View source for Reproducing-kernel Hilbert space

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r1300701.png" /> be a [[Hilbert space|Hilbert space]] of functions defined on an abstract set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r1300702.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r1300703.png" /> denote the [[Inner product|inner product]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r1300704.png" /> be the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r1300705.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r1300706.png" /> is called a reproducing-kernel Hilbert space if there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r1300707.png" />, the [[Reproducing kernel|reproducing kernel]], on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r1300708.png" /> such that:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r1300709.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007010.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007011.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007012.png" /> (the reproducing property). From this definition it follows that the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007013.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007014.png" /> is a continuous [[Linear functional|linear functional]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007015.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007016.png" /></td> </tr></table>

The converse is also true. The following theorem holds: A Hilbert space of functions on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007017.png" /> is a reproducing-kernel Hilbert space if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007019.png" />.

By the [[Riesz theorem(2)|Riesz theorem]], the above assumption implies the existence of a linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007020.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007021.png" />. By the construction, the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007022.png" /> is the reproducing kernel for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007023.png" />.

An example of a construction of a reproducing-kernel Hilbert space is the rigged triple of Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007024.png" />, which is defined as follows [[#References|[a5]]] (cf. also [[Rigged Hilbert space|Rigged Hilbert space]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007025.png" /> be a Hilbert space of functions, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007026.png" /> be a linear densely defined [[Self-adjoint operator|self-adjoint operator]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007028.png" /> (the eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007029.png" /> are counted according to their multiplicities) and assume that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007030.png" /></td> </tr></table>

Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007031.png" /> to be the Hilbert space with inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007032.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007033.png" /> is the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007034.png" /> in the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007035.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007036.png" /> be the dual space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007037.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007038.png" />. Then the inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007039.png" /> is defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007041.png" />, equipped with the inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007042.png" />, is a Hilbert space.

Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007043.png" />, where the overline stands for complex conjugation. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007044.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007045.png" />. Indeed,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007046.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007047.png" /></td> </tr></table>

Furthermore,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007048.png" /></td> </tr></table>

so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007049.png" /> is the [[Reproducing kernel|reproducing kernel]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007050.png" />. Moreover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007051.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007052.png" /> is a constant independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007053.png" />. Indeed, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007055.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007057.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007058.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007059.png" />.

Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007060.png" /> is a reproducing kernel Hilbert space with the reproducing kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007061.png" /> defined above. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007062.png" /> is a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007063.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007064.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

then one can define a [[Pre-Hilbert space|pre-Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007065.png" /> of functions of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007066.png" /></td> </tr></table>

The inner product of two functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007067.png" /> is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007068.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007069.png" /></td> </tr></table>

This definition makes sense because of (a1) and because of reproducing property 2). In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007070.png" />, as follows from (a1), and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007071.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007072.png" />, as follows from property 2).

Indeed,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007073.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007074.png" /></td> </tr></table>

Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007077.png" />, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007078.png" /> as claimed.

Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007079.png" /> the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007080.png" /> in the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007081.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007082.png" /> is a reproducing-kernel Hilbert space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007083.png" /> is its reproducing kernel.

A reproducing-kernel Hilbert space is uniquely defined by its reproducing kernel. Indeed, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007084.png" /> is another reproducing-kernel Hilbert space with the same reproducing kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007085.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007087.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007088.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007090.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007091.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007092.png" />. Using this and the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007093.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007094.png" />, one can check that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007095.png" /> and vice versa, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007096.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007098.png" /> consist of the same set of elements. Moreover, the norms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r13007099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070100.png" /> are equal. Indeed, take an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070101.png" /> and a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070103.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070104.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070105.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070106.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070107.png" /></td> </tr></table>

Thus, the norms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070109.png" /> are equal, as claimed, and so are the inner products (by the polarization identity).

Define a [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070111.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070113.png" /> is the range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070114.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070115.png" />, which will be equipped with the structure of a Hilbert space below:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070116.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070117.png" /> is a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070119.png" /> is a positive [[Measure|measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070122.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070123.png" />, and it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070124.png" /> is injective, that is, the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070125.png" /> is total in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070126.png" /> (cf. also [[Total set|Total set]]).

Define

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070127.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070128.png" /></td> </tr></table>

This kernel clearly satisfies condition (a1) and therefore is a reproducing kernel for the reproducing-kernel Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070129.png" /> which it generates. Clearly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070130.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070131.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070132.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070134.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070135.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070136.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070137.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070138.png" /></td> </tr></table>

if one equips <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070139.png" /> with the inner product such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070140.png" />. This requirement is formally equivalent to the following one: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070141.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070142.png" />, so that the distributional kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070143.png" /> is not the usual [[Delta-function|delta-function]], but the one which acts by the rule

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070144.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070145.png" /></td> </tr></table>

and formally one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070146.png" />.

With the inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070147.png" />, the linear set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070148.png" /> becomes a Hilbert space:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070149.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070150.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070151.png" /></td> </tr></table>

Thus, this inner product makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070152.png" /> an [[Isometric operator|isometric operator]] defined on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070153.png" /> and makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070154.png" /> a (complete) Hilbert space, namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070155.png" />, a reproducing-kernel Hilbert space. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070156.png" /> is assumed injective, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070157.png" /> is defined on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070158.png" /> and, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070159.png" /> is complete in the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070160.png" />, one concludes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070161.png" /> is continuous (by the Banach theorem). Consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070162.png" /> is a co-isometry, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070163.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070164.png" /> is the [[Adjoint operator|adjoint operator]] to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070165.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070166.png" />, then one can write an inversion formula for the linear transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070167.png" /> similar to the well-known inversion formula for the [[Fourier transform|Fourier transform]]. Formally one has:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070168.png" /></td> </tr></table>

The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070169.png" /> is the reproducing-kernel Hilbert space generated by kernel (a3) which is the reproducing kernel for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070170.png" />. The above formal inversion formulas may be of practical interest if the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070171.png" /> is a standard one. In this case the second formula should be suitably interpreted, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070172.png" /> is defined at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070173.png" />-almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070174.png" />.

In [[#References|[a6]]] it is claimed that the characterization of the range of the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070175.png" />, defined in (a3), can be given as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070176.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070177.png" /> is the reproducing-kernel Hilbert space generated by kernel (a3).

However, in fact such a characterization does not give, in general, practically useful necessary and sufficient conditions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070178.png" /> because the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070179.png" /> is not defined in terms of standard norms such as Sobolev or Hölder ones (see [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]]). However, when the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130070/r130070180.png" /> is equivalent to a standard norm, the above characterization becomes efficient (see [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], and also [[#References|[a6]]]).

Many concrete examples of reproducing-kernel Hilbert spaces can be found in [[#References|[a1]]], [[#References|[a2]]] and [[#References|[a6]]].

The papers [[#References|[a1]]] and [[#References|[a7]]] are important in this area, the book [[#References|[a6]]] contains many references, while [[#References|[a2]]] is an earlier book important for the development of the theory of reproducing-kernel Hilbert spaces.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Aronszajn,   "Theory of reproducing kernels"  ''Trans. Amer. Math. Soc.'' , '''68'''  (1950)  pp. 337–404</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Bergman,   "The kernel function and conformal mapping" , Amer. Math. Soc.  (1950)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.G. Ramm,   "On the theory of reproducing kernel Hilbert spaces"  ''J. Inverse Ill-Posed Probl.'' , '''6''' :  5  (1998)  pp. 515–520</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.G. Ramm,   "On Saitoh's characterization of the range of linear transforms"  A.G. Ramm (ed.) , ''Inverse Problems, Tomography and Image Processing'' , Plenum  (1998)  pp. 125–128</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.G. Ramm,   "Random fields estimation theory" , Longman/Wiley  (1990)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Saitoh,   "Integral transforms, reproducing kernels and their applications" , ''Pitman Res. Notes'' , Longman  (1997)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L. Schwartz,   "Sous-espaces hilbertiens d'espaces vectoriels topologiques et noyaux associès (noyaux reproduisants)"  ''J. Anal. Math.'' , '''13'''  (1964)  pp. 115–256</TD></TR></table>