# Spline wavelet

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Wavelet constructed using a spline function

Animation showing the compactly supported cardinal B-spline wavelets of orders 1, 2, 3, 4 and 5.

In the [mathematical theory](/source/Mathematics) of [wavelets](/source/Wavelet), a **spline wavelet** is a wavelet constructed using a [spline function](/source/Spline_function).[1] There are different types of spline wavelets. The interpolatory spline wavelets introduced by C.K. Chui and J.Z. Wang are based on a certain [spline](/source/Spline_(mathematics)) [interpolation](/source/Interpolation) formula.[2] Though these wavelets are [orthogonal](/source/Orthogonality), they do not have [compact](/source/Compact_space) [supports](/source/Support_(mathematics)). There is a certain class of wavelets, unique in some sense, constructed using [B-splines](/source/B-spline) and having compact supports. Even though these wavelets are not orthogonal they have some special properties that have made them quite popular.[3] The terminology *spline wavelet* is sometimes used to refer to the wavelets in this class of spline wavelets. These special wavelets are also called **B-spline wavelets** and **cardinal B-spline wavelets**.[4] The Battle-Lemarie wavelets are also wavelets constructed using spline functions.[5]

## Cardinal B-splines

Let *n* be a fixed non-negative [integer](/source/Integer). Let *C**n* denote the set of all [real-valued functions](/source/Real-valued_function) defined over the set of [real numbers](/source/Real_number) such that each function in the set as well its first *n* [derivatives](/source/Derivative) are [continuous](/source/Continuous_function) everywhere. A [bi-infinite sequence](/source/Bi-infinite_sequence) . . . *x*−2, *x*−1, *x*0, *x*1, *x*2, . . . such that *x**r* < *x**r*+1 for all *r* and such that *x**r* approaches ±∞ as r approaches ±∞ is said to define a set of knots. A *spline* of order *n* with a set of knots {*x**r*} is a function *S*(*x*) in *C**n* such that, for each *r*, the restriction of *S*(*x*) to the interval [*x*r, *x**r*+1) coincides with a [polynomial](/source/Polynomial) with real coefficients of degree at most *n* in *x*.

If the separation *x**r*+1 - *x**r*, where *r* is any integer, between the successive knots in the set of knots is a constant, the spline is called a *cardinal spline*. The set of integers *Z* = {. . ., -2, -1, 0, 1, 2, . . .} is a standard choice for the set of knots of a cardinal spline. Unless otherwise specified, it is generally assumed that the set of knots is the set of integers.

A cardinal B-spline is a special type of cardinal spline. For any positive integer *m* the cardinal B-spline of order *m*, denoted by *N**m*(*x*), is defined recursively as follows.

- N 1 ( x ) = { 1 0 ≤ x < 1 0 otherwise {\displaystyle N_{1}(x)={\begin{cases}1&0\leq x<1\\0&{\text{otherwise}}\end{cases}}}

- N m ( x ) = ∫ 0 1 N m − 1 ( x − t ) d t {\displaystyle N_{m}(x)=\int _{0}^{1}N_{m-1}(x-t)dt} , for m > 1 {\displaystyle m>1} .

Concrete expressions for the cardinal B-splines of all orders up to 5 and their graphs are given later in this article.

## Properties of the cardinal B-splines

### Elementary properties

1. The [support](/source/Support_(mathematics)) of N m ( x ) {\displaystyle N_{m}(x)} is the closed interval [ 0 , m ] {\displaystyle [0,m]} .

1. The function N m ( x ) {\displaystyle N_{m}(x)} is non-negative, that is, N m ( x ) > 0 {\displaystyle N_{m}(x)>0} for 0 < x < m {\displaystyle 0<x<m} .

1. ∑ k = − ∞ ∞ N m ( x − k ) = 1 {\displaystyle \sum _{k=-\infty }^{\infty }N_{m}(x-k)=1} for all x {\displaystyle x} .

1. The cardinal B-splines of orders *m* and *m-1* are related by the identity: N m ( x ) = x m N m − 1 ( x ) + m + 1 − x m N m − 1 ( x − 1 ) {\displaystyle N_{m}(x)={\frac {x}{m}}N_{m-1}(x)+{\frac {m+1-x}{m}}N_{m-1}(x-1)} .

1. The function N m ( x ) {\displaystyle N_{m}(x)} is symmetrical about x = m 2 {\displaystyle x={\frac {m}{2}}} , that is, N m ( m 2 − x ) = N m ( m 2 + x ) {\displaystyle N_{m}\left({\frac {m}{2}}-x\right)=N_{m}\left({\frac {m}{2}}+x\right)} .

1. The derivative of N m ( x ) {\displaystyle N_{m}(x)} is given by N m ′ ( x ) = N m − 1 ( x ) − N m − 1 ( x − 1 ) {\displaystyle N_{m}^{\prime }(x)=N_{m-1}(x)-N_{m-1}(x-1)} .

1. ∫ − ∞ ∞ N m ( x ) d x = 1 {\displaystyle \int _{-\infty }^{\infty }N_{m}(x)\,dx=1}

### Two-scale relation

The cardinal B-spline of order *m* satisfies the following two-scale relation:

- N m ( x ) = ∑ k = 0 m 2 − m + 1 ( m k ) N m ( 2 x − k ) {\displaystyle N_{m}(x)=\sum _{k=0}^{m}2^{-m+1}{m \choose k}N_{m}(2x-k)} .

### Riesz property

The cardinal B-spline of order *m* satisfies the following property, known as the Riesz property: There exists two positive real numbers A {\displaystyle A} and B {\displaystyle B} such that for any square summable two-sided sequence { c k } k = − ∞ ∞ {\displaystyle \{c_{k}\}_{k=-\infty }^{\infty }} and for any *x*,

- A ‖ { c k } ‖ 2 ≤ ‖ ∑ k = − ∞ ∞ c k N m ( x − k ) ‖ 2 ≤ B ‖ { c k } ‖ 2 {\displaystyle A\left\Vert \{c_{k}\}\right\Vert ^{2}\leq \left\Vert \sum _{k=-\infty }^{\infty }c_{k}N_{m}(x-k)\right\Vert ^{2}\leq B\left\Vert \{c_{k}\}\right\Vert ^{2}}

where ‖ ⋅ ‖ {\displaystyle \Vert \cdot \Vert } is the norm in the ℓ2-space.

## Cardinal B-splines of small orders

The cardinal B-splines are defined recursively starting from the B-spline of order 1, namely N 1 ( x ) {\displaystyle N_{1}(x)} , which takes the value 1 in the interval [0, 1) and 0 elsewhere. Computer algebra systems may have to be employed to obtain concrete expressions for higher order cardinal B-splines. The concrete expressions for cardinal B-splines of all orders up to 6 are given below. The graphs of cardinal B-splines of orders up to 4 are also exhibited. In the images, the graphs of the terms contributing to the corresponding two-scale relations are also shown. The two dots in each image indicate the extremities of the interval supporting the B-spline.

### Constant B-spline

The B-spline of order 1, namely N 1 ( x ) {\displaystyle N_{1}(x)} , is the constant B-spline. It is defined by

- N 1 ( x ) = { 1 0 ≤ x < 1 0 otherwise {\displaystyle N_{1}(x)={\begin{cases}1&0\leq x<1\\0&{\text{otherwise}}\end{cases}}}

The two-scale relation for this B-spline is

- N 1 ( x ) = N 1 ( 2 x ) + N 1 ( 2 x − 1 ) {\displaystyle N_{1}(x)=N_{1}(2x)+N_{1}(2x-1)}

Constant B-spline N 1 ( x ) {\displaystyle N_{1}(x)}

### Linear B-spline

The B-spline of order 2, namely N 2 ( x ) {\displaystyle N_{2}(x)} , is the linear B-spline. It is given by

- N 2 ( x ) = { x 0 ≤ x < 1 − x + 2 1 ≤ x < 2 0 otherwise {\displaystyle N_{2}(x)={\begin{cases}x&0\leq x<1\\-x+2&1\leq x<2\\0&{\text{otherwise}}\end{cases}}}

The two-scale relation for this wavelet is

- N 2 ( x ) = 1 2 N 2 ( 2 x ) + N 2 ( 2 x − 1 ) + 1 2 N 2 ( 2 x − 2 ) {\displaystyle N_{2}(x)={\frac {1}{2}}N_{2}(2x)+N_{2}(2x-1)+{\frac {1}{2}}N_{2}(2x-2)}

Linear B-spline N 2 ( x ) {\displaystyle N_{2}(x)}

### Quadratic B-spline

The B-spline of order 3, namely N 3 ( x ) {\displaystyle N_{3}(x)} , is the quadratic B-spline. It is given by

- N 3 ( x ) = { 1 2 x 2 0 ≤ x < 1 − x 2 + 3 x − 3 2 1 ≤ x < 2 1 2 x 2 − 3 x + 9 2 2 ≤ x < 3 0 otherwise {\displaystyle N_{3}(x)={\begin{cases}{\frac {1}{2}}x^{2}&0\leq x<1\\-x^{2}+3x-{\frac {3}{2}}&1\leq x<2\\{\frac {1}{2}}x^{2}-3x+{\frac {9}{2}}&2\leq x<3\\0&{\text{otherwise}}\end{cases}}}

The two-scale relation for this wavelet is

- N 3 ( x ) = 1 4 N 3 ( 2 x ) + 3 4 N 3 ( 2 x − 1 ) + 3 4 N 3 ( 2 x − 2 ) + 1 4 N 3 ( 2 x − 3 ) {\displaystyle N_{3}(x)={\frac {1}{4}}N_{3}(2x)+{\frac {3}{4}}N_{3}(2x-1)+{\frac {3}{4}}N_{3}(2x-2)+{\frac {1}{4}}N_{3}(2x-3)}

Quadratic B-spline N 3 ( x ) {\displaystyle N_{3}(x)}

### Cubic B-spline

The cubic B-spline is the cardinal B-spline of order 4, denoted by N 4 ( x ) {\displaystyle N_{4}(x)} . It is given by the following expressions:

- N 4 ( x ) = { 1 6 x 3 0 ≤ x < 1 − 1 2 x 3 + 2 x 2 − 2 x + 2 3 1 ≤ x < 2 1 2 x 3 − 4 x 2 + 10 x − 22 3 2 ≤ x < 3 − 1 6 x 3 + 2 x 2 − 8 x + 32 3 3 ≤ x < 4 0 otherwise {\displaystyle N_{4}(x)={\begin{cases}{\frac {1}{6}}x^{3}&0\leq x<1\\-{\frac {1}{2}}x^{3}+2x^{2}-2x+{\frac {2}{3}}&1\leq x<2\\{\frac {1}{2}}x^{3}-4x^{2}+10x-{\frac {22}{3}}&2\leq x<3\\-{\frac {1}{6}}x^{3}+2x^{2}-8x+{\frac {32}{3}}&3\leq x<4\\0&{\text{otherwise}}\end{cases}}}

The two-scale relation for the cubic B-spline is

- N 4 ( x ) = 1 8 N 4 ( 2 x ) + 1 2 N 4 ( 2 x − 1 ) + 3 4 N 4 ( 2 x − 2 ) + 1 2 N 4 ( 2 x − 3 ) + 1 8 N 4 ( 2 x − 4 ) {\displaystyle N_{4}(x)={\frac {1}{8}}N_{4}(2x)+{\frac {1}{2}}N_{4}(2x-1)+{\frac {3}{4}}N_{4}(2x-2)+{\frac {1}{2}}N_{4}(2x-3)+{\frac {1}{8}}N_{4}(2x-4)}

Cubic B-spline N 4 ( x ) {\displaystyle N_{4}(x)}

### Bi-quadratic B-spline

The bi-quadratic B-spline is the cardinal B-spline of order 5 denoted by N 5 ( x ) {\displaystyle N_{5}(x)} . It is given by

- N 5 ( x ) = { 1 24 x 4 0 ≤ x < 1 − 1 6 x 4 + 5 6 x 3 − 5 4 x 2 + 5 6 x − 5 24 1 ≤ x < 2 1 4 x 4 − 5 2 x 3 + 35 4 x 2 − 25 2 x + 155 24 2 ≤ x < 3 − 1 6 x 4 + 5 2 x 3 − 55 4 x 2 + 65 2 x − 655 24 3 ≤ x < 4 1 24 x 4 − 5 6 x 3 + 25 4 x 2 − 125 6 x + 625 24 4 ≤ x < 5 0 otherwise {\displaystyle N_{5}(x)={\begin{cases}{\frac {1}{24}}x^{4}&0\leq x<1\\-{\frac {1}{6}}x^{4}+{\frac {5}{6}}x^{3}-{\frac {5}{4}}x^{2}+{\frac {5}{6}}x-{\frac {5}{24}}&1\leq x<2\\{\frac {1}{4}}x^{4}-{\frac {5}{2}}x^{3}+{\frac {35}{4}}x^{2}-{\frac {25}{2}}x+{\frac {155}{24}}&2\leq x<3\\-{\frac {1}{6}}x^{4}+{\frac {5}{2}}x^{3}-{\frac {55}{4}}x^{2}+{\frac {65}{2}}x-{\frac {655}{24}}&3\leq x<4\\{\frac {1}{24}}x^{4}-{\frac {5}{6}}x^{3}+{\frac {25}{4}}x^{2}-{\frac {125}{6}}x+{\frac {625}{24}}&4\leq x<5\\0&{\text{otherwise}}\end{cases}}}

The two-scale relation is

- N 5 ( x ) = 1 16 N 5 ( 2 x ) + 5 16 N 5 ( 2 x − 1 ) + 10 16 N 5 ( 2 x − 2 ) + 10 16 N 5 ( 2 x − 3 ) + 5 16 N 5 ( 2 x − 4 ) + 1 16 N 5 ( 2 x − 5 ) {\displaystyle N_{5}(x)={\frac {1}{16}}N_{5}(2x)+{\frac {5}{16}}N_{5}(2x-1)+{\frac {10}{16}}N_{5}(2x-2)+{\frac {10}{16}}N_{5}(2x-3)+{\frac {5}{16}}N_{5}(2x-4)+{\frac {1}{16}}N_{5}(2x-5)}

### Quintic B-spline

The quintic B-spline is the cardinal B-spline of order 6 denoted by N 6 ( x ) {\displaystyle N_{6}(x)} . It is given by

- N 6 ( x ) = { 1 120 x 5 0 ≤ x < 1 − 1 24 x 5 + 1 4 x 4 − 1 2 x 3 + 1 2 x 2 − 1 4 x + 1 20 1 ≤ x < 2 1 12 x 5 − x 4 + 9 2 x 3 − 19 2 x 2 + 39 4 x − 79 20 2 ≤ x < 3 − 1 12 x 5 + 3 2 x 4 − 21 2 x 3 + 71 2 x 2 − 231 4 x + 731 20 3 ≤ x < 4 1 24 x 5 − x 4 + 19 2 x 3 − 89 2 x 2 + 409 4 x − 1829 20 4 ≤ x < 5 − 1 120 x 5 + 1 4 x 4 − 3 x 3 + 18 x 2 − 54 x + 324 5 5 ≤ x < 6 0 otherwise {\displaystyle N_{6}(x)={\begin{cases}{\frac {1}{120}}x^{5}&0\leq x<1\\-{\frac {1}{24}}x^{5}+{\frac {1}{4}}x^{4}-{\frac {1}{2}}x^{3}+{\frac {1}{2}}x^{2}-{\frac {1}{4}}x+{\frac {1}{20}}&1\leq x<2\\{\frac {1}{12}}x^{5}-x^{4}+{\frac {9}{2}}x^{3}-{\frac {19}{2}}x^{2}+{\frac {39}{4}}x-{\frac {79}{20}}&2\leq x<3\\-{\frac {1}{12}}x^{5}+{\frac {3}{2}}x^{4}-{\frac {21}{2}}x^{3}+{\frac {71}{2}}x^{2}-{\frac {231}{4}}x+{\frac {731}{20}}&3\leq x<4\\{\frac {1}{24}}x^{5}-x^{4}+{\frac {19}{2}}x^{3}-{\frac {89}{2}}x^{2}+{\frac {409}{4}}x-{\frac {1829}{20}}&4\leq x<5\\-{\frac {1}{120}}x^{5}+{\frac {1}{4}}x^{4}-3x^{3}+18x^{2}-54x+{\frac {324}{5}}&5\leq x<6\\0&{\text{otherwise}}\end{cases}}}

## Multi-resolution analysis generated by cardinal B-splines

The cardinal B-spline N m ( x ) {\displaystyle N_{m}(x)} of order *m* generates a [multi-resolution analysis](/source/Multi-resolution_analysis). In fact, from the elementary properties of these functions enunciated above, it follows that the function N m ( x ) {\displaystyle N_{m}(x)} is [square integrable](/source/Square_integrable) and is an element of the space L 2 ( R ) {\displaystyle L^{2}(R)} of square integrable functions. To set up the multi-resolution analysis the following notations used.

- - For any integers k , j {\displaystyle k,j} , define the function N m , k j ( x ) = N m ( 2 k x − j ) {\displaystyle N_{m,kj}(x)=N_{m}(2^{k}x-j)} . - For each integer k {\displaystyle k} , define the subspace V k {\displaystyle V_{k}} of L 2 ( R ) {\displaystyle L^{2}(R)} as the [closure](/source/Closure_(mathematics)) of the [linear span](/source/Linear_span) of the set { N m , k j ( x ) : j = ⋯ , − 2 , − 1 , 0 , 1 , 2 , ⋯ } {\displaystyle \{N_{m,kj}(x):j=\cdots ,-2,-1,0,1,2,\cdots \}} .

That these define a multi-resolution analysis follows from the following:

1. The spaces V k {\displaystyle V_{k}} satisfy the property: ⋯ ⊂ V − 2 ⊂ V − 1 ⊂ V 0 ⊂ V 1 ⊂ V 2 ⊂ ⋯ {\displaystyle \cdots \subset V_{-2}\subset V_{-1}\subset V_{0}\subset V_{1}\subset V_{2}\subset \cdots } .

1. The closure in L 2 ( R ) {\displaystyle L^{2}(R)} of the union of all the subspaces V k {\displaystyle V_{k}} is the whole space L 2 ( R ) {\displaystyle L^{2}(R)} .

1. The intersection of all the subspaces V k {\displaystyle V_{k}} is the singleton set containing only the zero function.

1. For each integer k {\displaystyle k} the set { N m , k j ( x ) : j = ⋯ , − 2 , − 1 , 0 , 1 , 2 , ⋯ } {\displaystyle \{N_{m,kj}(x):j=\cdots ,-2,-1,0,1,2,\cdots \}} is an unconditional basis for V k {\displaystyle V_{k}} . (A sequence {*x**n*} in a Banach space *X* is an unconditional basis for the space *X* if every permutation of the sequence {*x**n*} is also a basis for the same space *X*.[6])

## Wavelets from cardinal B-splines

Let *m* be a fixed positive integer and N m ( x ) {\displaystyle N_{m}(x)} be the cardinal B-spline of order *m*. A function ψ m ( x ) {\displaystyle \psi _{m}(x)} in L 2 ( R ) {\displaystyle L^{2}(R)} is a basic wavelet relative to the cardinal B-spline function N m ( x ) {\displaystyle N_{m}(x)} if the closure in L 2 ( R ) {\displaystyle L^{2}(R)} of the linear span of the set { ψ m ( x − j ) : j = ⋯ , − 2 , − 1 , 0 , 1 , 2 , ⋯ } {\displaystyle \{\psi _{m}(x-j):j=\cdots ,-2,-1,0,1,2,\cdots \}} (this closure is denoted by W 0 {\displaystyle W_{0}} ) is the [orthogonal complement](/source/Orthogonal_complement) of V 0 {\displaystyle V_{0}} in V 1 {\displaystyle V_{1}} . The subscript *m* in ψ m ( x ) {\displaystyle \psi _{m}(x)} is used to indicate that ψ m ( x ) {\displaystyle \psi _{m}(x)} is a basic wavelet relative the cardinal B-spline of order *m*. There is no unique basic wavelet ψ m ( x ) {\displaystyle \psi _{m}(x)} relative to the cardinal B-spline N m ( x ) {\displaystyle N_{m}(x)} . Some of these are discussed in the following sections.

## Wavelets relative to cardinal B-splines using fundamental interpolatory splines

### Fundamental interpolatory spline

#### Definitions

Let *m* be a fixed positive integer and let N m ( x ) {\displaystyle N_{m}(x)} be the cardinal B-spline of order *m*. Given a sequence { f j : j = ⋯ , − 2 , − 1 , 0 , 1 , 2 , ⋯ } {\displaystyle \{f_{j}:j=\cdots ,-2,-1,0,1,2,\cdots \}} of real numbers, the problem of finding a sequence { c m , k : k = ⋯ , − 2 , − 1 , 0 , 1 , 2 , ⋯ } {\displaystyle \{c_{m,k}:k=\cdots ,-2,-1,0,1,2,\cdots \}} of real numbers such that

- ∑ k = − ∞ ∞ c m , k N m ( j + m 2 − k ) = f j {\displaystyle \sum _{k=-\infty }^{\infty }c_{m,k}N_{m}\left(j+{\frac {m}{2}}-k\right)=f_{j}} for all j {\displaystyle j} ,

is known as the *cardinal spline interpolation problem*. The special case of this problem where the sequence { f j } {\displaystyle \{f_{j}\}} is the sequence δ 0 j {\displaystyle \delta _{0j}} , where δ i j {\displaystyle \delta _{ij}} is the [Kronecker delta](/source/Kronecker_delta) function δ i j {\displaystyle \delta _{ij}} defined by

- δ i j = { 1 , if i = j 0 , if i ≠ j {\displaystyle \delta _{ij}={\begin{cases}1,&{\text{ if }}i=j\\0,&{\text{ if }}i\neq j\end{cases}}} ,

is the *fundamental cardinal spline interpolation problem*. The solution of the problem yields the *fundamental cardinal interpolatory spline* of order *m*. This spline is denoted by L m ( x ) {\displaystyle L_{m}(x)} and is given by

- L m ( x ) = ∑ k = − ∞ ∞ c m , k N m ( x + m 2 − k ) {\displaystyle L_{m}(x)=\sum _{k=-\infty }^{\infty }c_{m,k}N_{m}\left(x+{\frac {m}{2}}-k\right)}

where the sequence { c m , k } {\displaystyle \{c_{m,k}\}} is now the solution of the following system of equations:

- ∑ k = − ∞ ∞ c m , k N m ( j + m 2 − k ) = δ 0 j {\displaystyle \sum _{k=-\infty }^{\infty }c_{m,k}N_{m}\left(j+{\frac {m}{2}}-k\right)=\delta _{0j}}

#### Procedure to find the fundamental cardinal interpolatory spline

The fundamental cardinal interpolatory spline L m ( x ) {\displaystyle L_{m}(x)} can be determined using [Z-transforms](/source/Z-transform). Using the following notations

- A ( z ) = ∑ k = − ∞ ∞ δ k 0 z k = 1 , {\displaystyle A(z)=\sum _{k=-\infty }^{\infty }\delta _{k0}z^{k}=1,}

- B m ( z ) = ∑ k = − ∞ ∞ N m ( k + m 2 ) z k , {\displaystyle B_{m}(z)=\sum _{k=-\infty }^{\infty }N_{m}\left(k+{\frac {m}{2}}\right)z^{k},}

- C m ( z ) = ∑ k = − ∞ ∞ c m , k z k , {\displaystyle C_{m}(z)=\sum _{k=-\infty }^{\infty }c_{m,k}z^{k},}

it can be seen from the equations defining the sequence c m , k {\displaystyle c_{m,k}} that

- B m ( z ) C m ( z ) = A ( z ) {\displaystyle B_{m}(z)C_{m}(z)=A(z)}

from which we get

- C m ( z ) = 1 B m ( z ) {\displaystyle C_{m}(z)={\frac {1}{B_{m}(z)}}} .

This can be used to obtain concrete expressions for c m , k {\displaystyle c_{m,k}} .

#### Example

As a concrete example, the case L 4 ( x ) {\displaystyle L_{4}(x)} may be investigated. The definition of B m ( z ) {\displaystyle B_{m}(z)} implies that

- B 4 ( x ) = ∑ k = − ∞ ∞ N 4 ( 2 + k ) z k {\displaystyle B_{4}(x)=\sum _{k=-\infty }^{\infty }N_{4}(2+k)z^{k}}

The only nonzero values of N 4 ( k + 2 ) {\displaystyle N_{4}(k+2)} are given by k = − 1 , 0 , 1 {\displaystyle k=-1,0,1} and the corresponding values are

- N 4 ( 1 ) = 1 6 , N 4 ( 2 ) = 4 6 , N 4 ( 3 ) = 1 6 . {\displaystyle N_{4}(1)={\frac {1}{6}},N_{4}(2)={\frac {4}{6}},N_{4}(3)={\frac {1}{6}}.}

Thus B 4 ( z ) {\displaystyle B_{4}(z)} reduces to

- B 4 ( z ) = 1 6 z − 1 + 4 6 z 0 + 1 6 z 1 = 1 + 4 z + z 2 6 z {\displaystyle B_{4}(z)={\frac {1}{6}}z^{-1}+{\frac {4}{6}}z^{0}+{\frac {1}{6}}z^{1}={\frac {1+4z+z^{2}}{6z}}}

This yields the following expression for C 4 ( z ) {\displaystyle C_{4}(z)} .

- C 4 ( z ) = 6 z 1 + 4 z + z 2 {\displaystyle C_{4}(z)={\frac {6z}{1+4z+z^{2}}}}

Splitting this expression into partial fractions and expanding each term in powers of *z* in an annular region the values of c 4 , k {\displaystyle c_{4,k}} can be computed. These values are then substituted in the expression for L 4 ( x ) {\displaystyle L_{4}(x)} to yield

- L 4 ( x ) = ∑ k = − ∞ ∞ ( − 1 ) k 3 ( 2 − 3 ) | k | N 4 ( x + 2 − k ) {\displaystyle L_{4}(x)=\sum _{k=-\infty }^{\infty }(-1)^{k}{\sqrt {3}}(2-{\sqrt {3}})^{|k|}N_{4}(x+2-k)}

### Wavelet using fundamental interpolatory spline

For a positive integer *m*, the function ψ m ( x ) {\displaystyle \psi _{m}(x)} defined by

- ψ I , m ( x ) = d m d x m L 2 m ( 2 x − 1 ) {\displaystyle \psi _{I,m}(x)={\frac {d^{m}}{dx^{m}}}L_{2m}(2x-1)}

is a basic wavelet relative to the cardinal B-spline of order N m ( x ) {\displaystyle N_{m}(x)} . The subscript *I* in ψ I , m {\displaystyle \psi _{I,m}} is used to indicate that it is based in the interpolatory spline formula. This basic wavelet is not compactly supported.

### Example

The wavelet of order 2 using interpolatory spline is given by

- ψ I , 2 ( x ) = d 2 d x 2 L 4 ( 2 x − 1 ) {\displaystyle \psi _{I,2}(x)={\frac {d^{2}}{dx^{2}}}L_{4}(2x-1)}

The expression for L 4 ( x ) {\displaystyle L_{4}(x)} now yields the following formula:

- ψ I , 2 ( x ) = d 2 d x 2 ∑ k = − ∞ ∞ ( − 1 ) k 3 ( 2 − 3 ) | k | N 4 ( 2 x + 1 − k ) {\displaystyle \psi _{I,2}(x)={\frac {d^{2}}{dx^{2}}}\sum _{k=-\infty }^{\infty }(-1)^{k}{\sqrt {3}}(2-{\sqrt {3}})^{|k|}N_{4}(2x+1-k)}

Now, using the expression for the derivative of N m ( x ) {\displaystyle N_{m}(x)} in terms of N m − 1 ( x ) {\displaystyle N_{m-1}(x)} the function ψ 2 ( x ) {\displaystyle \psi _{2}(x)} can be put in the following form:

- ψ I , 2 ( x ) = ∑ k = − ∞ ∞ ( − 1 ) k 4 3 ( 2 − 3 ) | k | ( ( N 2 ( 2 x + k − 1 ) − 2 N 2 ( 2 x + k − 2 ) + N 2 ( 2 x + k − 3 ) ) {\displaystyle \psi _{I,2}(x)=\sum _{k=-\infty }^{\infty }(-1)^{k}4{\sqrt {3}}(2-{\sqrt {3}})^{|k|}{\Big (}(N_{2}(2x+k-1)-2N_{2}(2x+k-2)+N_{2}(2x+k-3){\Big )}}

The following [piecewise linear function](/source/Piecewise_linear_function) is the approximation to ψ 2 ( x ) {\displaystyle \psi _{2}(x)} obtained by taking the sum of the terms corresponding to k = − 3 , … , 3 {\displaystyle k=-3,\ldots ,3} in the infinite series expression for ψ 2 ( x ) {\displaystyle \psi _{2}(x)} .

- ψ I , 2 ( x ) ≈ { 0.07142668 x + 0.17856670 − 2.5 < x ≤ − 2 − 0.48084803 x − 0.92598272 − 2 < x ≤ − 1.5 2.0088293 x + 2.8085333 − 1.5 < x ≤ − 1 − 7.5684795 x − 6.7687755 − 1 < x ≤ − 0.5 28.245949 x + 11.138439 − 0.5 < x ≤ 0 − 57.415316 x + 11.138439 0 < x ≤ 0.5 57.415316 x − 46.276878 0.5 < x ≤ 1 − 28.245949 x + 39.384388 1 < x ≤ 1.5 7.5684795 x − 14.337255 1.5 < x ≤ 2 − 2.0088293 x + 4.8173625 2 < x ≤ 2.5 0.48084803 x − 1.4068308 2.5 < x ≤ 3 − 0.07142668 x + 0.24999338 3 < x ≤ 3.5 0 o t h e r w i s e {\displaystyle \psi _{I,2}(x)\approx {\begin{cases}0.07142668x+0.17856670&-2.5<x\leq -2\\-0.48084803x-0.92598272&-2<x\leq -1.5\\2.0088293x+2.8085333&-1.5<x\leq -1\\-7.5684795x-6.7687755&-1<x\leq -0.5\\28.245949x+11.138439&-0.5<x\leq 0\\-57.415316x+11.138439&0<x\leq 0.5\\57.415316x-46.276878&0.5<x\leq 1\\-28.245949x+39.384388&1<x\leq 1.5\\7.5684795x-14.337255&1.5<x\leq 2\\-2.0088293x+4.8173625&2<x\leq 2.5\\0.48084803x-1.4068308&2.5<x\leq 3\\-0.07142668x+0.24999338&3<x\leq 3.5\\0&{otherwise}\end{cases}}}

### Two-scale relation

The two-scale relation for the wavelet function ψ m ( x ) {\displaystyle \psi _{m}(x)} is given by

- ψ I , m ( x ) = ∑ − ∞ ∞ q n N m ( 2 x − n ) {\displaystyle \psi _{I,m}(x)=\sum _{-\infty }^{\infty }q_{n}N_{m}(2x-n)} where q n = ∑ j = 0 m ( − 1 ) j ( m j ) c m + n − j − 1 . {\displaystyle q_{n}=\sum _{j=0}^{m}(-1)^{j}{m \choose j}c_{m+n-j-1}.}

## Compactly supported B-spline wavelets

The spline wavelets generated using the interpolatory wavelets are not compactly supported. Compactly supported B-spline wavelets were discovered by Charles K. Chui and Jian-zhong Wang and published in 1991.[3][7] The compactly supported B-spline wavelet relative to the cardinal B-spline N m ( x ) {\displaystyle N_{m}(x)} of order *m* discovered by Chui and Wang and denoted by ψ C , m ( x ) {\displaystyle \psi _{C,m}(x)} , has as its support the interval [ 0 , 2 m − 1 ] {\displaystyle [0,2m-1]} . These wavelets are essentially unique in a certain sense explained below.

### Definition

The compactly supported B-spline wavelet of order *m* is given by

- ψ C , m ( x ) = 1 2 m − 1 ∑ j = 0 2 m − 2 ( − 1 ) j N 2 m ( j + 1 ) d m d x m N 2 m ( 2 x − j ) {\displaystyle \psi _{C,m}(x)={\frac {1}{2^{m-1}}}\sum _{j=0}^{2m-2}(-1)^{j}N_{2m}(j+1){\frac {d^{m}}{dx^{m}}}N_{2m}(2x-j)}

This is an *m*-th order spline. As a special case, the compactly supported B-spline wavelet of order 1 is

- ψ C , 1 ( x ) = N 2 ( 1 ) d d x N 2 ( 2 x ) = { 1 0 ≤ x < 1 2 − 1 1 2 ≤ x < 1 0 otherwise {\displaystyle \psi _{C,1}(x)=N_{2}(1){\frac {d}{dx}}N_{2}(2x)={\begin{cases}1&0\leq x<{\frac {1}{2}}\\-1&{\frac {1}{2}}\leq x<1\\0&{\text{otherwise}}\end{cases}}}

which is the well-known [Haar wavelet](/source/Haar_wavelet).

### Properties

1. The support of ψ C , m ( x ) {\displaystyle \psi _{C,m}(x)} is the closed interval [ 0 , 2 m − 1 ] {\displaystyle [0,2m-1]} .

1. The wavelet ψ C , m ( x ) {\displaystyle \psi _{C,m}(x)} is the unique wavelet with minimum support in the following sense: If η ( x ) ∈ W 0 {\displaystyle \eta (x)\in W_{0}} generates W 0 {\displaystyle W_{0}} and has support not exceeding 2 m − 1 {\displaystyle 2m-1} in length then η ( x ) = c 0 ψ C , m ( x − n 0 ) {\displaystyle \eta (x)=c_{0}\psi _{C,m}(x-n_{0})} for some nonzero constant c 0 {\displaystyle c_{0}} and for some integer n 0 {\displaystyle n_{0}} .[8]

1. ψ C , m ( x ) {\displaystyle \psi _{C,m}(x)} is symmetric for even *m* and antisymmetric for odd *m*.

### Two-scale relation

ψ m ( x ) {\displaystyle \psi _{m}(x)} satisfies the two-scale relation:

- ψ C , m ( x ) = ∑ n = 0 3 m − 2 q n N m ( 2 x − n ) {\displaystyle \psi _{C,m}(x)=\sum _{n=0}^{3m-2}q_{n}N_{m}(2x-n)} where q n = ( − 1 ) n 2 m − 1 ∑ j = 0 m ( m j ) N 2 m ( n − j + 1 ) {\displaystyle q_{n}={\frac {(-1)^{n}}{2^{m-1}}}\sum _{j=0}^{m}{m \choose j}N_{2m}(n-j+1)} .

### Decomposition relation

The decomposition relation for the compactly supported B-spline wavelet has the following form:

- N m ( 2 x − l ) = ∑ k = − ∞ ∞ [ a m , l − 2 k N m ( x − k ) + b m , l − 2 k ψ C , m ( x − k ) ] {\displaystyle N_{m}(2x-l)=\sum _{k=-\infty }^{\infty }\left[a_{m,l-2k}N_{m}(x-k)+b_{m,l-2k}\psi _{C,m}(x-k)\right]}

where the coefficients a m , j {\displaystyle a_{m,j}} and b m , j {\displaystyle b_{m,j}} are given by

- a m , j = − ( − 1 ) j 2 ∑ l = − ∞ ∞ q − j + 2 m − 2 l + 1 c 2 m , l , {\displaystyle a_{m,j}=-{\frac {(-1)^{j}}{2}}\sum _{l=-\infty }^{\infty }q_{-j+2m-2l+1}c_{2m,l},}

- b m , j = ( − 1 ) j 2 ∑ l = − ∞ ∞ p − j + 2 m − 2 l + 1 c 2 m , l . {\displaystyle b_{m,j}={\frac {(-1)^{j}}{2}}\sum _{l=-\infty }^{\infty }p_{-j+2m-2l+1}c_{2m,l}.}

Here the sequence c 2 m , l {\displaystyle c_{2m,l}} is the sequence of coefficients in the fundamental interpolatoty cardinal spline wavelet of order *m*.

## Compactly supported B-spline wavelets of small orders

### Compactly supported B-spline wavelet of order 1

The two-scale relation for the compactly supported B-spline wavelet of order 1 is

- ψ C , 1 ( x ) = N 1 ( 2 x ) − N 1 ( 2 x − 1 ) {\displaystyle \psi _{C,1}(x)=N_{1}(2x)-N_{1}(2x-1)}

The closed form expression for compactly supported B-spline wavelet of order 1 is

- ψ C , 1 ( x ) = { 1 0 ≤ x < 1 2 − 1 1 2 ≤ x < 1 0 otherwise {\displaystyle \psi _{C,1}(x)={\begin{cases}1&0\leq x<{\frac {1}{2}}\\-1&{\frac {1}{2}}\leq x<1\\0&{\text{otherwise}}\end{cases}}}

### Compactly supported B-spline wavelet of order 2

The two-scale relation for the compactly supported B-spline wavelet of order 2 is

- ψ C , 2 ( x ) = 1 12 ( N 2 ( 2 x ) − 6 N 2 ( 2 x − 1 ) + 10 N 2 ( 2 x − 2 ) − 6 N 2 ( 2 x − 3 ) + N 2 ( 2 x − 4 ) ) {\displaystyle \psi _{C,2}(x)={\frac {1}{12}}\left(N_{2}(2x)-6N_{2}(2x-1)+10N_{2}(2x-2)-6N_{2}(2x-3)+N_{2}(2x-4)\right)}

The closed form expression for compactly supported B-spline wavelet of order 2 is

- ψ C , 2 ( x ) = { 1 6 x 0 ≤ x < 1 2 − 7 6 x + 2 3 1 2 ≤ x < 1 8 3 x − 19 6 1 ≤ x < 3 2 − 8 3 x + 29 6 3 2 ≤ x < 2 7 6 x − 17 6 2 ≤ x < 5 2 − 1 6 x + 1 2 5 2 ≤ x < 3 0 otherwise {\displaystyle \psi _{C,2}(x)={\begin{cases}{\frac {1}{6}}x&0\leq x<{\frac {1}{2}}\\-{\frac {7}{6}}x+{\frac {2}{3}}&{\frac {1}{2}}\leq x<1\\{\frac {8}{3}}x-{\frac {19}{6}}&1\leq x<{\frac {3}{2}}\\-{\frac {8}{3}}x+{\frac {29}{6}}&{\frac {3}{2}}\leq x<2\\{\frac {7}{6}}x-{\frac {17}{6}}&2\leq x<{\frac {5}{2}}\\-{\frac {1}{6}}x+{\frac {1}{2}}&{\frac {5}{2}}\leq x<3\\0&{\text{otherwise}}\end{cases}}}

### Compactly supported B-spline wavelet of order 3

The two-scale relation for the compactly supported B-spline wavelet of order 3 is

- ψ C , 3 ( x ) = 1 480 [ ( N 3 ( 2 x ) − 29 N 3 ( 2 x − 1 ) + 147 N 3 ( 2 x − 2 ) − 303 N 3 ( 2 x − 3 ) + {\displaystyle \psi _{C,3}(x)={\frac {1}{480}}{\Big [}(N_{3}(2x)-29N_{3}(2x-1)+147N_{3}(2x-2)-303N_{3}(2x-3)+} - - - - 303 N 3 ( 2 x − 4 ) − 147 N 3 ( 2 x − 5 ) + 29 N 3 ( 2 x − 6 ) − N 3 ( 2 x − 7 ) ] {\displaystyle 303N_{3}(2x-4)-147N_{3}(2x-5)+29N_{3}(2x-6)-N_{3}(2x-7){\Big ]}}

The closed form expression for compactly supported B-spline wavelet of order 3 is

- ψ C , 3 ( x ) = { 1 240 x 2 0 ≤ x < 1 2 − 31 240 x 2 + 2 15 x − 1 30 1 2 ≤ x < 1 103 120 x 2 − 221 120 x + 229 240 1 ≤ x < 3 2 − 313 120 x 2 + 1027 120 x − 1643 240 3 2 ≤ x < 2 22 5 x 2 − 779 40 x + 339 16 2 ≤ x < 5 2 − 22 5 x 2 + 981 40 x − 541 16 5 2 ≤ x < 3 313 120 x 2 − 701 40 x + 2341 80 3 ≤ x < 7 2 − 103 120 x 2 + 809 120 x − 3169 240 7 2 ≤ x < 4 31 240 x 2 − 139 120 x + 623 240 4 ≤ x < 9 2 − 1 240 x 2 + 1 24 x − 5 48 9 2 ≤ x < 5 0 otherwise {\displaystyle \psi _{C,3}(x)={\begin{cases}{\frac {1}{240}}x^{2}&0\leq x<{\frac {1}{2}}\\-{\frac {31}{240}}x^{2}+{\frac {2}{15}}x-{\frac {1}{30}}&{\frac {1}{2}}\leq x<1\\{\frac {103}{120}}x^{2}-{\frac {221}{120}}x+{\frac {229}{240}}&1\leq x<{\frac {3}{2}}\\-{\frac {313}{120}}x^{2}+{\frac {1027}{120}}x-{\frac {1643}{240}}&{\frac {3}{2}}\leq x<2\\{\frac {22}{5}}x^{2}-{\frac {779}{40}}x+{\frac {339}{16}}&2\leq x<{\frac {5}{2}}\\-{\frac {22}{5}}x^{2}+{\frac {981}{40}}x-{\frac {541}{16}}&{\frac {5}{2}}\leq x<3\\{\frac {313}{120}}x^{2}-{\frac {701}{40}}x+{\frac {2341}{80}}&3\leq x<{\frac {7}{2}}\\-{\frac {103}{120}}x^{2}+{\frac {809}{120}}x-{\frac {3169}{240}}&{\frac {7}{2}}\leq x<4\\{\frac {31}{240}}x^{2}-{\frac {139}{120}}x+{\frac {623}{240}}&4\leq x<{\frac {9}{2}}\\-{\frac {1}{240}}x^{2}+{\frac {1}{24}}x-{\frac {5}{48}}&{\frac {9}{2}}\leq x<5\\0&{\text{otherwise}}\end{cases}}}

### Compactly supported B-spline wavelet of order 4

The two-scale relation for the compactly supported B-spline wavelet of order 4 is

- ψ C , 4 ( x ) = 1 40320 [ N 4 ( 2 x ) − 124 N 4 ( 2 x − 1 ) + 1677 N 4 ( 2 x − 2 ) − 7904 N 4 ( 2 x − 3 ) + 18482 N 4 ( 2 x − 4 ) − {\displaystyle \psi _{C,4}(x)={\frac {1}{40320}}{\Big [}N_{4}(2x)-124N_{4}(2x-1)+1677N_{4}(2x-2)-7904N_{4}(2x-3)+18482N_{4}(2x-4)-} - - - - 24264 N 4 ( 2 x − 5 ) + 18482 N 4 ( 2 x − 6 ) − 7904 N 4 ( 2 x − 7 ) + 1677 N 4 ( 2 x − 8 ) − 124 N 4 ( 2 x − 9 ) + N 4 ( 2 x − 10 ) ] {\displaystyle 24264N_{4}(2x-5)+18482N_{4}(2x-6)-7904N_{4}(2x-7)+1677N_{4}(2x-8)-124N_{4}(2x-9)+N_{4}(2x-10){\Big ]}}

The closed form expression for compactly supported B-spline wavelet of order 4 is

- ψ C , 4 ( x ) = { 1 30240 x 3 0 ≤ x < 1 2 − 127 30240 x 3 + 2 315 x 2 − 1 315 x + 1 1890 1 2 ≤ x < 1 19 280 x 3 − 47 224 x 2 + 2147 10080 x − 103 1440 1 ≤ x < 3 2 − 1109 2520 x 3 + 465 224 x 2 − 32413 10080 x + 16559 10080 3 2 ≤ x < 2 5261 3360 x 3 − 33463 3360 x 2 + 42043 2016 x − 145193 10080 2 ≤ x < 5 2 − 35033 10080 x 3 + 93577 3360 x 2 − 148517 2016 x + 216269 3360 5 2 ≤ x < 3 4832 945 x 3 − 27691 560 x 2 + 113923 720 x − 28145 168 3 ≤ x < 7 2 − 4832 945 x 3 + 58393 1008 x 2 − 52223 240 x + 2048227 7560 7 2 ≤ x < 4 35033 10080 x 3 − 75827 1680 x 2 + 981101 5040 x − 234149 840 4 ≤ x < 9 2 − 5261 3360 x 3 + 38509 1680 x 2 − 112487 1008 x + 30347 168 9 2 ≤ x < 5 1109 2520 x 3 − 24077 3360 x 2 + 78311 2016 x − 141311 2016 5 ≤ x < 11 2 − 19 280 x 3 + 1361 1120 x 2 − 14617 2016 x + 4151 288 11 2 ≤ x < 6 127 30240 x 3 − 55 672 x 2 + 5359 10080 x − 11603 10080 6 ≤ x < 13 2 − 1 30240 x 3 + 1 1440 x 2 − 7 1440 x + 49 4320 13 2 ≤ x < 7 0 otherwise {\displaystyle \psi _{C,4}(x)={\begin{cases}{\frac {1}{30240}}x^{3}&0\leq x<{\frac {1}{2}}\\-{\frac {127}{30240}}x^{3}+{\frac {2}{315}}x^{2}-{\frac {1}{315}}x+{\frac {1}{1890}}&{\frac {1}{2}}\leq x<1\\{\frac {19}{280}}x^{3}-{\frac {47}{224}}x^{2}+{\frac {2147}{10080}}x-{\frac {103}{1440}}&1\leq x<{\frac {3}{2}}\\-{\frac {1109}{2520}}x^{3}+{\frac {465}{224}}x^{2}-{\frac {32413}{10080}}x+{\frac {16559}{10080}}&{\frac {3}{2}}\leq x<2\\{\frac {5261}{3360}}x^{3}-{\frac {33463}{3360}}x^{2}+{\frac {42043}{2016}}x-{\frac {145193}{10080}}&2\leq x<{\frac {5}{2}}\\-{\frac {35033}{10080}}x^{3}+{\frac {93577}{3360}}x^{2}-{\frac {148517}{2016}}x+{\frac {216269}{3360}}&{\frac {5}{2}}\leq x<3\\{\frac {4832}{945}}x^{3}-{\frac {27691}{560}}x^{2}+{\frac {113923}{720}}x-{\frac {28145}{168}}&3\leq x<{\frac {7}{2}}\\-{\frac {4832}{945}}x^{3}+{\frac {58393}{1008}}x^{2}-{\frac {52223}{240}}x+{\frac {2048227}{7560}}&{\frac {7}{2}}\leq x<4\\{\frac {35033}{10080}}x^{3}-{\frac {75827}{1680}}x^{2}+{\frac {981101}{5040}}x-{\frac {234149}{840}}&4\leq x<{\frac {9}{2}}\\-{\frac {5261}{3360}}x^{3}+{\frac {38509}{1680}}x^{2}-{\frac {112487}{1008}}x+{\frac {30347}{168}}&{\frac {9}{2}}\leq x<5\\{\frac {1109}{2520}}x^{3}-{\frac {24077}{3360}}x^{2}+{\frac {78311}{2016}}x-{\frac {141311}{2016}}&5\leq x<{\frac {11}{2}}\\-{\frac {19}{280}}x^{3}+{\frac {1361}{1120}}x^{2}-{\frac {14617}{2016}}x+{\frac {4151}{288}}&{\frac {11}{2}}\leq x<6\\{\frac {127}{30240}}x^{3}-{\frac {55}{672}}x^{2}+{\frac {5359}{10080}}x-{\frac {11603}{10080}}&6\leq x<{\frac {13}{2}}\\-{\frac {1}{30240}}x^{3}+{\frac {1}{1440}}x^{2}-{\frac {7}{1440}}x+{\frac {49}{4320}}&{\frac {13}{2}}\leq x<7\\0&{\text{otherwise}}\end{cases}}}

### Compactly supported B-spline wavelet of order 5

The two-scale relation for the compactly supported B-spline wavelet of order 5 is

- ψ C , 5 ( x ) = 1 5806080 [ N 5 ( 2 x ) − 507 N 5 ( 2 x − 1 ) + 17128 N 5 ( 2 x − 2 ) − 166304 N 5 ( 2 x − 3 ) + 748465 N 5 ( 2 x − 4 ) {\displaystyle \psi _{C,5}(x)={\frac {1}{5806080}}{\Big [}N_{5}(2x)-507N_{5}(2x-1)+17128N_{5}(2x-2)-166304N_{5}(2x-3)+748465N_{5}(2x-4)} - - - - − 1900115 N 5 ( 2 x − 5 ) + 2973560 N 5 ( 2 x − 6 ) − 2973560 N 5 ( 2 x − 7 ) + 1900115 N 5 ( 2 x − 8 ) {\displaystyle -1900115N_{5}(2x-5)+2973560N_{5}(2x-6)-2973560N_{5}(2x-7)+1900115N_{5}(2x-8)} - − 748465 N 5 ( 2 x − 9 ) + 166304 N 5 ( 2 x − 10 ) − 17128 N 5 ( 2 x − 11 ) + 507 N 5 ( 2 x − 12 ) − N 5 ( 2 x − 13 ) ] {\displaystyle -748465N_{5}(2x-9)+166304N_{5}(2x-10)-17128N_{5}(2x-11)+507N_{5}(2x-12)-N_{5}(2x-13){\Big ]}}

The closed form expression for compactly supported B-spline wavelet of order 5 is

- ψ C , 5 ( x ) = { 1 8709120 x 4 0 ≤ x < 1 2 − 73 1244160 x 4 + 1 8505 x 3 − 1 11340 x 2 + 1 34020 x − 1 272160 1 2 ≤ x < 1 9581 4354560 x 4 − 19417 2177280 x 3 + 1303 96768 x 2 − 19609 2177280 x + 6547 2903040 1 ≤ x < 3 2 − 118931 4354560 x 4 + 366119 2177280 x 3 − 186253 483840 x 2 + 121121 311040 x − 427181 2903040 3 2 ≤ x < 2 759239 4354560 x 4 − 3146561 2177280 x 3 + 6466601 1451520 x 2 − 13202873 2177280 x + 26819897 8709120 2 ≤ x < 5 2 − 2980409 4354560 x 4 + 5183893 725760 x 3 − 13426333 483840 x 2 + 426589 8960 x − 12635243 414720 5 2 ≤ x < 3 7873577 4354560 x 4 − 16524079 725760 x 3 + 7385369 69120 x 2 − 17868671 80640 x + 497668543 2903040 3 ≤ x < 7 2 − 14714327 4354560 x 4 + 108543091 2177280 x 3 − 56901557 207360 x 2 + 1454458651 2177280 x − 5286189059 8709120 7 2 ≤ x < 4 15619 3402 x 4 − 33822017 435456 x 3 + 15828929 32256 x 2 − 597598433 435456 x + 277413649 193536 4 ≤ x < 9 2 − 15619 3402 x 4 + 38150335 435456 x 3 − 20157247 32256 x 2 + 859841695 435456 x − 64472345 27648 9 2 ≤ x < 5 14714327 4354560 x 4 − 4466137 62208 x 3 + 165651247 290304 x 2 − 875490655 435456 x + 4614904015 1741824 5 ≤ x < 11 2 − 7873577 4354560 x 4 + 30717383 725760 x 3 − 179437319 483840 x 2 + 16606729 11520 x − 869722273 414720 11 2 ≤ x < 6 2980409 4354560 x 4 − 12698561 725760 x 3 + 16211669 96768 x 2 − 19138891 26880 x + 3289787993 2903040 6 ≤ x < 13 2 − 759239 4354560 x 4 + 10519741 2177280 x 3 − 10403603 207360 x 2 + 71964499 311040 x − 3481646837 8709120 13 2 ≤ x < 7 118931 4354560 x 4 − 1774639 2177280 x 3 + 630259 69120 x 2 − 14096161 311040 x + 245108501 2903040 7 ≤ x < 15 2 − 9581 4354560 x 4 + 21863 311040 x 3 − 407387 483840 x 2 + 9758873 2177280 x − 25971499 2903040 15 2 ≤ x < 8 73 1244160 x 4 − 4343 2177280 x 3 + 5273 207360 x 2 − 313703 2177280 x + 380873 1244160 8 ≤ x < 17 2 − 1 8709120 x 4 + 1 241920 x 3 − 1 17920 x 2 + 3 8960 x − 27 35840 17 2 ≤ x < 9 0 otherwise {\displaystyle \psi _{C,5}(x)={\begin{cases}{\frac {1}{8709120}}x^{4}&0\leq x<{\frac {1}{2}}\\-{\frac {73}{1244160}}x^{4}+{\frac {1}{8505}}x^{3}-{\frac {1}{11340}}x^{2}+{\frac {1}{34020}}x-{\frac {1}{272160}}&{\frac {1}{2}}\leq x<1\\{\frac {9581}{4354560}}x^{4}-{\frac {19417}{2177280}}x^{3}+{\frac {1303}{96768}}x^{2}-{\frac {19609}{2177280}}x+{\frac {6547}{2903040}}&1\leq x<{\frac {3}{2}}\\-{\frac {118931}{4354560}}x^{4}+{\frac {366119}{2177280}}x^{3}-{\frac {186253}{483840}}x^{2}+{\frac {121121}{311040}}x-{\frac {427181}{2903040}}&{\frac {3}{2}}\leq x<2\\{\frac {759239}{4354560}}x^{4}-{\frac {3146561}{2177280}}x^{3}+{\frac {6466601}{1451520}}x^{2}-{\frac {13202873}{2177280}}x+{\frac {26819897}{8709120}}&2\leq x<{\frac {5}{2}}\\-{\frac {2980409}{4354560}}x^{4}+{\frac {5183893}{725760}}x^{3}-{\frac {13426333}{483840}}x^{2}+{\frac {426589}{8960}}x-{\frac {12635243}{414720}}&{\frac {5}{2}}\leq x<3\\{\frac {7873577}{4354560}}x^{4}-{\frac {16524079}{725760}}x^{3}+{\frac {7385369}{69120}}x^{2}-{\frac {17868671}{80640}}x+{\frac {497668543}{2903040}}&3\leq x<{\frac {7}{2}}\\-{\frac {14714327}{4354560}}x^{4}+{\frac {108543091}{2177280}}x^{3}-{\frac {56901557}{207360}}x^{2}+{\frac {1454458651}{2177280}}x-{\frac {5286189059}{8709120}}&{\frac {7}{2}}\leq x<4\\{\frac {15619}{3402}}x^{4}-{\frac {33822017}{435456}}x^{3}+{\frac {15828929}{32256}}x^{2}-{\frac {597598433}{435456}}x+{\frac {277413649}{193536}}&4\leq x<{\frac {9}{2}}\\-{\frac {15619}{3402}}x^{4}+{\frac {38150335}{435456}}x^{3}-{\frac {20157247}{32256}}x^{2}+{\frac {859841695}{435456}}x-{\frac {64472345}{27648}}&{\frac {9}{2}}\leq x<5\\{\frac {14714327}{4354560}}x^{4}-{\frac {4466137}{62208}}x^{3}+{\frac {165651247}{290304}}x^{2}-{\frac {875490655}{435456}}x+{\frac {4614904015}{1741824}}&5\leq x<{\frac {11}{2}}\\-{\frac {7873577}{4354560}}x^{4}+{\frac {30717383}{725760}}x^{3}-{\frac {179437319}{483840}}x^{2}+{\frac {16606729}{11520}}x-{\frac {869722273}{414720}}&{\frac {11}{2}}\leq x<6\\{\frac {2980409}{4354560}}x^{4}-{\frac {12698561}{725760}}x^{3}+{\frac {16211669}{96768}}x^{2}-{\frac {19138891}{26880}}x+{\frac {3289787993}{2903040}}&6\leq x<{\frac {13}{2}}\\-{\frac {759239}{4354560}}x^{4}+{\frac {10519741}{2177280}}x^{3}-{\frac {10403603}{207360}}x^{2}+{\frac {71964499}{311040}}x-{\frac {3481646837}{8709120}}&{\frac {13}{2}}\leq x<7\\{\frac {118931}{4354560}}x^{4}-{\frac {1774639}{2177280}}x^{3}+{\frac {630259}{69120}}x^{2}-{\frac {14096161}{311040}}x+{\frac {245108501}{2903040}}&7\leq x<{\frac {15}{2}}\\-{\frac {9581}{4354560}}x^{4}+{\frac {21863}{311040}}x^{3}-{\frac {407387}{483840}}x^{2}+{\frac {9758873}{2177280}}x-{\frac {25971499}{2903040}}&{\frac {15}{2}}\leq x<8\\{\frac {73}{1244160}}x^{4}-{\frac {4343}{2177280}}x^{3}+{\frac {5273}{207360}}x^{2}-{\frac {313703}{2177280}}x+{\frac {380873}{1244160}}&8\leq x<{\frac {17}{2}}\\-{\frac {1}{8709120}}x^{4}+{\frac {1}{241920}}x^{3}-{\frac {1}{17920}}x^{2}+{\frac {3}{8960}}x-{\frac {27}{35840}}&{\frac {17}{2}}\leq x<9\\0&{\text{otherwise}}\end{cases}}}

### Images of compactly supported B-spline wavelets

B-spline wavelet of order 1 B-spline wavelet of order 2 B-spline wavelet of order 3 B-spline wavelet of order 4 B-spline wavelet of order 5

## Battle-Lemarie wavelets

The Battle-Lemarie wavelets form a class of orthonormal wavelets constructed using the class of cardinal B-splines. The expressions for these wavelets are given in the [frequency domain](/source/Frequency_domain); that is, they are defined by specifying their Fourier transforms. The Fourier transform of a function of *t*, say, F ( t ) {\displaystyle F(t)} , is denoted by F ^ ( ω ) {\displaystyle {\hat {F}}(\omega )} .

### Definition

Let *m* be a positive integer and let N m ( x ) {\displaystyle N_{m}(x)} be the cardinal B-spline of order *m*. The Fourier transform of N m ( x ) {\displaystyle N_{m}(x)} is N ^ m ( ω ) {\displaystyle {\hat {N}}_{m}(\omega )} . The scaling function ϕ m ( t ) {\displaystyle \phi _{m}(t)} for the *m*-th order Battle-Lemarie wavelet is that function whose Fourier transform is

- ϕ ^ m ( ω ) = N ^ m ( ω ) ( ∑ k = − ∞ ∞ | N ^ m ( ω + 2 π k ) | 2 ) 1 / 2 . {\displaystyle {\hat {\phi }}_{m}(\omega )={\frac {{\hat {N}}_{m}(\omega )}{\left(\sum _{k=-\infty }^{\infty }\vert {\hat {N}}_{m}(\omega +2\pi k)\vert ^{2}\right)^{1/2}}}.}

The *m*-th order Battle-Lemarie wavelet is the function ψ B L , m ( t ) {\displaystyle \psi _{BL,m}(t)} whose Fourier transform is

- ψ ^ B L , m ( ω ) = − e − i ω / 2 ϕ ^ m ( ω + 2 π ) ¯ ϕ ^ m ( ω 2 ) ϕ ^ m ( ω 2 + π ) ¯ {\displaystyle {\hat {\psi }}_{BL,m}(\omega )=-{\frac {e^{-i\omega /2}\,\,{\overline {{\hat {\phi }}_{m}(\omega +2\pi )}}\,\,{\hat {\phi }}_{m}\left({\frac {\omega }{2}}\right)}{\overline {{\hat {\phi }}_{m}\left({\frac {\omega }{2}}+\pi \right)}}}}

## References

1. **[^](#cite_ref-ten_1-0)** Michael Unser (1997). ["Ten good reasons for using spline wavelets"](http://bigwww.epfl.ch/publications/unser9702.pdf) (PDF). In Aldroubi, Akram; Laine, Andrew F.; Unser, Michael A. (eds.). *Wavelet Applications in Signal and Image Processing V*. Vol. 3169. pp. 422–431. [Bibcode](/source/Bibcode_(identifier)):[1997SPIE.3169..422U](https://ui.adsabs.harvard.edu/abs/1997SPIE.3169..422U). [doi](/source/Doi_(identifier)):[10.1117/12.292801](https://doi.org/10.1117%2F12.292801). [S2CID](/source/S2CID_(identifier)) [12705597](https://api.semanticscholar.org/CorpusID:12705597). Retrieved 21 December 2014.

1. **[^](#cite_ref-2)** Chui, Charles K, and Jian-zhong Wang (1991). ["A cardinal spline approach to wavelets"](http://www.ams.org/journals/proc/1991-113-03/S0002-9939-1991-1077784-X/S0002-9939-1991-1077784-X.pdf) (PDF). *Proceedings of the American Mathematical Society*. **113** (3): 785–793. [doi](/source/Doi_(identifier)):[10.2307/2048616](https://doi.org/10.2307%2F2048616). [JSTOR](/source/JSTOR_(identifier)) [2048616](https://www.jstor.org/stable/2048616). Retrieved 22 January 2015.{{[cite journal](https://en.wikipedia.org/wiki/Template:Cite_journal)}}: CS1 maint: multiple names: authors list ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_multiple_names:_authors_list))

1. ^ [***a***](#cite_ref-ChuiCompact_3-0) [***b***](#cite_ref-ChuiCompact_3-1) Charles K. Chui and Jian-Zhong Wang (April 1992). ["On Compactly Supported Spline Wavelets and a Duality Principle"](http://www.shsu.edu/~mth_jxw/pdfflies/CWTRAN.pdf) (PDF). *Transactions of the American Mathematical Society*. **330** (2): 903–915. [doi](/source/Doi_(identifier)):[10.1090/s0002-9947-1992-1076613-3](https://doi.org/10.1090%2Fs0002-9947-1992-1076613-3). Retrieved 21 December 2014.

1. **[^](#cite_ref-4)** Charles K Chui (1992). *An Introduction to Wavelets*. Academic Press. p. 177.

1. **[^](#cite_ref-Daubechies_5-0)** Ingrid Daubechies (1992). [*Ten Lectures on Wavelets*](https://archive.org/details/tenlecturesonwav0000daub). Philadelphia: Society for Industrial and Applied Mathematics. pp. [146–153](https://archive.org/details/tenlecturesonwav0000daub/page/146). [ISBN](/source/ISBN_(identifier)) [9780898712742](https://en.wikipedia.org/wiki/Special:BookSources/9780898712742).

1. **[^](#cite_ref-6)** Christopher Heil (2011). [*A Basis Theory Primer*](https://archive.org/details/basistheoryprime00chei). Birkhauser. pp. [177](https://archive.org/details/basistheoryprime00chei/page/n203)–188. [ISBN](/source/ISBN_(identifier)) [9780817646868](https://en.wikipedia.org/wiki/Special:BookSources/9780817646868).

1. **[^](#cite_ref-7)** Charles K Chui (1992). *An Introduction to Wavelets*. Academic Press. p. 249.

1. **[^](#cite_ref-8)** Charles K Chui (1992). *An Introduction to Wavelets*. Academic Press. p. 184.

## Further reading

- Amir Z Averbuch and Valery A Zheludev (2007). ["Wavelet transforms generated by splines"](http://www.cs.tau.ac.il/~zhel/PS/splitr3AA.pdf) (PDF). *International Journal of Wavelets, Multiresolution and Information Processing*. **257** (5). Retrieved 21 December 2014.

- Amir Z. Averbuch, Pekka Neittaanmaki, and Valery A. Zheludev (2014). *Spline and Spline Wavelet Methods with Applications to Signal and Image Processing Volume I*. Springer. [ISBN](/source/ISBN_(identifier)) [978-94-017-8925-7](https://en.wikipedia.org/wiki/Special:BookSources/978-94-017-8925-7).{{[cite book](https://en.wikipedia.org/wiki/Template:Cite_book)}}: CS1 maint: multiple names: authors list ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_multiple_names:_authors_list))

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