index.md

NormalSplines.jl

1-D Normal Splines implementation in Julia

Pages = [
      "index.md",
      "Usage.md",
      "Public-API.md",
      "Interpolating-Normal-Splines.md",
      "Reproducing-Kernel-of-Bessel-Potential-space.md"
]
Depth = 3

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