$\setCounter{0}$
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) $.
Below is a definition of Hölder space $C^t_b(R^n)$ from [9]:
The kernel $K_{\gamma}$ becomes especially simple when $\gamma$ is half-integer.
[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.
[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 Unpublished manuscript. 1996.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.
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]).
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].
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].
&& 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
\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.
