In the mathematical theory of wavelets, a spline wavelet is a wavelet constructed using a spline function. 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 interpolation formula. Though these wavelets are orthogonal, they do not have compact supports. There is a certain class of wavelets, unique in some sense, constructed using B-splines and having compact supports. Even though these wavelets are not orthogonal they have some special properties that have made them quite popular. 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. The Battle-Lemarie wavelets are also wavelets constructed using spline functions.
Cardinal B-splines Let n be a fixed non-negative integer. Let Cn denote the set of all real-valued functions defined over the set of real numbers such that each function in the set as well its first n derivatives are continuous everywhere. A bi-infinite sequence . . . x−2, x−1, x0, x1, x2, . . . such that xr < xr+1 for all r and such that xr approaches ±∞ as r approaches ±∞ is said to define a set of knots. A spline of order n with a set of knots {xr} is a function S(x) in Cn such that, for each r, the restriction of S(x) to the interval [xr, xr+1) coincides with a polynomial with real coefficients of degree at most n in x. If the separation xr+1 - xr, 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 Nm(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 The support of N m ( x ) {\displaystyle N_{m}(x)} is the closed interval [ 0 , m ] {\displaystyle [0,m]} . 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} .
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