An important result can be derived by computing the inverse Fourier transform of the Fourier transform of a delta function

	$\delta (x-x_0)$	$={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}\delta (x-x_0)e^{-ikx}𝑑x{e}^{ikx}𝑑k$
		$={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{e}^{-ik{x}_0}{e}^{ikx}𝑑k={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{e}^{ik(x-x_0)}𝑑k.$

From this we get that the inverse Fourier transform of a constant is the delta function

${\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{e}^{ik(x-x_0)}𝑑k=\sqrt{2\pi }\delta (x-x_0).$

Taking the complex conjugate of this equation and making use of the fact that ${\delta }^{*}(x-x_0)=\delta (x-x_0)$ we get as a definition for the delta function

$\delta (x-x_0)={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{e}^{\pm ik(x-x_0)}𝑑k.$

(D.5)

Using this we can derive the astonishing result

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(x)𝑑x=\sqrt{2\pi }F(0)$

(D.6)

as can be seen from

	${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(x)𝑑x$	$={\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}F(k)e^{ikx}𝑑k𝑑x={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}F(k)\underset{2\pi \delta (k)}{\underbrace{{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{e}^{ikx}}}𝑑x𝑑k$
		$=\sqrt{2\pi }{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}F(k)\delta (k)𝑑k=\sqrt{2\pi }F(0).$

The Fourier transform of the product of two function in $k$-space is

	${\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}F(k)G(k)e^{ikx}𝑑k=$
	$={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(x^{\prime })e^{-ik{x}^{\prime }}𝑑x^{\prime }{\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}g(x^{\prime \prime })e^{-ik{x}^{\prime \prime }}𝑑x^{\prime \prime }{e}^{ikx}𝑑k$
	$={\displaystyle \frac{1}{{(2\pi )}^{3/2}}}{\displaystyle {\iiint }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(x^{\prime })e^{-ik{x}^{\prime }}g(x^{\prime \prime })e^{-ik{x}^{\prime \prime }}{e}^{ikx}𝑑x^{\prime }𝑑x^{\prime \prime }𝑑k$
	$={\displaystyle \frac{1}{{(2\pi )}^{3/2}}}{\displaystyle {\iint }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(x^{\prime })g(x^{\prime \prime })\underset{2\pi \delta (x^{\prime \prime }-(x-x^{\prime }))}{\underbrace{{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{e}^{-ik(x^{\prime }+x^{\prime \prime }-x)}𝑑k}}𝑑x^{\prime }𝑑x^{\prime \prime }$
	$={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(x^{\prime }){\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}g(x^{\prime \prime })\delta (x^{\prime \prime }-(x-x^{\prime }))𝑑x^{\prime \prime }𝑑x^{\prime }$
	$={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(x^{\prime })g(x-x^{\prime })𝑑x^{\prime }=h(x).$

The integral $h(x)$ is called convolution of the functions $f(x)$ and $g(x)$. So the convolution theorem says that

$h(x)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(x^{\prime })g(x-x^{\prime })𝑑x^{\prime }={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}F(k)G(k)e^{ikx}𝑑k.$

(D.7)

The autocorrelation of a function is defined as⁵⁷⁵⁷Note that with our definition of the Fourier transform we cannot define the autocorrelation function as $h_{AC}(x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x^{\prime })f^{*}(x+x^{\prime })𝑑x^{\prime }$, because we could then not derive the Wiener-Khinchin theorem.

$h_{AC}(x)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{f}^{*}(x^{\prime })f(x+x^{\prime })𝑑x^{\prime }.$

(D.8)

The Wiener-Khinchin Theorem states that

${\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{f}^{*}(x^{\prime })f(x+x^{\prime })𝑑x^{\prime }={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|F(k)\right|}^2{e}^{ikx}𝑑k,$

(D.9)

which can be proved in analogy to the convolution theorem

	${\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{f}^{*}(x^{\prime })f(x+x^{\prime })𝑑x^{\prime }=$
	$={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{f}^{*}(x^{\prime }){\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(x^{\prime \prime })\delta (x^{\prime \prime }-(x+x^{\prime }))𝑑x^{\prime \prime }𝑑x^{\prime }$
	$={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{f}^{*}(x^{\prime }){\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(x^{\prime \prime }){\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{e}^{-ik(x^{\prime }+x^{\prime \prime }-x)}𝑑k𝑑x^{\prime \prime }𝑑x^{\prime }$
	$={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{f}^{*}(x^{\prime })e^{ik{x}^{\prime }}𝑑x^{\prime }{\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(x^{\prime \prime })e^{-ik{x}^{\prime \prime }}𝑑x^{\prime \prime }{e}^{ikx}𝑑k$
	$={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{F}^{*}(k)F(k)e^{ikx}𝑑k={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|F(k)\right|}^2{e}^{ikx}𝑑k.$

A special case of the Wiener-Khinchin theorem is Parseval’s theorem

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|f(x)\right|}^2𝑑x={\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|F(k)\right|}^2𝑑k,$

(D.10)

which can be obtained from the Wiener-Khinchin theorem for $x=0$

	$h_{AC}(0)$	$={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{f}^{*}(x^{\prime })f(x^{\prime })𝑑x^{\prime }={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|f(x)\right|}^2𝑑x$
		$={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|F(k)\right|}^2{e}^{ik0}𝑑k={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|F(k)\right|}^2𝑑k.$

A structure function of order $p$ is defined as⁵⁸⁵⁸See Pope (2000).

$S_p(f(x))=⟨{\left[f(x+x^{\prime })-f(x^{\prime })\right]}^p⟩={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left[f(x+x^{\prime })-f(x^{\prime })\right]}^p𝑑x^{\prime }.$

(D.11)

The second order structure function is related to the spectrum $\left|F(k)\right|$ of the function $f$ like

${\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left[f(x+x^{\prime })-f(x^{\prime })\right]}^2𝑑x^{\prime }={\displaystyle \frac{2}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}(1-e^{ikx}){\left|F(k)\right|}^2𝑑k,$

(D.12)

which can be proved⁵⁹⁵⁹A sketch of this prove can also be found in Pope (2000, Appendix G). by expanding the second order structure function

	$S_2(f(x))={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left[f(x+x^{\prime })-f(x^{\prime })\right]}^2𝑑x^{\prime }$
	$={\displaystyle \frac{1}{\sqrt{2\pi }}}\left[{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|f(x+x^{\prime })\right|}^2𝑑x^{\prime }-2{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{f}^{*}(x^{\prime })f(x+x^{\prime })𝑑x^{\prime }+{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|f(x^{\prime })\right|}^2𝑑x^{\prime }\right].$

Substituting $x^{\prime \prime }=x+x^{\prime }$ in the first term we get

	$S_2(f(x))$	$={\displaystyle \frac{1}{\sqrt{2\pi }}}\left[{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|f(x^{\prime \prime })\right|}^2𝑑x^{\prime \prime }-2{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{f}^{*}(x^{\prime })f(x+x^{\prime })𝑑x^{\prime }+{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|f(x^{\prime })\right|}^2𝑑x^{\prime }\right]$
		$={\displaystyle \frac{2}{\sqrt{2\pi }}}\left[{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|f(x^{\prime })\right|}^2𝑑x^{\prime }-{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{f}^{*}(x^{\prime })f(x+x^{\prime })𝑑x^{\prime }\right].$

Using Parseval’s and the Wiener-Khinchin theorem we obtain the final result

	$S_2(f(x))={\displaystyle \frac{2}{\sqrt{2\pi }}}\left[{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|F(k)\right|}^2𝑑k-{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\left|F(k)\right|}^2{e}^{ikx}𝑑k\right]$
	$={\displaystyle \frac{2}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}(1-e^{ikx}){\left|F(k)\right|}^2𝑑k.$

The structure functions used in the theory of Kolmogorov are the so called longitudinal structure functions of the velocity, which are defined as

$S_2(v_{\parallel }(l))=⟨{\left(\left[𝒗(𝒙+𝒍)-𝒗(𝒙)\right]\cdot {\displaystyle \frac{𝒍}{l}}\right)}^p⟩=⟨{\left(v_{\parallel }(𝒙+𝒍)-v_{\parallel }(𝒙)\right)}^p⟩.$

(D.13)

They are related to the longitudinal velocity spectrum⁶⁰⁶⁰In the literature this is often called kinetic energy spectrum, but this is only true for incompressible flows. ${\left|V_{\parallel }(k)\right|}^2$ via equation (D.12). Sometimes also second order transverse structure functions are measured. These are defined as

$S_2(v_{⟂}(l))=⟨{\left({\displaystyle \frac{\left|\left[𝒗(𝒙+𝒍)-𝒗(𝒙)\right]\times 𝒍\right|}{l}}\right)}^p⟩.$

(D.14)

The behavior of the second order transverse structure functions for homogeneous turbulence is uniquely determined by the longitudinal structure function (Pope, 2000, p. 192, Eqs. (6.28)). They also show the characteristic $2/3$-slope as predicted for the longitudinal structure functions (Frisch, 1995, p.60).

In general structure functions of vectorial quantities like the velocity are tensors, e.g. the general second order structure function of the velocity can be defined as

$S_{ij}(𝒙,𝒍)=⟨\left[v_i(𝒙+𝒍)-v_i(𝒙)\right]\left[v_j(𝒙+𝒍)-v_j(𝒙)\right]⟩.$

(D.15)

But it can be shown that for local isotropy only the longitudinal structure function $S_2(v_{\parallel }(l))=S_{11}$ and the transversal structure $S_2(v_{⟂}(l))=S_{22}=S_{33}$ are unequal zero (Pope, 2000). Since the transverse structure function is determined by the longitudinal structure function in case of local homogeneity, for homogeneous and isotropic turbulence $S_{ij}$ is determined by the single scalar function $S_{11}=S_2(v_{\parallel }(l))$ (Pope, 2000).

The third order structure function used in Kolmogorov theory is defined as

$S_{111}(𝒙,𝒍)=⟨{\left[v_1(𝒙+𝒍)-v_1(𝒙)\right]}^3⟩,$

(D.16)

which is often simply called $S_3(v(l))$. So the famous four-fifths law of Kolmogorov is actually true only for one component of the third order structure function tensor, but again for homogeneous and isotropic turbulence the third order structure function tensor $S_{ijk}$ is uniquely determined by the single scalar function $S_{111}=S_3(v(l))$.