\section{introduction}

introduction

From calculus of a single variable, the formal definition of the derivative $f'(x)$ of some function $f(x)$ of a single variable $x$, at $x_0$, requires of taking the limit $x\to x_0$ of a ratio of differences

%

\[

f'(x)

:=\lim_{x\to x_0}

\frac{f(x)-f(x_0)}{x-x_0}.

\]

%

However, there are at least two situations in which we need to compute the derivative when it is not possible to take the mentioned limit $x\to x_0$. One situation is numerical computation. When doing numerical calculations, it is not possible to perform a limit like the infinitesimal separation between the discrete set of points of a mesh, and then an approximation to the derivative is used.

Other situation is in the theory of operators. there are operators with a discrete spectrum. The values of the spectrum of the operator are fixed number and there is no meaning in taking a limit like $x\to x_0$. Thus, an exact finite differences, a finite differences which will give the exact derivative even if $\Delta$ has any finite value, is desirable.

Usual finite differences calculus, for equally spaced points in a mesh, is well developed^[1][2][3][4], Here we introduce finite differences calculus which is exact for the first derivative. We specialize the results to finite intervals, first order finte differences and to the real exponential function, the real eigenfunction of the derivative.

\section{The partition}

The Partitio

Let $\{q_{i}\}_{i=-N}^{N}$ be a partition of the real interval $[-a,a]$, that is

%

\begin{align*}

a=q_{-N} < q_{-N+1} < \dots < q_{N-1} < q_{N}=b.

\end{align*}

%

The separations between mesh points are

%

\begin{align*}

\Delta_j = q_{j+1}-q_{j}, \quad j = -N, \dots, N-1.

\end{align*}

%

For a given $v\in\mathbb{R}$, a forward finite differences, first order, first derivative, of a vector $g=(g_{-N}, g_{-N-1},\dots,g_{N-1},g_N)^T$, at $q_j$, defined on the partition, is

%

\[ (D_fg)_j=\frac{g_{j+1}-g_j}{\chi_{f,j}},

\quad

\chi_{f,j}=\frac{e^{v\Delta_j}-1}{v}

=\Delta_j+\frac{v}{2}\Delta^2_j+{\cal O}(\Delta^3_j). \]

%

The function in the denominator of this expression, the function $\chi_{f,j}$, makes sure that the real exponential function $e^{vq}$ be, exactly, an eigenfunction of the derivative operation with real eigenvalue $v$,

%

\begin\begi{align*}

(D_fe^{vq})_j=v\,e^{vq_j},

\end{align*}

%

which is the same property asf the continuous variable derivative has.

When $|\chi_{f,j}-\Delta_j|$ is less than some small $\epsilon$, our method and the usual finite-differences derivative will give similar results for any vector defined on the mesh.

A backwards version of the finite differences derivative, at $q_j$, is

%

\[ (D_bg)_j=\frac{g_j-g_{j-1}}{\chi_{b,j}},

\quad

\chi_{b,j}=\frac{1-e^{-v\Delta_{j-1}}}{v}

=\Delta_{j-1}-\frac{v}{2}\Delta^2_{j-1}

+{\cal O}(\Delta^3_{j-1}), \]

%

which also complies with

%

\begin{align*}

(D_be^{vq})_j=v\,e^{vq_j}.

\end{align*}

\subsection{Properties}

Propertie

Some properties of the exact finite-differences derivative are

{\em The equality\/}

%

$ \frac{\chi_{b,j+1}}{\chi_{f,j}}

=e^{-v\Delta_j}

$ implies that

%

\[ (D_fg)_j

=e^{-v\Delta_j}\,(D_bg)_{j+1}

. \]

{\em Two discrete versions of the summation of the derivative.\/} The finite differences versions of $\int_a^xdy\,g'(y)=g(x)-g(a)$ are

%

\begin{align*}

&\sum_{j=-N}^n\Delta_j(D_fg)_j

%

&=

-\frac{\Delta_{-N}}{\chi_{f,-N}}g_{-N}

+\sum_{j=-N+1}^n\left(

\frac{\Delta_{j-1}}{\chi_{f,j-1}}

-\frac{\Delta_j}{\chi_{f,j}}\right)g_j

+\frac{\Delta_n}{\chi_{f,n}}g_{n+1}

,

\end{align*}

%

and

%

\begin{align*}

&\Delta_{-N}(D_fg)_{-N}

+\sum_{j=-N+1}^n\Delta_{j-1}(D_bg)_j

%

&=

-\left(\frac{1}{\chi_{f,-N}}

+\frac{1}{\chi_{b,-N+1}}\right)\Delta_{-N}g_{-N}

+\frac{\Delta_{-N}}{\chi_{f,-N}}g_{-N+1}

+\sum_{j=-N+1}^{n-1}\left(

\frac{\Delta_{j-1}}{\chi_{b,j}}

-\frac{\Delta_j}{\chi_{b,{j+1}}}\right)g_j

+\frac{\Delta_{n-1}}{\chi_{b,n}}g_n

%,

\end{align*}

%

where $n<N$. The summation term at the right hand side of this equality involve interference terms that vanish when the separation between mesh points is the same. For the whole interval, we have

%

\begin{align*}

&

\sum_{j=-N}^{N-1}\Delta_j(D_fg)_j

+\Delta_{N-1}(D_bg)_N

\\

%

&=

-\frac{\Delta_{-N}}{\chi_{f,-N}}g_{-N}

+\sum_{j=-N+1}^{N-1}\left(

\frac{\Delta_{j-1}}{\chi_{f,j-1}}

-\frac{\Delta_j}{\chi_{f,j}}\right)g_j

-\frac{\Delta_{N-1}}{\chi_{b,N}}g_{N-1}+\left(\frac{1}

{\chi_{f,N-1}}

+\frac{1}{\chi_{b,N}}\right)\Delta_{N-1}g_N

,

\end{align*}

%

\begin{align*}

&

\Delta_{-N}(D_fg)_{-N}

+\sum_{j=-N+1}^N\Delta_{j-1}(D_bg)_j\\

&=-\left(\frac{1}{\chi_{f,-N}}

+\frac{1}{\chi_{b,-N+1}}\right)

\Delta_{-N}g_{-N}

+\frac{\Delta_{-N}}{\chi_{f,-N}}g_{-N+1}

+\sum_{j=-N+1}^{N-1}\left(

\frac{\Delta_{j-1}}{\chi_{b,j}}

-\frac{\Delta_j}{\chi_{b,j+1}}\right)g_j

+\frac{\Delta_{N-1}}{\chi_{b,N}}g_N

,

\end{align*}

{\em The eigenfunction of the summation.\/} The finite differences versions of $\int_a^xdx\,v\,e^{vx}=e^{va}-e^{vx}$ are

%

\begin{align*}

\sum_{j=-N}^n\Delta_jv\,e^{vq_j}

&=\sum_{j=-N}^n\Delta_j(D_f\,e^{vq})_j\\

%

&=-\frac{\Delta_{-N}}{\chi_{f,-N}}e^{vq_{-N}}

+\sum_{-N+1}^n\left(

\frac{\Delta_{j-1}}{\chi_{f,j-1}}

-\frac{\Delta_j}{\chi_{f,j}}\right)e^{vq_j}

+\frac{\Delta_n}{\chi_{f,n}}e^{vq_{n+1}},

\end{align*}

%

and

%

\begin{align*}

&\Delta_{-N}v\,e^{vq_{-N}}

+\sum_{j=-N+1}^n\Delta_{j-1}v\,e^{vq_j}

%=\Delta_{-N}(D_f\,e^{vq})_{-N}

% +\sum_{j=-N+1}^n\Delta_{j-1}(D_b\,e^{vq})_j\\

%

&=-\left(\frac{1}{\chi_{f,-N}}

+\frac{1}{\chi_{b,-N+1}}\right)

\Delta_{-N}e^{vq_{-N}}

+\frac{\Delta_{-N}}{\chi_{f,-N}}e^{vq_{-N+1}}

+\sum_{j=-N+1}^{n-1}\left(

\frac{\Delta_{j-1}}{\chi_{b,j}}

-\frac{\Delta_j}{\chi_{b,j+1}}\right)e^{vq_j}

+\frac{\Delta_{n-1}}{\chi_{b,n}}e^{vq_n},

\end{align*}

%

where $-N<n<N$.

{\em The derivative of one.\/}

%

\begin{align*}

(D_f\,1)_j=0,

\quad\text{and}\quad

(D_b\,1)_j=0,

\end{align*}

{\em The derivatives of $q$.\/}

%

\begin{align*}

(D_f\,q)_j

=\frac{q_{j+1}-q_j}{\chi_{f,j}}

=\frac{\Delta_j}{\chi_{f,j}},

\quad\text{and}\quad

(D_b\,q)_j

=\frac{q_j-q_{j-1}}{\chi_{b,j}}

=\frac{\Delta_{j-1}}{\chi_{b,j}}.

\end{align*}

%

These derivatives will approach to one in the limit of small $\Delta_j$.

{\em The chain rule.\/}

%

\begin{align*}

(D_fg(h(q)))_j

=({\cal D}_fg)_j\,(D_fh)_j,

\end{align*}

%

where

%

\begin{align*}

({\cal D}_fg)_j

=\frac{g(h(q_{j+1}))-g(h(q_j))}{h(q_{j+1})-h(q_j)}.

\end{align*}

%

and

%

\begin{align*}

(D_bg(h(q)))_j

=({\cal D}_bg)_j\,(D_bh)_j,

\end{align*}

%

where

%

\begin{align*}

({\cal D}_bg)_j

=\frac{g(h(q_j))-g(h(q_{j-1}))}{\chi_{b,j}}.

\end{align*}

{\em The derivative of a product of vectors.\/}

%

\begin{align*}

(D_f\,gh)_j

=g_{j+1}(D_fh)_j+(D_fg)_jh_j

.

\end{align*}

%

\begin{align*}

(D_f\,gh)_j

=(D_fg)_jh_{j+1}+g_j(D_fh)_j

.

\end{align*}

%

\begin{align*}

(D_b\,gh)_j

=g_j(D_bh)_j+(D_bg)_jh_{j-1}

.

\end{align*}

%

\begin{align*}

(D_b\,gh)_j

=(D_bg)_jh_j+g_{j-1}(D_bh)_j

.

\end{align*}

{\em The derivative of the inverse of a vector.\/}

%

\begin{align*}

\left(D_f\,\frac{1}{h}\right)_j

=-\frac{(D_fh)_j}{h_jh_{j+1}},

%

\quad\text{and}\quad

%

\left(D_b\,\frac{1}{h}\right)_j

=-\frac{(D_bh)_j}{h_jh_{j-1}}.

\end{align*}

{\em The derivative of a ratio of vectors.\/}

%

\begin{align*}

\left(D_f\,\frac{g}{h}\right)_j

&=\frac{(D_f\,g)_j}{h_j}

-\frac{(D_fh)_j}{h_jh_{j+1}}g_{j+1}

%=\frac{h_{j+1}(D_f\,g)_j-(D_fh)_jg_{j+1}}{h_jh_{j+1}}

.

%

\quad\text{or}\quad

%

\left(D_f\,\frac{g}{h}\right)_j

=\frac{(D_f\,g)_j}{h_{j+1}}

-\frac{(D_fh)_j}{h_jh_{j+1}}g_j\\

.

\end{align*}

%

and

%

\begin{align*}

\left(D_b\,\frac{g}{h}\right)_j

=\frac{(D_b\,g)_j}{h_{j-1}}

-\frac{(D_bh)_j}{h_jh_{j-1}}g_j

,

%

\quad\text{or}\quad

%

\left(D_b\,\frac{g}{h}\right)_j

=\frac{(D_b\,g)_j}{h_j}

-\frac{(D_bh)_j}{h_jh_{j-1}}g_{j-1}

.

\end{align*}

{\em Summation by parts.\/}

%

\begin{align*}

&\sum_{j=-N}^{N-1}\Delta_jg_{j+1}(D_fh)_j

+\Delta_{N-1}g_N(D_bh)_N

+\sum_{j=-N}^{N-1}\Delta_j(D_fg)_jh_j

+\Delta_{N-1}(D_bg)_Nh_{N-1}

=\text{b.i.t},

\end{align*}

%

where

%

\begin{align*}

&\text{b.i.t.}

=\sum_{j=-N}^{N-1}\Delta_j(D_f\,gh)_j

+\Delta_{N-1}(D_b\,gh)_N\\

&=-\frac{\Delta_{-N}}{\chi_{f,-N}}g_{-N}h_{-N}

+\sum_{-N+1}^{N-1}\left(

\frac{\Delta_{j-1}}{\chi_{f,j-1}}

-\frac{\Delta_j}{\chi_{f,j}}\right)h_jg_j

-\frac{\Delta_{N-1}}{\chi_{b,N}}h_{N-1}g_{N-1}\\

&\quad+\left(\frac{1}{\chi_{b,{N-1}}}

+\frac{1}{\chi_{b,N}}\right)\Delta_{N-1}h_Ng_N.

\end{align*}

%

The terms in the summation vanish when $\Delta_j\to0$ or when the separation between the mesh points is the same.

Also

%

\begin{align*}

&\Delta_{-N}g_{-N+1}(D_fh)_{-N}

+\Delta_{-N}(D_fg)_{-N}h_{-N}

+\sum_{j=-N+1}^n\Delta_{j-1}g_j(D_bh)_j

+\sum_{j=-N+1}^n\Delta_{j-1}(D_bg)_jh_{j-1}

\\

\\

%

&=

-\left(\frac{1}{\chi_{f,-N}}

+\frac{1}{\chi_{b,-N+1}}\right)\Delta_{-N}g_{-N}h_{-N}

+\frac{\Delta_{-N}}{\chi_{f,-N}}g_{-N+1}h_{-N+1}

+\sum_{j=-N+1}^{N-1}\left(

\frac{\Delta_{j-1}}{\chi_{b,j}}

-\frac{\Delta_j}{\chi_{b,{j+1}}}\right)g_jh_j\\

&\quad+\frac{\Delta_{N-1}}{\chi_{b,N}}g_Nh_N

,

\end{align*}

{\em The commutator between $q$ and $D$.\/} The discrete version of $\frac{d}{dq}qh(q)-q\frac{d}{dq}h(q)=h(q)$ becomes

%

\begin{align*}

(D_f\,qh)_j-q_{j+1}(D_f\,h)_j

=\frac{\Delta_j}{\chi_{f,j}}q_j,

\end{align*}

%

\begin{align*}

(D_b\,qh)_j-q_j(D_b\,h)_j

=\frac{\Delta_j}{\chi_{b,j}}q_{j-1}

\end{align*}

{\em Translation of the exponential function.\/}

%

\begin{align*}

(e^{s\,D_{b,f}}e^{vq})_j

=e^{v(q_j+s)}

\end{align*}

{\em The bounded Fourier transform\/} of a given function $g(v)$ is defined as

%

\begin{align*}

\tilde g_j=(Fg)_j

=\frac{1}{\sqrt{2a}}

\int_{-{\cal V}/2}^{{\cal V}/2}e^{ivq_j}g(v)dv,

\end{align*}

{\em The discrete Fourier transform\/} of a vector $h_j$ is defined as

%

\begin{align*}

\tilde h(v)=({\cal F}h)(v)

=\sum_{j=-N}^N\sqrt{\frac{\Delta_j}{\cal V}}\,e^{-ivq_j}h_j.

\end{align*}

%

These transformations preserve the norm of vectors and functions.

{\em The conjugate of $D_f$}. Consider the equality

%

\begin{align*}

p({\cal F}h)_j

&=\sum_{j=-N}^{N-1}\sqrt{\frac{\Delta_j}{\cal V}}\,

e^{-ipq_{j+1}}(-iD_fh)_j

+\sqrt{\frac{\Delta_{N-1}}{\cal V}}e^{-ipq_{N-1}}(-iD_bh)_N

\\

&\quad +i\sum_{j=-N}^N\sqrt{\frac{\Delta_j}{\cal V}}\,

(D_fe^{-ipq}h)_j

+i\sqrt{\frac{\Delta_{N-1}}{\cal V}}\,(D_be^{-ipq}h)_N.

\end{align*}

%

Then, there is the relationship

%

\begin{align*}

p\leftrightarrow

\begin{cases}

e^{-ip\Delta_j}(-iD_fh)_j,& j=-N,\dots,N-1,\\

e^{ip\Delta_{N-1}}(-iD_bh)_N,& j=N.

\end{cases}

\end{align*}

%

provided interference and boundary terms vanish.

{\em The conjugate of $q_j$.} The relationship

%

\begin{align*}

q_j\tilde g_j

&=i\frac{1}{\sqrt{2a}}

\int_{-{\cal V}/2}^{{\cal V}/2}

e^{ivq_j}\frac{d\,g(v)}{dv}dv

-i\frac{1}{\sqrt{2a}}

e^{ivq_j}g(v)\Big|_{v=-{\cal V}/2}^{{\cal V}/2}

\end{align*}

%

indicates that

%

\begin{align*}

q_j\leftrightarrow

i\,\frac{d}{dv}

\end{align*}

%

provided boundary terms vanish.

{\em The normalized eigenvector\/} of the coordinate operator with eigenvalue $q_n$ in the $v$ representation is $e_{q_n}(v)=e^{-ivq_n}/\sqrt{\cal V}$.

Orthonormality $\int_{-{\cal V}/2}^{{\cal V}/2}dv\,

e^*_{q_m}(v)

e_{q_n}(v)

=\text{sinc}

\left[\frac{\cal V}{2}(q_m-q_n)\right]$

{\em Conjugate vectors.} The conjugate vector to $e_{q_n}(v)=e^{-ivq_n}/\sqrt{\cal V}$ is

%

\begin{align*}

e_{q_n}(m)

=\frac{1}{\sqrt{2a}}\int_{-{\cal V}/2}^{{\cal V}/2}

e^{ivq_m}\frac{e^{-ivq_n}}{\sqrt{\cal V}}dv

=\sqrt{\frac{{\cal V}}{2a}}\,

\text{sinc}\left[\frac{\cal V}{2}(q_m-q_n)\right],

\end{align*}

{\em The normalized} eigenvector, conjugate to $e_{q_n}(v)=e^{-ivq_n}/\sqrt{\cal V}$ is

%

\begin{align*}

e_{q_n}(m)

=\frac{1}{\sqrt{\Delta_m}}\,

\text{sinc}\left[\frac{\cal V}{2}(q_m-q_n)\right],

\end{align*}

{\em The orthonormality\/} between these vectors read

%

\begin{align*}

\sum_{m=-N}^N\Delta_me^*_{q_j}(m)e_{q_n}(m)

&=\sum_{m=-N}^N\Delta_m\frac{1}{\Delta_m}

\text{sinc}\left(\frac{\cal V}{2}(q_m-q_j)\right)

\text{sinc}\left(\frac{\cal V}{2}(q_m-q_n)\right)\\

%

&\mathop{\longrightarrow}_{{\cal V}\to\infty}

\delta_{jn}

,

\end{align*}

{\em The normalized eigenvector\/} of the discrete derivative with eigenvalue $v$ in the coordinate representation is $e_{v}(n)=e^{ivq_n}/\sqrt{\Delta_n(2N+1)}$

{\em Orthogonality}

%

\begin{align*}

\sum_{n=-N}^N\Delta_ne^*_{v'}(n)e_{v}(n)

=\frac{1}{2N+1}\sum_{n=-N}^Ne^{i(v-v')q_n}

\mathop{\longrightarrow}_{N\to\infty}

\delta(v-v')

.

\end{align*}

{\em The conjugate\/} function to $e_{v}(n)=e^{ivq_n}/\sqrt{2N+1}$ is

%

\begin{align*}

e_{v}(v')

=\sum_{j=-N}^N\sqrt{\frac{\Delta_j}{\cal V}}

e^{-ivq_j}\frac{e^{iv'q_j}}{\sqrt{2N+1}}

=\sum_{j=-N}^N\sqrt{\frac{\Delta_j}{{\cal V}(2N+1)}}

e^{i(v'-v)q_j},

\end{align*}

{\em The normalized vector is}

%

\begin{align*}

e_{v}(v')

=\sum_{j=-N}^N\frac{e^{i(v'-v)q_j}}

{\sqrt{{\cal V}(2N+1)}}

,

\end{align*}

This is a local approach to the finite differences first derivative. Another point of view is collecting the finite differences at each point of the mesh in a single matrix, the subject of another article.

Another approach to the derivative on a mesh make use of second order finite differences. This subject is deal with in another articles.^[5][6]