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

$$ 
K ( x , s )  =  K ( s , x ) .
$$

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:

$$ \tag{1 }
\phi ( x) - \lambda
\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s  =  f ( x) ,\ \ 
x \in [ a , b ] .
$$

In the construction of the theory of integral equations with symmetric kernel, it suffices to suppose that the symmetric kernel  $  K $
is measurable on the square  $  [ a , b ] \times [ a , b ] $
and that

$$ \tag{2 }
\int\limits _ { a } ^ { b }  \int\limits _ { a } ^ { b } 
| K ( x , s ) |  ^ {2}  d x  d s  <  \infty ,
$$

while the free term  $  f $
and the unknown function  $  \phi $
are square-integrable functions on  $  [ a , b ] $(
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

$$ \tag{3 }
\phi ( x) - \lambda
\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s) d s  =  0 ,\  x
\in [ a , b ] .
$$

Thus it is proved that: equation (3) has at least one eigen value (when  $  K $
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  $  \lambda $.

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  $  \{ \mu _ {1} , \mu _ {2} ,\dots \} $; 
to each  $  \mu _ {n} $
of the spectrum corresponds a finite set of linearly independent eigen functions. The eigen values and eigen functions can be arranged as two sequences:

$$ \tag{4 }
\left . 
\begin{array}{l}
\lambda _ {1} , \lambda _ {2} \dots  \\
\phi _ {1} , \phi _ {2} \dots  \\
\end{array}
 \right \}
$$

such that the absolute values of the eigen values are non-decreasing:  $  | \lambda _ {k} | \leq  | \lambda _ {k+} 1 | $, 
and each eigen value is repeated as many times as there are eigen functions corresponding to it. Thus, to each  $  \lambda _ {k} $
in (4) corresponds just one eigen function. The system of eigen functions  $  \{ \phi _ {k} \} $
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  $  K $
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  $  K ( x , s ) $, 
regarded as a function of  $  s $, 
with respect to the orthonormal system  $  \{ \phi _ {k} ( s) \} $
is

$$ \tag{5 }
\sum _{k=1} ^  \infty  

\frac{\phi _ {k} ( x) \phi _ {k} ( s) }{\lambda _ {k} }
 .
$$

This series formed from the eigen system of the symmetric kernel  $  K $
is called the bilinear series of  $  K $, 
or the bilinear expansion of  $  K $
in eigen functions. This series converges in the mean to  $  K $, 
that is,

$$ 
\lim\limits _ {n\rightarrow \infty } \ 
\int\limits _ { a } ^ { b }  \int\limits _ { a } ^ { b } 
\left [
K ( x , s ) -
\sum _{k=1} ^ { n } 

\frac{\phi _ {k} ( x) \phi _ {k} ( s) }{\lambda _ {k} }

\right ]  d x  d s  =  0 .
$$

If, in addition, the bilinear series (5) converges uniformly, then

$$ 
K ( x , s )  = \ 
\sum _{k=1} ^  \infty  

\frac{\phi _ {k} ( x) \phi _ {k} ( s) }{\lambda _ {k} }
 .
$$

In particular, the latter equality always holds if the kernel has only a finite number of eigen values. In this case the kernel  $  K $
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  $  K $
that is continuous on the square  $  [ a , b ] \times [ a , b ] $
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  $  \lambda $
is not an eigen value of  $  K $, 
then the symmetric integral equation (1) has a unique solution  $  \phi $, 
given by the formula

$$ \tag{6 }
\phi ( x)  =  f ( x) + \lambda
\sum _{k=1} ^  \infty  

\frac{f _ {k} }{\lambda _ {k} - \lambda }

\phi _ {k} ( x) ,
$$

where  $  \lambda _ {k} $
are the eigen values and  $  f _ {k} $
are the Fourier coefficients of  $  f $
with respect to the orthogonal system  $  \{ \phi _ {m} \} $
of eigen functions of the kernel, in other words,

$$ 
f _ {k}  =  \int\limits _ { a } ^ { b }  f ( s) \phi _ {k} ( s) \ 
d s ,\  k = 1 , 2 \dots
$$

Let  $  \lambda = \lambda _ {1} $
be an eigen value of  $  K $. 
Then the symmetric integral equation (1) is solvable if and only if the following conditions hold:

$$ 
f _ {k}  = \ 
\int\limits _ { a } ^ { b } 
f ( s) \phi _ {k} ( s)  d s  =  0 ,\  k = 1 \dots q ,
$$

where  $  \phi _ {1} \dots \phi _ {q} $
are the eigen functions corresponding to  $  \lambda _ {1} $. 
If these conditions are fulfilled, then all solutions of (1) are expressed by the formula

$$ \tag{7 }
\phi ( x)  =  f ( x) + \lambda
\sum _{k=q+1} ^  \infty  

\frac{f _ {k} }{\lambda _ {k} - \lambda }

\phi _ {k} ( x) +
\sum _ {k=1} ^ { q } 
c _ {k} \phi _ {k} ( x) ,
$$

where  $  c _ {1} \dots c _ {q} $
are arbitrary constants.

If  $  K $
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  $  K $
satisfies

$$ 
\int\limits _ { a } ^ { b } 
| K ( x , s ) |  ^ {2} \ 
d s  \leq   \textrm{ const } ,\ \ 
x \in [ a , b ] ,
$$

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:  $  \overline{ {K ( x , s ) }}\; = K ( s , x ) $.

If the eigen system (4) of a symmetric kernel  $  K $
is known, then it is easy to study the symmetric Fredholm equation of the first kind

$$ \tag{8 }
\int\limits _ { a } ^ { b } 
K ( x , s ) \phi ( s)  d s  =  f ( x) ,\ \ 
x \in [ a , b ] .
$$

Suppose again that the symmetric kernel  $  K $
of equation (8) is square integrable on the square  $  [ a , b ] \times [ a , b ] $
and that the right-hand side  $  f $
and the unknown function  $  \phi $
are square-integrable functions on  $  [ a , b ] $.

The symmetric kernel is called complete it its system of eigen functions  $  \{ \phi _ {n} \} $
is complete (closed) (cf. [[Complete system of functions|Complete system of functions]]).

Picard's theorem. Let  $  K ( x , s ) $
be a complete kernel. Then equation (8) is a solvable if and only if the series

$$ 
\sum _{k=1} ^  \infty  
\lambda _ {k}  ^ {2}
| f _ {k} |  ^ {2}
$$

converges, where  $  f _ {k} $
are the Fourier coefficients of  $  f $. 
When the condition holds, the (unique) solution is expressible in the form

$$ \tag{9 }
\phi ( x)  = \ 
\sum _{k=1} ^  \infty  
\lambda _ {k} f _ {k} \phi _ {k} ( x) ,
$$

and this series converges in the mean.

Carleman [[#References|[5]]] has a constructed a theory under less restrictive conditions on the symmetric kernel  $  K $
than those of Hilbert and Schmidt. These conditions are as follows: 1)  $  \int _ {a}  ^ {b} K  ^ {2} ( x , s )  d s $
exists in the sense of Lebesgue and

$$ 
\lim\limits _ {y \rightarrow x } \ 
\int\limits _ { a } ^ { b } 
[ K ( y , s ) - K ( x , s ) ]  ^ {2}  d s  =  0
$$

for any  $  x \neq x _ {i} $, 
where  $  \{ x _ {i} \} $
is some sequence of points which may have a finite number of limit points; 2) there may exist a finite set of points  $  s _ {1} \dots s _ {n} \in \{ x _ {i} \} $
in a neighbourhood of which the function

$$ 
\sigma  ^ {2} ( x)  = \ 
\int\limits _ { a } ^ { b }  K  ^ {2} ( x , s )  d s
$$

is not integrable, but it must be integrable on the set  $  \Delta _  \epsilon  $
obtained by removing from the interval  $  [ a , b ] $
the intervals  $  ( s _ {i} - \epsilon , s _ {i} + \epsilon ) $, 
$  i = 1 \dots n $, 
where  $  \epsilon $
is an arbitrarily small positive number.

Let  $  K _  \epsilon  ( x , s ) $
be the function that vanishes on the set of points  $  \{ ( x , s ) \} $
for which  $  | x - s _ {i} | \leq  \epsilon $, 
$  | s - s _ {i} | \leq  \epsilon $, 
$  i = 1 \dots n $, 
and that is equal to the kernel  $  K $
of equation (1) on the set of points of  $  [ a , b ] \times [ a , b ] $
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  $  K _  \epsilon  $. 
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  $  \epsilon \rightarrow 0 $.

====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 willkürlicher Funktionen nach Systemen 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  $  [ a , b ] \times [ a , b ] $
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  $  | \lambda _ {k} | \rightarrow \infty $
as  $  k \rightarrow \infty $, 
an arbitrarily small error in  $  f $
can result in an arbitrarily large error in  $  \phi $, 
if it appears in a Fourier component  $  f _ {k} $
with sufficiently large  $  k $(
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>