ in L 2(Q). Proof: From above || đ??šđ?‘› ⊗ Ď•âˆ’F ⊗ Ď•||22 = ≤ ≤

1 1 ∞ | −∞ đ??šđ?‘› 0 0 1 1 ∞ ( |đ??š 0 0 −∞ đ?‘› 1 1 ∞ 0 0 −∞

t, w − đ?‘Ľ − F t, w − đ?‘Ľ )Ď• đ?‘Ľ dđ?‘Ľ| 2 dt dw t, w − đ?‘Ľ − F t, w − đ?‘Ľ | |Ď•(x) |dx) 2 dt dw 2

đ??šđ?‘› t, w − đ?‘Ľ − F t, w − x

| Ď•(x) |dx dt dw

(By Jensen’s inequality) ≤

∞ |Ď• −∞

đ?‘Ľ | dđ?‘Ľ

≤

∞ |Ď• −∞

đ?‘Ľ | dđ?‘Ľ . ||đ??šđ?‘› − F||22 ≤ C ||đ??šđ?‘› − F||22 → 0 as n → ∞,

1 1 |đ??šđ?‘› 0 0

t, w − đ?‘Ľ − F t, w − đ?‘Ľ |2 dt dw

for some suitable constant C. | đ??šđ?‘› ⊗ Ď• − F ⊗ Ď• |→ 0 as n → ∞ Therefore we get đ??šđ?‘› ⊗ Ď• → F ⊗ Ď• in L2 (Q) as n → ∞. Lemma 3.6: If đ??šđ?‘› → F as n → ∞ in L2 (Q) and (Ď•n ) ∈ ∆ then đ??šđ?‘› ⊗ Ď•n →F as n → ∞ in L2 (Q). Proof: Consider ||đ??šđ?‘› ⊗ Ď•n – F||2 ≤ ||(đ??šđ?‘› – F) ⊗ Ď•n || 2 + ||F⊗ Ď•n – F||2 Now, ||(đ??šđ?‘› − F) ⊗ Ď•n ||22 = = ≤ ≤

1 1 | 0 0

đ??šđ?‘› – F ⊗ Ď•n t, w |2 dt dw

1 1 ∞ | −∞(đ??šđ?‘› 0 0 1 1 ∞ ( 0 0 −∞ 1 1 ∞ 0 0 −∞

t, w − đ?‘Ľ − F t, w − đ?‘Ľ ) Ď•n đ?‘Ľ dđ?‘Ľ| 2 dt dw

đ??šđ?‘› t, w − đ?‘Ľ − F t, w − đ?‘Ľ . |Ď•n đ?‘Ľ | dx) 2 dt dw đ??šđ?‘› t, w − đ?‘Ľ − F t, w − đ?‘Ľ

2

|Ď•n đ?‘Ľ | dx dt

(By Jensen’s inequality) =

∞ |Ď•n −∞

đ?‘Ľ | dđ?‘Ľ

1 1 |đ??šđ?‘› 0 0

t, w − đ?‘Ľ − F t, w − đ?‘Ľ |2 dt dw

(1)

37

Zak Transform for Boehmians ∞ |Ď•n −∞

=

đ?‘Ľ | dđ?‘Ľ . ||đ??šđ?‘› − F||22 ≤ M . ||đ??šđ?‘› − F||22 → 0 as n → ∞ (By ∆2 and đ??šđ?‘› → F as n → ∞)

⇒ ||(đ??šđ?‘› – F) ⊗ Ď•n ||2 → 0 as n → ∞

(2)

Also by Lemma 3.3 we get ||F⊗ Ď•n - F||2 → 0 as n → ∞

(3)

Using (2) and (3) in (1) we get, ||đ??šđ?‘› ⊗ Ď•n - F||2 → 0 as n → ∞ therefore đ??šđ?‘› ⊗ Ď•n → F as n → ∞ in L2 (Q). Now the Boehmian space Bđ?‘„2 can be constructed in a canonical way. The notation of đ?›ż - convergence on Bđ?‘„2 is đ?›ż

defined as follows đ?‘Śđ?‘› → y as n → ∞ if there exists (Ď•k ) ∈ ∆ such that đ?‘Śđ?‘› ⊗ Ď•k and y⊗ Ď•k ∈ L2 (Q) and đ?‘Śđ?‘› ⊗ Ď•k → y ⊗ Ď•k in L2 (Q). We shall use the following lemma; whose proof can be taken from [5]. đ?›ż

Lemma 3.7: đ?‘Śđ?‘› → y as n → ∞ in Bđ?‘„2 if and only if there exists đ??šđ?‘› đ?‘˜ , đ??šđ?‘˜ ∈ L2 (Q), and (Ď•k) ∈ ∆, such that đ?‘Śđ?‘› = đ??šđ?‘˜ đ?œ™đ?‘˜

đ??šđ?‘› đ?‘˜ Ď•k

, y=

and đ??šđ?‘› đ?‘˜ → đ??šđ?‘˜ in L2 (Q) as n →∞, for each k ∈ N. Where

đ??šđ?‘› đ?‘˜ Ď•k

and

đ??šđ?‘˜

denotes equivalence classes containing the quotients of sequences

Ď•k

đ??šđ?‘› đ?‘˜ Ď•k

���

đ??šđ?‘˜ đ?œ™đ?‘˜

đ?‘&#x;đ?‘’đ?‘ đ?‘?đ?‘’đ?‘?đ?‘Ąđ?‘–đ?‘Łđ?‘’đ?‘™đ?‘Ś.

ZAK TRANSFORM Definition 4.1: Define the Zak transform đ?‘?đ?‘Ž : Bđ?‘…2 → Bđ?‘„2 by đ?‘?đ?‘Ž

đ?‘“đ?‘›

≜

Ď•n

the quotient of sequence

ÎŚ( đ?‘“đ?‘› ) Ď•n

, where ÎŚ(đ?‘“đ?‘› ) is a Zak transform of đ?‘“đ?‘› and

ÎŚ( đ?‘“đ?‘› ) Ď•n

đ?‘‘đ?‘’đ?‘›đ?‘œđ?‘Ąđ?‘’ the equivalence class containing

ÎŚ( đ?‘“đ?‘› ) Ď•n

.

Lemma 4.1: If f ∈ L2(R) and Ď• ∈ D(R) then ÎŚ (f∗ Ď• ) (t, w) = (ÎŚf⊗ Ď• ) (t, w) Proof: ÎŚ (f ∗ Ď• ) (t, w) = =

đ?‘Ž

=

∞ [ −∞

=

đ?‘Ž

∞ đ?‘˜=−∞

∞ ∞ đ?‘˜=−∞ −∞ f [a ∞ đ?‘˜=−∞ f

a

∞ ÎŚf −∞

f ∗ Ď• (at + ak) đ?‘’ −2đ?œ‹ikw t − đ?‘Ľ + ak] Ď• (x) dx đ?‘’ −2đ?œ‹ikw

a(t − đ?‘Ľ + ak) đ?‘’ −2đ?œ‹ikw ] Ď• (x) dx (Since Ď• ∈ D(R))

(t-x, w) Ď• (x) dx

= (ÎŚf ⊗ Ď• ) (t, w) Therefore ÎŚ(f ∗ Ď• ) (t, w) = (ÎŚ f ⊗ Ď• ) (t, w) ∀ (t, w)∈ Q. Lemma 4.2: The Zak transform đ?‘?đ?‘Ž : Bđ?‘…2 → Bđ?‘„2 is well defined. Proof: Let

đ?‘“đ?‘› Ď•n

∈ Bđ?‘…2 then đ?‘“đ?‘› ∈ L2(R) and (Ď•n ) ∈ ∆. This implies that ÎŚ(đ?‘“đ?‘› ) ∈ L2 (Q).

We shall show that

ÎŚ( đ?‘“đ?‘› ) Ď•n

is a quotient. Since

đ?‘“đ?‘› Ď•n

∈ Bđ?‘…2 ⇒

đ?‘“đ?‘› Ď•n

is a quotient we have, đ?‘“đ?‘› ∗ Ď•m = đ?‘“đ?‘š ∗ Ď•n ∀ m, n ∈

N. Applying the classical Zak transform on both sides and by Lemma 4.1 we get

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Vasant Gaikwad & M. S. Chaudhary

ÎŚ(đ?‘“đ?‘› )⊗ Ď•m = ÎŚ(đ?‘“đ?‘š )⊗ Ď•n ∀ m, n ∈ N ⇒

ÎŚ( đ?‘“đ?‘› ) Ď•n

is a quotient.

Next we show that the definition of đ?‘?đ?‘Ž is independent of the choice of the representative. đ?‘“đ?‘›

Let

=

Ď•n

�� ѹn

, then we have đ?‘“đ?‘› ∗ Ńąm = đ?‘”đ?‘š ∗ Ď•n ∀ m, n ∈ N.

Again by applying classical Zak transform and by using Lemma 4.1, we get ÎŚ(đ?‘“đ?‘› ) ⊗ Ńąm = ÎŚ(đ?‘”đ?‘š )⊗ Ď•n ∀ m, n ∈ N. Hence the lemma follows. Theorem 4.1: The Zak transform đ?‘?đ?‘Ž : Bđ?‘…2 → Bđ?‘„2 is a linear map. Proof: Let

đ?‘“đ?‘› Ď•n

��

,

Ńąđ?‘›

2 ∈ đ??ľđ?‘…, Îą, β∈ â‚ľ Now đ?‘?đ?‘Ž (âˆ?

=

ÎŚ(âˆ? đ?‘“đ?‘› âˆ—Ńąn +βđ?‘”đ?‘› âˆ—Ď• n Ď• n âˆ—Ńąn

âˆ?ÎŚ( đ?‘“đ?‘› )

=

Ď•n

+

ÎŚ( đ?‘“đ?‘› )

=�

= � Za

�� ѹn

âˆ?đ?‘“đ?‘› âˆ—Ńąn +βđ?‘”đ?‘› âˆ—Ď• n

) =đ?‘?đ?‘Ž

Ď• n âˆ—Ńąn

,

(By Lemma 4.1)

βΌ(đ?‘”đ?‘› ) Ńąn

+β

Ď•n

+β

, (By definition)

âˆ?ÎŚ( đ?‘“đ?‘› )⊗ѹn +βΌ(đ?‘”đ?‘› )⊗ϕ n Ď• n âˆ—Ńąn

=

đ?‘“đ?‘› Ď•n

đ?‘“đ?‘›

Ό(�� ) ѹn

+ β Za

Ď•n

�� ѹn

In this proof we have used the fact that ÎŚ is linear wherever it is required. Theorem 4.2: The Zak transform đ?‘?đ?‘Ž :B2đ?‘… → B2đ?‘„ is one-one. Proof: Let

đ?‘“đ?‘› Ď•n

,

�� ѹn

đ?‘“đ?‘›

∈ B2đ?‘… . If đ?‘?đ?‘Ž

Ď•n

= Za

��

then we have

Ńąn

ÎŚ( đ?‘“đ?‘› ) Ď•n

=

Ό(�� ) ѹn

and hence we get

ÎŚ(đ?‘“đ?‘› )⊗ Ńąm = ÎŚ(đ?‘”đ?‘š ) ⊗ Ď•n ∀ m, n ∈ N. Using Lemma 4.1 We get ÎŚ (đ?‘“đ?‘› ∗ Ńąm ) = ÎŚ(đ?‘”đ?‘š ∗ Ď•n ) Since, ÎŚ is one-to-one we get, đ?‘“đ?‘› ∗ Ńąm = đ?‘”đ?‘š ∗ Ď•n

∀ m, n ∈ N. ∀ m, n ∈ N

(Since Zak transform is isometric from L2(R) onto L2 (Q)). This implies that Theorem 4.3: Proof: Since

đ??šđ?‘› Ď•n đ??šđ?‘› Ď•n

đ?‘“đ?‘› Ď•n đ?‘“đ?‘›

= đ?‘?đ?‘Ž = đ?‘?đ?‘Ž

=

đ?‘“đ?‘›

. Therefore, đ?‘?đ?‘Ž is one-to-one.

if and only if Fn ∈ L2 (Q) ∀ n ∈ N, in particular đ?‘?đ?‘Ž : B2đ?‘… → B2đ?‘„ is onto.

Ď•n

=

Ď•n

�� ѹn

ÎŚ( đ?‘“đ?‘› ) Ď•n

Conversely if there exists

đ??šđ?‘› Ď•n

, this implies that Fn ∈ L2 (Q) ∀ n ∈ N. ∈ B2đ?‘„ such that đ??šđ?‘› ∈ L2 (Q) , then we can choose đ?‘“đ?‘› ∈ L2(R) Such that ÎŚ(đ?‘“đ?‘› ) = Fn

∀ n ∈ N. First we show that

đ?‘“đ?‘› Ď•n

is a quotient. Since

đ??šđ?‘› Ď•n

is a quotient we have

đ??šđ?‘› ⊗ Ď•m = đ??šđ?‘š ⊗ Ď•n ∀ đ?‘š, n ∈ N i.e. ÎŚ(đ?‘“đ?‘› )⊗ Ď•m = ÎŚ(đ?‘“đ?‘š )⊗ Ď•n ∀ m, n ∈ N.

39

Zak Transform for Boehmians

By Lemma 4.1, we have ÎŚ ( đ?‘“đ?‘› ∗ Ď•m ) = ÎŚ( đ?‘“đ?‘š ∗ Ď•n ) ∀ m, n ∈ N Since ÎŚ is one-one we get, đ?‘“đ?‘› ∗ Ď•m = đ?‘“đ?‘š ∗ Ď•n ∀ m, n ∈ N⇒ Therefore

đ?‘“đ?‘› Ď•n

đ?‘“đ?‘›

∈ B2đ?‘… , Such that đ?‘?đ?‘Ž

Ď•n

=

ÎŚ( đ?‘“đ?‘› ) Ď•n

=

đ?‘“đ?‘› Ď•n

is a quotient.

Fn Ď•n

Hence the theorem follows. Theorem 4.4: The Zak transform đ?‘?đ?‘Ž : B2đ?‘… â&#x;ś B2đ?‘„ is consistent with ÎŚ : L2(R) â&#x;ś L2 (Q). Proof: Let f ∈ L2(R). Then by Lemma 4.1, we have đ?‘?đ?‘Ž

đ?‘“ ∗ Ď•n Ď•n

=

ÎŚ f ∗ Ď•n Ď•n

=

Ό(f)⊗ϕ n ) ϕn

Hence the theorem proof is completed. Theorem 4.5: The Zak transform đ?‘?đ?‘Ž : B2đ?‘… â&#x;ś B2đ?‘„ is consistent with respect to the đ?›ż - convergence. đ?›ż

Proof: Let đ?‘Ľđ?‘› → x as n â&#x;ś ∞ in B2đ?‘… , then by Lemma 3.7 there exist, đ?‘“đ?‘› đ?‘˜ , đ?‘“đ?‘˜ ∈ L2(R), and (Ď•n ) ∈ ∆ such that đ?‘Ľđ?‘› = x=

đ?‘“đ?‘˜ đ?œ™đ?‘˜

đ?‘“đ?‘› đ?‘˜ Ď•k

,

and đ?‘“đ?‘› đ?‘˜ → đ?‘“đ?‘˜ as n → ∞ in L2(R) for each k ∈ N. Since ÎŚ is continuous from L2(R) â&#x;ś L2 (Q), this implies ÎŚ(đ?‘“đ?‘› đ?‘˜ )

â&#x;ś ÎŚ(đ?‘“đ?‘˜ ) as n â&#x;ś ∞ in L2 (Q) for each k ∈ N. This shows that

ÎŚ(đ?‘“đ?‘› đ?‘˜ ) Ď•k

�

→

ÎŚ(đ?‘“ đ?‘˜ ) Ď•k

as n â&#x;ś ∞ in B2đ?‘„ .

Therefore đ?‘?đ?‘Ž is consistent with respect to the δ – convergence.

A COMPARATIVE STUDY We know that L2(R) is properly contained in B2đ?‘… (for example compactly supported distributions belong to B2đ?‘… but do not represent L2 – function) and we have proved that đ?‘?đ?‘Ž is consistent with ÎŚ. Thus our theory extends the classical Zak transform on L2(R) to a Boehmian space as a continuous linear map of B2đ?‘… into B2đ?‘… whose image can be completely characterized.

CONCLUSIONS In this paper, we extend the definition of classical Zak transform to larger class and this is consistent with the classical definition. There are some functions which are not Zak transformable in classical sense but for which we can obtain the generalized Zak transform.

ACKNOWLEDGEMENTS The authors thanks to Shivaji University, Kolhapur to provide their Library facility for the research work.

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