Glossary—Splines

Polynomial Spline For the degree \(n=0,\) the piecewise polynomial \(p_{{\mathbb{S}},{\mathbb{F}},{\mathbb{P}}}^{0}\) is called a piecewise-constant spline when \({\mathbb{F}}({\mathbb{S}})\) is such that \(\forall k\in{\mathbb{Z}}:f_{k}=\frac{p_{{\mathbf{a}}_{k-1}^{0}}((s_{k-1}+s_{k})/2)+p_{{\mathbf{a}}_{k}^{0}}((s_{k}+s_{k+1})/2)}{2}.\) For a positive integer degree \(n,\) the piecewise polynomial \(p_{{\mathbb{S}},{\mathbb{F}},{\mathbb{P}}}^{n}\) is called a spline when it happens to be of differentiability class \({\mathcal{C}}^{n-1}\) over \({\mathbb{R}}.\)

Uniform Polynomial Spline A polynomial spline is said to be uniform when it admits the uniform split \({\mathbb{S}}(s_{0})\) as its split of the real numbers.

Polynomial B-Spline A polynomial B-spline of nonnegative integer degree is uniquely defined as a uniform spline of minimal support that is pointwise even-symmetric and that satisfies the pointwise partition of unity. It is notated \(\beta^{n}:{\mathbb{R}}\rightarrow{\mathbb{R}},\) where \(n\) is the degree.

Cardinal Polynomial B-Spline A cardinal polynomial B-spline of nonnegative integer degree is uniquely defined as a uniform spline that is interpolating, pointwise even-symmetric, and that satisfies the pointwise partition of unity. It is notated \(\eta^{n}:{\mathbb{R}}\rightarrow{\mathbb{R}},\) where \(n\) is the degree.

Polynomial Dual B-Spline A polynomial dual B-spline is notated \(\mathring{\beta}^{m,n}.\) It is a real function indexed by a nonnegative integer dual degree \(m\) and a nonnegative integer primal degree \(n.\) It is uniquely defined as the spline of dual degree \(m\) that is integer-shift-orthonormal to a polynomial B-spline of primal degree \(n,\) so that \({\mathbf{[\![}}k=0\,{\mathbf{]\!]}}=\int_{-\infty}^{\infty}\,\mathring{\beta}^{m,n}(x)\,\beta^{n}(x+k)\,{\mathrm{d}}x,\) where the notation \({\mathbf{[\![}}P\,{\mathbf{]\!]}}\) is that of the Iverson bracket.

Polynomial Orthonormal B-Spline A polynomial orthonormal B-spline is notated \(\phi^{n}.\) It is a real function indexed by a nonnegative integer degree \(n.\) It is uniquely defined as the uniform spline of degree \(n\) that is integer-shift-orthonormal to itself, so that \({\mathbf{[\![}}k=0\,{\mathbf{]\!]}}=\int_{-\infty}^{\infty}\,\phi^{n}(x)\,\phi^{n}(x+k)\,{\mathrm{d}}x,\) along with \(\phi^{n}(x)=\sum_{k\in{\mathbb{Z}}}\,p[k]\,\beta^{n}(x-k)\) for some well-chosen sequence \(p,\) where the notation \({\mathbf{[\![}}P\,{\mathbf{]\!]}}\) is that of the Iverson bracket.

Knots The knots of a polynomial spline \(f\) of nonnegative integer degree \(n\) are those arguments \(x\) at which \(\frac{{\mathrm{d}}^{n+1}f(x)}{{\mathrm{d}}x^{n+1}}\not\in{\mathbb{R}}.\) When the spline is uniform, they are a finite subset of the uniform split \({\mathbb{S}}(s_{0}+\frac{n+1}{2})\) associated to the spline.

M-Scale Relation The M-scale relation expresses a B-spline of nonnegative integer degree as a sum of translated and rescaled (minified) B-splines of same degree.