Spline Wavelets: Construction, Implication, and Uses - Bezier and Splines in Image Processing and Machine Vision - page 163

∞ n− 1

2 λ r cos ω + λ r
|

1 1 −

| N n ( ω +2 πr )

2 =

| ,

(2 n − 1!) λ r |

r = −∞ r =1

since λ r s are negative and − 1 ≤ cos ω ≤ 1wehave,

∞

| N n ( ω +2 πr )

2

A n ≤ | ≤ 1 .

r = −∞

Hence, for any integer n ≥ 2 the cardinal B-spline basis B = { N n ( x − r ) } ,r ∈

Z is a Riesz basis of V 0

with bounds A = A n and B =1.

Example: Compute the optimal Riesz bounds for the first and second order
cardinal B-splines, N ! and N 2 . From equation (8.18), we have

∞

| N 1 (2 x +2 πr )

sin 2 x
1!

2 = d

dx cot x,

| −

r = −∞

sin 2 x (

cosec 2 x ) ,

= − −

=1 ,

or,

∞

| N 1 ( ω +2 πr )

2 =1 ,

|

r = −∞

and,

∞

| N 2 (2 x +2 πr )

sin 4 x
3!

d 3

2 =

| −

dx 3 cot x,

r = −∞

sin 4 x
6

2( cosec 4 x +2 cosec x cot x )

= − {− } ,

2

6 (1 + 2 cos 2 x ) ,

=

1

3 (1 + 2 cos 2 x ) ,

=

or,

∞

3 cos 2 ( ω

2 = 1

3 + 2

| N 2 ( ω +2 πr ) |

2 ) .

r = −∞

{ N 1 ( . − r ) }

Hence, is orthonormal and

∞

1
3 ≤

| N 2 ( ω +2 πr )

2

| ≤ 1 .

r = −∞

8.2.2 Scaling and Cardinal B-Spline Functions

Since the cardinal B-spline basis B is a Riesz basis of V 0 , one can conclude
that

2 j/ 2 N n (2 j x

{ − r ) ,r ∈ Z }

(8.19)