This topic is concerned with numerical solution of the following interpolation problems:
Problem 1. Given points x1<x2<⋯<xn1 find a function f such that
f(xi)=ui,i=1,2,…,n1 ,
Problem 2. Given points x1<x2<⋯<xn1, s1<s2<⋯<sn2 find a function f such that
f(xi)=ui,i=1,2,…,n1 ,f′(sj)=vj,j=1,2,…,n2 ,n1≥0, n2≥0,
Problem 3. Given points x1<x2<⋯<xn1, s1<s2<⋯<sn2 and t1<t2<⋯<tn3 find a function f such that
f(xi)=ui,i=1,2,…,n1 ,f′(sj)=vj,j=1,2,…,n2 ,f″ 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).
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
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
\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:
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
\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}
- 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*} 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
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:
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*} 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
\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.
[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.