NormalSplines.jl

1-D Normal Splines implementation in Julia

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Public API

NormalSplines.jl implements the normal splines method for solving following interpolation problem:

Problem: Given points $x_1 \lt x_2 \lt \dots \lt x_{n_1}$, $s_1 \lt s_2 \lt \dots \lt s_{n_2}$ and $t_1 \lt t_2 \lt \dots \lt t_{n_3}$ find a function $f$ such that

\[\tag{1} \begin{aligned} & f(x_i) = u_i \, , \quad i = 1, 2, \dots, n_1 \, , \\ & f'(s_j) = v_j \, , \quad j = 1, 2, \dots, n_2 \, , \\ & f''(t_k) = w_k \, , \quad k = 1, 2, \dots, n_3 \, , \\ & n_1 \gt 0 \, , \ \ n_2 \ge 0 \, , \ \ n_3 \ge 0 \, . \end{aligned}\]

Knots $\{x_i\}$, $\{s_j\}$ and $\{t_k\}$ may coincide. We assume that function $f$ is an element of an appropriate reproducing kernel Hilbert space $H$.

The normal splines method consists in finding a solution of system (1) having minimal norm in Hilbert space $H$, thus the interpolation normal spline $\sigma$ is defined as follows:

\[\tag{2} \sigma = {\rm arg\,min}\{ \| f \|^2_H : (1), \forall f \in H \} \, .\]

Normal splines method is based on the following functional analysis results:

Embedding theorems (Sobolev embedding theorem and Bessel potential space embedding theorem)
The Riesz representation theorem for Hilbert spaces
Reproducing kernel properties.

Using these results it is possible to reduce task (2) to solving a system of linear equations with symmetric positive definite Gram matrix.

Normal splines are constructed in Sobolev space $W^l_2 [a, b]$ with norm:

\[\| f \|_{W_2^l} = \left ( \sum_{i=0}^{l-1} (f^{(i)}(a))^2 + \int_a^b (f^{(l)}(s))^2 ds \right )^{1/2} \, , \quad l = 3, 2, \text{or} \ 1,\]

and in Bessel potential space $H_\varepsilon^l(R)$ defined as

\[ H^l_\varepsilon (R) = \left\{ f | f \in S' , ( \varepsilon ^2 + | s |^2 )^{l/2}{\mathcal F} [f] \in L_2 (R) \right\} \, , \varepsilon \gt 0 , \quad l = 3, 2, \text{or} \ 1,\]

where $S' (R)$ is space of L. Schwartz tempered distributions and $\mathcal F [f]$ is a Fourier transform of the $f$. Space $H^l_\varepsilon (R)$ is Hilbert space with norm

\[\| f \|_ {H^l_\varepsilon} = \| ( \varepsilon ^2 + | s |^2 )^{l/2} {\mathcal F} [\varphi ] \|_{L_2} \ .\]

If $n_3 > 0$ then value of $l$ is $3$, in a case when $n_3 = 0$ and $n_2 > 0$ the value of $l$ must be $2$ or $3$.

Detailed explanation is given in Interpolating Normal Splines