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The continuous quaternion wavelet transform (CQWT) is a generalization of the classical continuous wavelet transform within the context of quaternion algebra. First of all, we show that the directional quaternion Fourier transform (QFT) uncertainty principle can be obtained using the component-wise QFT uncertainty principle. Based on this method, the directional QFT uncertainty principle using representation of polar coordinate form is easily derived. We derive a variation on uncertainty principle related to the QFT. We state that the CQWT of a quaternion function can be written in terms of the QFT and obtain a variation on uncertainty principle related to the CQWT. Finally, we apply the extended uncertainty principles and properties of the CQWT to establish logarithmic uncertainty principles related to generalized transform.

As it is known, the classical wavelet transform (WT) is a very useful mathematical tool. It has been discussed extensively in the literature and has been proven to be powerful and useful in the communication theory, quantum mechanics, and many other fields [

Motivated by the authors in [

The concept of the quaternion algebra [

Let

Now we notice that

Any quaternion

It is convenient to introduce an inner product for two quaternion functions

A couple

and, for

Derivatives are conveniently expressed by multi-indices:

Next, we obtain the Schwartz space as (compared to [

In the following, we introduce the (right-sided) QFT and some of its fundamental properties such as Riemann-Lebesgue lemma and continuity.

The (right-sided) quaternion Fourier transform (QFT) of

Suppose that

Since

Applying (

It is worth noting here that if

Some important properties of the QFT are stated in the following lemmas.

If

We prove expression (

Equation (

If

By Riesz’s interpolation theorem, we get that the Hausdorff-Young inequality (see [

The following theorem is an extension of the Riemann-Lebesgue lemma in the QFT domain.

For a function in

Notice first that

If

From the QFT definition (

In what follows, we investigate the uncertainty principles associated with the QFT. These results will be needed in the next section.

If

An alternative form of Theorem

Notice that for

Under the assumptions of Theorem

It is not difficult to check that

A generalized version of Theorem

If

We have

A different proof of Theorem

Using the polar coordinate form of quaternion function

If

Applying (

This section briefly introduces the continuous quaternion wavelet transform (CQWT). We shall derive two theorems of the CQWT which will be used in proving the main theorem. A more complete and detailed discussion of the properties of the CQWT can be found in [

A quaternion-valued function is admissible if and only if it satisfies the following admissibility condition:

Let

The relationship between (

Let

If we assume that

The CQWT of a quaternion function

As an easy consequence of the above definition, we further obtain the following useful theorem.

Let

We need the following two important results, which will be useful in proving the logarithmic uncertainty principle for the CQWT.

Let

From the definition of QFT (

Let

Applying Plancherel’s theorem for QFT (

The simplest formulation of the uncertainty principle in harmonic analysis is Heisenberg-Weyl inequality, which gives us the information that a nontrivial function and its Fourier transform cannot both be simultaneously sharply localized [

Due to the uncertainty principle for QFT (

For

Applying Plancherel’s theorem for QFT (

It is proved that, for every

Let

For the proof of Theorem

Suppose that

A straightforward computation yields

Using the uncertainty principle in Theorem

Let us derive a logarithmic uncertainty principle associated with the continuous quaternion wavelet transform (CQWT).

Let

Next, we need the following lemma to assist the proof of the above theorem.

Under the same conditions as in Theorem

A simple calculation reveals

It is known that

It is worth nothing that, following the steps of the proof of the above theorem, we can also obtain Theorem

Based on the logarithmic uncertainty principle in the quaternion Fourier domain, we have established a logarithmic uncertainty principle related to the CQWT. It is a more general form of component-wise uncertainty principle associated with the CQWT, which describes the relationship between the QFT of a quaternion function and its CQWT. We also presented a variation on uncertainty principle related to the QFT and then found the variation on uncertainty principle related to the CQWT.

The authors declare that there are no competing interests regarding the publication of this paper.

This work is partially supported by WCU Percepatan Publikasi Internasional Tahun 2016 (no. 3834/UN4.21/PL.09/2016) from the Hasanuddin University, Indonesia. The second author is partially supported by JSPS KAKENHI (C)25400202 of Japan.