Example 2. Find the first characteristic number of the kernel by the method of traces

$$
K(x, t)=\left\{\begin{array}{ll}
t, & x \geqslant t, \\
x, & x \leqslant t,
\end{array} \quad a=0, b=1\right.
$$
Solution. Since the kernel $K(x, t)$ is symmetric, it is sufficient to find $K_{2}(x, t)$ only for $t<x$.

We have

$$
\begin{aligned}
& K_{2}(x, t)=\int_{0}^{1} K(x, z) K(z, t) d z= \\
& =\int_{0}^{t} z^{2} d z+\int_{t}^{x} z t d z+\int_{x}^{1} x t d z=x t-\frac{x^{2} t}{2}-\frac{t^{3}}{6}
\end{aligned}
$$

Next, using formula (5) for $\boldsymbol{m}=1$ and $\boldsymbol{m}=2$ respectively, we find

$$
\begin{aligned}
A_{2} & =2 \int_{0}^{1} d x \int_{0}^{x} K_{1}^{2}(x, t) d t=2 \int_{0}^{1} d x \int_{0}^{x} t^{2} d t=2 \int_{0}^{1} \frac{x^{3}}{3} d x=\frac{1}{6} \\
A_{4} & =2 \int_{0}^{1} d x \int_{0}^{x} K_{2}^{2}(x, t) d t=2 \int_{0}^{1} d x \int_{0}^{x}\left(x^{2} t^{2}+\frac{x^{4} t^{2}}{4}+\frac{t^{6}}{36}-x^{3} t^{2}-\frac{x t^{4}}{3}+\frac{x^{2} t^{4}}{6}\right) d t= \\
& =\left.2 \int_{0}^{1}\left(\frac{t^{3} x^{2}}{3}+\frac{t^{3} x^{4}}{12}+\frac{t^{7}}{7 \cdot 36}-\frac{x^{3} t^{3}}{3}-\frac{x t^{5}}{15}+\frac{x^{2} t^{5}}{30}\right)\right|_{t=0} ^{t=x} d x= \\
& =2 \int_{0}^{1}\left(\frac{x^{5}}{3}+\frac{x^{7}}{12}+\frac{x^{7}}{7 \cdot 36}-\frac{x^{6}}{3}-\frac{x^{6}}{15}+\frac{x^{7}}{30}\right) d x=\frac{17}{630}
\end{aligned}
$$

According to formula (4)

$$
\lambda_{1} \approx \sqrt{\frac{\frac{1}{6}}{\frac{17}{630}}}=2.48
$$

## Problems for Independent Solution

Using the method of traces, find the first characteristic number of the following kernels ( $a=0$, $b=1):$

343. $K(x, t)=x t . \quad$ 344. $K(x, t)=x^{2} t^{2}$.
344. $K(x, t)= \begin{cases}\frac{1}{2} x(2-t), & x \leqslant t, \\ \frac{1}{2} t(2-x), & x \geqslant t .\end{cases}$
345. $K(x, t)= \begin{cases}-\sqrt{x t} \ln t, & x \leqslant t, \\ -\sqrt{x t} \ln x, & x \geqslant t .\end{cases}$
$3^{\circ}$. Kellogg's Method. Let $K(x, t)$ be a symmetric kernel, which for definiteness we will assume to be positive definite, and let $\omega(x)$ be an arbitrary function from $L_{2}(a, b)$. We construct a sequence of functions

$$
\left\{\begin{array}{l}
\omega_{1}(x)=\int_{a}^{b} K(x, t) \omega(t) d t \\
\omega_{2}(x)=\int_{a}^{b} K(x, t) \omega_{1}(t) d t \\
\ldots \ldots \ldots \ldots \ldots \ldots \ldots \\
\omega_{n}(x)=\int_{a}^{b} K(x, t) \omega_{n-1}(t) d t \\
\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots
\end{array}\right.
$$

and consider the numerical sequence

$$
\left\{\frac{\left\|\omega_{n-1}\right\|}{\left\|\omega_{n}\right\|}\right\}
$$

Let $\varphi_{1}(x), \varphi_{2}(x), \ldots$ be the orthonormal eigenfunctions of the kernel $K(x, t)$ and $\lambda_{1} \leqslant \lambda_{2} \leqslant \ldots-$ the corresponding characteristic numbers. Suppose, further, that the function $\omega(x)$ is orthogonal to the functions $\varphi_{1}(x)$, $\varphi_{2}(x), \ldots, \varphi_{k-1}(x)$, but not orthogonal to the eigenfunction $\varphi_{k}(x)$. Then the sequence (6) has as its limit the $k$-th characteristic number $\lambda_{k}$.

The sequence of functions $\left\{\frac{\omega_{n}(x)}{\left\|\omega_{n}(x)\right\|}\right\}$ converges in this case to some function, which is a linear combination of the eigenfunctions corresponding to the characteristic number $\lambda_{k} \cdot \mathrm{K}$. The same limit, to which the sequence (6) converges, converges the sequence

$$
\left\{\frac{1}{\sqrt[n]{\left\|\omega_{n}\right\|}}\right\}
$$

If $\left(\omega, \varphi_{1}\right) \neq 0$, we obtain two approximate formulas for the smallest characteristic number:

$$
\begin{aligned}
& \lambda_{1} \approx \frac{\left\|\omega_{n-1}\right\|}{\left\|\omega_{n}\right\|} \\
& \lambda_{1} \approx \frac{1}{\sqrt[n]{\left\|\omega_{n}\right\|}}
\end{aligned}
$$

where formula (7) gives the value of $\lambda_{1}$ with a surplus. If the kernel $K(x, t)$ is positive definite, then formulas (7), (8) give an approximate value of the smallest absolute value of the characteristic number of the given kernel. With a fortunate choice of $\omega(x)$, Kellogg's method is relatively simple for computations.

The drawback of the method is that it is not known in advance which of the characteristic numbers has been determined.