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

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[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.

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[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.