Spline Wavelets: Construction, Implication, and Uses - Bezier and Splines in Image Processing and Machine Vision - page 162

since, e −i 2 nx =(cos2 nx − i sin 2 nx )=1.

Now,

l

1
( x + πr ) ,

cot x = lim
l→∞

r = − l

and hence

∞

d 2 n− 1
dx 2 n− 1 cot x.

1
( x + πr ) 2 n

1

= −

(2 n − 1)!

r = −∞

This provides

∞

sin 2 n x
(2 n

d 2 n− 1
dx 2 n− 1 cot x.

| N n (2 x +2 πr )

2 =

| − (8.18)

− 1)!

r = −∞

Equation (8.18) helps to compute optimal Riesz bounds. For smaller values,
the computation of spline order is straightforward, while for larger values,
algebraic exercise to some extent is needed.
There could be other approaches. One such approach establishes [37]

∞ n − 1

| N n ( ω +2 πr )

2 =

N 2 n ( n + r ) e −irω ,

|

r = −∞ r = −n +1

and using the properties of cardinal B-splines, one can show

∞

| N n ( ω +2 πr )

2

| ≤ 1 .

r = −∞

The Riesz basis bound B = 1. To get the greatest lower bound, one can
consider “Euler-Frobenius polynomials:”

n− 1

1)! z n− 1 N 2 n ( n + r ) z r E 2 n− 1 ( z )=(2 n −

r = − n +1

of order 2 n − 1. Since its degree is 2 n − 2, it has 2 n − 2 roots. All these 2 n − 2

roots, say, λ 1 ,λ 2 , ··· ,λ 2 n− 2 are negative, simple, real, and are found to hold

the relation

0 >λ 1 >λ 2 > ··· >λ 2 n− 2 ,

and

λ 1 λ 2 n− 2 = ··· = λ n− 1 λ n =1 .

This provides,

n − 1

(1 + λ r ) 2
|

1

A n = > 0 .

(2 n − 1!) λ r |

r =1

Also using the properties of Euler-Frobenius polynomial, one can show