Abstract
This paper is an exposition of some estimates which have a number of applications to interpolation theory. In particular some recent problems in image processing and singular integral operators require the computation of suitable estimates. In Abilov et al. (Comput Math Math Phys 48:2146, 2008) , Abilov et al. proved two useful estimates for the Fourier transform in the space of square integral multivariable functions on certain classes of functions characterized by the generalized continuity modulus, and these estimates are proved by Abilov for only two variables, using a translation operator. The purpose of this paper is to study these estimates for measurable sets from complex domain to hyper complex domain by using quaternion algebras, associated with the quaternion linear canonical transform, constructed by the generalized Steklov function.
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1 Introduction
The integral Fourier transform, as well as Fourier series, are widely used in various fields of calculus, computational mathematics, mathematical physics, etc. Certain applications of this transform are described in a number of fundamental monographs (for example, see [25, 26]). Numerical estimates of the Fourier transform are presented in [30].
The classical linear canonical transform (LCT) is considered as a generalization of the Fourier transform (FT), and was first proposed in the 1970 s by Collins [9] and Moshinsky and Quesne [21]. It is an effective processing tool for chirp signal analysis, such as the parameter estimation, sampling progress for non bandlimited signals with nonlinear Fourier atoms [20], and the LCT filtering [27].
In this paper we will give our results in a more general context, that of quaternion linear canonical transform (QLCT). There are many studies in the literature that are concerned with the QLCT (see, for example, [5, 7, 16,17,18, 29]). They established some important properties of the QLCT, such as the uncertainty principle, the inversion formula and the study of generalized swept-frequency filters.
Recently several results of estimation have been proved in several different versions and for several different types of transforms (for example for the Fourier transform [3, 13], for the Bessel transform, [10], for the Dunkl transform [11], for the Laguerre Hypergroup transform [22]). In [1] the authors estimated the integral \(\displaystyle \int _{|x|\ge R}|{\mathcal {F}}\{f\}(x)|^2dx \) in certain classes of functions in \(L^2({\mathbb {R}}^n)\) where \({\mathcal {F}}\{f\}\) is the Fourier transform (FT) of f. Since the QLCT is a generalization of the FT, so for this reason we want in this paper, to estimate the integral \(\displaystyle \int _{|\omega |\ge N^2}|{\mathcal {L}}_{A_{1},A_{2}}\{f\}(\omega )|^2d^2\omega \), where \({\mathcal {L}}_{A_{1},A_{2}}\{f\}\) stands for the QLCT transform of f and \(N \ge 1\).
In order to describe our results, we first need to introduce some facts about harmonic analysis related to the QLCT. We cite here, as briefly as possible, some properties. For more details we refer to [2, 6, 8, 12, 14, 16,17,18, 28, 29].
The quaternion algebra \({\mathcal {H}}\) was first invented by W. R. Hamilton in 1843 for extending complex numbers to a 4D algebra [24]. A quaternion \(q \in {\mathcal {H}}\) can be written in this form
where i, j, k satisfy Hamilton’s multiplication rules
Using Hamilton’s multiplication rules, the multiplication of two quaternions \(p=p_{0}+{\underline{p}}\) and \(q=q_{0}+{\underline{q}}\) can be expressed as
We define the conjugation of \(q\in {\mathcal {H}}\) by \({\overline{q}}=q_{0}-iq_{1}-jq_{2}-kq_{3}.\) Clearly, \(q{\overline{q}}=q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2}.\) So the modulus of a quaternion q is defined by
In this paper, we study the quaternion-valued signal \(f: {\mathbb {R}}^{2} \rightarrow {\mathcal {H}}\) that can be expressed as
where \(x=x_{1}e_{1}+x_{2}e_{2}\in {\mathbb {R}}^{2}\) and \(f_{0},~f_{1},~f_{2}~\text {and}~f_{3}\) are real-valued functions. For \(1\le r<\infty ,\) the quaternion modulus \(L^{r}({\mathbb {R}}^{2},{\mathcal {H}})\) is defined as
Let \(f\in L^{r}({\mathbb {R}}^{2},{\mathcal {H}}).\) The quaternion Fourier transform (QFT) of f is defined by
The inner product of f, \(g\in L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\) is defined by
Clearly, \(\Vert f\Vert _{2}^2=\langle f,f\rangle \). For all \(\theta \in {\mathbb {R}}\) we have
Now, we define a norm of \({\mathcal {F}}(f)\) as
Furthermore, we obtain the \(L^{r}({\mathbb {R}}^{2},{\mathcal {H}})\)-norm
For \(f\in L_{}^{1}({\mathbb {R}}^{2},{\mathcal {H}})\), we have
(QFT Plancherel) If \(f\in L^{1}({\mathbb {R}}^{2},{\mathcal {H}})\cap L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\), then
Moreover,
then we can rewrite the QFT Plancherel as follows
Indeed, we have
Applying 3 into the right-hand side of the above identity gives
Since \(f_i(x), i=0,1,2,3\), is real-valued, the above equation can be written in the form
Suppose that \({\mathcal {F}}(f)\in L^{1}({\mathbb {R}}^{2},{\mathcal {H}})\) and \({\mathcal {F}}(\frac{\partial ^n f}{\partial x_{1}^{n}})\in L^{1}({\mathbb {R}}^{2},{\mathcal {H}})\). Then
Moreover, if \({\mathcal {F}}(f)\in L^{1}({\mathbb {R}}^{2},{\mathcal {H}})\) and \({\mathcal {F}}(\frac{\partial ^m f}{\partial x_{2}^{m}})\in L^{1}({\mathbb {R}}^{2},{\mathcal {H}})\), then
Let \(A_{s}=\begin{pmatrix} a_s &{} b_s \\ c_s &{} d_s \end{pmatrix}\in {\mathbb {R}}^{2\times 2}\) be a real matrix parameter such that \(\det (A_s)=1\), for \(s=1,2.\) The two-sided (sandwich) QLCT of \(f\in L^{1}({\mathbb {R}}^{2},{\mathcal {H}})\) is defined by
where the kernel functions of the QLCT above are given by
Let the kernel function \(K_{A}\) be defined by (7) or (8). Then
-
\(K_{A}(-x,\omega )=K_{A}(x,-\omega ).\)
-
\(K_{A}(-x,-\omega )=K_{A}(x,\omega ).\)
-
\(\overline{K_{A}(x,\omega )}=K_{A}^{-1}(\omega ,x).\)
From the definition of the QLCT, we can easily see that when \(b_{1}b_{2} = 0\) and \(b_{1}=b_{2}=0\), the QLCT of a signal is essentially a quaternion chirp multiplication. Therefore, in this work, we always assume \(b_{1}b_{2} \ne 0\).
(Inversion formula) The (Two-sided) inverse quaternion linear canonical transform of \(g \in L^{1}({\mathbb {R}}^{2},{\mathcal {H}})\)
(Hausdorff–Young inequality) If \(1\le r< 2\) and letting \(r'\) be such that \(1/r+1/r'=1\) then for all \(f\in L^r({\mathbb {R}}^{2},{\mathcal {H}})\) it holds that
(Plancherel theorem of QLCTs) Let \(f\in L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\). Then
(Shift property) For a quaternion function \(f\in L^{1}({\mathbb {R}}^{2},{\mathcal {H}})\), we denote by \(\tau _kf(x)\) the shifted (translated) function defined by \(\tau _kf(x) = f(x-k)\), where \(k=k_{1}e_{1}+k_{1}e_{1}\in {\mathbb {R}}^2\). Then we obtain
(Modulation property). We define a modulation operator \({\mathbb {M}}_{\omega _0}f\) by
with \(\omega _0=u_0e_{1}+v_0e_{1}\). So
(Time-frequency shift). Let a quaternion function \(f\in L^{1}({\mathbb {R}}^{2},{\mathcal {H}})\). Then we obtain that
For a function f on \(L^{1}({\mathbb {R}}^{2},{\mathcal {H}})\) and for any \(h_{1}, h_{2}\in {\mathbb {R}}\), we define the operator \(\Delta _{h_{1},h_{2}}\) by
Definition 1.1
Let \(f(x)=f(x_{1},x_{2})\) belongs to \(L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\). We say that f is in the Lipschitz space \(\text {Lip}_{A_{1},A_{2}}(\alpha _{1}, \alpha _{2} )\) if
as \(h_{1},h_{2}\) tend to zero, \(0 <\alpha _{1}, \alpha _{2}\le 1\).
In \(L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\), consider the operator
Observe that if \(a_{1}=a_{2}=0\), then
This is analogous to the Steklov operator.
Let the function \(f\in L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\). The finite differences of the order m (\(m\in {1,2,3,\ldots }\)) are defined as follows:
here I is the unit operator, and the mth order generalized continuity modulus of the function f is defined by the formula
where \(\delta > 0\).
For a function f on \(L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\), we define the function \(g_f\) by
From the definition of \(g_f\), we easily obtain
Consider in \(L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\) the operator
where \(D=\dfrac{\partial ^2}{\partial x_{1}^{2}}+\dfrac{\partial ^2}{\partial x_{2}^{2}},\) \(D_{A_{1},A_{2}}^0f=f\), \(D_{A_{1},A_{2}}^rf=D_{A_{1},A_{2}}(D_{A_{1},A_{2}}^{r-1}f)\), \(r=1,2,\ldots \).In view of formulas (5), (6) and (18), we have
and hence
Denote by \(W_{2,\phi }^{2,k}({\mathbb {R}})\) the class of functions \(f\in L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\) having the generalized derivatives \(\dfrac{\partial f}{\partial x_1}\), \(\dfrac{\partial ^2 f}{\partial x_1\partial x_2}\),...in the sense of Levi (see [19, 23]) in \(L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\) are estimated by
where \(\phi (t)\) is a continuous steadily increasing function on \([0,+\infty )\) and \(\phi (0)=0\).
2 Some new estimates for quaternion linear canonical transform
In order to prove the main result, we shall need some preliminary results.
Lemma 2.1
If f belongs to \(L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\), then
Proof
Using the inequality (16), we have \(\Vert F_{h}^{A_{1}, A_{2}}(f)\Vert _{2}\le \Vert f\Vert _{2}\). Then \(\Vert \Delta _{h}^{1}f\Vert _{2}\le 2\Vert f\Vert _{2}\). Thus the result follows easily by using the recurrence for m. \(\square \)
Lemma 2.2
If quaternion function \(f\in L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\), then
Proof
Let \(f\in L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\). Taking into account the formula (1), we have
It is easily seen that
and
So that the transform of \( F_{h}^{A_{1}, A_{2}}f(x)\) is given as
This completes the proof. \(\square \)
Corollary 2.3
For any function f in \(L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\), we have
In the next, in order to describe our results we will use the following notation:
-
\(\displaystyle G=\{(w_1,w_2):(\frac{w_{1}}{b_1})^2+(\frac{w_{2}}{b_2})^2\ge N^2\}\).
-
\(\displaystyle |w|=(\frac{w_{1}}{b_1})^2+(\frac{w_{2}}{b_2})^2.\)
-
\(\displaystyle \varphi _{h}(\frac{w_{1}}{b_1},\frac{w_{2}}{b_2}) =\frac{\sin (w_{1}h/b_1)}{w_{1}h/b_1}\frac{\sin (w_{2}h/b_2)}{w_{2}h/b_2}.\)
-
\(\displaystyle I_G=\left( \int _{G}|{\mathcal {L}}_{A_{1},A_{2}} \{f\}(\omega )|^2d^2\omega \right) .\)
In the following result, we estimate the integral
in certain classes of functions in \(L^2({\mathbb {R}}^2,{\mathcal {H}})\).
Theorem 2.4
For functions \(f\in L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\) in the class \(W_{2,\phi }^{2,k}\),
where \(r=1,\ldots \); \(k=1,2,\ldots \); and \(\phi (t)\) is any nonnegative function defined on the interval \([0,\infty )\).
Proof
Let \(f\in W_{2,\phi }^{2,k}\). Thanks to Hölder’s inequality, we obtain
where \(\displaystyle I_G=\left( \int _{G}|{\mathcal {L}}_{A_{1},A_{2}} \{f\}(\omega )|^2d^2\omega \right) \).
In view of Holder inequality, we have that
Using the inequalities (23) and (24),
Now, let us estimate the integral
It is easy to see that, for this purpose, it is sufficient to consider the domain of integration
Divide the domain E into the two subdomains
Then,
Since \(\displaystyle |\sin x| \le |x|\), \(\displaystyle \frac{w_{1}}{b_1}\ge \frac{N}{\sqrt{2}}((w_1,w_2)\in E_1)\) and \(\displaystyle \frac{w_{2}}{b_2}\ge \frac{N}{\sqrt{2}}((w_1,w_2)\in E_2)\) it is clear that
Consequently,
Setting \(\displaystyle h=\frac{\pi ^2}{N}\).
Hence,
and we have
which yields the desired result. \(\square \)
Theorem 2.5
Let \(\phi (t)=t^\alpha \) (\(\alpha >0\)). Then the next conditions are equivalent:
and
where \(m=1,2\ldots .\); \(r=1,\ldots \); \(k=1,2,\ldots \); and \(0<\alpha <m\).
Proof
It follows from Theorem 2.4 that (25) entails (26).Suppose now that
Thus, by Parseval’s identity, we have
Divide this integral into two, \(\underbrace{\int _{{\mathbb {R}}^{2}}}=\underbrace{\int _{|\omega |<N^2}}_{I_{1}}+\underbrace{\int _{|\omega |>N^2}}_{I_{2}},\) where \(N =[h^{-1}] \), and estimate each of them. Firstly, we estimate \(I_2\), since
it follows that
i.e.,
Secondly, we estimate \(I_{1}\), since
and
i.e.,
Finally, combining the estimates for \(I_1\) and \(I_2\) gives
which means that \(f\in W_{2,t^{\alpha }}^{2,k}\). Hence the conditions (25) and (26) are equivalent. This proves the Theorem 2.5. \(\square \)
Remark
As in the article [15], the previous definition can be generalized as follows: For any two pure quaternions \(\alpha \) and \(\beta \) such that \(\alpha ^2 = \beta ^2 = -1\) used for re placing i and j in (7) and (8), and f in \(L^{1}({\mathbb {R}}^{2},{\mathcal {H}})\)
From linearity of \({\mathcal {L}}_{A_{1},A_{2}}^{\alpha ,\beta }\) we obtain the QLCT for the OPS split \(f=f_{+}+f_{-}\) where \(f_{ \pm }=\frac{1}{2}\left( f \pm \alpha f \beta \right) \)
so what was done above for the integral \(\displaystyle \int _{|\omega |\ge N^2}|{\mathcal {L}}_{A_{1},A_{2}}\{f\}(\omega )|^2d^2\omega \) can be done again for the two integrals \(\displaystyle \int _{|\omega |\ge N^2}|{\mathcal {L}}_{A_{1},A_{2}}^{\alpha ,\beta }\{f_{+}\}(\omega )|^2d^2\omega \) and \(\displaystyle \int _{|\omega |\ge N^2}|{\mathcal {L}}_{A_{1},A_{2}}^{\alpha ,\beta }\{f_{-}\}(\omega )|^2d^2\omega \) so that we can find a generalization of our results.
3 Conclusion
In this paper after having given two estimates for the quaternion linear canonical transform which generalize those of Abilov [1] we note so far the difficulty lies only in the fact that for quaternion fields we have no commutativity, whereas for the Fourier transform this does not pose a problem, but even for quaternion fields there is a question which arises, if we can have the same approximation for the right-sided Quaternion linear canonical transform [4], the answer is positive although there will be a slight difference concerning the calculations of certain steps.
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Achak, A., Akhlidj, A., Daher, R. et al. On estimates for the quaternion linear canonical transform in the space \(L^{2}({\mathbb {R}}^{2},{\mathcal {H}})\). Rend. Circ. Mat. Palermo, II. Ser 73, 1701–1714 (2024). https://doi.org/10.1007/s12215-024-01010-w
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DOI: https://doi.org/10.1007/s12215-024-01010-w