$\setCounter{0}$
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 $\varphi \in H$:
\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:
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
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*}
\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
[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.
[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).
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 $H$).
&& \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
&& 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.
\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
\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
\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}\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:
\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:
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&& \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}
&& 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}
- for every $\xi \in X$, $V(\eta, \xi)$ as function of $\eta$ belongs to $H$
- for every $\xi \in X$ and every function $\varphi \in H$
\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]):&& 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}
\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_{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]):
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
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}
&& {\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
\| 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
\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 $\bar x_i, \bar s_j, \bar t_k$:
&& \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&& \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
\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
\| 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
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
&& 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*}
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
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
&& \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&& 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}
\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
&& \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}
&& \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.
[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.
