For 2D real-valued functions

C.1 For 2D real-valued functions

The formula for expanding a real-valued two-dimensional function G(x,y) in terms of basis functions cos ( mπx- nπy)
Lx + Ly and sin (m-πx nπy)
Lx + Ly can be readily recovered from Eqs. (120) and (119). For notation ease, deﬁne

m πx nπy α = -Lx- + Ly-.

(121)

Then Eq. (120) is written as

∑∞ ∞∑ G(x,y) = cmn [cosα + isinα ] m= −∞ n= −∞ ( ) ∑∞ ∞∑ ---1--∫ Ly ∫ Lx = 4LxLy − Ly −Lx G (x,y)cos αdxdy [cosα + isinα] m= −∞ n= −∞ ( ∫ ∫ ) ∑∞ ∞∑ ---1-- Ly Lx + 4LxLy − Ly −Lx G (x,y)sinαdxdy [− icosα + sin α], m= −∞ n= −∞

Sine G(x,y) is assumed to be real-valued, the imaginary parts of the above expression will cancel each other. Therefore, the above expansion is simpliﬁed to

∞∑ ∑∞ G (x,y) = m= −∞ n=−∞ ( 1 ∫ Ly∫ Lx ) [ ------ G(x,y)cosαdxdy cosα ( 4LxLy −Ly −Lx ) 1 ∫ Ly ∫ Lx + 4LxLy- G (x,y)sinαdxdy sin α]. (122) −Ly −Lx

This is a compact Fouier expansion for 2D real-valued function. Furthermore, noting that (m,n) term is equal to (−m,−n) term, the above expansion can be further reduced to

∑∞ ∑∞ G(x,y) = Gcnm cosα +Gsnm sinα, (123) n=−∞ m=0

with

∫ ∫ Gc = --1--- Ly Lx G(x,y)dxdy. 00 4LxLy −Ly − Lx

(124)

and the other coeﬃcients given by

c --1---∫ Ly∫ Lx G mn = 2LxLy −L −L G (x,y)cosαdxdy. y x

(125)

s 1 ∫ Ly ∫ Lx G mn = 2LxLy- G (x,y)sinαdxdy. −Ly −Lx

(126)

Here the range of m is reduced to [0 : +∞]. In this case, we have an edge case, G₀₀^c, that needs special treatment. We see that allowing the index runing from −∞ to +∞ has the advantage of that there are no edge cases that needs special treatment.

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