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Tuesday, January 29, 2019

Hermite-Birkhoff Interpolation of Scattered Data


Suppose P1={pi,piRn}M1i=1 and P2={pi,piRn}M2i=1 are sets of data locations lying in a domain ΩRn.
Let functional fi is a value of a function φ at a node piP1 and functional fi is a value of that function directional derivative (a slope of the function) at a node piP2
Assume the values of functionals fi and fi at nodes belonging to P1 and P2 are known:
(fi,φ)=φ(pi)=ui ,piP1 ,1iM1,(fi,φ)=φei(pi)=ui ,piP2 ,1iM2,
where φei(pi)=nk=1φxk(pi)cosθik, and cosθik,1kn directional cosines of a unit vector ei.

Consider a problem of the function φ reconstruction. Let function to be approximated φ belongs to Hilbert space Hsε(Rn) where s>r+n/2,r=1,2,. In that case space Hsε is continuously embedded in Hölder space Crb(Rn), φ(x)Cr(Rn) and values of function φ(x) and the function partial derivatives at the fixed points are linear continuous functionals in Hsε.
We assume that P1 does not contain coincide nodes and there is no directional derivatives defined in a node in P2 which directions are linearly dependent, also a node from P1 may coincide with a node from P2. Under this restriction functionals fi,fi are linearly independent.
In accordance with Riesz representation theorem [1] linear continuous functionals fi,fi can be represented in the form of inner product of some elements hi,hiHsε and  φHsε, for any φHsε.
(fi,φ)=hi,φHsrε,(fi,φ)=hi,φHsrε,φHsε .
The notation ,Hsrε in (3) denotes the inner product in space Hsrε (see the previous post for details). It is easy to see that under above assumptions elements hi,hi are linearly independent.
Let Vsr(η,x,ε),sr=r+n/2+1/2,r=1,2, is a reproduction kernel of Hilbert space Hsrε(Rn) (see the previous post). The kernel Vsr(η,x,ε)Cr(Rn) as a function of ηRn. In accordance with reproducing property of the reproducing kernel for any φHsrε(Rn) and xRn the identity
φ(x)=Vsr(,x,ε),φHsrε
holds. This means that Vsr(,x) is the Riesz representer of the evaluation functional δx defined as value of a function φHsrε at a given point xRn. As Vsr is a radial basis continuously differentiable function we can differentiate the identity (4) and receive the following identity
φ(x)xi=Vsr(,x,ε)xi,φHsrε
which holds for any φHsrε and xRn. It means that function Vsr(,x)xi represents a point-wise functional defined as value of function φ()xi at a point x.
Thereby
hi(η,ε)=Vsr(η,pi,ε),piP1 ,1iM1, hi(η,ε)=Vsr(η,pi,ε)ei, piP2 ,1iM2, ηRn.
and system of constraints (1), (2) can be presented as a system of linear equations in Hilbert space Hsrε:
hi,φHsrε=ui,1iM1, hi,φHsrε=ui,  1iM2,hi,hi,φHsrε(Rn).
This system of constraints defines a convex and closed set in Hilbert space — as an intersection of a finite number of hyperplanes (this set is not an empty set because of linear independence of functionals hi,hi).

The problem of reconstruction of function φ satisfying system of constraints (8) and (9) is an undetermined problem. We reformulate it as the problem of finding a solution of the system (8), (9) that has minimal norm:
σ1=argmin{
such solution exists and it is unique [1, 6]. The element \sigma_1 is an interpolating normal spline. 
In accordance with generalized Lagrange method ([5], [6]) the solution of the problem (10) can be written as follows:
\begin{eqnarray} \sigma_1 =  \sum _{j=1}^{M_1} \mu_j  h_j + \sum _{j=1}^{M_2} \mu'_j  h'_j \ , \end{eqnarray}
here coefficients \mu_i,  \mu'_i  are defined by the system
\begin{eqnarray}  \sum _{j=1}^{M_1} g_{ij} \mu_j + \sum _{j=1}^{M_2} g'_{ij} \mu'_j &=& u_i  \, , \quad 1 \le i \le M_1 \, , \\  \sum _{j=1}^{M_1} g'_{ji} \mu_j + \sum _{j=1}^{M_2} g''_{ij} \mu'_j &=& u'_i  \, , \quad 1 \le i \le M_2 \, , \end{eqnarray}
where g_{ij}, g'_{ij},  g''_{ij} are coefficients of the positive definite symmetric Gram matrix of the set of linearly independent elements h_i, h'_i \in H^{s_r}_\varepsilon (R^n):
\begin{eqnarray}  g_{ij} = {\langle h_i,  h_j \rangle}_{H^{s_r}_\varepsilon} = (f_i, h_j) \, , \quad 1 \le i \le M_1 \, , \ 1 \le j \le M_1 \, , \\ g'_{ij} = {\langle h_i,  h'_j \rangle}_{H^{s_r}_\varepsilon} = (f_i, h'_j) \, , \quad 1 \le i \le M_1 \, , \ 1 \le j \le M_2 \, , \\ g''_{ij} = {\langle h'_i,  h'_j \rangle}_{H^{s_r}_\varepsilon} = (f'_i, h'_j) \, , \quad 1 \le i \le M_2 \, ,  \ 1 \le j \le M_2 \, , \end{eqnarray} Taking into account (3), (6), (7) we receive:
\begin{eqnarray}  g_{ij} &=& V_{s_r}(p_i , p_j, \varepsilon)  \, , \quad p_i, p_j \in P_1 , \  1 \le i \le M_1 , \ 1 \le j \le M_1 , \\ \nonumber \\ g'_{ij} &=&  \frac{ \partial {V_{s_r}(p_i, p'_j, \varepsilon)} }{\partial{e_j}}  =  \sum_{k=1}^n  \frac{ \partial {V_{s_r}(p_i, p'_j, \varepsilon)} }{\partial{x_k}} \cos \theta _{jk}  \, , \\ \nonumber && p_i \in P_1 \, , \ 1 \le i \le M_1 , \  p'_j \in P_2 , \ 1 \le j \le M_2 \, , \\ g''_{ij} &=&  \frac{ \partial^2 {V_{s_r}(p'_i, p'_j, \varepsilon)} } {\partial{e_i} \partial{e_j}} =  \sum_{l=1}^n \sum_{k=1}^n \frac{ \partial^2 {V_{s_r}(p'_i, p'_j, \varepsilon)} } {\partial{\eta_l} \partial{x_k}} \cos \theta _{il} \cos \theta _{jk} \, ,\\ \nonumber  &&  p'_i, p'_j \in P_2 ,  \ 1 \le i \le M_2 \, , \  1 \le j \le M_2 \, . \end{eqnarray}
Thus the problem (10) is reduced to solving a system of linear equations (12), (13) with Gram matrix defined by (18) — (19) and constructing the normal spline (11).

Assume the function to be approximated \varphi belongs to Hilbert space H^{s_1}_\varepsilon (R^n)s_1 = 1 + n/2 + 1/2 (see the previous post). It is a continuously differentiable function, \varphi \in C^1 (R^n). Write down the expressions for h_j, h'_j, g_{ij}, g'_{ij}, g''_{ij} in space  H^{s_1}_\varepsilon (R^n):
\begin{eqnarray}   h_j (\eta, \varepsilon) &=&  \exp (-\varepsilon |\eta - p_j |) (1 + \varepsilon |\eta - p_j |) \ , \\ \nonumber &&  p_j \in P_1 \ , \ 1 \le j \le M_1 \, , \\   h'_j (\eta, \varepsilon) &=& \varepsilon^2 \exp (-\varepsilon | \eta - p'_j | ) \sum _{k=1}^n (\eta_k - p'_{jk})\cos \theta_{jk} \ , \\ \nonumber  &&   p'_j \in P_2 \ , \ 1 \le j \le M_2 \, , \quad  \eta \in R^n \,  , \end{eqnarray}
then
\begin{eqnarray}   g_{ij} &=& \exp (-\varepsilon | p_i - p_j | )(1 + \varepsilon | p_i - p_j | ) \ ,  \  p_i, p_j \in P_1 \ , 1 \le i \le M_1 \, , \\   g'_{ij} &=& \varepsilon^2 \exp (-\varepsilon | p_i - p'_j | ) \sum _{k=1}^n (p_{ik} - p'_{jk})\cos \theta_{jk}  \ , \\ \nonumber &&   p_i \in P_1 \, , \  p'_j \in P_2 \ , 1 \le i \le M_1 \, , \ 1 \le j \le M_2 \, ,  \\ g''_{ij} &=&  \sum_{l=1}^n \sum_{k=1}^n \frac{ \partial^2 {V_{s_1}(p'_i, p'_j, \varepsilon)} } {\partial{\eta_l} \partial{x_k}} \cos \theta _{il} \cos \theta _{jk} \, ,\\ \nonumber  &&  p'_i, p'_j \in P_2 ,  \ 1 \le i \le M_2 \, , \  1 \le j \le M_2 \, ,  i \ne j \, , \\ \nonumber && \text{where} \\ \nonumber   && \frac{ \partial^2 {V_{s_1}(p'_i, p'_j, \varepsilon)} } {\partial{\eta_k} \partial{x_k}} = \varepsilon^2 \exp (-\varepsilon | p'_i - p'_j | ) \left( 1 - \varepsilon \frac {(p'_{ik} - p'_{jk})^2}{| p'_i - p'_j |} \right) \, , \\ \nonumber   && \frac{ \partial^2 {V_{s_1}(p'_i, p'_j, \varepsilon)} } {\partial{\eta_l} \partial{x_k}} = -\varepsilon^3 \exp (-\varepsilon | p'_i - p'_j | )  \frac {(p'_{ik} - p'_{jk})(p'_{il} - p'_{jl})}{| p'_i - p'_j |} \, , \ \  l \ne k \, , \\  g''_{ii} &=& \varepsilon^2 \sum _{l=1}^n\  (\cos \theta_{il})^2  = \varepsilon^2 \, ,  \quad 1 \le i \le M_2 \, . \end{eqnarray}

Consider a case when the function to be approximated \varphi may be treated as an element of Hilbert space H^{s_2}_\varepsilon (R^n)s_2 = 2 + n/2 + 1/2. In this case it is a twice continuously differentiable function, \varphi \in C^2 (R^n). Let's write down the expressions for h_j, h'_j in space  H^{s_2}_\varepsilon (R^n):
\begin{eqnarray}   h_j (\eta, \varepsilon) &=&  \exp (-\varepsilon |\eta - p_j |) (3 + 3 \varepsilon |\eta - p_j | +  \varepsilon^2  |\eta - p_j |^2) ) \ , \\ \nonumber && p_j \in P_1 \ , 1 \le j \le M_1 \, , \\   h'_j (\eta, \varepsilon) &=&  \varepsilon^2 \exp (-\varepsilon |\eta - p'_j | ) (1 + \varepsilon |\eta - p'_j |) \sum _{k=1}^n (\eta_k - p'_{jk})\cos \theta_{jk}  \ , \\ \nonumber  &&  p'_j \in P_2 \, , \ \ 1 \le j \le M_2 \, , \ \eta \in R^n \, , \end{eqnarray}
and corresponding Gram matrix elements g_{ij}, g'_{ij}, g''_{ij} are as follows
\begin{eqnarray}   g_{ij} &=& \exp (-\varepsilon |p_i - p_j |) (3 + 3 \varepsilon |p_i - p_j | +  \varepsilon^2  |p_i - p_j |^2) ) \ ,  \\ \nonumber &&  p_i, p_j \in P_1 \ , 1 \le i \le M_1 \, , \\   g'_{ij} &=& \varepsilon^2 \exp (-\varepsilon |p_i - p'_j | ) (1 + \varepsilon |p_i - p'_j |) \sum _{k=1}^n (p_{ik} - p'_{jk})\cos \theta_{jk}  \ , \\ \nonumber &&   p_i \in P_1 \, , \  p'_j \in P_2 \ , 1 \le i \le M_1 \, , 1 \le j \le M_2 \, , \\ g''_{ij} &=&  \sum_{l=1}^n \sum_{k=1}^n \frac{ \partial^2 {V_{s_2}(p'_i, p'_j, \varepsilon)} } {\partial{\eta_l} \partial{x_k}} \cos \theta _{il} \cos \theta _{jk} \, ,\\ \nonumber  &&  p'_i, p'_j \in P_2 ,  \ 1 \le i \le M_2 \, , \  1 \le j \le M_2 \, ,  i \ne j \, , \\ \nonumber && \text{where} \\ \nonumber   && \frac{ \partial^2 {V_{s_2}(p'_i, p'_j, \varepsilon)} } {\partial{\eta_k} \partial{x_k}} = \varepsilon^2 \exp (-\varepsilon | p'_i - p'_j | ) (1 + \varepsilon | p'_i - p'_j | - \varepsilon^2 (p'_{ik} - p'_{jk})^2) \, , \\ \nonumber   && \frac{ \partial^2 {V_{s_2}(p'_i, p'_j, \varepsilon)} } {\partial{\eta_l} \partial{x_k}} = -\varepsilon^4 \exp (-\varepsilon | p'_i - p'_j | )  (p'_{ik} - p'_{jk})(p'_{il} - p'_{jl}) \, , \ \  l \ne k \, , \\  g''_{ii} &=& \varepsilon^2 \sum _{l=1}^n\  (\cos \theta_{il})^2  = \varepsilon^2 \, ,  \quad 1 \le i \le M_2 \, , \end{eqnarray}
If the original task does not contain constraints including function \varphi derivatives (P_2 is an empty set, thereby there are no constraints (2) in (1)—(2)), we may construct an interpolating normal spline assuming \varphi is an element of Hilbert space H^{s_0}_\varepsilon (R^n)s_0 =  n/2 + 1/2. In this case it is a continuous function, \varphi \in C (R^n), and expressions for h_i, g_{ij} are defined by:
\begin{eqnarray}   h_i (\eta, \varepsilon) &=&  \exp (-\varepsilon |\eta - p_i |) \ , \eta \in R^n, \  p_i \in P_1 \ , 1 \le i \le M_1 \, , \\  g_{ij} &=& \exp (-\varepsilon | p_j - p_i | )) \ ,  \\ \nonumber && \  p_i, p_j \in P_1 \ , 1 \le i \le M_1 \, ,  \  1 \le j \le M_1 \, , \end{eqnarray}
When value of the parameter \varepsilon is small this spline is similar to multivariate generalization of the one dimensional linear spline.

We now consider the choice of value for parameters s and \varepsilon. The parameter s defines smoothness of the spline —  if s >  n/2 then spline belongs to C^{[s - n/2 ]}, where [\cdot] denotes integer part of number. Moreover, discontinuities of derivatives of a greater order are observed only at the nodes p_i, p'_i, at all other points of R^n the spline is infinitely differentiable. With increasing s the condition number of the Gram matrix increases and computational properties of the problem of the function \varphi approximation are worsening. Thereby it make sense to choose a reasonably small value for the parameter s.
A normal spline is similar to Duchon's D^m -spline [3] when value of parameter \varepsilon is small. Approximating properties of the spline are getting better with smaller values of \varepsilon, however with decreasing value of \varepsilon the condition number of Gram matrix increases. Therefore, when choosing the value of parameter \varepsilon, a compromise is needed. In practice, it is necessary to choose such value of the \varepsilon that condition number of Gram matrix is small enough. Numerical procedures of the matrix condition number estimation are well known (e.g. [4]).
As well, as a rule, it is useful to preprocess the source data of the problem by scaling the domain \Omega \subset R^n in which the interpolation nodes p_i, p'_i are located into a unit hypercube.

Often accurate values of u_i, u'_i in (1), (2) are not known. Assuming that uniform error bounds \delta_i, \delta'_i are given, we can formulate the problem of function \varphi approximation by constructing a uniform smoothing normal spline \sigma_2
\begin{eqnarray} &&  \sigma_2 = {\rm arg\,min}\{  \| \varphi \|^2_{H^{s_r}_\varepsilon} : (35), (36), \forall \varphi \in {H^{s_r}_\varepsilon} (R^n) \} \, , \\ && | \varphi (p_i) - u_i  | \le \delta _i \, ,  \quad  p_i \in P_1 \ , 1 \le i \le M_1,                                              \\ && \left|  \frac{ \partial{\varphi} }{ \partial{e_i} } (p'_i) -  u'_i  \right| \le \delta' _i \, , \  p'_i \in P_2 \ , 1 \le i \le M_2 \, . \end{eqnarray}
In a case when a mean squared error \delta^2 is known an approximating mse-smoothing normal spline \sigma_3 can be constructed as a solution of the following variational problem:
\begin{eqnarray} &&  \sigma_3 = {\rm arg\,min}\{  \| \varphi \|^2_{H^{s_r}_\varepsilon} : (38)\, , \forall \varphi \in {H^{s_r}_\varepsilon} (R^n) \} \, , \\ && \sum _{i \in P_1} ( \varphi (p_i) - u_i )^2 +   \sum _{i \in P_2}\left( \frac{ \partial{\varphi} }{ \partial{e_i} } (p'_i) - u'_i \right)^2 \le \delta ^2 \, . \end{eqnarray} Constraints (35)—(36) and (38) define convex closed sets in Hilbert space H^{s_r}_\varepsilon (R^n), s_r  = r + n/2 + 1/2, \,  r=1, 2, \dots. Under assumptions made earlier these sets are not empty, thus problems (34) and (37) are problems of finding a projection of zero element of Hilbert space to a non-empty closed convex set. A solution of such problem exists and it is unique.

Algorithms of solving problems (34) and (37) will be described in next posts.

Earlier a problem of multivariate normal splines constructing was treated in ([2], [7], [8]) and presented at [9] and [10].

References

[1] A. Balakrishnan, Applied Functional Analysis, New York, Springer-Verlag, 1976.
[2] 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)
[3] J. Duchon, Splines minimizing rotation-invariant semi-norms in Sobolev spaces, Lect. Notes in Math., Vol. 571, Springer, Berlin, 1977
[4] W.Hager, Condition Estimates, SIAM Journal on Scientific and Statistical Computing, Vol.5, N.2, 1984
[5] A. Ioffe, V. Tikhomirov, Theory of extremal problems, North-Holland, Amsterdam, 1979.
[6] P.-J. Laurent, Approximation et optimization, Paris, 1972.
[7] 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)
[8] I. Kohanovsky, Data approximation using multidimensional normal splines. Unpublished manuscript. 1996.
[9] I. Kohanovsky, Multidimensional Normal Splines and Problem of Physical Field Approximation, International Conference on Fourier Analysis and its Applications, Kuwait, 1998.
[10] I. Kohanovsky, Normal splines in fractional order Sobolev spaces and some of its
applications, The Third Siberian Congress on Applied and Industrial mathematics (INPRIM-98), Novosibirsk, 1998.

Friday, January 4, 2019

Reproducing Kernel of Bessel Potential space

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The standard definition of Bessel potential space H^s can be found in ([1], [2], [6], [11]). Here the normal splines will be constructed in the Bessel potential space H^s_\varepsilon defined as:
\begin{eqnarray}    H^s_\varepsilon (R^n) = \left\{ \varphi | \varphi \in S' ,   ( \varepsilon ^2 + | \xi |^2 )^{s/2}{\cal F} [\varphi ] \in L_2 (R^n) \right\} ,   \varepsilon > 0 ,  s > \frac{n}{2} . \end{eqnarray} where S'  (R^n) is space of L. Schwartz tempered distributions, parameter s may be treated as a fractional differentiation order and \cal F [\varphi ] is a Fourier transform of the \varphi. 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).
    Theoretical properties of spaces H^s_\varepsilon at \varepsilon > 0 are identical — they are Hilbert spaces with inner product
\begin{eqnarray}  \langle \varphi , \psi \rangle _{H^s_\varepsilon} =  \int ( \varepsilon ^2  + | \xi |^2 )^s       \cal F [\varphi ] \overline{\cal F [\psi ] } \, d \xi \end{eqnarray} and  norm
\begin{eqnarray}  \| \varphi \|_ {H^s_\varepsilon} = \left( \langle \varphi , \varphi \rangle _{H^s_\varepsilon} \right)^{1/2} =     \| (  \varepsilon ^2 + | \xi |^2 )^{s/2} {\cal F} [\varphi ] \|_{L_2} \ . \end{eqnarray} It is easy to see that all \| \varphi \|_{H^s_\varepsilon} norms are equivalent. It means that space H^s_\varepsilon (R^n) is equivalent to H^s (R^n) =  H^s_1 (R^n) .

Let's describe the Hölder spaces C_b^t(R^n), t > 0 ([9], [2]).
Definition 1. We denote the space
\begin{eqnarray}    S(R^n) = \left\{ f | f \in C^\infty (R^n) ,  \sup_{x \in R^n} | x^\alpha D^\beta f(x) |  < \infty , \forall \alpha, \beta \in \mathbb{N}^n  \right\} \end{eqnarray} as Schwartz space (or space of complex-valued rapidly decreasing infinitely differentiable functions defined on R^n) ([6], [7]).

Below is a definition of Hölder space C^t_b(R^n) from [9]:
Definition 2: If 0 < t = [t] + \{t\}, [t] is non-negative integer, 0 < \{t\} < 1, then C^t_b(R^n) denotes the completion of S(R^n) in the norm 
\begin{eqnarray}    C^t_b (R^n) &=& \left\{ f | f \in C^{[t]}_b (R^n) ,  \| f  \|_{C^t_b}  < \infty \right\} , \\   \| f  \|_{C^t_b} &=& \| f  \|_{C_b^{[t]}} \, + \, \sum _{|\alpha | = [t]} \sup _{x \ne y} \frac {| D^\alpha  f(x)  - D^\alpha  f(y) | } { | x - y |^{\{t\}}} \\   \| f  \|_{C_b^{[t]}} &=& \sup _{x \in R^n} | D^\alpha f(x) |, \forall \alpha : | \alpha | \le [t]. \end{eqnarray} Here space C^{[t]}_b (R^n) consists of all functions having bounded continuous derivatives up to order [t]. It is easy to see that C_b^t(R^n) is Banach space [9].

Connection of Bessel potential spaces H^s(R^n) with the spaces C_b^t(R^n) is expressed in theorem ([9], [2]):
Embedding Theorem: If s = n/2+t, where t non-integer, t > 0, then space H^s(R^n) is continuously embedded in C_b^t(R^n)

Particularly from this theorem follows that if f \in H^{n/2 + 1/2}_\varepsilon (R^n), corrected if necessary on a set of Lebesgue measure zero, then it is uniformly continuous and bounded. Further if
f \in H^{n/2 + 1/2 + r}_\varepsilon (R^n), r — integer non-negative number, then it can be treated as f \in C^r (R^n), where C^r (R^n) is a class of functions with r continuous derivatives.

It can be shown ([3], [11], [8], [4], [5]) that function 
\begin{eqnarray}  && V_s ( \eta , x, \varepsilon ) = c_V (n,s,\varepsilon) (\varepsilon |\eta - x | )^{s - \frac{n}{2}}           K_{s - \frac{n}{2}} (\varepsilon |\eta - x | )     \ , \\  && c_V (n,s,\varepsilon) = \frac{\varepsilon ^{n-2s}} { 2^{s-1} (2 \pi )^{n/2} \Gamma (s) }, \ \eta \in R^n, \  x \in R^n, \ \varepsilon > 0 ,  s > \frac{n}{2} \end{eqnarray} is a reproducing kernel of H^s_\varepsilon (R^n) space. Here K_{\gamma} is modified Bessel function of the second kind [10]. The exact value of c_V (n,s,\varepsilon) is not important here and will be set to \sqrt{\frac{2}{\pi}} for ease of further calculations. This reproducing kernel sometimes is called as Matérn kernel [12].

The kernel K_{\gamma} becomes especially simple when \gamma  is half-integer. 
\begin{eqnarray}  \gamma =  r  + \frac{1}{2} \ , (r = 0, 1, \dots ). \end{eqnarray} In this case it is expressed via elementary functions (see [10]):
\begin{eqnarray} K_{r+1/2}(t) &=&    \sqrt{\frac{\pi} {2t}} t^{r+1} \left (                    - \frac{1}{t} \frac{d}{dt} \right )^{r+1} \exp (-t) \ , \\ K_{r+1/2}(t) &=&    \sqrt{\frac{\pi} {2t}} \exp (-t) \sum_{k=0}^r                    \frac{(r+k)!}{k! (r-k)! (2t)^k} \ ,    \  (r = 0, 1, \dots ) \  . \end{eqnarray}
Let s_r =  r + \frac{n}{2} + \frac{1}{2}, \  r = 0, 1, \dots, then H^{s_r}_\varepsilon(R^n) is continuously embedded in C_b^r(R^n) and its reproducing kernel with accuracy to constant multiplier can be presented as follows
\begin{eqnarray}  V_{r + \frac{n}{2} + \frac{1}{2}}(\eta , x, \varepsilon) &=& \exp (-\varepsilon |\eta - x |) \      \sum_{k=0}^{r} \frac{(r+k)!}{2^k k! (r-k)!} (\varepsilon |\eta - x |)^{r-k} \ , \\ \nonumber  &&     (r = 0, 1, \dots ) \  . \end{eqnarray} Particularly (with accuracy to constant multiplier):
\begin{eqnarray} &&  V_{\frac{n}{2} + \frac{1}{2}}(\eta , x, \varepsilon) = \exp (-\varepsilon |\eta - x |) \ , \\ &&   V_{1 + \frac{n}{2} + \frac{1}{2}}(\eta , x, \varepsilon) = \exp (-\varepsilon |\eta - x |)                              (1 + \varepsilon |\eta - x |) \ , \\ &&   V_{2 + \frac{n}{2} + \frac{1}{2}}(\eta , x, \varepsilon) = \exp (-\varepsilon |\eta - x |)              (3 + 3\varepsilon |\eta - x | + \varepsilon ^2 |\eta - x | ^2 ) \ . \end{eqnarray}

References

[1] D.R. Adams, L.I. Hedberg, Function spaces and potential theory. Berlin, Springer, 1996.
[2] M.S. Agranovich, Sobolev Spaces, Their Generalizations and Elliptic Problems in
Smooth and Lipschitz Domains, Springer, Switzerland, 2015.
[3] N. Aronszajn, K.T. Smith, Theory of bessel potentials I, Ann.Inst.Fourier, 11,  1961.
[4] A. Imamov, M. Jurabaev, Splines in Sobolev spaces. Deposited report. UzNIINTI, 24.07.89, No 880.
[5] I. Kohanovsky, Data approximation using multidimensional normal splines. Unpublished manuscript. 1996.
[6] S. M. Nikol'skiĭ, Approximation of functions of several variables and imbedding theorems, Grundl. Math. Wissensch., 205, Springer-Verlag, New York, 1975.
[7] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, 1972.
[8] R. Schaback, Kernel-based Meshless Methods, Lecture Notes, Goettingen, 2011.
[9] H. Triebel, Interpolation. Function Spaces. Differential Operators. North-Holland, Amsterdam, 1978.
[10] G.N. Watson, A Treatise on the Theory of Bessel Functions ( 2nd.ed.), Cambridge University Press, 1966.
[11] H. Wendland, Scattered Data Approximation. Cambridge University Press, 2005.
[12] G.E. Fasshauer, Green’s Functions: Taking Another Look at Kernel Approximation, Radial Basis Functions, and Splines. In: Neamtu M., Schumaker L. (eds) Approximation Theory XIII: San Antonio 2010. Springer Proceedings in Mathematics, vol 13. Springer, New York, 2012.