The first summary is given in four tables of the basic relationships and equations, primarily developed in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients , for the scaling function φ(t), scaling coefficients h(n), and their Fourier transforms Φ(ω) and H(ω) for the multiplier M=2 or two-band multiresolution system. The various assumptions and conditions are omitted in order to see the “big picture" and to see the effects of increasing constraints.

Table 12.1. Properties of M=2 Scaling Functions (SF) and their Fourier Transforms

Case	Condition	φ ( t )	Φ ( ω )	Signal Space
1	Multiresolution			distribution
2	Partition of 1	∑ φ ( t – n ) = 1	Φ ( 2 π k ) = δ ( k )	distribution
3	Orthogonal		∑ | Φ ( ω + 2 π k ) | 2 = 1	L2
5	SF Smoothness			poly ∈Vj
6	SF Moments			Coiflets

Table 12.2. Properties of M=2 Scaling Coefficients and their Fourier Transforms

Case	Condition	h ( n )	H ( ω )	Eigenval.{T}
1	Existence
2	Fundamental	∑ h ( 2 n ) = ∑ h ( 2 n + 1 )	H ( π ) = 0	EV = 1
3	QMF		| H ( ω ) | 2 + | H ( ω + π ) | 2 = 2	EV ≤ 1
4	Orthogonal		| H ( ω ) | 2 + | H ( ω + π ) | 2 = 2	one EV =1
	L2 Basis		and H(ω)≠0,|ω|≤π/3	others <1
6	Coiflets	∑ nkh ( n ) = 0

Table 12.3. Properties of M=2 Wavelets (W) and their Fourier Transforms

Case	Condition	ψ ( t )	Ψ ( ω )	Signal Space
1	MRA			distribution
3	Orthogonal			L2
3	Orthogonal			L2
5	W Moments			poly not∈Wj