Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

Table 2

Comparison of convolution theorems of the QFT and classical FT for 𝑓,𝑔∈𝐿²(ℝ²;ℍ).

Assumptions on quaternion functions	QFT of convolution
ℱ𝑞{𝑓},ℱ𝑞{𝑔}∈𝐿^2(ℝ^2;ℍ)	QFT ≠ classical FT
ℱ𝑞{𝑓}∈𝐿^2(ℝ^2;ℝ) and ℱ𝑞{𝑔}∈𝐿^2(ℝ^2;ℍ)	QFT ≠ classical FT
ℱ𝑞{𝑓}∈𝐿^2(ℝ^2;ℍ) and ℱ𝑞{𝑔}∈𝐿^2(ℝ^2;ℝ)	QFT ≠ classical FT
𝑓=𝑓0+𝐢𝑓1 and ℱ𝑞{𝑓},ℱ𝑞{𝑔}∈𝐿^2(ℝ^2;ℝ)	QFT = classical FT
𝑓=𝐣𝑓2+𝐤𝑓3 and ℱ𝑞{𝑓},ℱ𝑞{𝑔}∈𝐿^2(ℝ^2;ℝ)	QFT = classical FT