J₀(y)=Joy

This expression captures the idea that the Bessel function of the first kind of order zero, \( J_0(y) \), represents more than just a mathematical function. It symbolizes the joy of discovery, the beauty of mathematical solutions, and the profound satisfaction that comes from understanding the intricate patterns of the universe.

Khinchin's theorem

Summarizing an excerpt about Khinchin's theorem from

An Introduction to the Theory of Stationary Random Functions

Khinchin's theorem is a simple consequence of the following two statements,
taken together:

(a) The class of functions $B (t)$, which are correlation functions of
stationary random processes, coincides with the class of positive definite
functions of the variable $t$ (see above, Sec. 4 for a real case and Sec. 5
for a complex case).

(b) A continuous function $B (t)$ of the real variable $t$ is positive
definite if, and only if, it can be represented in the form (2.52), where $F
(\omega)$ is bounded and nondecreasing (this statement was proved
independently by Bochner and Khinchin, but was first published by Bochner and
therefore is known as Bochner's theorem; see, e.g., Bochner (1959) and also
Note 3 to Introduction).

In the preceding section it was emphasized that Khinchin's theorem lies at the
basis of almost all the proofs of the spectral representation theorem for
stationary random processes. It is, however, obvious that if we proved the
spectral representation theorem without using Khinchin's theorem, this would
also clearly imply the possibility of representing $B (t)$ in the form (2.52).
Indeed, replacing $X (t + \tau)$ and $X (t)$ in the formula $B (t) = \langle X
(t + \tau) X (t) \rangle$ by their spectral representation (2.61) and then
using (2.1) by definition (2.62) of the corresponding Fourier--Stieltjes
integral and the property (b') of the random function $Z (\omega)$, we obtain
at once (2.52), where
\begin{equation}
  F (\omega + \Delta \omega) - F (\omega) = |Z (\omega + \Delta \omega) - Z
  (\omega) |^2
\end{equation}
so that $F (\omega)$ is clearly a nondecreasing function. Formula (2.76) can
also be written in the differential form:
\begin{equation}
  \langle dZ (\omega)^2 \rangle = dF (\omega)
\end{equation}
Moreover, $(2.77)$ can be combined with the property $(b')$ of $Z (\omega)$ in
the form of a single symbolic relation
\begin{equation}
  \langle dZ (\omega) dZ (\omega') \rangle = \delta (\omega - \omega') dF
  (\omega) d \omega'
\end{equation}
where $\delta (\omega)$ is the Dirac delta-function. It is easy to see that
the substitution of $(2.78)$ into the expression for the mean value of any
double integral with respect to $dZ (\omega)$ and $dZ (\omega')$ gives the
correct result. As the simplest example we consider the following derivation
of Khinchin's formula $(2.52)$:
\begin{equation}
  \begin{array}{ll}
    \langle X (t + \tau) X (t) \rangle & = \left\langle \int_{-
    \infty}^{\infty} e^{i \omega (t + \tau)} dZ (\omega) \int_{-
    \infty}^{\infty} e^{- i \omega' t} dZ (\omega') \right\rangle\\
    & = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} e^{i \omega (t +
    \tau) - i \omega' t}  \langle dZ (\omega) dZ (\omega') \rangle\\
    & = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} e^{i \omega (t +
    \tau) - i \omega' t} \delta (\omega - \omega') dF (\omega) d \omega'\\
    & = \int_{- \infty}^{\infty} e^{i \omega \tau} dF (\omega)
  \end{array}
\end{equation}
Quite similarly, the following more general result can be derived:
\begin{equation}
  \int_{- \infty}^{\infty} g (\omega) dZ (\omega)  \int_{- \infty}^{\infty} h
  (\omega') dZ (\omega') = \int_{- \infty}^{\infty} g (\omega) h (\omega')
  \delta (\omega - \omega') dF (\omega)
\end{equation}
where $g (\omega)$ and $h (\omega)$ are any two complex functions whose
squared absolute values are integrable with respect to $dF (\omega)$. Note
also that if the spectral density $f (\omega)$ exists, then the relations
$(2.77)$ and $(2.78)$ obviously take the form
\begin{equation}
  \langle dZ (\omega)^2 \rangle = f (\omega) d \omega
\end{equation}
\begin{equation}
  \langle dZ (\omega) dZ (\omega') \rangle = \delta (\omega - \omega') f
  (\omega) d \omega d \omega'
\end{equation}
Formulae $(2.76) (2.78)$ and $(2.80)  (2.81)$
establish the relationship between the spectral representation of the
correlation function (determined by the functions $F (\omega)$ and $f
(\omega)$) and the spectral representation of the stationary random process $X
(t)$ itself, which includes the random point function $Z (\omega)$ or the
random interval function
\begin{equation}
  Z (\Delta \omega) = Z (\omega_2) - Z (\omega_1)
\end{equation}
where $\Delta \omega = [\omega_1, \omega_2]$. We shall see in Sec. 11 that
this relationship gives physical meaning to Khinchin's mathematical theorem
and permits one to verify it experimentally when the stationary process $X
(t)$ is realized in the form of oscillations of some measurable physical
quantity $X$.