Time-Space Fourier Transforms

The time-space Fourier transform is defined as

Figure 1. EQUATION_DISPLAY
FTts[h(x;t)]=∫s(∫−∞∞h(x,t)e−i(ωt+k˙⋅s)ⅆt)|dx=f(k;ω)⁢     ∀k∈K;     ω∈[−∞, ∞]
(495)

The inverse time-space Fourier transform is

Figure 2. EQUATION_DISPLAY
FTts−1[f(k;ω)]=1(2π)3∫K(∫−∞∞f(k;ω)e−i(ωt+k˙⋅s)ⅆω)|dx|=h(x;t)⁢     ∀x∈S; t∈[−∞, ∞]
(496)

The following relationships hold between the transforms:

Figure 3. EQUATION_DISPLAY
FTts[h(x;t)]=f(k;ω)=FTs[g(x;ω)]=FTs[FTt[h(x;t)]]
(497)
Figure 4. EQUATION_DISPLAY
FTts−1[f(k;ω)]=h(x;t)=FTt−1[g(x;ω)]=FTt−1[FTs−1[f(k;ω)]]
(498)