Pitt's inequality and the uncertainty principle associated with the quaternion Fourier transform

The quaternion Fourier transform – a generalized form of the classical Fourier transform – has been shown to be a powerful analyzing tool in image and signal processing. This paper investigates Pitt's inequality and uncertainty principle associated with the two-sided quaternion Fourier transform. It is shown that by applying the symmetric form f=f1+if2+f3j+if4jf=f1+if2+f3j+if4j of quaternion from Hitzer and the novel module or LpLp-norm of the quaternion Fourier transform fˆ, then any nonzero quaternion signal and its quaternion Fourier transform cannot both be highly concentrated. Two part results are provided, one part is Heisenberg–Weyl's uncertainty principle associated with the quaternion Fourier transform. It is formulated by using logarithmic estimates which may be obtained from a sharp of Pitt's inequality; the other part is the uncertainty principle of Donoho and Stark associated with the quaternion Fourier transform.