Interpolating Normal Splines: One-dimensional case

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This topic is concerned with numerical solution of the following interpolation problems:

Problem 1. Given points $x_1 \lt  x_2 \lt \dots \lt x_{n_1}$ find a function $f$ such that
$$\begin{eqnarray} f(x_i) = u_i , \qquad  i = 1, 2, \dots, n_1 \ , \end{eqnarray}$$
Problem 2. Given points $x_1 \lt  x_2 \lt \dots \lt x_{n_1}$, $s_1 \lt s_2 \lt \dots \lt s_{n_2}$ find a function $f$ such that
$$\begin{eqnarray} f(x_i) &=& u_i , \qquad  i  = 1, 2, \dots, n_1 \ , \\ \nonumber f'(s_j) &=& v_j , \qquad  j = 1, 2, \dots, n_2 \ , \quad n_1 \ge 0 \, , \  \ \  n_2 \ge 0 \, , \end{eqnarray}$$
Problem 3. 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
$$\begin{eqnarray} f(x_i) &=& u_i , \qquad  i = 1, 2, \dots, n_1 \ , \\ \nonumber f'(s_j) &=& v_j , \qquad  j = 1, 2, \dots, n_2 \ , \\ \nonumber f''(t_k) &=& w_k , \qquad  k = 1, 2, \dots, n_3 \, , \quad n_1 \ge 0 \, ,  \ \  n_2 \ge 0 \, , \ \  n_3 \ge 0 \, . \end{eqnarray}$$ Note that knots $\{x_i\}$, $\{s_j\}$ and $\{t_k\}$ may coincide.

Assume that function $f$ is an element of Hilbert space $H = H(X)$ over a set $X$ (here $X$ is $R$ or an interval from $R$) and Hilbert space is selected in a such way that it is continuously embedded in the space $C^2(X)$ of functions continuous with their second derivatives and therefore functionals $F_i$, $F'_j$, and $F''_k$
$$\begin{eqnarray*} && F_i(\varphi) = \varphi (x_i) \, , \ \ F'_j(\varphi) = \varphi' (s_j) \, , \ \  F''_k(\varphi) = \varphi''(t_k) \, ,  \forall \varphi \in H \, , \\  \nonumber &&  x_i, s_j, t_k \in X \, , \ \  i = 1, 2, \dots, n_1 \, , \ \  j = 1, 2, \dots, n_2 \, ,   \ \  k = 1, 2, \dots, n_3 \, . \end{eqnarray*}$$ are linear continuous functionals in $H$. It is obvious that all these functionals are linear independent. In accordance with Riesz representation theorem [1] these linear continuous functionals can be represented in the form of inner product of some elements $h_i, h′_j, h''_k \in H$ and $\varphi \in H$, for any $\varphi \in H$:

$$\begin{eqnarray*} && f(x_i) =  F_i(\varphi) = {\langle h_i,  \varphi \rangle}_H \, ,  \quad  F'_j(\varphi) = {\langle h'_j,  \varphi \rangle}_H  \, , \quad  F''_k(\varphi) = {\langle h''_k,  \varphi \rangle}_H \, , \\  \nonumber &&  \forall \varphi \in H \, , \ \  i = 1, 2, \dots, n_1 \, , \ \  j = 1, 2, \dots, n_2 \, ,  \ \ k = 1, 2, \dots, n_3 \, . \end{eqnarray*}$$ Elements $h_i, h′_j$ and $h''_k$ are twice continuously differentiable functions.

Original constraint systems of Problems 1—3 can be written in form:
$$\begin{eqnarray} && f(x_i) = F_i(f) = {\langle h_i,  f \rangle}_H  = u_i \, ,  \\  \nonumber  && f \in H \, , \ \  i = 1, 2, \dots, n_1 \, ,  \\ \nonumber \\ && f(x_i) =  F_i(f) = {\langle h_i,  f \rangle}_H  = u_i \, , \\ \nonumber &&  f'(s_j) = F'_j(f) = {\langle h'_j,  f \rangle}_H = v_j \, ,   \\  \nonumber && f \in H \, , \ \  i = 1, 2, \dots, n_1 \, , \ \  j = 1, 2, \dots, n_2 \, , \\ \nonumber \\ &&  f(x_i) = F_i(f) = {\langle h_i,  f \rangle}_H  = u_i \, , \\ \nonumber &&  f'(s_j) = F'_j(f) = {\langle h'_j,  f \rangle}_H = v_j \, ,  \\ \nonumber &&  f''(t_k) = F''_k(f) = {\langle h''_k,  f \rangle}_H = w_k\, ,  \\  \nonumber && f \in H \, , \ \  i = 1, 2, \dots, n_1 \, , \ \  j = 1, 2, \dots, n_2 \, , \\ \nonumber && k = 1, 2, \dots, n_3 \, , \end{eqnarray}$$ here all functions $h_i, h'_j, h''_k \in H$ are linear independent and every system of constrains (4), (5), (6) defines a nonempty convex and closed set (as intersection of hyper-planes in the Hilbert space $H$).

Problem of reconstruction of function $f$ satisfying system of constraints (4); (5) or (6) is undetermined. We reformulate it as the problem of finding solution of the corresponding system of constraints that has minimal norm:

$$\begin{eqnarray} && \sigma_1 = {\rm arg\,min}\{  \| f - z \|^2_H : (4), z \in H, \forall f \in H \} \, , \\  && \sigma_2 = {\rm arg\,min}\{  \| f - z \|^2_H : (5), z \in H, \forall f \in H \} \, , \\  && \sigma_3 = {\rm arg\,min}\{  \| f - z \|^2_H : (6), z \in H, \forall f \in H \} \, , \end{eqnarray}$$ where $z, z \in H$ is a "prototype" function. Solution of every problem (7), (8) and (9) exists and it is unique [1, 5] as a projection of element $z$ on the nonempty convex closed set in Hilbert space $H$. Elements $\sigma_1, \sigma_2$ and $\sigma_3$ are interpolating normal splines.

In accordance with generalized Lagrange method ([4], [5]) solution of the problem (7) can be written as follows:
$$\begin{eqnarray} \sigma_1 =  z + \sum _{i=1}^{n_1} \mu_i  h_i  \ , \end{eqnarray}$$ where coefficients $\mu_i$ are defined by the system

$$\begin{eqnarray}  \sum _{l=1}^{n_1} g_{il} \mu_l &=& u_i  - {\langle h_i,  z \rangle}_H \, , \quad 1 \le i \le n_1 \, , \end{eqnarray}$$ solution of the problem (8) can be written as follows:
$$\begin{eqnarray} \sigma_2 =  z + \sum _{i=1}^{n_1} \mu_i  h_i  + \sum _{j=1}^{n_2} \mu'_j  h'_j \ , \end{eqnarray}$$ where coefficients $\mu_i$ and $\mu'_j$ are defined by the system

$$\begin{eqnarray}  \sum _{l=1}^{n_1} g_{il} \mu_l + \sum _{j=1}^{n_2} g'_{ij} \mu'_j &=& u_i - {\langle h_i,  z \rangle}_H \, , \quad 1 \le i \le n_1 \, , \\ \nonumber \sum _{i=1}^{n_1} g'_{ij} \mu_i + \sum _{l=1}^{n_2} g''_{jl} \mu'_l &=& v_j - {\langle h'_j,  z \rangle}_H  \, , \quad 1 \le j \le n_2 \, , \end{eqnarray}$$ and solution of the problem (9) can be presented as:

$$\begin{eqnarray} \sigma_3 =  z + \sum _{i=1}^{n_1} \mu_i  h_i  + \sum _{j=1}^{n_2} \mu'_j  h'_j  + \sum _{k=1}^{n_3} \mu''_k  h''_k \ , \end{eqnarray}$$ where coefficients $\mu_i$, $\mu'_j$ and $\mu''_k$ are defined by the system

$$\begin{eqnarray}  && \sum _{l=1}^{n_1} g_{il} \mu_l + \sum _{j=1}^{n_2} g'_{ij} \mu'_j  + \sum _{k=1}^{n_3} g''_{ik} \mu''_k &=& u_i - {\langle h_i,  z \rangle}_H \, , \quad 1 \le i \le n_1 \, , \\ \nonumber && \sum _{i=1}^{n_1} g'_{ij} \mu_i + \sum _{l=1}^{n_2} g''_{jl} \mu'_l  + \sum _{k=1}^{n_3} g'''_{jk} \mu''_k &=& v_j - {\langle h'_j,  z \rangle}_H  \, , \quad 1 \le j \le n_2 \, ,  \\ \nonumber && \sum _{i=1}^{n_1} g''_{ik} \mu_i + \sum _{j=1}^{n_2} g'''_{jk} \mu'_j  + \sum _{l=1}^{n_3} g^{\rm iv}_{kl} \mu''_l &=& w - {\langle h''_k,  z \rangle}_H  \, , \quad 1 \le k \le n_3 \, , \end{eqnarray}$$ Matrix of every system (11), (13) and (15) is a positive definite symmetric Gram matrix of the corresponding set of linearly independent elements $\{h_i\}$, $\{h_i\}, \{h'_j\}$ and $\{h_i\}, \{h'_j\}$, $\{h''_k\}$ and coefficients $g_{il}, g'_{ij}, g''_{ik}, g''_{jl}, g'''_{jk}, g^{\rm iv}_{kl}$ are defined as follows:

$$\begin{eqnarray} && g_{il} = {\langle h_i,  h_l \rangle}_H \, , \ \ g'_{ij} = {\langle h_i,  h'_j \rangle}_H \, , \ \ g''_{ik} = {\langle h_i,  h''_k \rangle}_H \\  \nonumber && g''_{jl} = {\langle h'_j,  h'_l \rangle}_H \, , \ \ g'''_{jk} = {\langle h'_j,  h''_k \rangle}_H \, , \ \ g^{\rm iv}_{kl} = {\langle h''_k,  h''_l \rangle}_H \, .  \end{eqnarray}$$

Let $H = H(X)$ be a reproducing kernel Hilbert space with reproducing kernel $V(\eta, \xi)$. Remind ([2], [3]) the definition of the reproducing kernel. The reproducing kernel is a such function $V(\eta, \xi)$ that

• for every $\xi \in X$, $V(\eta, \xi)$ as function of $\eta$ belongs to $H$
• for every $\xi \in X$ and every function $\varphi \in H$

$$\begin{eqnarray} \varphi(\xi) = {\langle V(\eta, \xi),  \varphi(\eta) \rangle}_H \end{eqnarray}$$ Reproducing kernel is a symmetric function:
$$\begin{eqnarray*} V(\eta, \xi) = V(\xi, \eta) \, , \end{eqnarray*}$$ also, in the considered here case it is twice continuously differentiable function by $\xi$ and by $\eta$. Differentiating the identity (17) allows to get the identities for derivatives:
$$\begin{eqnarray} \frac {d \varphi(\xi)}{d \xi} = {\left \langle \frac{\partial V(\cdot, \xi)} {\partial \xi}, \varphi \right \rangle}_H , \ \frac {d^2 \varphi(\xi)}{d \xi^2} = {\left \langle \frac{\partial^2 V(\cdot, \xi)} {\partial \xi^2}, \varphi \right \rangle}_H  . \end{eqnarray}$$ Now it is possible to express functions $h_i, h'_j, h''_k$ via the reproducing kernel $V$. Comparing (4), (5), (6) with (17) and (18) we receive:

$$\begin{eqnarray} && h_i (\eta) =  V(\eta, x_i) \, ,  \qquad  i = 1, 2, \dots, n_1 \, \\ \nonumber && h'_j (\eta) =  \frac{\partial V(\eta, s_j)}{\partial \xi}  \, , \quad  j = 1, 2, \dots, n_2 \ , \\ \nonumber && h''_k (\eta) =  \frac{\partial^2 V(\eta, t_k)}{\partial \xi^2}  \, , \ \  k = 1, 2, \dots, n_3 \ . \end{eqnarray}$$

The coefficients (16) of the Gram matrices can be presented as ([3], [11], [12]):

$$\begin{eqnarray} && g_{il} = {\langle h_i,  h_l \rangle}_H = {\langle V(\cdot, x_i),  V(\cdot, x_l) \rangle}_H = V(x_i, x_l) \, , \\  && g'_{ij} = {\langle h_i,  h'_j \rangle}_H = {\left \langle V(\cdot, x_i), \frac{\partial V(\cdot, s_j)}{\partial \xi} \right \rangle}_H =  \frac{\partial V(x_i, s_j)}{\partial \xi} \,  , \\  && g''_{ik} = {\langle h_i,  h''_k \rangle}_H = {\left \langle V(\cdot, x_i), \frac{\partial^2 V(\cdot, t_k)}{\partial \xi^2} \right \rangle}_H =  \frac{\partial^2 V(x_i, t_k)}{\partial \xi^2} \, . \end{eqnarray}$$  With the help of (17) and (21), we can also calculate $g''_{jl}$ ([11], [12]):

$$\begin{eqnarray} g''_{jl} = {\langle h'_j,  h'_l \rangle}_H &=& {\left \langle \frac{\partial V(\cdot, s_j)}{\partial \xi}, \frac{\partial V(\cdot, s_l)}{\partial \xi} \right \rangle}_H = \\ \nonumber && \left . \frac {d} {d \xi} {\left \langle V(\cdot, \xi), \frac{\partial V(\cdot, s_l)}{\partial \xi} \right \rangle}_H \right \vert_{\xi = s_j} = \\ \nonumber && \left . \frac {d} {d \xi} \left ( \frac{\partial V(\xi, s_l)} {\partial \xi} \right ) \right |_{\xi = s_j} = \frac {\partial^2 V(s_j, s_l)} {\partial \eta \partial \xi} \, . \end{eqnarray}$$ Further

$$\begin{eqnarray} g'''_{jk} = {\langle h'_j,  h''_k \rangle}_H  =  \frac {\partial^3 V(s_j, t_k)} {\partial \eta \partial \xi^2} \, , \\ g^{\rm iv}_{kl} = {\langle h''_k,  h''_l \rangle}_H  =  \frac {\partial^4 V(t_k, t_l)}{\partial \eta^2 \partial \xi^2}  \, , \end{eqnarray}$$ and

$$\begin{eqnarray} && {\langle h_i,  z \rangle}_H  =  {\langle V(\cdot, x_i),  z \rangle}_H  = z(x_i)  \, , \\ \nonumber && {\langle h'_j,  z \rangle}_H  =  z'(s_j) \, , \\ \nonumber &&{\langle h''_k,  z \rangle}_H  =  z''(t_k) \, . \end{eqnarray}$$

Here normal splines will be constructed in Sobolev spaces $W^3_2 [a, b]$, $W^3_2 [a, \infty)$  and in Bessel potential space $H^3_\varepsilon (R)$ (See [6, 7, 8, 9] for details). Elements of these spaces can be treated as twice continuously differentiable functions.
Reproducing kernel for Sobolev spaces $W^l_2 [0,1]$ (here $l$ — any positive integer) was constructed in work [10]. Thus, reproducing kernel for Sobolev space $W^3_2 [0, 1]$ with norm

$$\begin{eqnarray*} \| f \| = \left ( \sum_{i=0}^2 (f^{(i)}(0))^2 + \int_0^1 (f^{(3)}(s))^2 ds \right )^{1/2} \, , \end{eqnarray*}$$ can be presented as

$$\nonumber V(\eta, \xi) =   \begin{cases}        \sum_{i=0}^2 \frac{\xi^i}{!i} \left ( \frac{\eta^i}{!i} + (-1)^{i} \frac {\eta^{5 - i}}{(5 - i)!}  \right )  \,  , &  0 \le \eta \le \xi \le 1         \\        \sum_{i=0}^2 \frac{\eta^i}{!i} \left ( \frac{\xi^i}{!i} + (-1)^{i} \frac {\xi^{5 - i}}{(5 - i)!}  \right ) \,  , &  0 \le \xi \le \eta \le 1   \end{cases}$$
or

$$V(\eta, \xi) =   \begin{cases}        1 + \eta \xi + \frac{(\eta^5 - 5 \eta^4 \xi + 10 \eta^3 \xi^2 + 30 \eta^2 \xi^2 )}{120}  \,  , &  0 \le \eta \le \xi \le 1 \\        1 + \eta \xi + \frac{(\xi^5 - 5 \xi^4 \eta + 10 \xi^3 \eta^2 + 30 \xi^2 \eta^2 )}{120} \,  , &  0 \le \xi \le \eta \le 1 \, .   \end{cases}$$ Correspondingly
$$\begin{eqnarray} && \frac{\partial V(\eta, \xi)}{\partial \xi}  = \frac{\eta (4 \eta \xi (\eta+3) - \eta^3)}{24} + \eta \, ,   \\ \nonumber && \frac{\partial^2 V(\eta, \xi)}{\partial \eta \partial \xi} = -\frac{\eta^3}{6} + \frac{\eta \xi (\eta + 2)}{2} + 1 \, , \\  \nonumber  && \frac{\partial^2 V(\eta, \xi)}{\partial \xi^2}  =  \frac{\eta^2 (\eta + 3)}{6}  \, ,  \\ \nonumber  && \frac{\partial^3 V(\eta, \xi)}{\partial \eta \partial \xi^2} =  \frac{\eta^2}{2} +  \eta \, , \\ && \nonumber \frac{\partial^4 V(\eta, \xi)}{\partial \eta^2 \partial \xi^2} =  \eta  +  1 \, , \\ \nonumber &&  0 \le \eta \le \xi \le 1 \, .                                                                      \end{eqnarray}$$ In addition, the following formulae are required for computing the normal spline derivatives

$$\begin{eqnarray} && \frac{\partial V(\eta, \xi)}{\partial \eta}  = \frac{\eta^4 - 4\xi (\eta^3 - 6) + 6 \eta \xi^2 (\eta + 2)}{24} \, ,   \\ \nonumber && \frac{\partial^2 V(\eta, \xi)}{\partial \eta^2} = \frac{\eta ^3 - 3 \eta^2 \xi + 3 \xi^2 (\eta + 1)}{6} \, ,  \\ \nonumber  && \frac{\partial^3 V(\eta, \xi)}{\partial \eta^2 \partial \xi} = -\frac{\eta^2}{2} +  \eta \xi + \xi \, , \\ \nonumber &&  0 \le \eta \le \xi \le 1 \, .                            \end{eqnarray}$$ Thereby we can construct a normal interpolating spline in interval $[0, 1]$. Solving the interpolating Problems 1 — 3 in an arbitrary interval can be done by mapping the latter to $[0, 1]$ through an affine change of variable.  Let's do it for the Problem 3.
Define constants $a$ and $b$ as
$$\begin{eqnarray*} a =\min (x_1, s_1, t_1),  \qquad  b = \max (x_{n_1}, s_{n_2}, t_{n_3}) \, , \end{eqnarray*}$$ and introduce values $\bar x_i, \bar s_j, \bar t_k$:

$$\begin{eqnarray*} && \bar x_i = \frac{x_i - a}{b - a}, \quad \bar s_j = \frac{s_j - a}{b - a}, \quad \bar t_k = \frac{t_k - a}{b - a} \, , \\  \nonumber  && i = 1, \dots, {n_1}, \quad j = 1, \dots, n_2, \quad k = 1, \dots, n_3 \, . \end{eqnarray*}$$ Then original Problem 3 is transformed to

Problem $\bar 3$. Given points $0 \le \bar x_1 \lt   \bar x_2 \lt \dots \lt \bar x_{n_1} \le 1$, $0 \le \bar s_1 \lt \bar s_2 \lt \dots \lt \bar s_{n_2} \le 1$ and  $0 \le \bar t_1 \lt \bar t_2 \lt \dots \lt \bar t_{n_3} \le 1$ find a smooth function $\bar f$ such that
$$\begin{eqnarray*} \bar f(\bar x_i) &=& u_i , \qquad \qquad \  i = 1, 2, \dots, n_1 \ , \\ \nonumber \bar f'(\bar s_j) &=& v_j (b - a) , \quad \ \  j = 1, 2, \dots, n_2 \ , \\ \nonumber \bar f''(\bar t_k) &=& w_k (b - a)^2 , \quad  k = 1, 2, \dots, n_3 \ . \end{eqnarray*}$$ Assuming $\bar \sigma_3 (\bar \eta)$ is a normal spline constructed for the Problem $\bar 3$, the normal spline $\sigma_3 (\eta)$  can be received as
$$\begin{eqnarray*} \sigma_3 (\eta) = \bar \sigma_3 \left( \frac{\eta - a}{b - a} \right),  \qquad a \le \eta \le b \, . \end{eqnarray*}$$
Reproducing kernel for Sobolev spaces $W^3_2 [0, \infty)$ with norm

$$\begin{eqnarray*} \| f \| = \left ( \int_0^\infty \left[ (f(s))^2  + \left(f^{(3)}(s) \right)^2 \right] ds \right )^{1/2} \, , \end{eqnarray*}$$ was received in [11]. It is the symmetric function

$$\begin{eqnarray} V(\eta, \xi) = \begin{cases} \sum_{i=1}^6 y_i(\eta) c_i(\xi) \,  , \quad 0 \le \eta \le \xi \lt \infty \, ,  \\     \sum_{i=1}^6 y_i(\xi) c_i(\eta) \,  , \quad 0 \le \xi \le \eta \lt \infty \, , \end{cases}                 \end{eqnarray}$$ where

$$\begin{eqnarray} && y_1(\eta) = \exp (\eta) \, , \quad  y_2(\eta) = \exp (-\eta) \,  , \\ \nonumber &&   y_3(\eta) = \exp (-\eta / 2) \cos (\sqrt 3 \eta /2) \, , \\ \nonumber && y_4(\eta) = \exp (-\eta / 2) \sin (\sqrt 3 \eta /2) \, ,  \\ \nonumber &&   y_5(\eta) = \exp (\eta / 2) \cos (\sqrt 3 \eta /2) \, , \\ \nonumber &&  y_6(\eta) = \exp (\eta / 2) \sin (\sqrt 3 \eta /2) \, , \\ \nonumber && c_1(\xi) = -\exp(-\xi) / 6 \, , \\ \nonumber && c_2(\xi) = \exp(-\xi /2) ( \sqrt 3 \sin (\sqrt 3 \xi /2) - \cos(\sqrt 3 \xi /2) ) / 3 - \exp(-\xi) / 2 \, , \\ \nonumber  && c_3(\xi) = \exp(-\xi /2) ( \sin (\sqrt 3 \xi /2) / \sqrt 3  - \cos(\sqrt 3 \xi /2) ) / 2 - \exp(-\xi) / 3 \, , \\ \nonumber  && c_4(\xi) = \exp(-\xi /2) (\sqrt 3  \cos (\sqrt 3 \xi /2)  - 5 \sin(\sqrt 3 \xi /2) ) / 6 \, +  \\ \nonumber && \qquad \qquad \exp(-\xi) / \sqrt 3 \, ,  \\ \nonumber  && c_5(\xi) = -\exp(-\xi /2) (\cos (\sqrt 3 \xi /2)  - \sqrt 3 \sin(\sqrt 3 \xi /2) ) / 6 \, ,  \\ \nonumber  && c_6(\xi) = -\exp(-\xi /2) (\sin (\sqrt 3 \xi /2)  + \sqrt 3 \cos(\sqrt 3 \xi /2) ) / 6 \, .        \end{eqnarray}$$ Correspondingly

$$\begin{eqnarray} && \frac{\partial V(\eta, \xi)}{\partial \xi}  = \sum_{i=1}^6 y_i(\eta) c'_i(\xi) \, ,   \quad  \frac{\partial^2 V(\eta, \xi)}{\partial \eta \partial \xi} = \sum_{i=1}^6 y'_i(\eta) c'_i(\xi) \, , \\ \nonumber && \frac{\partial^2 V(\eta, \xi)}{\partial \xi^2}  =  \sum_{i=1}^6 y_i(\eta) c''_i(\xi) \, ,  \quad \frac{\partial^3 V(\eta, \xi)}{\partial \eta \partial \xi^2} =  \sum_{i=1}^6 y'_i(\eta) c''_i(\xi)  \, , \\ && \nonumber \frac{\partial^4 V(\eta, \xi)}{\partial \eta^2 \partial \xi^2} =  \sum_{i=1}^6 y''_i(\eta) c''_i(\xi)  \, ,  \quad \frac{\partial V(\eta, \xi)}{\partial \eta}  =  \sum_{i=1}^6 y'_i(\eta) c_i(\xi)  \, , \\ \nonumber  && \frac{\partial^2 V(\eta, \xi)}{\partial \eta^2} = \sum_{i=1}^6 y''_i(\eta) c_i(\xi)  \, , \quad \frac{\partial^3 V(\eta, \xi)}{\partial \eta^2 \partial \xi} = \sum_{i=1}^6 y''_i(\eta) c'_i(\xi)  \, ,  \\ \nonumber && 0 \le \eta \le \xi \le 1 \, . \end{eqnarray}$$ Where

$$\begin{eqnarray} && y'_1(\eta) = \exp (\eta) \, , \quad  y'_2(\eta) = -\exp (-\eta) \,  , \\ \nonumber &&   y'_3(\eta) = -\exp (-\eta / 2) \sin (\sqrt 3 \eta /2 + \pi/6) \, , \\ \nonumber && y'_4(\eta) = \exp (-\eta / 2) \cos (\sqrt 3 \eta /2 + \pi/6)  \, ,  \\ \nonumber &&   y'_5(\eta) = \exp (\eta / 2) \sin (\pi/6 - \sqrt 3 \eta /2)  \, , \\ \nonumber &&  y'_6(\eta) =  \exp (\eta / 2) \cos (\pi/6 - \sqrt 3 \eta /2)  \, , \\ \nonumber  && y''_1(\eta) = \exp (\eta) \, , \quad  y''_2(\eta) = \exp (-\eta) \,  , \\ \nonumber &&   y''_3(\eta) = \exp (-\eta / 2) (\sin (\sqrt 3 \eta /2 + \pi/6) - \sqrt 3 \cos (\sqrt 3 \eta /2 + \pi/6))/2 \, , \\ \nonumber && y''_4(\eta) = -\exp (-\eta / 2) (\sin (\sqrt 3 \eta /2 + \pi/6) + \sqrt 3 \cos (\sqrt 3 \eta /2 + \pi/6))/2 \, ,  \\ \nonumber &&   y''_5(\eta) = -\exp (\eta / 2) (\sqrt 3 \sin (\sqrt 3 \eta /2) + \cos (\sqrt 3 \eta /2))/2 \, , \\ \nonumber &&  y''_6(\eta) = \exp (\eta / 2) (\sqrt 3 \cos (\sqrt 3 \eta /2) - \sin (\sqrt 3 \eta /2))/2 \, ,  \\ \nonumber && c'_1(\xi) = \exp(-\xi) / 6 \, , \\ \nonumber && c'_2(\xi) = 2\exp(-\xi /2) \cos (\sqrt 3 \xi /2)/3 + \exp(-\xi) / 2 \, , \\ \nonumber  && c'_3(\xi) = \exp(-\xi /2) \cos(\pi/6 - \sqrt 3 \xi /2) / \sqrt 3 + \exp(-\xi) / 3 \, , \\ \nonumber  && c'_4(\xi) = \exp(-\xi /2) (-3\sqrt 3  \cos (\sqrt 3 \xi /2) + 5 \sin(\sqrt 3 \xi /2) ) / 6 \, -  \\ \nonumber && \qquad \qquad \exp(-\xi) / \sqrt 3 \, ,  \\ \nonumber  && c'_5(\xi) = \exp(-\xi /2) \cos (\sqrt 3 \xi /2) / 3 \, ,  \\ \nonumber  && c'_6(\xi) = \exp(-\xi /2) \sin (\sqrt 3 \xi /2) / 3  \, ,  \\ \nonumber && c''_1(\xi) = -\exp(-\xi) / 6 \, , \\ \nonumber && c''_2(\xi) = -2\exp(-\xi /2) \sin (\sqrt 3 \xi /2 + \pi/6)/3 - \exp(-\xi) / 2 \, , \\ \nonumber  && c''_3(\xi) =  -\exp(-\xi /2) \sin (\sqrt 3 \xi /2)/\sqrt 3 - \exp(-\xi) / 3 \, , \\ \nonumber  && c''_4(\xi) = \exp(-\xi /2) (\sqrt 3  \cos (\sqrt 3 \xi /2) + 2 \sin(\sqrt 3 \xi /2) ) / 3 \, +  \\ \nonumber && \qquad \qquad \exp(-\xi) / \sqrt 3 \, ,  \\ \nonumber  && c''_5(\xi) = -\exp(-\xi /2) (\cos (\sqrt 3 \xi /2)  + \sqrt 3 \sin(\sqrt 3 \xi /2) ) / 6 \, ,  \\ \nonumber  && c''_6(\xi) =  \exp(-\xi /2) (\sqrt 3 \cos (\sqrt 3 \xi /2) - \sin(\sqrt 3 \xi /2) ) / 6 \, . \end{eqnarray}$$

Reproducing kernel for Bessel potential space was presented in [8] and its simplified variant in [16, 15, 17,18]. Normal splines will be constructed in Bessel potential space $H_\varepsilon^3(R)$ defined as
$$\begin{eqnarray*}    H^3_\varepsilon (R) = \left\{ f | f \in S' ,   ( \varepsilon ^2 + | s |^2 )^{3/2}{\cal F} [f] \in L_2 (R) \right\} \, ,  \varepsilon \gt 0 , \end{eqnarray*}$$ where $S'  (R)$ is space of L. Schwartz tempered distributions and $\cal F [f]$ is a Fourier transform of the $f$ [8, 19, 20]. The parameter $\varepsilon$ introduced here may be treated as a "scaling parameter". It allows to control approximation properties of the normal spline which usually are getting better with smaller values of $\varepsilon$, also it may be used to reduce the illconditioness of the related computational problem (in traditional theory $\varepsilon = 1$) (see [15, 9]). This is a Hilbert space with norm
$$\begin{eqnarray*} \| f \|_ {H^3_\varepsilon} =  \| (  \varepsilon ^2 + | s |^2 )^{3/2} {\cal F} [\varphi ] \|_{L_2} \ . \end{eqnarray*}$$ It is continuously embedded in the Hölder space $C_b^3(R)$ that consists of all functions having bounded continuous derivatives up to order $2$ ([19]). The reproducing kernel of this space is defined up to a constant as follows ([15, 9])
$$\begin{eqnarray*} V(\eta , \xi, \varepsilon) = \exp (-\varepsilon |\xi - \eta|)              (3 + 3\varepsilon |\xi  - \eta| + \varepsilon ^2 |\xi - \eta| ^2 ) \, . \end{eqnarray*}$$ Correspondingly

$$\begin{eqnarray} && \frac{\partial V(\eta , \xi, \varepsilon)}{\partial \xi}  = - \varepsilon^2  \exp (-\varepsilon |\xi - \eta|)(\xi - \eta)(\varepsilon |\xi - \eta) + 1| \, ,  \\ \nonumber && \frac{\partial^2 V(\eta , \xi, \varepsilon)}{\partial \eta \partial \xi} = -\varepsilon^2  \exp (-\varepsilon |\xi - \eta|)(\varepsilon |\xi - \eta|(\varepsilon |\xi - \eta| - 1) - 1)  \, ,  \\  \nonumber  && \frac{\partial^2 V(\eta , \xi, \varepsilon)}{\partial \xi^2}  =  \varepsilon^2  \exp (-\varepsilon |\xi - \eta|)(\varepsilon |\xi - \eta|(\varepsilon |\xi - \eta| - 1) - 1) \, , \\ \nonumber  && \frac{\partial^3 V(\eta , \xi, \varepsilon)}{\partial \eta \partial \xi^2} =  \varepsilon^4  \exp (-\varepsilon |\xi - \eta|)(\xi - \eta)(\varepsilon |\xi - \eta| - 3) \, , \\ && \nonumber \frac{\partial^4 V(\eta , \xi, \varepsilon)}{\partial \eta^2 \partial \xi^2} =  \varepsilon^4  \exp (-\varepsilon |\xi - \eta|)(\varepsilon |\xi - \eta|(\varepsilon |\xi - \eta| - 5) +3) \, .\end{eqnarray}$$ In addition, the following formulae are required for computing the normal spline derivatives

$$\begin{eqnarray} && \frac{\partial V(\eta , \xi, \varepsilon)}{\partial \eta}  = \varepsilon^2  \exp (-\varepsilon |\xi - \eta|)(\xi - \eta)(\varepsilon |\xi - \eta| + 1) \, ,  \\ \nonumber && \frac{\partial^2 V(\eta , \xi, \varepsilon)}{\partial \eta^2} = \varepsilon^2  \exp (-\varepsilon |\xi - \eta|)(\varepsilon |\xi - \eta|(\varepsilon |\xi - \eta| - 1) - 1) \, ,  \\ \nonumber  && \frac{\partial^3 V(\eta , \xi, \varepsilon)}{\partial \eta^2 \partial \xi} = -\varepsilon^4  \exp (-\varepsilon |\xi - \eta|)(\xi - \eta)(\varepsilon |\xi - \eta| - 3) \, .                        \end{eqnarray}$$

The normal splines method for one-dimensional function interpolation and linear ordinary differential and integral equations was proposed in [13] and developed in [10, 11, 12].  General formula of reproducing kernel for Bessel potential spaces was published in [8] and its simplified version was given in works [15, 17, 18]. Multidimensional normal splines method was developed for two-dimensional problem of low-range computerized tomography in [16] and applied for solving a mathematical economics problem in [14]. Further results were reported on seminars and conferences.

References

[1] A. Balakrishnan, Applied Functional Analysis, New York, Springer-Verlag, 1976.
[2] N. Aronszajn, Theory of reproducing kernels, Tranzactions of the AMS.– 950 – Vol.68.
[3] A. Bezhaev, V. Vasilenko, Variational Theory of Splines, Springer US, 2001.
[4] A. Ioffe, V. Tikhomirov, Theory of extremal problems, North-Holland, Amsterdam, 1979.
[5] P.-J. Laurent, Approximation et optimization, Paris, 1972.
[6]  R. Adams, J. Fournier, Sobolev Spaces. Pure and Applied Mathematics. (2nd ed.). Boston, MA: Academic Press, 2003.
[7] D.R. Adams, L.I. Hedberg, Function spaces and potential theory. Berlin, Springer, 1996.

[8] N. Aronszajn, K.T. Smith, Theory of bessel potentials I, Ann.Inst.Fourier, 11,  1961.

[9] Reproducing Kernel of Bessel Potential space
[10] V. Gorbunov, V. Petrishchev, Improvement of the normal spline collocation method for linear differential equations, Comput. Math. Math. Phys., 43:8 (2003), 1099–1108
[11] V. Gorbunov, V. Sviridov, The method of normal splines for linear DAEs on the number semi-axis. Applied Numerical Mathematics, 59(3-4), 2009, 656–670.
[12] V. Gorbunov, Extremum Problems of Measurements Data Processing, Ilim, 1990 (in Russian).

[13] V. Gorbunov, The method of normal spline collocation, USSR Computational Mathematics and Mathematical Physics,Volume 29, Issue 1, 1989, Pages 145-154.

[14] V. Gorbunov, I. Kohanovsky, K. Makedonsky, Normal splines in reconstruction of multi-dimensional dependencies. Papers of WSEAS International Conference on Applied Mathematics, Numerical Analysis Symposium, Corfu, 2004. (http://www.wseas.us/e-library/conferences/corfu2004/papers/488-312.pdf)

[15] I. Kohanovsky, Multidimensional Normal Splines and Problem of Physical Field Approximation, International Conference on Fourier Analysis and its Applications, Kuwait, 1998.

[16] I. Kohanovsky, Normal Splines in Computing Tomography, Avtometriya, 1995, N 2. (https://www.iae.nsk.su/images/stories/5_Autometria/5_Archives/1995/2/84-89.pdf)

[17] H. Wendland, Scattered Data Approximation. Cambridge University Press, 2005.

[18] R. Schaback, Kernel-based Meshless Methods, Lecture Notes, Goettingen, 2011.
[19] M.S. Agranovich, Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains, Springer, Switzerland, 2015.

[20] H. Triebel, Interpolation. Function Spaces. Differential Operators. North-Holland, Amsterdam, 1978.