On uncertainty principle of the local polynomial Fourier transform

EURASIP Journal on Advances in Signal Processing

Table 1 Expressions of 〈ω〉_t and $B_{t}^{2}$ for LPFTs of order 2, 3 and 4

M
	〈ω〉_t
2	a₀ + ω₁t + (a₁ - ω₁) 〈τ〉_t
3	$a_{0} + ω_{1} t - \frac{ω_{2} t^{2}}{2} + (a_{1} - ω_{1} + ω_{2} t) {⟨τ⟩}_{t} + \frac{1}{2} (a_{2} - ω_{2}) {⟨τ^{2}⟩}_{t}$
4	$\begin{gathered} a_{0} + ω_{1} t - \frac{ω_{2}}{2} t^{2} + \frac{ω_{3}}{6} t^{3} + (a_{1} - ω_{1} + ω_{2} t - \frac{ω_{3}}{2} t^{2} {⟨τ⟩}_{t} \\ + \frac{1}{2} (a_{2} - ω_{2} + ω_{3} t) {⟨τ^{2}⟩}_{t} + \frac{1}{6} (a_{3} - ω_{3}) {⟨τ^{3}⟩}_{t} \\ B_{t}^{2} \end{gathered}$
2	$\frac{a + α}{2} + \frac{1}{2 (a + α)} {(a_{1} + ω_{1})}^{2}$
3	$\frac{a + α}{2} + \frac{1}{2 (a + α)} {[(a_{1} - ω_{1} + ω_{2} t) + (a_{2} - ω_{2}) {⟨τ⟩}_{t}]}^{2} + \frac{1}{8 {(a + α)}^{2}} {(a_{2} - ω_{2})}^{2}$
4	$\begin{gathered} \frac{a + α}{2} + \frac{1}{2 (a + α)} [(a_{1} - ω_{1} + ω_{2} t - \frac{ω_{3}}{2} t^{2}) + (a_{2} - ω_{2} + ω_{3} t) {⟨τ⟩}_{t} \\ {+ \frac{1}{2} (a_{3} - ω_{3}) {⟨τ^{2}⟩}_{t}]}^{2} + \frac{1}{8 {(a + α)}^{2}} (a_{2} - ω_{2} + ω_{3} t) + {(a_{3} - ω_{3}) {⟨τ⟩}_{t}]}^{2} \\ + \frac{1}{48 {(a + α)}^{3}} {(a_{3} - ω_{3})}^{2} \end{gathered}$