The quaternion Fourier and wavelet transforms on spaces of functions and distributions
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RESEARCH
The quaternion Fourier and wavelet transforms on spaces of functions and distributions Drema Lhamu1 and Sunil Kumar Singh2* * Correspondence:
[email protected] Department of Mathematics, Mahatma Gandhi Central University, Motihari, Bihar 845401, India Full list of author information is available at the end of the article 2
Abstract In this paper, the right-sided quaternion Fourier transform (right-sided QFT), which is a non-trivial generalization of the real and complex Fourier transform, is studied on spaces of test functions and distributions. The continuous quaternion wavelet transform of periodic functions is also defined and its quaternion Fourier representation form is established. The Plancherel and inversion formulas for the continuous quaternion wavelet transform are established. Keywords: quaternion algebra, quaternion Fourier transform, quaternion wavelet transform Mathematics Subject Classification: 11R52, 42C40, 46F12
1 Introduction Irish mathematician Sir W. R. Hamilton was the first to introduce the quaternion algebra in 1843. The quaternions, denoted by H, are an extension of complex numbers to a four-dimensional associative non-commutative algebra. Every element of H is a linear combination of real scalars and three imaginary units denoted by i, j and k with real coefficients written as [3,6]
H = q = q0 + iq1 + jq2 + kq3 ; q0 , q1 , q2 , q3 ∈ R ,
(1.1)
which satisfy Hamilton’s multiplication rules
ij = −ji = k;
jk = −kj = i;
ki = −ik = j;
i2 = j 2 = k 2 = ijk = −1.
The quaternion conjugate of q is denoted by q and given by q = q0 − iq1 − jq2 − kq3 ;
123
q 0 , q1 , q2 , q3 ∈ R
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and q = q.
(1.2)
It is an anti-involution, i.e. p q = q p.
(1.3)
The norm of q ∈ H is defined as |q| =
qq =
q02 + q12 + q22 + q32 .
It is easy to see that |q p| = |q||p| for all p, q ∈ H. According to (1.1), a quaternion-valued function f : R2 → H can be expressed as f (x) = f0 (x) + if1 (x) + jf2 (x) + kf3 (x);
f0 , f1 , f2 , f3 ∈ R.
(1.4)
Definition 1.1 (The space Lp (R2 ; H)) The space Lp (R2 ; H), 1 ≤ p < ∞, is the class of all measurable quaternion-valued functions f defined on R2 for which R2
|f (x)|p d2 x < ∞,
where d2 x = dx1 dx2 . The space Lp (R2 ; H) is a normed linear space with the norm defined by f p 2 = L (R ;H)
R2
1/p |f (x)|p d2 x .
Definition 1.2 The right-sided quaternion Fourier transform (right-sided QFT) of f ∈ L1 (R2 ; H) is denoted by Fq f and defined as Fq f (ω) := fˆ (ω) =
R2
f (x)e−2π iω1 x1 e−2π jω2 x2 d2 x,
(1.5)
where x = x1 e1 + x2 e2 , ω = ω1 e1 + ω2 e2 , e1 = (1, 0) and e2 = (0, 1), and the quaternion exponential product e−2π iω1 x1 e−2π jω2 x2 is called the quaternion Fourier kernel.
Theorem 1.3 ([3]) For two functions f, g ∈ L2 R2 ; H , we obtain Plancherel’s formula, specific to the right-sided QFT f, g L2 (R2 ;H) = Fq f , Fq g L2 (R2 ;H) .
(1.6)
In particular, if f = g then we get Par
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