View source for Integral equation with symmetric kernel

An integral equation with a real symmetric kernel (cf. [[Kernel of an integral operator|Kernel of an integral operator]]):

<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/i/i051/i051420/i0514201.png" /></td> </tr></table>

The theory of linear equations with real symmetric kernel was first constructed by D. Hilbert (1904) by drawing upon the theory of symmetric quadratic forms and going over from a finite to an infinite number of variables. Shortly after this, E. Schmidt (1907) put forward a more elementary method of substantiating Hilbert's results. For this reason, the theory of integral equations with symmetric kernel is also called the Hilbert–Schmidt theory. A significant weakening of the restrictions imposed in this theory on the data and unknown elements was achieved by T. Carleman (see below).

Consider an integral equation of the second kind with real symmetric kernel:

<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/i/i051/i051420/i0514202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

In the construction of the theory of integral equations with symmetric kernel, it suffices to suppose that the symmetric kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i0514203.png" /> is measurable on the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i0514204.png" /> and 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/i/i051/i051420/i0514205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

while the free term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i0514206.png" /> and the unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i0514207.png" /> are square-integrable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i0514208.png" /> (the integral being understood in the sense of Lebesgue).

The development of the theory of integral equations with symmetric kernel begins with the study of a number of general properties of the eigen values and functions of the homogeneous symmetric integral equation

<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/i/i051/i051420/i0514209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

Thus it is proved that: equation (3) has at least one eigen value (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142010.png" /> is almost-everywhere non-zero); eigen functions corresponding to distinct eigen values are orthogonal; the eigen values are real; since the kernel is real, it can be assumed without loss of generality that the eigen functions are real; and there are only a finite number of eigen values on any finite interval of values for the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142011.png" />.

The set of eigen values of (3) is called the spectrum of the equation. The spectrum is a non-empty finite or countable set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142012.png" />; to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142013.png" /> of the spectrum corresponds a finite set of linearly independent eigen functions. The eigen values and eigen functions can be arranged as two sequences:

<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/i/i051/i051420/i05142014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

such that the absolute values of the eigen values are non-decreasing: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142015.png" />, and each eigen value is repeated as many times as there are eigen functions corresponding to it. Thus, to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142016.png" /> in (4) corresponds just one eigen function. The system of eigen functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142017.png" /> can be assumed to be orthonormal. The sequence (4) is called a system of eigen values and eigen functions (or an eigen system) of the symmetric kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142018.png" /> or of the equation (3). The determination of this system is equivalent to the complete solution of the homogeneous symmetric integral equation (3).

The Fourier series of the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142019.png" />, regarded as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142020.png" />, with respect to the orthonormal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142021.png" /> is

<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/i/i051/i051420/i05142022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

This series formed from the eigen system of the symmetric kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142023.png" /> is called the bilinear series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142024.png" />, or the bilinear expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142025.png" /> in eigen functions. This series converges in the mean to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142026.png" />, that is,

<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/i/i051/i051420/i05142027.png" /></td> </tr></table>

If, in addition, the bilinear series (5) converges uniformly, 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/i/i051/i051420/i05142028.png" /></td> </tr></table>

In particular, the latter equality always holds if the kernel has only a finite number of eigen values. In this case the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142029.png" /> is degenerate (cf. [[Degenerate kernel|Degenerate kernel]]).

The converse statement also holds: A degenerate symmetric kernel has a finite number of eigen values (and, hence, a finite set of eigen functions). The bilinear series of a kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142030.png" /> that is continuous on the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142031.png" /> and has positive eigen values converges uniformly.

Knowledge of the eigen system (4) enables one to construct the solution of the inhomogeneous equation (1). The following theorems hold.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142032.png" /> is not an eigen value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142033.png" />, then the symmetric integral equation (1) has a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142034.png" />, given by the formula

<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/i/i051/i051420/i05142035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142036.png" /> are the eigen values and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142037.png" /> are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142038.png" /> with respect to the orthogonal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142039.png" /> of eigen functions of the kernel, in other words,

<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/i/i051/i051420/i05142040.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142041.png" /> be an eigen value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142042.png" />. Then the symmetric integral equation (1) is solvable if and only if the following conditions hold:

<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/i/i051/i051420/i05142043.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142044.png" /> are the eigen functions corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142045.png" />. If these conditions are fulfilled, then all solutions of (1) are expressed by the formula

<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/i/i051/i051420/i05142046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142047.png" /> are arbitrary constants.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142048.png" /> has an infinite number of eigen values and, consequently, there are infinite series on the right-hand sides of the formulas (6), (7), then these series converge in the mean. If it is further required that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142049.png" /> satisfies

<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/i/i051/i051420/i05142050.png" /></td> </tr></table>

then the above series converge absolutely and uniformly.

Formulas (6) and (7) are called Schmidt's formulas. Much of the theory of integral equations with symmetric kernel extends easily to complex-valued functions. In this case, the analogue of the real symmetric kernel is a Hermitian kernel: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142051.png" />.

If the eigen system (4) of a symmetric kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142052.png" /> is known, then it is easy to study the symmetric Fredholm equation of the first kind

<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/i/i051/i051420/i05142053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

Suppose again that the symmetric kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142054.png" /> of equation (8) is square integrable on the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142055.png" /> and that the right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142056.png" /> and the unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142057.png" /> are square-integrable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142058.png" />.

The symmetric kernel is called complete it its system of eigen functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142059.png" /> is complete (closed) (cf. [[Complete system of functions|Complete system of functions]]).

Picard's theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142060.png" /> be a complete kernel. Then equation (8) is a solvable if and only if the series

<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/i/i051/i051420/i05142061.png" /></td> </tr></table>

converges, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142062.png" /> are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142063.png" />. When the condition holds, the (unique) solution is expressible in 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/i/i051/i051420/i05142064.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

and this series converges in the mean.

Carleman [[#References|[5]]] has a constructed a theory under less restrictive conditions on the symmetric kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142065.png" /> than those of Hilbert and Schmidt. These conditions are as follows: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142066.png" /> exists in the sense of Lebesgue and

<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/i/i051/i051420/i05142067.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142068.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142069.png" /> is some sequence of points which may have a finite number of limit points; 2) there may exist a finite set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142070.png" /> in a neighbourhood of which the function

<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/i/i051/i051420/i05142071.png" /></td> </tr></table>

is not integrable, but it must be integrable on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142072.png" /> obtained by removing from the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142073.png" /> the intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142075.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142076.png" /> is an arbitrarily small positive number.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142077.png" /> be the function that vanishes on the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142078.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142081.png" />, and that is equal to the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142082.png" /> of equation (1) on the set of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142083.png" /> outside this set. The idea of Carleman's method is as follows:

Instead of equation (1) one considers the linear integral equation of the second kind with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142084.png" />. One studies the spectrum and solutions of this equation and then one investigates the spectrum and solution of equation (1) by passing to the limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142085.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Hilbert, "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen" , Chelsea, reprint (1953) {{MR|0056184}} {{ZBL|0050.10201}} </TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> E. Schmidt, "Zur Theorie der linearen und nichtlinearen Integralgleichungen. 1. Entwicklung willküriger Funktionen nach Systeme vorgeschriebener" ''Math. Ann.'' , '''63''' (1907) pp. 433–476 {{MR|1511415}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> E. Schmidt, "Zur Theorie der linearen und nichtlinearen Integralgleichungen. 2. Auflösung der allgemeinen linearen Integralgleichung" ''Math. Ann.'' , '''64''' (1907) pp. 162–174 {{MR|1511415}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2c]</TD> <TD valign="top"> E. Schmidt, "Zur Theorie der linearen und nichtlinearen Integralgleichungen. 3. Ueber die Auflösung der nichtlinearen Integralgleichung und die Verzweigung ihrer Lösungen" ''Math. Ann.'' , '''65''' (1908) pp. 370–399 {{MR|1511472}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G. Wiarda, "Integralgleichungen unter besonderer Berücksichtigung der Anwendungen" , Teubner (1930) {{MR|1522749}} {{ZBL|56.0334.01}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S.G. Mikhlin, "Linear integral equations" , Hindushtan Publ. Comp. , Delhi (1960) (Translated from Russian) {{MR|1534167}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> T. Carleman, "Sur les équations intégrales singulières à noyau réel et symmétrique" ''Univ. Årsskrift'' : 3 , Uppsala (1923)</TD></TR></table>



====Comments====
The fact that the bilinear series of a continuous kernel on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142086.png" /> with positive eigen values converges uniformly is usually called the [[Mercer theorem|Mercer theorem]].

Formula (9) shows the ill-posedness of an integral equation of the first kind: Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142087.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142088.png" />, an arbitrarily small error in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142089.png" /> can result in an arbitrarily large error in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142090.png" />, if it appears in a Fourier component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142091.png" /> with sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051420/i05142092.png" /> (see [[Ill-posed problems|Ill-posed problems]]).

For additional references see [[Integral equation|Integral equation]]. See also [[Hermitian kernel|Hermitian kernel]] for the symmetricity condition.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T. Carleman, "Sur la théorie des équations intégrales et ses applications" , ''Verh. Internat. Mathematiker Kongress. Zürich, 1932'' , '''1''' , O. Füssli (1932) pp. 138–151 {{MR|}} {{ZBL|0006.40001}} {{ZBL|58.0403.02}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Tricomi, "Integral equations" , Interscience (1985) {{MR|0809184}} {{ZBL|0208.54601}} {{ZBL|0092.10803}} {{ZBL|0078.09404}} </TD></TR></table>