We investigate the Krätzel transform on certain class of generalized functions. We propose operations that lead to the construction of desired spaces of generalized functions. The Krätzel transform is extended and some of its properties are obtained.

1. Introduction

In recent years, integral transforms of Bohemian have comprised an active area of research. Several integral transforms are extended to various spaces of Bohemian, especially, that permit a factorization property of Fourier convolution type.

On the other hand, several integral transforms that have not permitted a factorization property of Fourier convolution type are also extended to various spaces of Bohemian. In the sequence of these integral transforms, the Krätzel transform [1] where , , is extended to certain space of ultra-Bohemian, denoted by and , , respectively.

Another form of the Krätzel transform was initially introduced in [2], and defined as a generalization of the Laplace transform in [3] as where for and . The transform was extended to generalized functions in [3] and to distributions in [4]. By , we denote the space of equivalence classes of measurable functions such that where two measurable functions are equivalent when they are equal to a.e.

For , the -norm is defined as The Banach space of Lebesgue measurable functions is defined by [3, page 446] Due to [3, Proposition 2.1], we state the following theorem.

Theorem 1. Let , , , and and then is a continuous linear mapping from into .

Note that we always assume that the hypothesis of Theorem 1 is satisfied. By we denote the Mellin-type convolution product of first order defined by [5] By , or simply , denote the Schwartz space of test functions of compact support defined on . Then we have the following definition.

Definition 2. Let . For and , we define the operation given by

Then we easily see that the operations in (8) and (9) are very basic for the next construction of the desired Bohemian spaces.

Theorem 3. Let and , then .

Proof. Let . For and we, by (3) and (8), get that A change of variables in (10) implies Hence, by (9), we have This completes the proof of theorem.

2. General Bohemian

The structure necessary for the construction of Bohemian consists of the following:(1)a group and a commutative semigroup ;(2)a operation such that , for all and ;(3)a collection such that(i)for all , we have ,(ii)If , then where , and .

Then the set that satisfies (i) and (ii) is called the set of all delta sequences.

Let . Then we say if there are , such that , for all . The relation is an equivalence relation in . The space of equivalence classes in is called the space of Bohemian and denoted by . Each element of is called Bohemian. Then the convergence in is defined as follows.(1) is said to be -convergent to , denoted by as , if a delta sequence such that , and as , for all .(2) is said to be -convergent to , denoted by as , if a such that , for all , and as .The following theorem is equivalent to the statement of -convergence.

Theorem 4. as if and only if there are and such that and, for all , as . See [617] for more details.

3. The Spaces and

In this section we construct the space and the space of Bohemian and give their properties.

At the first step, we prove the following connecting theorem.

Theorem 5. Let and ; then, .

Proof. Let and ; then, applying the Fubinitz theorem yields The change of variables implies and further Therefore, Hence, (15) implies This completes the proof of the theorem.

Next, forthcoming theorems prove the existence of the space .

Theorem 6. Let and and then .

Proof. For each and , we have By Jensen’s theorem, we write where , is a compact subset containing the support of . Hence, from (18), we get where is certain positive real number. This completes the proof of the theorem.

Theorem 7. Let and and then(i);(ii);(iii)let and as then for , and we have as ;(iv).

Proof of Part (i), (ii), and (iii) follows from properties of integral operators . Similarly, the proof of Part (iv) is straightforward from the properties of proved in [5].

Following theorem is straightforward, and detailed proof is omitted.

Theorem 8. Let and then as .

Proof. Since is a dense subspace of then it is a dense subspace of . Hence, there can be found such that for . Also, from (18), we have Let then; and hence uniformly continuous on .
Therefore, there is such that Thus, using (22), we get By (22), as implies that there can be found , such that , for all . Further, the fact that the function is of compact support thus this implies that , where is a compact subset of . Thus, from (23), we write Now, we have On using (20), (21) and (24) prove that Thus the theorem is proved. Then the Bohemian space is therefore constructed. The operations such as sum and multiplication by a scalar of two Bohemian in are defined in a natural way where α  is a complex number.
Similarly, the operation and differentiation are defined by Now constructing the space follows from theorems which were used for constructing the space . Therefore, the corresponding proofs of Theorems 10 and 11 are omitted.

Theorem 9. Let and and then .

Theorem 10. Let , and then the following hold:(i); (ii); (iii)if in , as then as ;(iv)if , then as .

Theorem 11. Let and and then .

Proof of this theorem is similar to that of Theorem 6. Thus the space can be regarded as Bohemian space.

4. Krätzel Transform of Bohemian

By aid of Theorem 4, we have the right to define the Krätzel transform of as the Bohemian is embedded in the space .

Theorem 12. The mapping is(i)well defined,(ii)linear.

Proof. Let be such that , and then . Using Theorem 4 implies , for all . The idea of quotient of sequences in implies that That is, To prove part (ii) of the theorem, if , then Hence Also, if , then Hence This completes the proof.

Definition 13. Let . We define the inverse of transform of the Bohemian as for each .

Theorem 14. is an isomorphism.

Proof. Let and then by (29) we get . Therefore, Theorem 4 implies Thus , for all . The concept of quotients of equivalent classes of then gives This proves that is injective.
To show that is surjective, let . Then is a quotient of sequences in Hence, , for all . Once again, Theorem 4 implies that . Hence satisfies This completes the proof of the theorem.

Theorem 15. Let and then one has(i);(ii).

Proof. By using (29), we have Theorem 4 then gives Hence the part (i) of the theorem is proved. The proof of part (ii) is similar thus we omit the details. This completes the proof of the theorem.

Theorem 16. The mappings(i) are continuous with respect to and -convergence.(ii) are continuous with respect to and -convergence.

Proof. At first, let us show that and are continuous with respect to -convergence.
Let in as and then we show that as . By virtue of Theorem 5, we can find and in such that such that as for every . Hence, as . Thus, as .
To prove the Part (ii), Let be such that as . Then, once again, by Theorem 5, and for some and as . Hence as .
Using (36), we get Now, we establish that and are continuous with respect to -convergence.
Let in as . Then, there exist and such that and as . By applying (29) we get Hence, we have as in .
Therefore Hence, as .
Finally, let as and then we find such that and as for some .
Now, using (36), we obtain Theorem 4 implies Thus From this, we find that as in .
This completes the proof of the theorem.

Theorem 17. The transform is consistent with the classical transform .

Proof. For every , let be its representative in and then where , for all . Since is independent from the representative, for all , therefore which is the representative of . Thus this completes the proof.

Theorem 18. Let and then a necessary and sufficient condition for to be in the range of is that belongs to range of , for all .

Proof. Let be in the range of and then it is clear that belongs to the range of , for all .
To establish the converse, let be in the range of , for all . Then there is such that , . Since , for all , therefore, where and , for all . Then it follows that we get , for all . Thus is a quotient of sequences in . Therefore, we have and Hence the theorem is proved.

Theorem 19. If and , then one has

Proof. Let the hypothesis of the theorem be satisfied for some and . Therefore, This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme having project no. 5527068.