Spline wavelet

1. The support of ${\displaystyle N_{m}(x)}$ is the closed interval ${\displaystyle [0,m]}$.
2. The function ${\displaystyle N_{m}(x)}$ is non-negative, that is, ${\displaystyle N_{m}(x)>0}$ for ${\displaystyle 0<x<m}$.
3. ${\displaystyle \sum _{k=-\infty }^{\infty }N_{m}(x-k)=1}$ for all ${\displaystyle x}$.
4. The cardinal B-splines of orders m and m-1 are related by the identity: ${\displaystyle N_{m}(x)={\frac {x}{m-1}}N_{m-1}(x)+{\frac {m-x}{m-1}}N_{m-1}(x-1)}$.
5. The function ${\displaystyle N_{m}(x)}$ is symmetrical about ${\displaystyle x={\frac {m}{2}}}$, that is, ${\displaystyle N_{m}\left({\frac {m}{2}}-x\right)=N_{m}\left({\frac {m}{2}}+x\right)}$.
6. The derivative of ${\displaystyle N_{m}(x)}$ is given by ${\displaystyle N_{m}^{\prime }(x)=N_{m-1}(x)-N_{m-1}(x-1)}$.
7. ${\displaystyle \int _{-\infty }^{\infty }N_{m}(x)\,dx=1}$

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

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

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

${\displaystyle A\left\Vert \{c_{k}\}\right\Vert ^{2}\leq \left\Vert \sum _{k=-\infty }^{\infty }c_{k}N_{m}(x-k)\right\Vert ^{2}\leq B\left\Vert \{c_{k}\}\right\Vert ^{2}}$

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

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

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

The two-scale relation for this B-spline is

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

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

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

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

The two-scale relation for this wavelet is

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

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

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

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

The two-scale relation for this wavelet is

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

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

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

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

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

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

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

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

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

The two-scale relation is

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

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

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

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

${\displaystyle \sum _{k=-\infty }^{\infty }c_{m,k}N_{m}\left(j+{\frac {m}{2}}-k\right)=f_{j}}$ for all ${\displaystyle j}$,

is known as the cardinal spline interpolation problem. The special case of this problem where the sequence ${\displaystyle \{f_{j}\}}$ is the sequence ${\displaystyle \delta _{0j}}$, where ${\displaystyle \delta _{ij}}$ is the Kronecker delta function ${\displaystyle \delta _{ij}}$ defined by

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

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 ${\displaystyle L_{m}(x)}$ and is given by

${\displaystyle L_{m}(x)=\sum _{k=-\infty }^{\infty }c_{m,k}N_{m}\left(x+{\frac {m}{2}}-k\right)}$

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

${\displaystyle \sum _{k=-\infty }^{\infty }c_{m,k}N_{m}\left(j+{\frac {m}{2}}-k\right)=\delta _{0j}}$

The fundamental cardinal interpolatory spline ${\displaystyle L_{m}(x)}$ can be determined using Z-transforms. Using the following notations

${\displaystyle A(z)=\sum _{k=-\infty }^{\infty }\delta _{k0}z^{k}=1,}$

${\displaystyle B_{m}(z)=\sum _{k=-\infty }^{\infty }N_{m}\left(k+{\frac {m}{2}}\right)z^{k},}$

${\displaystyle C_{m}(z)=\sum _{k=-\infty }^{\infty }c_{m,k}z^{k},}$

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

${\displaystyle B_{m}(z)C_{m}(z)=A(z)}$

from which we get

${\displaystyle C_{m}(z)={\frac {1}{B_{m}(z)}}}$.

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

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

${\displaystyle B_{4}(x)=\sum _{k=-\infty }^{\infty }N_{4}(2+k)z^{k}}$

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

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

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

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

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

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

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

${\displaystyle L_{4}(x)=\sum _{k=-\infty }^{\infty }(-1)^{k}{\sqrt {3}}(2-{\sqrt {3}})^{|k|}N_{4}(x+2-k)}$

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

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

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

The wavelet of order 2 using interpolatory spline is given by

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

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

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

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

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

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

${\displaystyle \psi _{I,2}(x)\approx {\begin{cases}0.07142668x+0.17856670&-2.5<x\leq -2\\-0.48084803x-0.92598272&-2<x\leq -1.5\\2.0088293x+2.8085333&-1.5<x\leq -1\\-7.5684795x-6.7687755&-1<x\leq -0.5\\28.245949x+11.138439&-0.5<x\leq 0\\-57.415316x+11.138439&0<x\leq 0.5\\57.415316x-46.276878&0.5<x\leq 1\\-28.245949x+39.384388&1<x\leq 1.5\\7.5684795x-14.337255&1.5<x\leq 2\\-2.0088293x+4.8173625&2<x\leq 2.5\\0.48084803x-1.4068308&2.5<x\leq 3\\-0.07142668x+0.24999338&3<x\leq 3.5\\0&{otherwise}\end{cases}}}$

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

${\displaystyle \psi _{I,m}(x)=\sum _{-\infty }^{\infty }q_{n}N_{m}(2x-n)}$ where ${\displaystyle q_{n}=\sum _{j=0}^{m}(-1)^{j}{m \choose j}c_{m+n-j-1}.}$

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

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

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

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

which is the well-known Haar wavelet.

1. The support of ${\displaystyle \psi _{C,m}(x)}$ is the closed interval ${\displaystyle [0,2m-1]}$.
2. The wavelet ${\displaystyle \psi _{C,m}(x)}$ is the unique wavelet with minimum support in the following sense: If ${\displaystyle \eta (x)\in W_{0}}$ generates ${\displaystyle W_{0}}$ and has support not exceeding ${\displaystyle 2m-1}$ in length then ${\displaystyle \eta (x)=c_{0}\psi _{C,m}(x-n_{0})}$ for some nonzero constant ${\displaystyle c_{0}}$ and for some integer ${\displaystyle n_{0}}$.[8]
3. ${\displaystyle \psi _{C,m}(x)}$ is symmetric for even m and antisymmetric for odd m.

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

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

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

${\displaystyle N_{m}(2x-l)=\sum _{k=-\infty }^{\infty }\left[a_{m,l-2k}N_{m}(x-k)+b_{m,l-2k}\psi _{C,m}(x-k)\right]}$

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

${\displaystyle a_{m,j}=-{\frac {(-1)^{j}}{2}}\sum _{l=-\infty }^{\infty }q_{-j+2m-2l+1}c_{2m,l},}$

${\displaystyle b_{m,j}={\frac {(-1)^{j}}{2}}\sum _{l=-\infty }^{\infty }p_{-j+2m-2l+1}c_{2m,l}.}$

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

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

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

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

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

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

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

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

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

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

${\displaystyle \psi _{C,3}(x)={\frac {1}{480}}{\Big [}(N_{3}(2x)-29N_{3}(2x-1)+147N_{3}(2x-2)-303N_{3}(2x-3)+}$

${\displaystyle 303N_{3}(2x-4)-147N_{3}(2x-5)+29N_{3}(2x-6)-N_{3}(2x-7){\Big ]}}$

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

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

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

${\displaystyle \psi _{C,4}(x)={\frac {1}{40320}}{\Big [}N_{4}(2x)-124N_{4}(2x-1)+1677N_{4}(2x-2)-7904N_{4}(2x-3)+18482N_{4}(2x-4)-}$

${\displaystyle 24264N_{4}(2x-5)+18482N_{4}(2x-6)-7904N_{4}(2x-7)+1677N_{4}(2x-8)-124N_{4}(2x-9)+N_{4}(2x-10){\Big ]}}$

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

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

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

${\displaystyle \psi _{C,5}(x)={\frac {1}{5806080}}{\Big [}N_{5}(2x)-507N_{5}(2x-1)+17128N_{5}(2x-2)-166304N_{5}(2x-3)+748465N_{5}(2x-4)}$

${\displaystyle -1900115N_{5}(2x-5)+2973560N_{5}(2x-6)-2973560N_{5}(2x-7)+1900115N_{5}(2x-8)}$

${\displaystyle -748465N_{5}(2x-9)+166304N_{5}(2x-10)-17128N_{5}(2x-11)+507N_{5}(2x-12)-N_{5}(2x-13){\Big ]}}$

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

${\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}}}$

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