#  Four Properties of Reproducing Kernel Hilbert Spaces††thanks: Approved for
public release; distribution is unlimited.

Alan Rufty

(November 28, 2007)

###### Abstract

A reproducing kernel Hilbert space (RKHS) has four well-known easily derived
properties. Since these properties are usually not emphasized as a simple
means of gaining insight into RKHS structure, they are singled out and proved
here.

  * Key words: reproducing kernel, Dirichlet form

  * AMS subject classification (2000): Primary 46E22.

##  1 Introduction

A recent article by the author built on the concept of a reproducing kernel
Hilbert space (RKHS) [5] and the present article provides additional standard
background material on RKHSs. For concreteness, since the primary norms and
inner products of interest in [5] were related to Dirichlet integrals defined
over some connected $\mathbb{R}^{3}$ region $\Omega$, the same overall setting
and notation is assumed here, although the properties and proofs given apply
to more general settings; hence, the admissible functions and inner products
are assumed to be real valued and vectors denote points of $\mathbb{R}^{3}$.
While general introductions to RKHS theory can be found in [1], [4] or [3],
the present discussion is limited in scope, but is more-or-less self contained
and should be accessible to a wide readership.

##  2 Reproducing Kernel Properties

First consider RKHS theory where, as it will be presently shown, all
reproducing kernels are symmetric. Let points $\vec{P}$ and $\vec{Q}$ be
$\mathbb{R}^{3}$ points in the same connected region ($\Omega$) and let
$\mathscr{H}$ denote the associated Hilbert space of real valued functions
defined over $\Omega$ with a real valued inner product,
$(\,\,\cdot\,\,,\,\,\cdot\,\,)$. A reproducing kernel $K(\vec{P},\,\vec{Q})$
associated with $\mathscr{H}$ can be compactly characterized by the following
two requirements [4]:

  1. (I)

$K(\vec{P},\,\,\cdot\,)\in\mathscr{H}\ $ and
$K(\,\cdot\,,\,\vec{Q})\in\mathscr{H}\,\,$, which is to say $\|K\|$ must be
bounded when treated as a function of either argument.

  2. (II)

$\boldsymbol{(}K{(\vec{P},\vec{X})},\,f(\vec{X})\,\boldsymbol{)}\equiv
f(\vec{P})$.

As an aside, observe that while a Dirac delta function satisfies the second
requirement, it fails to satisfy the first one so it is not a reproducing
kernel. The actual form and existence of a closed form reproducing kernel is
closely tied to the shape of the region of interest. Reproducing kernels and
their associated spaces (RKHSs) were first studied by Bergman and others and
then brought to a mature state of development over fifty years ago (for a
summary of this work and historical comments see Aronszajn [1], or [7] {which
contains [1]}).

Next consider the following four important properties of a reproducing kernel:

  * • Property (1)

$K{(\vec{P},\vec{Q})}\equiv K{(\vec{Q},\vec{P})}$; i.e., all real reproducing
kernels are symmetric.

  * • Property (2)

$K(\vec{P},\vec{Q})$ is bounded.

  * • Property (3)

Two different reproducing kernels over $\Omega$ cannot exist for the same
norm.

  * • Property (4)

For norms that can be expressed as integrals over $\Omega$, two different
norms cannot share the same reproducing kernel.

Let $\vec{X}$ denote the dummy integration or inner-product variables.
Property (1) follows immediately from the fact that
$(\,K(\vec{P},\,\vec{X})\,,K(\vec{Q},\,\vec{X})\,)=K(\vec{Q},\,\vec{P})$ ,
$(\,K(\vec{Q},\,\vec{X})\,,K(\vec{P},\,\vec{X})\,)=K(\vec{P},\,\vec{Q})$ and
that $(f,\,g)=(g,\,f)$. Since this property means that every reproducing
kernel is symmetric, the adjective symmetric will normally be used. Next
consider Property (2). Let $\mathscr{H}$ be a space of square integrable
functions so that $\|f\|$ is bounded for all $f\in\mathscr{H}$, then
$\|K(\vec{P},\,\,\cdot\,)\,\|^{2}:=(\,K(\vec{P},\,\,\cdot\,),\,K(\vec{P},\,\,\cdot\,)\,)=K(\vec{P},\,\vec{P})$
is bounded. Because $\|K(\vec{P},\,\,\cdot\,)\,\|$ and
$\|K(\vec{Q},\,\,\cdot\,)\,\|$ are bounded, it follows that
$K(\vec{P},\,\vec{Q})$ is bounded since
$K(\vec{P},\,\vec{Q})=(\,K(\vec{P},\,\,\cdot\,),\,\,K(\vec{Q},\,\,\cdot\,)\,\,)\leqq\|K(\vec{P},\,\,\cdot\,)\|\,\|K(\vec{Q},\,\,\cdot\,)\|$.
To prove Property (3) assume to the contrary that some $K_{A}$ and $K_{B}$
exists for a given norm with $K_{A}\neq K_{B}$. Then
$K_{A}{(\vec{P},\vec{Q})}=(K_{A}{(\vec{P},\vec{X})},\,K_{B}{(\vec{X},\vec{Q})})=K_{B}{(\vec{P},\vec{Q})}$;
consequently, Property (3) must hold.

To prove Property (4) some additional notation is required. First consider
only inner-products defined without embedded operators that can be described
in terms of weighted integrals: $\boldsymbol{(}f,\,g\boldsymbol{)}=\int
fg\,{\mu}(\vec{X})\,\,d^{3}\,\vec{X}$, where $\mu$ is the weight. Let two such
different norms or inner products exist and label them
$\boldsymbol{(}\,\cdot\,,\,\cdot\,\boldsymbol{)}{\\!\vphantom{\big{)}}}_{A}$
and $\boldsymbol{(}\,\cdot\,,\,\cdot\,\boldsymbol{)}{\vphantom{\big{)}}}_{B}$.
The statement that these yield different norms then means that $f$ and $g$
always exist such that
$\boldsymbol{(}f,\,g\boldsymbol{)}{\\!\vphantom{\big{)}}}_{A}\neq\boldsymbol{(}f,\,g\boldsymbol{)}_{B}$
, which, in turn, means that since the regions are the same they have
different weight functions: ${\mu}_{A}\neq{\mu}_{B}$. Ordinarily a vector
dummy argument is tied to the implementation of the norm and the choice of
symbol for it does not matter, but here the choice of dummy arguments must be
tracked, so for $\nu=A$ or $B$ let
$\boldsymbol{(}f,\,g\boldsymbol{)}_{\nu(\vec{Q})}$ indicate that vector field
variable $\vec{Q}$ fills this role. Then, contrary to our assumption, observe
that

$\boldsymbol{(}f,\,g\boldsymbol{)}{\\!\vphantom{\big{)}}}_{A}\equiv\boldsymbol{(}f,\,g\boldsymbol{)}_{\\!A(\vec{X})}=\boldsymbol{(}\,\boldsymbol{(}K{(\vec{P},\vec{X})},\,f\boldsymbol{)}_{B(\vec{P})}\,,\,g\boldsymbol{)}_{\\!A(\vec{X})}=\boldsymbol{(}\,\boldsymbol{(}K{(\vec{P},\vec{X})},\,g\boldsymbol{)}_{\\!A(\vec{X})}\,,\,f\boldsymbol{)}_{\\!B(\vec{P})}\\\
=\boldsymbol{(}g,\,f\boldsymbol{)}_{B(\vec{P})}\equiv\boldsymbol{(}f,\,g\boldsymbol{)}{\vphantom{\big{)}}}_{B}$
(1)

(where the step in the middle corresponds to rearranging the integrals of the
associated inner product expressions), which proves the desired result for
weighted norms. This line of argument can be immediately generalized to
include inner products with intrinsic differential operators, such as the
gradient terms occurring in the (weighted) Dirichlet integral.

Property (4) is usually proved in terms of Hilbert space inner products that
have a quadratic form [1, 2], but the derivation just given in terms of inner
products with an integral form is more natural in the present context since it
clearly generalizes all the inner products encountered in [5] and it does not
explicitly require the assumption of a countable basis. (Other standard
features of RKHS theory were also derived in [6] without resorting to the
usual technical functional analysis assumptions.) Property (4) means that
there are usually infinitely many possible symmetric reproducing kernels for a
given region since there a like number of possible norms. Finally, routinely
some norm and associated symmetric reproducing kernel can be transformed by
using the action of a positive definite linear differential operator (or
linear representer) to yield a new norm and associated symmetric reproducing
kernel (see, for example, Moritz [4]).

## References

  * [1] N. Aronszajn, _Theory of Reproducing Kernels_ , Am. Math. Soc. Trns. 68 (1950), 337–404.
  * [2] Haakan Hedenmalm, Boris Korenblum and Kehe Zhu, _Theory of Bergman Spaces_ , Springer-Verlag, New York, N.Y., 2000.
  * [3] László Máté, _Hilbert Space Methods in Science and Engineering_ , Adam Hilger imprint by IOP Publishing Ltd, Bristol, England, 1989.
  * [4] Helmut Moritz, _Advanced Physical Geodesy_ , Abacus Press, Tunbridge Wells, Kent, England, 1980.
  * [5] Alan Rufty, _A Dirichlet-Integral Based Dual-Access Collocation-Kernel Approach to Point-Source Gravity-Field Modeling_ , SIAM Journal on Applied Mathematics, 68, No. 1, 199–221.
  * [6] Alan Rufty, _Dirichlet-integral point-source harmonic interpolation over ${\mathbb{R}}^{3}$ spherical interiors:
DIDACKS II_, [arxiv:math-ph/0702063].

  * [7] Howard L. Weinert (editor), _Reproducing Kernel Hilbert Spaces Applications in Statistical Signal Processing_ , Benchmark Papers in Electrical Engineering and computer Science / 25, Hutchinson Ross Publishing Company, Stroudsburg, Pennsylvania, 1982.


m_{\tilde{\chi}_{2}^{\pm}}^{\mathrm{OS}}-U_{22}^{*}V_{22}^{*}\delta
m_{\tilde{\chi}_{1}^{\pm}}^{\mathrm{OS}}+$
$\displaystyle\sqrt{2}\left(c_{\beta}\left(k_{d}-k_{o}\right)V_{12}^{*}V_{22}^{*}+s_{\beta}\left(r_{o}-r_{d}\right)U_{12}^{*}U_{22}^{*}\right)M_{W}c_{\beta}^{2}\delta
t_{\beta}+$
$\displaystyle\left(s_{\beta}\left(k_{d}-k_{o}\right)V_{12}^{*}V_{22}^{*}-c_{\beta}\left(r_{o}-r_{d}\right)U_{12}^{*}U_{22}^{*}\right)\frac{\delta
M_{W}^{2}}{\sqrt{2}M_{W}}\biggr{\\}},$ $\displaystyle\delta\mu=$
$\displaystyle\frac{1}{k_{o}r_{o}-k_{d}r_{d}}\biggl{\\{}U_{21}^{*}V_{21}^{*}\delta
m_{\tilde{\chi}_{1}^{\pm}}^{\mathrm{OS}}-U_{11}^{*}V_{11}^{*}\delta
m_{\tilde{\chi}_{2}^{\pm}}^{\mathrm{OS}}-$
$\displaystyle\sqrt{2}\left(s_{\beta}\left(k_{d}-k_{o}\right)V_{11}^{*}V_{21}^{*}+c_{\beta}\left(r_{o}-r_{d}\right)U_{11}^{*}U_{21}^{*}\right)M_{W}c_{\beta}^{2}\delta
t_{\beta}+$
$\displaystyle\left(c_{\beta}\left(k_{d}-k_{o}\right)V_{11}^{*}V_{21}^{*}-s_{\beta}\left(r_{o}-r_{d}\right)U_{11}^{*}U_{21}^{*}\right)\frac{\delta
M_{W}^{2}}{\sqrt{2}M_{W}}\biggr{\\}},$

where we have used the short-hands

$k_{d}=U_{11}^{*}U_{22}^{*},\quad k_{o}=U_{12}^{*}U_{21}^{*},\quad
r_{d}=V_{11}^{*}V_{22}^{*},\quad r_{o}=V_{12}^{*}V_{21}^{*},$

and,

$\displaystyle\delta N_{n}=$ $\displaystyle 2c_{\beta}^{2}\delta
t_{\beta}\left(s_{\beta}N_{n3}^{*}+c_{\beta}N_{n4}^{*}\right)\left(M_{W}N_{n2}^{*}-M_{Z}s_{\mathrm{w}}N_{n1}^{*}\right)$
$\displaystyle+\left(c_{\beta}N_{n3}^{*}-s_{\beta}N_{n4}^{*}\right)\left(N_{n1}^{*}\left[\frac{\delta
M_{Z}^{2}}{M_{Z}}s_{\mathrm{w}}+2M_{Z}\delta
s_{\mathrm{w}}\right]-N_{n2}^{*}\frac{\delta M_{W}^{2}}{M_{W}}\right),$
$\displaystyle\delta m_{\tilde{\chi}_{n}^{0}}^{\mathrm{OS}}=$
$\displaystyle\widetilde{\operatorname{Re}}\left[m_{\tilde{\chi}_{n}^{0}}\Sigma_{\tilde{\chi}^{0}}^{L}\left(m_{\tilde{\chi}_{n}^{0}}^{2}\right)+\Sigma_{\tilde{\chi}^{0}}^{SL}\left(m_{\tilde{\chi}_{n}^{0}}^{2}\right)\right]_{nn},$
$\displaystyle\delta m_{\tilde{\chi}_{c}^{\pm}}^{\mathrm{OS}}=$
$\displaystyle\widetilde{\operatorname{Re}}\left[\frac{m_{\tilde{\chi}_{c}^{\pm}}}{2}\left(\Sigma_{\tilde{\chi}^{-}}^{L}\left(m_{\tilde{\chi}_{c}^{\pm}}^{2}\right)+\Sigma_{\tilde{\chi}^{-}}^{R}\left(m_{\tilde{\chi}_{c}^{\pm}}^{2}\right)\right)+\Sigma_{\tilde{\chi}^{-}}^{SL}\left(m_{\tilde{\chi}_{c}^{\pm}}^{2}\right)\right]_{cc}.$

We have used $n=4$ for BP1-4 and $n=3$ for BP5-6. A more thorough analysis can
be found in Fritzsche:2002bi  and Fritzsche:2013fta .
We have used an on-shell renormalization scheme, which has been implemented in
the Formcalc Fritzsche:2013fta , to evaluate these counterterms for the
benchmark scenarios.

Next, $\delta\mathscr{C}_{2}^{R}$ is given by,

$\delta\mathscr{C}_{2}^{R}=\frac{e}{4s_{\mathrm{W}}^{2}}\left(\frac{2}{c_{\mathrm{W}}^{3}}A_{2}+B_{2}\frac{s_{\mathrm{W}}}{c_{\mathrm{W}}}\right).$
(C.5)

In the above expression $A_{2}$ and $B_{2}$ are given by,

$\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle-2\left(c_{\alpha}Z_{13}-s_{\alpha}Z_{14}\right)\left(\left(\delta
Z_{\mathrm{e}}s_{\mathrm{W}}-\delta
s_{\mathrm{W}}\right)Z_{12}c_{\mathrm{W}}^{3}-Z_{11}\left(\delta
s_{\mathrm{W}}s_{\mathrm{W}}+\delta
Z_{\mathrm{e}}c_{\mathrm{W}}^{2}\right)s_{\mathrm{W}}^{2}\right)$
$\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{4}\left[\left(c_{\alpha}N_{i3}-s_{\alpha}N_{i4}\right)\left(s_{W}N_{11}-c_{W}N_{12}\right)+\left(s_{W}N_{i1}-c_{W}N_{i2}\right)\left(c_{\alpha}N_{13}-s_{\alpha}N_{14}\right)\right]\left(\delta\bar{Z}_{i1}+\delta
Z_{i1}^{*}\right)$ $\displaystyle+2(\delta Z_{22}^{H}-\delta
Z_{12}^{H})\left[s_{\alpha}(N_{13}-N_{14})+c_{\alpha}(N_{13}+N_{14})\right]\left(s_{W}N_{11}-c_{W}N_{12}\right).$

The wavefunction renormalization counterterms for the neutralino sector and
the CP-even Higgs sector have been determined using the on-shell
renormalization scheme Fritzsche:2002bi ; Frank:2006yh , following the
implementation in FormCalcFritzsche:2013fta . For different benchmark
scenarios, variants of the on-shell renormalization scheme have been adopted,
a detailed discussion may be found in Chatterjee:2011wc .

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