Multidimensional Wave Field Signal Theory: Transfer Function Relationships

Table 1

Dimension	Transforms	Pseudo-Green's (𝑔𝑛) function	Input/output relationship
3	Spherical Hankel and spherical harmonic	𝑔^3D𝑛(𝑟,𝑘)=−𝑖𝑘ℎ^𝑛(2)(𝑘𝑟)	𝑢^𝑘𝑙(𝑟,𝜔)=𝑔^3D𝑙(𝑟,𝑘)𝜂(𝜔)𝕊𝑙{𝑞^𝑘𝑙(𝑘)}
2	Hankel and Fourier Series	𝑔^2D𝑛(𝑟,𝑘)=−𝜋𝑖2𝐻^𝑛(2)(𝑘𝑟)	𝑢𝑛(𝑟,𝜔)=𝑔^2D𝑛(𝑟,𝑘)𝜂(𝜔)ℍ𝑛{𝑞𝑛(𝑘)}
1	1D Fourier	𝑔1D(𝑥)=−𝑖2𝑘𝑒^−𝑖𝑘|𝑥|	𝑢(𝑥,𝜔)=𝑔1D(𝑥)𝜂(𝜔)𝑄(−sgn(𝑥)𝑘)