6. Transformation between coordinate and derivative representations

Since one of the eigenvalues of the derivative matrix is a continuous variable, we can talk of conjugate functions with a continuous argument v. The relationship between discrete vectors on a partition qi � � and functions with a continuous argument v makes use of continuous and discrete Fourier type of transformations, a wavelet [12]. If we have a function h of continuous argument v, a conjugate vector on the partition qi � � is defined through the type of continuous Fourier transform F as

$$h(Fh)\left(\eta\_j\right) \coloneqq \frac{1}{L\sqrt{2\Delta}} \int\_{-L/2}^{L/2} e^{-i\eta\_j v} h(v) dv,\tag{65}$$

and vice-versa, a continuous variable function is defined with the help of a discrete type of Fourier transform F as

$$\chi(\text{Fg})(v) \coloneqq \frac{L}{\sqrt{2\Delta}} \sum\_{j=-N+1}^{N-1} 2\chi(v,\Delta)e^{i\boldsymbol{q}\cdot\boldsymbol{v}}\mathcal{g}\_{j}.\tag{66}$$

where

delta δx, 0.

inverse of each other.

Next, based on Eq. (28), we find that

If we sum this equality, we get

N X�1 j¼�Nþ1

J xð Þ ; <sup>N</sup> <sup>≔</sup> <sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup>

Δ

De �iqvg � �

<sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup> De�iqvg � �

N X�1 j¼�Nþ1 e

i qj ð Þ <sup>v</sup>�<sup>u</sup> <sup>¼</sup> <sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup>

Figure 1. A plot of the kernel function K xð Þ ; a; b with a ¼ 1 and b ¼ :1. This function is an approximation to the Kronecker

The ratio of sin functions, in this expression, is an approximation to a series of Dirac delta functions located at ð Þ v � u Δ ¼ kπ, k ∈ℕ. Thus, the operations F and F are finite differences

�iqv � �

<sup>j</sup> ¼ �iv

þ N X�1 j¼�Nþ1

�iqj <sup>v</sup> <sup>þ</sup> <sup>e</sup>

<sup>j</sup> þ e �iqj�1<sup>v</sup>

N X�1 j¼�Nþ1

�iqj�1<sup>v</sup>

2χð Þ v;Δ e

ð Þ Dg <sup>j</sup>

ð Þ Dg <sup>j</sup> :

<sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup> gjþ<sup>1</sup><sup>e</sup>

�iqj�1<sup>v</sup>

�iqj v

ð Þ Dg <sup>j</sup>

6.1. The discrete Fourier transform of the finite differences derivative of a vector

<sup>j</sup> <sup>¼</sup> gjþ<sup>1</sup> <sup>D</sup><sup>e</sup>

¼ �iv gjþ<sup>1</sup><sup>e</sup>

Δ

<sup>¼</sup> <sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup> Δ

N X�1 j¼�Nþ1 e ij vð Þ �u Δ

sinð Þ ð Þ N � 1=2 ð Þ v � u Δ sinð Þ ð Þ <sup>v</sup> � <sup>u</sup> <sup>Δ</sup>=<sup>2</sup> ,

Matrices Which are Discrete Versions of Linear Operations

http://dx.doi.org/10.5772/intechopen.74356

(70)

35

(71)

(72)

Assuming that the involved integrals converge absolutely, we can say that

$$F(\mathbf{F}g)\left(q\_j\right) \coloneqq \frac{1}{L\sqrt{2\Delta}}\int\_{-L/2}^{L/2} e^{-iq\_jv} \frac{L}{\sqrt{2\Delta}}\sum\_{k=-N+1}^{N-1} 2\chi(v,\Delta)e^{iq\_kv}g\_k dv$$

$$=\frac{1}{\Delta}\sum\_{k=-N+1}^{N-1} g\_k\int\_{-L/2}^{L/2} e^{i\left(q\_k-q\_j\right)v}\sinh(v\Delta)\frac{dv}{v}\tag{67}$$

$$=\sum\_{k=-N+1}^{N-1} g\_k K\left(q\_k - q\_j, L, \Delta\right).$$

where

$$\begin{split} K\left(q\_{k}-q\_{j},L,\Delta\right) &= \frac{1}{\Delta} \int\_{-L/2}^{L/2} e^{i\left(q\_{k}-q\_{j}\right)y} \sinh(v\Delta) \frac{dv}{v} \\ &= \frac{1}{2\Delta} \left\{ \sinh\left[\frac{L}{2}\left(i\left(q\_{k}-q\_{j}\right)+\Delta\right)\right] + i\sinh\left[\frac{L}{2}\left(q\_{k}-q\_{j}-i\Delta\right)\right] - 2i\sinh\left[\frac{L}{2}\left(q\_{k}-q\_{j}+i\Delta\right)\right] \right\}. \end{split} \tag{68}$$

The function K qk � qj ; L;Δ � � is an approximation to the Kronecker delta function <sup>δ</sup>k,j. The function shi is the hyperbolic sine integral shið Þ¼ <sup>z</sup> <sup>Ð</sup> <sup>z</sup> <sup>0</sup> dt sinhð Þt =t. A plot of it is shown in Figure 1.

Additionally,

$$\begin{aligned} \mathcal{F}(Fh)(v) &= \frac{L}{\sqrt{2\Delta}} \sum\_{j=-N+1}^{N-1} 2\chi(v,\Delta) e^{i\boldsymbol{\eta}\_{\parallel}v} \frac{1}{L\sqrt{2\Delta}} \int\_{-L/2}^{L/2} e^{-i\boldsymbol{\eta}\_{\parallel}u} h(u) du \\ &= \int\_{-L/2}^{L/2} du h(u) f(v-u, N), \end{aligned} \tag{69}$$

Figure 1. A plot of the kernel function K xð Þ ; a; b with a ¼ 1 and b ¼ :1. This function is an approximation to the Kronecker delta δx, 0.

where

on a partition qi

Fourier transform F as

34 Matrix Theory-Applications and Theorems

Fourier transform F as

where

K qk � qj

¼ 1

Figure 1.

Additionally,

; L;Δ � �

<sup>2</sup><sup>Δ</sup> shi <sup>L</sup>

�

The function K qk � qj

≔ 1 Δ ð<sup>L</sup>=<sup>2</sup> �L=2 e i qk�<sup>q</sup> ð Þ<sup>j</sup> <sup>v</sup>

<sup>2</sup> i qk � qj � �

� � � �

; L;Δ � �

Fð Þ Fh ð Þ¼ v

function shi is the hyperbolic sine integral shið Þ¼ <sup>z</sup> <sup>Ð</sup> <sup>z</sup>

¼ ð<sup>L</sup>=<sup>2</sup> �L=2

argument v, a conjugate vector on the partition qi

Fð Þ Fg qj � � ð Þ Fh qj � �

ð Þ <sup>F</sup><sup>g</sup> ð Þ<sup>v</sup> <sup>≔</sup> <sup>L</sup>

Assuming that the involved integrals converge absolutely, we can say that

N X�1 k¼�Nþ1

N X�1 k¼�Nþ1

ð<sup>L</sup>=<sup>2</sup> �L=2 e �iqj <sup>v</sup> L ffiffiffiffiffiffi <sup>2</sup><sup>Δ</sup> <sup>p</sup>

> gk ð<sup>L</sup>=<sup>2</sup> �L=2 e i qk�<sup>q</sup> ð Þ<sup>j</sup> <sup>v</sup>

sinhð Þ <sup>v</sup><sup>Δ</sup> dv

<sup>þ</sup> <sup>i</sup>shi <sup>L</sup>

v

2χð Þ v;Δ e

du h uð ÞJ vð Þ � u; N ,

iqj <sup>v</sup> 1 L ffiffiffiffiffiffi <sup>2</sup><sup>Δ</sup> <sup>p</sup>

gkK qk � qj

<sup>≔</sup> <sup>1</sup> L ffiffiffiffiffiffi <sup>2</sup><sup>Δ</sup> <sup>p</sup>

¼ 1 Δ

¼

þ Δ

L ffiffiffiffiffiffi <sup>2</sup><sup>Δ</sup> <sup>p</sup>

N X�1 j¼�Nþ1

<sup>≔</sup> <sup>1</sup> L ffiffiffiffiffiffi <sup>2</sup><sup>Δ</sup> <sup>p</sup>

ffiffiffiffiffiffi <sup>2</sup><sup>Δ</sup> <sup>p</sup>

� � and functions with a continuous argument v makes use of continuous and

2χð Þ v;Δ e

N X�1 k¼�Nþ1

; L;Δ � �

<sup>2</sup> qk � qj � <sup>i</sup><sup>Δ</sup> � � � �

:

iqj v gj

2χð Þ v;Δ e

v

�2ishi <sup>L</sup>

is an approximation to the Kronecker delta function δk,j. The

ð<sup>L</sup>=<sup>2</sup> �L=2 e �iqj u h uð Þdu

<sup>2</sup> qk � qj <sup>þ</sup> <sup>i</sup><sup>Δ</sup> � � � ��

<sup>0</sup> dt sinhð Þt =t. A plot of it is shown in

sinhð Þ <sup>v</sup><sup>Δ</sup> dv

iqkv gkdv

� � is defined through the type of continuous

h vð Þdv, (65)

: (66)

(67)

(68)

(69)

:

discrete Fourier type of transformations, a wavelet [12]. If we have a function h of continuous

ð<sup>L</sup>=<sup>2</sup> �L=2 e �iqj v

and vice-versa, a continuous variable function is defined with the help of a discrete type of

N X�1 j¼�Nþ1

$$\begin{split} J(\mathbf{x}, N) \coloneqq \frac{2\chi(\mathbf{v}, \Delta)}{\Delta} \sum\_{j=-N+1}^{N-1} e^{i\boldsymbol{q}\_j(\mathbf{v}-\boldsymbol{u})} &= \frac{2\chi(\mathbf{v}, \Delta)}{\Delta} \sum\_{j=-N+1}^{N-1} e^{i j \,(\mathbf{v}-\boldsymbol{u})\Delta} \\ &= \frac{2\chi(\mathbf{v}, \Delta)}{\Delta} \frac{\sin((N-1/2)(\mathbf{v}-\boldsymbol{u})\Delta)}{\sin((\mathbf{v}-\boldsymbol{u})\Delta/2)}, \end{split} \tag{70}$$

The ratio of sin functions, in this expression, is an approximation to a series of Dirac delta functions located at ð Þ v � u Δ ¼ kπ, k ∈ℕ. Thus, the operations F and F are finite differences inverse of each other.

#### 6.1. The discrete Fourier transform of the finite differences derivative of a vector

Next, based on Eq. (28), we find that

$$\begin{split} \left(\mathsf{D}e^{-iqv}\mathsf{g}\right)\_{j} &= \mathsf{g}\_{j+1}\left(\mathsf{D}e^{-iqv}\right)\_{j} + e^{-iq\_{j-1}v}(\mathsf{D}\mathsf{g})\_{j} \\ &= -i\upsilon\,\mathsf{g}\_{j+1}e^{-iq\_{j}v} + e^{-iq\_{j-1}v}(\mathsf{D}\mathsf{g})\_{j}. \end{split} \tag{71}$$

If we sum this equality, we get

$$\begin{aligned} \sum\_{j=-N+1}^{N-1} 2\chi(v,\Delta) \big(De^{-i\eta v} g\big)\_{j} &= -i v \sum\_{j=-N+1}^{N-1} 2\chi(v,\Delta) g\_{j+1} e^{-i\eta v} \\ &+ \sum\_{j=-N+1}^{N-1} 2\chi(v,\Delta) e^{-i\eta\_{j-1}v} (Dg)\_{j} \end{aligned} \tag{72}$$

i.e.,

$$\begin{aligned} (\mathbf{F}\_N(\mathbf{D}\mathbf{g}))(v) &= i v (\mathbf{F}\_{N+1}\mathbf{g})(v) \\ &+ e^{-iv\Delta} \left[ e^{-i\boldsymbol{q}\_j v} \mathbf{g}\_j \Big|\_{j=-N+2}^N + \frac{\sqrt{2}}{L} e^{-i\boldsymbol{q}\_j v} \mathbf{g}\_j \Big|\_{j=-N+1}^{N-1} \right] \end{aligned} \tag{73}$$

Therefore, the discrete Fourier transform of the derivative of a vector g is iv times the discrete Fourier transform of g, plus boundary terms.

The Fourier transform of the derivative of a continuous function of variable v is easily found if we consider the equality

$$\frac{d}{dv}e^{-i\boldsymbol{\eta}\_{\rangle}v} = -i\boldsymbol{\eta}\_{\rangle}e^{-i\boldsymbol{\eta}\_{\rangle}v}.\tag{74}$$

Let P<sup>N</sup> ¼ �iD<sup>N</sup> and v ¼ ix be the eigenvalue of DN, where x ∈ ℝ is a free parameter, the corresponding eigenvalue of �iD<sup>N</sup> is indeed the real value x; which is one of the properties of a Hermitian matrix, as is also the case of infinite-dimensional space (for the Hilbert space on a finite interval, these values are discrete, and for the Hilbert space on the real line, these values conform the continuous spectrum, instead of discrete eigenvalues). Other characteristic of �iD<sup>N</sup> is that the eigenvector corresponding to x is the same exponential function which is the

<sup>N</sup> denote the adjoint of PN. Thus, if we restrict our attention to the off-

N 

i,j ¼ �idj,i

Matrices Which are Discrete Versions of Linear Operations

http://dx.doi.org/10.5772/intechopen.74356

37

<sup>∗</sup> ¼ �idi,j <sup>¼</sup> ð Þ <sup>P</sup><sup>N</sup> i,j

, it is fulfilled that P†

(noticing that, with v ¼ ix then χð Þ¼ x;Δ sinð Þ x; Δ =x ∈ ℝ). Even more, if we do not care about the two entries di,i for i ¼ 1, N, we will have a Hermitian matrix. Finally, as it was seen in Section 4, we can say that P<sup>N</sup> can be considered as a suitable approximation to the conjugate

In conclusion, we have introduced a matrix with the properties that a Hermitian matrix should comply with, except for two of its entries. Besides, our partition provides congruency between discrete, continuous, and matrix treatments of the exponential function and of its properties.

[1] Boole G. A Treatise on the Calculus of Finite Differences. New York: Cambridge Univer-

[2] Harmuth HF, Meffert B. Dogma of the continuum and the calculus of finite differences in quantum physics. In: Advances in Imaging and Electron Physics. Vol. 137. San Diego:

[3] Jordan C. Calculus of Finite Differences. 2nd ed. New York: Chelsea Publishing Com-

[4] Richardson CH. An Introduction to the Calculus of Finite Differences. Toronto: D. Van

[5] Santhanam TS, Tekumalla AR. Quantum mechanics in finite dimensions. Foundations of

eigenfunction of �i d=dx (see Section 2).

diagonal entries ð Þ P<sup>N</sup> i,j ¼ �ið Þ D<sup>N</sup> i,j

matrix to the coordinate matrix.

Armando Martínez Pérez and Gabino Torres Vega\*

Physics Department, Cinvestav, México City, México

\*Address all correspondence to: gabino@fis.cinvestav.mx

Furthermore, let P†

Author details

References

pany; 1950

Nostrand; 1954

Physics. 1976;6:583

sity Press; 2009. p. 1860

Elsevier Academic Press; 2005

The integration of this equality with appropriate weights gives

$$-iq\_{\rangle} \int\_{-L/2}^{L/2} d\upsilon \, e^{-iq\_{\rangle}\upsilon} h(\upsilon) = -\int\_{-L/2}^{L/2} d\upsilon \, e^{-iq\_{\rangle}\upsilon} \frac{d h(\upsilon)}{d\upsilon} + e^{-iq\_{\rangle}\upsilon} h(\upsilon) \Big|\_{\upsilon=-L/2}^{L/2} \tag{75}$$

i.e.,

$$\dot{q}\_{\cdot}(Fh')\_{\dot{\jmath}} = \dot{q}\_{\dot{\jmath}}(Fh)\_{\dot{\jmath}} + \frac{1}{L\sqrt{2}} e^{-i\eta\_{\dot{\jmath}}v} h(v) \Big|\_{v=-L/2}^{L/2}.\tag{76}$$

Hence, as is usual, the Fourier transform of the derivative of a function h vð Þ of continuous variable v is equal to iqj times the Fourier transform of the function, plus boundary terms.
