On Uncertainty Principle for Quaternionic Linear Canonical Transform

Table 1

Properties of the QLCT.

Property	Function	QLCT
Real linearity	𝛼𝑓1𝑥1,𝑥2+𝛽𝑓2𝑥1,𝑥2	𝛼ℒ𝑟^𝐢,𝐣(𝑓)(𝑢1,𝑢2)+𝛽ℒ𝑟^𝐢,𝐣(𝑓2)(𝑢1,𝑢2)
Formula
Plancherel	𝑓1,𝑓2𝐿2(ℝ2;ℍ)=	ℒ𝑟^𝐢,𝐣𝑓1,ℒ𝑟^𝐢,𝐣𝑓2𝐿2(ℝ2;ℍ)
Parseval	𝑓𝐿2(ℝ2;ℍ)=	ℒ𝑟^𝐢,𝐣𝑓𝐿2(ℝ2;ℍ)
Derivatives	∫ℝ2𝑢^2𝑖ℒ𝑟^𝐢,𝐣(𝑓)(𝑢1,𝑢2)^2𝑑𝑢1𝑑𝑢2=	𝑏^2𝑖∫ℝ2𝜕𝜕𝑥𝑖𝑓(𝑥1,𝑥2)^2𝑑𝑥1𝑑𝑥2