# Coadjoint Formalism: Nonorthogonal Basis Problems

Authors: William Labecca; Osvaldo Guimarães; José Roberto C. Piqueira
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

Using nonorthogonal bases in spectral methods demands considerable effort, because applying the Gram-Schmidt process is a fundamental condition for calculations. However, operational matrices numerical methods are being used in an increasing way and extensions for nonorthogonal bases appear, requiring simplified procedures. Here, extending previous work, an efficient tensorial method is presented, in order to simplify the calculations related to the use of nonorthogonal bases in spectral numerical problems. The method is calledcoadjoint formalism and is based onbracket Dirac’s formulation of quantum mechanics. Some examples are presented, showing how simple it is to use the method.


---

## Body

## 1. Introduction

Spectral numerical methods are increasingly used to solve differential equations, even when fractional derivatives appear. Concerning these methods, they frequently use orthogonal functions [1], in order to ease the calculations and to preserve the basis elements when dimensional expansions are necessary.This is an important advantage when operational matrices are used, because if it is necessary to extend an orthogonal basis of functions from dimensionsm to m
+
1, only the new element must be calculated with all the former elements preserved [2, 3]. If the basis functions are not orthogonal, this assumption is not true.However, there are some situations that, even for numerical methods with operational matrices [4], solution of differential equations demands new basis of functions [5, 6], with some of them being nonorthogonal as, for instance, Bernstein’s polynomials, mainly in the cases where boundary or initial conditions must be considered [7–9].Some iterative methods, based on Krylov formulation, for instance, present the use of such bases and are implemented by using the Gram-Schmidt process [8, 10, 11] that, in most cases, is difficult to be operationalized.Trying to simplify this procedure, an alternative operational method is presented with the first ideas developed in [12] that is described by using tensorial language and calledcoadjoint formalism [13], allowing the direct use of nonorthogonal bases without any kind of previous conditioning process.In a nutshell,coadjoint formalism adapts Dirac’sbracket notation [14] to spectral methods considering finite dimension complex vectorial spaces. The methodology is applied to nonorthogonal bases in finite dimension function spaces with a generalized tensorial approach [15–17] that simplifies the operational conditions.In the next section, the theoretical fundamentals of thecoadjoint formalism are presented, trying to connect it with the well-known Diracbracket, followed by a section where two examples show how simple it is to apply the method developed here, with a conclusion section closing the work.


## 2. Coadjoint Formalism: Theoretical Foundations

This section presents the concepts and definitions used to build thecoadjoint formalism. The bases to be considered are finite sets of complex functions of a real variable; that is, the bases elements {
ϕ
k
}, k
=
1
,
…
,
n
<
∞, are given by ϕ
k
:
R
→
C, with ϕ
k
∈
H, where H is the function space, equipped as a Hilbert space.


### 2.1. Actuation Spaces

Here, it is considered that any operation regarding series expansion of a function occurs in two distinct spaces:(i)
A finite dimensional Hilbert spaceH.
(ii)
The finite dimension space with dimensionn, generated by the series expansion with n terms for the considered function, denoted by O
n and calledorder space.It is assumed thatkets are represented by column vectors and vice versa. It is the same forbras, represented by line vectors, understood as covectors, that is, dual space elements. Consequently, an equivalence relation can be established [14]:(1)
~
⋮
,
~
⋯
.Thekets can belong either to the function space, ∞
∈
H or to the order space, 0
n
∈
O
n. The same is valid forbras.Covariant and contravariant distinction are assumed [15], and the traditional notation of differential geometry and tensorial calculus is used:(i)
Contravariant components:u
i.
(ii)
Covariant components:u
i.Quantities described bykets are in theoriginal space; quantities described bybras are in thedual space, in the traditional way of linear algebra [14].Primitive space is related to the space where the quantity is defined; that is, if abra defines a quantity, thedual space assumes theprimitive space condition of the quantity.Given a covariant basis composed ofb
i
∈
V, a vector u belonging to the order space, O
n
(
V
), can be described by aket in two different ways:(i)
Invariant representation:u
0
n
=
∑
i
n
u
i
b
i.
(ii)
Coordinate representation:u
B
~
u
1
⋮
u
n
[
B
],with B
=
b
i. The spaces V where the spectral methods are generally applied are H, C and C
n.Given a reference basis[
ϵ
] from V
n, for a basis composed of b
i
∈
V
n and described in terms of the basis [
ϵ
], where V can be either the Hilbert space H or any other complex vectorial space, two representations are possible:(i)
B
.
[
ϵ
]
∈
M
n
V
:
B
.
[
ϵ
]
≃
b
1
[
ϵ
]
⋯
b
n
[
ϵ
].
(ii)
B
.
[
ϵ
]
∈
V
n
V
n
:
B
.
[
ϵ
]
≃
b
1
[
ϵ
]
⋮
b
n
[
ϵ
],with the inferior dot representing the covariant nature of the basis and M
n represents the set of the n-order square matrices. In this kind of disposition, the bases are described by their coordinates in the reference basis [
ϵ
], but the reference basis can be omitted and the invariant representation is assumed.Consequently, there are two isomorphic representations, given by(i)
B
.
∈
M
n
V
:
B
.
≃
b
1
⋯
b
n,
(ii)
B
.
∈
V
n
V
n
:
B
.
≃
b
1
⋮
b
n
.Given a vectoru with its components expressed in a generical basis e
i, the following operations are defined:(i)
Conjugation:u
∗
=
u
¯
,
e
i
∗
=
e
i
¯.
(ii)
Transposition:u
′
=
u
¯
,
e
i
′
=
e
i
¯
.
(iii)
Adjunction:u
†
=
u
,
e
i
†
=
e
i
.
(iv)
Duality:u
⋆
=
u
.
,
e
i
⋆
=
e
i
.


### 2.2. Inner Products

The inner products (IP),∣
:
O
n
(
V
)
×
O
n
(
V
)
→
C, are defined in several different ways, depending on the space and the representation, as it is shown below.Order Space
(1)
Hermitian product is as follows:
(i)
coordinate representation:u
∣
v
[
B
]
≔
∑
i
=
1
n
u
¯
i
v
i.
(2)
Dual product is as follows:
(i)
coordinate representation:u
.
∣
v
[
B
]
≔
∑
i
=
1
n
u
¯
i
v
i;
(ii)
invariant representation:u
.
∣
v
0
n
≔
∑
i
=
1
n
u
¯
i
v
i and u
∣
v
0
n
≔
∑
i
=
1
n
u
¯
i
v
i.Functions Space. Considering the space of the functions with domain a
,
b
⊂
R, the IP ∣
:
H
×
H
→
C is defined as(2)
f
∣
g
∞
≔
∫
a
b
f
¯
x
g
x
d
x
.Hybrid Spaces. The mixed inner product ∣
:
O
n
⋆
(
C
)
×
O
n
(
H
)
→
C is defined as(3)
c
∣
ϕ
≔
∑
r
n
c
r
ϕ
r
∞
.


### 2.3. Duality Relations

It is possible to obtain thedual basis
e
k from an original basis e
i by using the duality relations, given by(4)
e
k
∣
e
i
=
δ
k
i
,
e
i
∣
e
k
=
δ
i
k
.It is possible to express the same vector in several ways, considering the formalism described here, considering four distinct bases. Here, these representations are calledconnected representations and are shown in Table 1.Table 1
Connected representations.
Basis
Notation and correspondence
Invariant representations
Coordinate representations
Original
|
B
.
]
≃
B
|
u
〉
0
n
=
∑
i
n
u
i
|
e
i
〉
|
u
〉
[
B
]
Adjoint
[
B
.
|
≃
B
†
〈
u
|
0
n
=
∑
i
n
‍
〈
e
i
|
u
-
i
〈
u
|
[
B
]
Dual
[
B
˙
|
≃
B
⋆
〈
u
.
|
0
n
=
∑
i
n
‍
〈
e
i
|
u
-
i
〈
u
.
|
[
B
]
Coadjoint
|
B
˙
]
≃
(
B
⋆
)
†
|
u
.
〉
0
n
=
∑
i
n
u
i
|
e
i
〉
|
u
.
〉
[
B
]Considering that the covariant representation in the original space is considered to be natural, the dot below the basis symbol can be omitted and the transformation relations between the bases are shown in Table2.Table 2
Transformation relations between bases.
Basis
Notation
Transformation relations
Original
|
B
.
]
|
B
.
]
≃
B
Adjoint
[
B
.
|
[
B
.
|
≡
|
B
.
]
†
Dual
[
B
˙
|
[
B
˙
|
≡
|
B
.
]
-
1
Coadjoint
|
B
˙
]
|
B
˙
]
≡
[
B
˙
|
†
≃
(
B
-
1
)
†The covariant and contravariant components are obtained by applying(1)
contravariant:u
k
=
e
k
∣
u
0
n,
(2)
covariant:u
k
=
e
k
∣
u
0
n.


### 2.4. Series Expansions of Functions

Considering a basisϕ
i
∞ from O
n
(
H
) and elements ϕ
i
:
R
→
H, the infinite series expansion of a function f is given by(5)
f
∞
=
∑
r
∞
c
r
ϕ
r
∞
,with c
r being the expansion coefficients in that basis. Considering the mixed inner product definition, this expression can be modified as(6)
f
∞
=
c
∣
ϕ
r
,with the covector c
∈
O
n
⋆
(
C
) calledcoefficient covector.In the subspaceO
m
(
H
), projectors can be defined by(7)
P
m
≔
∑
r
m
ϕ
r
ϕ
r
,
P
m
†
≔
∑
r
m
ϕ
r
ϕ
r
,with the second one calledadjoint projector.Theeigenprojectors in the O
n
(
H
) space are defined as the projectors in the n-dimensional proper space; that is, P
=
P
n and P
†
=
P
n
†.In a way analogous to that followed in differential geometry,fundamental metric tensor of a basis, ϕ
i
∞ from O
n
(
H
), is defined as(8)
γ
r
s
≔
ϕ
r
∣
ϕ
s
∞
,with its r
eciprocal metric tensor given by(9)
γ
r
s
≔
ϕ
r
∣
ϕ
s
∞
.Consequently, the IPs between two functionsf, g represented by series are given by(10)
f
∣
g
0
n
=
∑
r
,
s
n
ϕ
r
f
^
∗
r
g
^
s
ϕ
s
=
∑
r
,
s
n
γ
r
s
f
^
∗
r
g
^
s
≡
∑
r
n
f
^
∗
r
g
^
r
,
f
.
∣
g
0
n
=
∑
r
,
s
n
ϕ
r
f
^
r
∗
g
^
s
ϕ
s
=
∑
r
,
s
n
δ
r
s
f
^
r
∗
g
^
s
≡
∑
s
n
f
^
s
∗
g
^
s
.with f
^ and g
^, respectively, representing the f and g expansion coefficients, in the basis ϕ.Themetric tensorial fundamental operator and its reciprocal are defined as(11)
γ
⃡
≡
P
†
P
≔
∑
r
,
s
n
ϕ
r
γ
r
s
ϕ
s
;
γ
⃡
⋆
≡
P
P
†
≔
∑
r
,
s
n
ϕ
r
γ
r
s
ϕ
s
.


### 2.5. Covariant-Contravariant Transformation Relations

Considering the covariant and contravariant matrix representations and definingB
=
ϕ
k, the transformation relations for bases and functions can be written as(12)
ϕ
k
=
∑
r
n
ϕ
r
γ
r
k
,
B
=
B
˙
γ
,
B
=
γ
′
B
˙
;
(13)
ϕ
i
=
∑
r
n
ϕ
r
γ
r
i
,
B
˙
=
B
γ
⋆
,
B
˙
=
γ
~
⋆
B
;
(14)
f
r
=
∑
k
n
γ
r
k
f
k
,
f
.
B
=
γ
f
B
;
(15)
f
i
=
∑
r
n
γ
i
r
f
r
,
f
B
=
γ
⋆
f
.
B
.Therefore,(16)
∑
r
n
γ
i
r
γ
r
k
≔
δ
i
k
,
∑
r
n
γ
k
r
γ
r
i
≔
δ
k
i
,meaning that the reciprocal tensor matrix is the inverse of the metrical tensor matrix. Calling the first one γ
⋆:(17)
γ
⋆
=
γ
-
1
.


### 2.6. Finite Expansions

Then-order finite expansion of a function f can be expressed by the action of the reciprocal tensorial metric operator over the function; that is,(18)
f
n
∞
=
γ
⃡
⋆
f
∞
=
∑
r
n
c
r
ϕ
r
∞
,
c
r
=
ϕ
r
∣
f
∞
≡
∑
s
n
γ
r
s
ϕ
s
∣
f
∞
.Under these conditions, the coefficient covector c
∈
O
n
⋆
(
C
) can be expressed as(19)
c
≃
c
1
⋯
c
n
.Then, the expansion is given by the mixed IP:f
n
∞
=
c
∣
ϕ, with ϕ
=
ϕ
1
⋮
ϕ
n being the vector associated with the second representation chosen basis.Equation (18) can be expressed by using matrices, giving a useful computational expression, in order to find the coefficient covector.Considering the coefficient covector(20)
c
r
=
∑
s
n
∫
a
b
ϕ
s
ξ
f
ξ
d
ξ
γ
s
r
′
,it is possible to write (21)
c
=
∫
a
b
ϕ
ξ
f
ξ
d
ξ
γ
⋆
′
.As a consequence, the expanded function becomes(22)
f
n
∞
=
∫
a
b
ϕ
ξ
f
ξ
d
ξ
γ
⋆
′
ϕ
∞
.


## 2.1. Actuation Spaces

Here, it is considered that any operation regarding series expansion of a function occurs in two distinct spaces:(i)
A finite dimensional Hilbert spaceH.
(ii)
The finite dimension space with dimensionn, generated by the series expansion with n terms for the considered function, denoted by O
n and calledorder space.It is assumed thatkets are represented by column vectors and vice versa. It is the same forbras, represented by line vectors, understood as covectors, that is, dual space elements. Consequently, an equivalence relation can be established [14]:(1)
~
⋮
,
~
⋯
.Thekets can belong either to the function space, ∞
∈
H or to the order space, 0
n
∈
O
n. The same is valid forbras.Covariant and contravariant distinction are assumed [15], and the traditional notation of differential geometry and tensorial calculus is used:(i)
Contravariant components:u
i.
(ii)
Covariant components:u
i.Quantities described bykets are in theoriginal space; quantities described bybras are in thedual space, in the traditional way of linear algebra [14].Primitive space is related to the space where the quantity is defined; that is, if abra defines a quantity, thedual space assumes theprimitive space condition of the quantity.Given a covariant basis composed ofb
i
∈
V, a vector u belonging to the order space, O
n
(
V
), can be described by aket in two different ways:(i)
Invariant representation:u
0
n
=
∑
i
n
u
i
b
i.
(ii)
Coordinate representation:u
B
~
u
1
⋮
u
n
[
B
],with B
=
b
i. The spaces V where the spectral methods are generally applied are H, C and C
n.Given a reference basis[
ϵ
] from V
n, for a basis composed of b
i
∈
V
n and described in terms of the basis [
ϵ
], where V can be either the Hilbert space H or any other complex vectorial space, two representations are possible:(i)
B
.
[
ϵ
]
∈
M
n
V
:
B
.
[
ϵ
]
≃
b
1
[
ϵ
]
⋯
b
n
[
ϵ
].
(ii)
B
.
[
ϵ
]
∈
V
n
V
n
:
B
.
[
ϵ
]
≃
b
1
[
ϵ
]
⋮
b
n
[
ϵ
],with the inferior dot representing the covariant nature of the basis and M
n represents the set of the n-order square matrices. In this kind of disposition, the bases are described by their coordinates in the reference basis [
ϵ
], but the reference basis can be omitted and the invariant representation is assumed.Consequently, there are two isomorphic representations, given by(i)
B
.
∈
M
n
V
:
B
.
≃
b
1
⋯
b
n,
(ii)
B
.
∈
V
n
V
n
:
B
.
≃
b
1
⋮
b
n
.Given a vectoru with its components expressed in a generical basis e
i, the following operations are defined:(i)
Conjugation:u
∗
=
u
¯
,
e
i
∗
=
e
i
¯.
(ii)
Transposition:u
′
=
u
¯
,
e
i
′
=
e
i
¯
.
(iii)
Adjunction:u
†
=
u
,
e
i
†
=
e
i
.
(iv)
Duality:u
⋆
=
u
.
,
e
i
⋆
=
e
i
.


## 2.2. Inner Products

The inner products (IP),∣
:
O
n
(
V
)
×
O
n
(
V
)
→
C, are defined in several different ways, depending on the space and the representation, as it is shown below.Order Space
(1)
Hermitian product is as follows:
(i)
coordinate representation:u
∣
v
[
B
]
≔
∑
i
=
1
n
u
¯
i
v
i.
(2)
Dual product is as follows:
(i)
coordinate representation:u
.
∣
v
[
B
]
≔
∑
i
=
1
n
u
¯
i
v
i;
(ii)
invariant representation:u
.
∣
v
0
n
≔
∑
i
=
1
n
u
¯
i
v
i and u
∣
v
0
n
≔
∑
i
=
1
n
u
¯
i
v
i.Functions Space. Considering the space of the functions with domain a
,
b
⊂
R, the IP ∣
:
H
×
H
→
C is defined as(2)
f
∣
g
∞
≔
∫
a
b
f
¯
x
g
x
d
x
.Hybrid Spaces. The mixed inner product ∣
:
O
n
⋆
(
C
)
×
O
n
(
H
)
→
C is defined as(3)
c
∣
ϕ
≔
∑
r
n
c
r
ϕ
r
∞
.


## 2.3. Duality Relations

It is possible to obtain thedual basis
e
k from an original basis e
i by using the duality relations, given by(4)
e
k
∣
e
i
=
δ
k
i
,
e
i
∣
e
k
=
δ
i
k
.It is possible to express the same vector in several ways, considering the formalism described here, considering four distinct bases. Here, these representations are calledconnected representations and are shown in Table 1.Table 1
Connected representations.
Basis
Notation and correspondence
Invariant representations
Coordinate representations
Original
|
B
.
]
≃
B
|
u
〉
0
n
=
∑
i
n
u
i
|
e
i
〉
|
u
〉
[
B
]
Adjoint
[
B
.
|
≃
B
†
〈
u
|
0
n
=
∑
i
n
‍
〈
e
i
|
u
-
i
〈
u
|
[
B
]
Dual
[
B
˙
|
≃
B
⋆
〈
u
.
|
0
n
=
∑
i
n
‍
〈
e
i
|
u
-
i
〈
u
.
|
[
B
]
Coadjoint
|
B
˙
]
≃
(
B
⋆
)
†
|
u
.
〉
0
n
=
∑
i
n
u
i
|
e
i
〉
|
u
.
〉
[
B
]Considering that the covariant representation in the original space is considered to be natural, the dot below the basis symbol can be omitted and the transformation relations between the bases are shown in Table2.Table 2
Transformation relations between bases.
Basis
Notation
Transformation relations
Original
|
B
.
]
|
B
.
]
≃
B
Adjoint
[
B
.
|
[
B
.
|
≡
|
B
.
]
†
Dual
[
B
˙
|
[
B
˙
|
≡
|
B
.
]
-
1
Coadjoint
|
B
˙
]
|
B
˙
]
≡
[
B
˙
|
†
≃
(
B
-
1
)
†The covariant and contravariant components are obtained by applying(1)
contravariant:u
k
=
e
k
∣
u
0
n,
(2)
covariant:u
k
=
e
k
∣
u
0
n.


## 2.4. Series Expansions of Functions

Considering a basisϕ
i
∞ from O
n
(
H
) and elements ϕ
i
:
R
→
H, the infinite series expansion of a function f is given by(5)
f
∞
=
∑
r
∞
c
r
ϕ
r
∞
,with c
r being the expansion coefficients in that basis. Considering the mixed inner product definition, this expression can be modified as(6)
f
∞
=
c
∣
ϕ
r
,with the covector c
∈
O
n
⋆
(
C
) calledcoefficient covector.In the subspaceO
m
(
H
), projectors can be defined by(7)
P
m
≔
∑
r
m
ϕ
r
ϕ
r
,
P
m
†
≔
∑
r
m
ϕ
r
ϕ
r
,with the second one calledadjoint projector.Theeigenprojectors in the O
n
(
H
) space are defined as the projectors in the n-dimensional proper space; that is, P
=
P
n and P
†
=
P
n
†.In a way analogous to that followed in differential geometry,fundamental metric tensor of a basis, ϕ
i
∞ from O
n
(
H
), is defined as(8)
γ
r
s
≔
ϕ
r
∣
ϕ
s
∞
,with its r
eciprocal metric tensor given by(9)
γ
r
s
≔
ϕ
r
∣
ϕ
s
∞
.Consequently, the IPs between two functionsf, g represented by series are given by(10)
f
∣
g
0
n
=
∑
r
,
s
n
ϕ
r
f
^
∗
r
g
^
s
ϕ
s
=
∑
r
,
s
n
γ
r
s
f
^
∗
r
g
^
s
≡
∑
r
n
f
^
∗
r
g
^
r
,
f
.
∣
g
0
n
=
∑
r
,
s
n
ϕ
r
f
^
r
∗
g
^
s
ϕ
s
=
∑
r
,
s
n
δ
r
s
f
^
r
∗
g
^
s
≡
∑
s
n
f
^
s
∗
g
^
s
.with f
^ and g
^, respectively, representing the f and g expansion coefficients, in the basis ϕ.Themetric tensorial fundamental operator and its reciprocal are defined as(11)
γ
⃡
≡
P
†
P
≔
∑
r
,
s
n
ϕ
r
γ
r
s
ϕ
s
;
γ
⃡
⋆
≡
P
P
†
≔
∑
r
,
s
n
ϕ
r
γ
r
s
ϕ
s
.


## 2.5. Covariant-Contravariant Transformation Relations

Considering the covariant and contravariant matrix representations and definingB
=
ϕ
k, the transformation relations for bases and functions can be written as(12)
ϕ
k
=
∑
r
n
ϕ
r
γ
r
k
,
B
=
B
˙
γ
,
B
=
γ
′
B
˙
;
(13)
ϕ
i
=
∑
r
n
ϕ
r
γ
r
i
,
B
˙
=
B
γ
⋆
,
B
˙
=
γ
~
⋆
B
;
(14)
f
r
=
∑
k
n
γ
r
k
f
k
,
f
.
B
=
γ
f
B
;
(15)
f
i
=
∑
r
n
γ
i
r
f
r
,
f
B
=
γ
⋆
f
.
B
.Therefore,(16)
∑
r
n
γ
i
r
γ
r
k
≔
δ
i
k
,
∑
r
n
γ
k
r
γ
r
i
≔
δ
k
i
,meaning that the reciprocal tensor matrix is the inverse of the metrical tensor matrix. Calling the first one γ
⋆:(17)
γ
⋆
=
γ
-
1
.


## 2.6. Finite Expansions

Then-order finite expansion of a function f can be expressed by the action of the reciprocal tensorial metric operator over the function; that is,(18)
f
n
∞
=
γ
⃡
⋆
f
∞
=
∑
r
n
c
r
ϕ
r
∞
,
c
r
=
ϕ
r
∣
f
∞
≡
∑
s
n
γ
r
s
ϕ
s
∣
f
∞
.Under these conditions, the coefficient covector c
∈
O
n
⋆
(
C
) can be expressed as(19)
c
≃
c
1
⋯
c
n
.Then, the expansion is given by the mixed IP:f
n
∞
=
c
∣
ϕ, with ϕ
=
ϕ
1
⋮
ϕ
n being the vector associated with the second representation chosen basis.Equation (18) can be expressed by using matrices, giving a useful computational expression, in order to find the coefficient covector.Considering the coefficient covector(20)
c
r
=
∑
s
n
∫
a
b
ϕ
s
ξ
f
ξ
d
ξ
γ
s
r
′
,it is possible to write (21)
c
=
∫
a
b
ϕ
ξ
f
ξ
d
ξ
γ
⋆
′
.As a consequence, the expanded function becomes(22)
f
n
∞
=
∫
a
b
ϕ
ξ
f
ξ
d
ξ
γ
⋆
′
ϕ
∞
.


## 3. Application Examples

In this section, two examples are developed showing the expansion of functions using nonorthogonal bases, one using the canonical basis and the other using a basis of nonorthogonal complex functions.


### 3.1. Canonical Basis

Here, the more common case of nonorthogonality is developed, considering the canonical basis of functions defined in the real interval, defined by the polynomials:(23)
ϵ
m
x
=
x
m
-
1
,
m
=
1,2
,
…
,
n
.The metric tensor can be calculated as(24)
γ
r
s
ϵ
=
ϵ
r
∣
ϵ
s
∞
≔
∫
-
1
1
x
r
+
s
-
2
d
x
=
x
r
+
s
-
1
r
+
s
-
1
-
1
1
=
1
-
-
1
r
+
s
-
1
r
+
s
-
1
.Therefore, the analytical expression for the metric tensor elements in the domainD
1 can be calculated, resulting in(25)
γ
r
s
=
2
p
a
r
r
+
s
r
+
s
-
1
,
1
≤
r
,
s
≤
n
,with(26)
par
k
≔
1
-
mod
⁡
k
,
2
=
1
,
i
f
k
i
s
e
v
e
n
;
0
,
i
f
k
i
s
o
d
d
.Following the calculations forn
=
5, the metric matrix is given by(27)
γ
ϵ
=
2
0
2
3
0
2
5
0
2
3
0
2
5
0
2
3
0
2
5
0
2
7
0
2
5
0
2
7
0
2
5
0
2
7
0
2
9
,allowing, by using expression (17), the calculation of the matrix representing its reciprocal tensor:(28)
γ
-
1
ϵ
=
225
128
0
-
525
64
0
945
128
0
75
8
0
-
105
8
0
-
525
64
0
2205
32
0
-
4725
64
0
-
105
8
0
175
8
0
945
128
0
-
4725
64
0
11025
128
.By using the methodology developed in the former section and applying the transformation relation (13), the reciprocal basis is composed of polynomials that, for n
=
5, are listed below: (29)
ϵ
1
x
=
225
128
-
525
x
2
64
+
945
x
4
128
;
ϵ
2
x
=
75
x
8
-
105
x
3
8
;
ϵ
3
x
=
-
525
64
+
2205
x
2
32
-
4725
x
4
64
;
ϵ
4
x
=
-
105
x
8
+
175
x
3
8
;
ϵ
5
x
=
945
128
-
4725
x
2
64
+
11025
x
4
128
.It is important to notice that, if the basis is orthogonal, to obtain its reciprocal is simply scale changing. Considering a nonorthogonal basis, the procedure is described here and changing order implies a whole recalculation of the basis elements.In order to illustrate the ideas in a particular case, the expansion of the functiong
(
x
)
=
2
cos
⁡
1
-
2
x
2
cos
⁡
x is considered.The development forn
=
5 gives(30)
g
5
∞
=
γ
⃡
⋆
g
∞
=
∑
r
=
1
5
∑
s
=
1
5
ϵ
r
γ
r
s
ϵ
s
∣
g
∞
=
∑
r
=
1
5
∑
s
=
1
5
2
p
a
r
r
+
s
r
+
s
-
1
x
r
-
1
∫
-
1
1
ξ
s
-
1
g
ξ
d
ξ
.According to (22), the matrix expression is(31)
g
5
=
∫
-
1
1
1
ξ
ξ
2
ξ
3
ξ
4
g
ξ
d
ξ
γ
⋆
T
1
x
x
2
x
3
x
4
,or(32)
g
5
=
c
∣
ϵ
=
c
1
c
2
c
3
c
4
c
5
1
x
x
2
x
3
x
4
.In these expressions, the coefficient covector isc
=
∫
-
1
1
1
ξ
ξ
2
ξ
3
ξ
4
g
(
ξ
)
d
ξ
[
γ
⋆
]
T, and, according to (21), can be calculated resulting in(33)
c
=
1.2047
0
0.1881
0
0.0683
1.7578
0
-
8.2031
0
7.3828
0
9.3750
0
-
13.125
0
-
8.2031
0
68.906
0
-
73.828
0
-
13.125
0
21.875
0
7.3828
0
-
73.828
0
86.133
,generating the expansion for the function(34)
g
5
x
=
1.0789
-
1.9639
x
2
+
0.89037
x
4
.Figure1(a) shows the almost perfect superposition of the function g
(
x
) and its expansion g
5
(
x
). Figure 1(b) shows the local error for the expansion.Figure 1
Approximation ofg
(
x
).
(a)
Superposition of the graphics ofg
(
x
) and g
5
x
(b)
Local error of the expansiong
5
(
x
) of g
(
x
)


### 3.2. Expansion in a Complex Nonorthogonal Basis

Here, considering a complex basisQ
i
∞, composed of elements Q
i
:
R
→
H, i
=
1
,
…
,
n, the real function f
(
x
)
=
exp
⁡
-
2
π
x
2 is expanded, showing efficient and concise results.Defining the basis(35)
Q
r
x
=
r
x
r
-
1
+
i
1
-
δ
1
r
x
r
-
2
,
r
=
1
,
…
,
n
,the metric tensor is expressed by(36)
γ
r
s
Q
=
Q
r
∣
Q
s
∞
=
∫
-
1
1
Q
r
∗
x
Q
s
x
d
x
=
∫
-
1
1
r
x
r
-
1
-
i
1
-
δ
1
r
x
r
-
2
s
x
s
-
1
+
i
1
-
δ
1
s
x
s
-
2
d
x
⟹
γ
r
s
Q
=
2
p
a
r
r
+
s
r
s
r
+
s
-
1
+
1
-
δ
1
r
1
-
δ
1
s
r
+
s
-
3
+
2
i
mod
⁡
r
+
s
,
2
r
1
-
δ
1
s
-
s
1
-
δ
1
r
r
+
s
-
2
.Forn
=
5
(37)
γ
Q
=
2
2
i
2
2
i
3
2
-
2
i
14
3
-
2
i
3
58
15
-
6
i
5
2
2
i
3
64
15
-
2
i
5
164
35
-
2
i
3
58
15
2
i
5
174
35
-
2
i
7
2
6
i
5
164
35
2
i
7
368
63
,and, for the reciprocal metrics,(38)
γ
⋆
Q
=
9.66896
-
6.48244
i
-
7.7305
2.88875
i
4.08105
6.48244
i
5.05372
-
4.87307
i
-
2.51961
2.60449
i
-
7.7305
4.87307
i
8.6524
-
1.75798
i
-
5.20898
-
2.88875
i
-
2.51961
1.75798
i
1.58252
-
0.861328
i
4.08105
-
2.60449
i
-
5.20898
0.861328
i
3.44531
.Expanding the given function forn
=
5,(39)
f
5
∞
=
γ
⃡
⋆
f
∞
=
∑
r
=
1
5
γ
k
r
μ
r
Q
k
∞
=
∑
r
=
1
5
μ
k
Q
k
∞
,where μ
k
=
∑
r
=
1
5
γ
k
r
Q
r
∣
f
∞ or(40)
f
5
=
μ
1
μ
2
μ
3
μ
4
μ
5
1
2
x
+
i
3
x
2
+
i
x
4
x
3
+
i
x
2
5
x
4
+
i
x
3
≡
μ
∣
Q
,with(41)
μ
=
∫
-
1
1
Q
1
∗
ξ
Q
2
∗
ξ
Q
3
∗
ξ
Q
4
∗
ξ
Q
5
∗
ξ
f
ξ
d
ξ
γ
⋆
Q
T
.Considering γ
⋆
[
Q
] formerly calculated, the expansion becomes(42)
f
5
x
=
1.38287
-
8.50196
×
10
-
16
i
Q
1
x
+
8.46164
×
10
-
16
+
0.502949
i
Q
2
x
-
1.0059
-
2.95875
×
10
-
16
i
Q
3
x
-
6.04446
×
10
-
16
+
0.110628
i
Q
4
x
+
0.442512
-
9.03615
×
10
-
16
i
Q
5
x
.Figure2 shows the superposition of the function f
(
x
) and its expansion f
5
(
x
). It can be noticed that the five-term approximation is not good.Figure 2
Superposition of the graphics off
(
x
) and f
5
(
x
).From the explicit expression (26), the metrical matrix for n
=
10 can be determined:(43)
γ
Q
=
2
2
i
2
2
i
3
2
2
i
5
2
2
i
7
2
2
i
9
-
2
i
14
3
-
2
i
3
58
15
-
6
i
5
134
35
-
10
i
7
242
63
-
14
i
9
382
99
2
2
i
3
64
15
-
2
i
5
164
35
-
6
i
7
104
21
-
10
i
9
508
99
-
14
i
11
-
2
i
3
58
15
2
i
5
174
35
-
2
i
7
118
21
-
2
i
3
598
99
-
10
i
11
906
143
2
6
i
5
164
35
2
i
7
368
63
-
2
i
9
652
99
-
6
i
11
1016
143
-
10
i
13
-
2
i
5
134
35
6
i
7
118
21
2
i
9
670
99
-
2
i
11
1082
143
-
6
i
13
106
13
2
10
i
7
104
21
2
i
3
652
99
2
i
11
1104
143
-
2
i
13
556
65
-
2
i
5
-
2
i
7
242
63
10
i
9
598
99
6
i
11
1082
143
2
i
13
1694
195
-
2
i
15
2434
255
2
14
i
9
508
99
10
i
11
1016
143
6
i
13
556
65
2
i
15
2464
255
-
2
i
17
-
2
i
9
382
99
14
i
11
906
143
10
i
13
106
13
2
i
5
2434
255
2
i
17
3438
323
,with the reciprocal matrix given by(44)
γ
⋆
Q
=
194.628
182.705
i
-
245.619
449.976
i
431.287
837.612
i
-
402.514
768.318
i
142.611
264.765
i
182.705
i
173.81
227.829
i
-
441.019
395.46
i
831.682
366.933
i
-
766.695
129.629
i
264.765
-
245.619
227.829
i
338.212
529.699
i
-
651.753
967.53
i
641.618
887.652
i
-
234.589
307.523
i
449.976
i
-
441.019
529.699
i
1306.76
825.923
i
-
2699.58
706.803
i
2604.53
236.383
i
-
922.568
431.287
395.46
i
-
651.753
825.923
i
1425.36
1406.66
i
-
1518.75
1258.88
i
583.739
434.151
i
837.612
i
831.682
967.53
i
-
2699.58
1406.66
i
5953.4
1114.41
i
-
5973.08
348.073
i
2170.75
-
402.514
366.933
i
641.618
706.803
i
-
1518.75
1114.41
i
1703.66
958.733
i
-
677.455
325.651
i
768.318
i
-
766.695
887.652
i
2604.53
1258.88
i
-
5973.08
958.733
i
6154.17
286.437
i
-
2279.55
142.611
129.629
i
-
234.589
236.383
i
583.739
348.073
i
-
677.455
286.437
i
275.799
95.1856
i
264.765
i
264.765
307.523
i
-
922.568
434.151
i
2170.75
325.651
i
-
2279.55
95.1856
i
856.67
.Consequently, the ten-term expansion is given by(45)
f
10
∞
=
γ
⃡
⋆
f
∞
=
∑
r
=
1
10
γ
k
r
μ
r
Q
k
∞
=
∑
r
=
1
10
μ
k
Q
k
∞
,where μ
k
=
∑
r
=
1
10
γ
k
r
Q
r
∣
f
∞ or, in matrix form,(46)
f
10
=
μ
1
μ
2
μ
3
μ
4
μ
5
μ
6
μ
7
μ
8
μ
9
μ
10
1
2
x
+
i
3
x
2
+
i
x
4
x
3
+
i
x
2
5
x
4
+
i
x
3
6
x
5
+
i
x
4
7
x
6
+
i
x
5
8
x
7
+
i
x
6
9
x
8
+
i
x
7
10
x
9
+
i
x
8
≡
μ
∣
Q
,with(47)
μ
=
∫
-
1
1
Q
1
∗
Q
2
∗
Q
3
∗
Q
4
∗
Q
5
∗
Q
6
∗
Q
7
∗
Q
8
∗
Q
9
∗
Q
10
∗
ξ
f
ξ
d
ξ
γ
⋆
Q
T
,where γ
⋆
[
Q
] is given by the new matrix Q or, analogously, calculating the coefficients by the equation(48)
μ
k
=
∑
r
=
1
n
∫
-
1
1
Q
r
⋆
ξ
f
ξ
d
ξ
γ
r
k
′
.Consequently, the new approximation is given by(49)
f
10
x
=
2.03777
-
1.8553
×
10
-
13
i
Q
1
x
+
1.82597
×
10
-
13
+
1.04832
i
Q
2
x
-
2.09663
-
2.15423
×
10
-
13
i
Q
3
x
-
5.49712
×
10
-
13
+
0.661302
i
Q
4
x
+
2.64521
-
3.28239
×
10
-
13
i
Q
5
x
+
1.1432
×
10
-
12
+
0.320826
i
Q
6
x
-
1.92496
-
2.75674
×
10
-
13
i
Q
7
x
-
1.10567
×
10
-
12
+
0.071728
i
Q
8
x
+
0.573824
-
9.09373
×
10
-
14
i
Q
9
x
+
3.92054
-
3.2685
i
×
10
-
13
Q
10
x
.Figure3(a) shows the almost perfect superposition of the function f
(
x
) and the real part of its expansion f
10
(
x
). Figure 3(b) shows the local rest for the expansion.Figure 3
Approximation off
(
x
).
(a)
Superposition of the graphics off
(
x
) and f
10
x
(b)
Local error of the expansionf
10
(
x
) of f
(
x
)


## 3.1. Canonical Basis

Here, the more common case of nonorthogonality is developed, considering the canonical basis of functions defined in the real interval, defined by the polynomials:(23)
ϵ
m
x
=
x
m
-
1
,
m
=
1,2
,
…
,
n
.The metric tensor can be calculated as(24)
γ
r
s
ϵ
=
ϵ
r
∣
ϵ
s
∞
≔
∫
-
1
1
x
r
+
s
-
2
d
x
=
x
r
+
s
-
1
r
+
s
-
1
-
1
1
=
1
-
-
1
r
+
s
-
1
r
+
s
-
1
.Therefore, the analytical expression for the metric tensor elements in the domainD
1 can be calculated, resulting in(25)
γ
r
s
=
2
p
a
r
r
+
s
r
+
s
-
1
,
1
≤
r
,
s
≤
n
,with(26)
par
k
≔
1
-
mod
⁡
k
,
2
=
1
,
i
f
k
i
s
e
v
e
n
;
0
,
i
f
k
i
s
o
d
d
.Following the calculations forn
=
5, the metric matrix is given by(27)
γ
ϵ
=
2
0
2
3
0
2
5
0
2
3
0
2
5
0
2
3
0
2
5
0
2
7
0
2
5
0
2
7
0
2
5
0
2
7
0
2
9
,allowing, by using expression (17), the calculation of the matrix representing its reciprocal tensor:(28)
γ
-
1
ϵ
=
225
128
0
-
525
64
0
945
128
0
75
8
0
-
105
8
0
-
525
64
0
2205
32
0
-
4725
64
0
-
105
8
0
175
8
0
945
128
0
-
4725
64
0
11025
128
.By using the methodology developed in the former section and applying the transformation relation (13), the reciprocal basis is composed of polynomials that, for n
=
5, are listed below: (29)
ϵ
1
x
=
225
128
-
525
x
2
64
+
945
x
4
128
;
ϵ
2
x
=
75
x
8
-
105
x
3
8
;
ϵ
3
x
=
-
525
64
+
2205
x
2
32
-
4725
x
4
64
;
ϵ
4
x
=
-
105
x
8
+
175
x
3
8
;
ϵ
5
x
=
945
128
-
4725
x
2
64
+
11025
x
4
128
.It is important to notice that, if the basis is orthogonal, to obtain its reciprocal is simply scale changing. Considering a nonorthogonal basis, the procedure is described here and changing order implies a whole recalculation of the basis elements.In order to illustrate the ideas in a particular case, the expansion of the functiong
(
x
)
=
2
cos
⁡
1
-
2
x
2
cos
⁡
x is considered.The development forn
=
5 gives(30)
g
5
∞
=
γ
⃡
⋆
g
∞
=
∑
r
=
1
5
∑
s
=
1
5
ϵ
r
γ
r
s
ϵ
s
∣
g
∞
=
∑
r
=
1
5
∑
s
=
1
5
2
p
a
r
r
+
s
r
+
s
-
1
x
r
-
1
∫
-
1
1
ξ
s
-
1
g
ξ
d
ξ
.According to (22), the matrix expression is(31)
g
5
=
∫
-
1
1
1
ξ
ξ
2
ξ
3
ξ
4
g
ξ
d
ξ
γ
⋆
T
1
x
x
2
x
3
x
4
,or(32)
g
5
=
c
∣
ϵ
=
c
1
c
2
c
3
c
4
c
5
1
x
x
2
x
3
x
4
.In these expressions, the coefficient covector isc
=
∫
-
1
1
1
ξ
ξ
2
ξ
3
ξ
4
g
(
ξ
)
d
ξ
[
γ
⋆
]
T, and, according to (21), can be calculated resulting in(33)
c
=
1.2047
0
0.1881
0
0.0683
1.7578
0
-
8.2031
0
7.3828
0
9.3750
0
-
13.125
0
-
8.2031
0
68.906
0
-
73.828
0
-
13.125
0
21.875
0
7.3828
0
-
73.828
0
86.133
,generating the expansion for the function(34)
g
5
x
=
1.0789
-
1.9639
x
2
+
0.89037
x
4
.Figure1(a) shows the almost perfect superposition of the function g
(
x
) and its expansion g
5
(
x
). Figure 1(b) shows the local error for the expansion.Figure 1
Approximation ofg
(
x
).
(a)
Superposition of the graphics ofg
(
x
) and g
5
x
(b)
Local error of the expansiong
5
(
x
) of g
(
x
)


## 3.2. Expansion in a Complex Nonorthogonal Basis

Here, considering a complex basisQ
i
∞, composed of elements Q
i
:
R
→
H, i
=
1
,
…
,
n, the real function f
(
x
)
=
exp
⁡
-
2
π
x
2 is expanded, showing efficient and concise results.Defining the basis(35)
Q
r
x
=
r
x
r
-
1
+
i
1
-
δ
1
r
x
r
-
2
,
r
=
1
,
…
,
n
,the metric tensor is expressed by(36)
γ
r
s
Q
=
Q
r
∣
Q
s
∞
=
∫
-
1
1
Q
r
∗
x
Q
s
x
d
x
=
∫
-
1
1
r
x
r
-
1
-
i
1
-
δ
1
r
x
r
-
2
s
x
s
-
1
+
i
1
-
δ
1
s
x
s
-
2
d
x
⟹
γ
r
s
Q
=
2
p
a
r
r
+
s
r
s
r
+
s
-
1
+
1
-
δ
1
r
1
-
δ
1
s
r
+
s
-
3
+
2
i
mod
⁡
r
+
s
,
2
r
1
-
δ
1
s
-
s
1
-
δ
1
r
r
+
s
-
2
.Forn
=
5
(37)
γ
Q
=
2
2
i
2
2
i
3
2
-
2
i
14
3
-
2
i
3
58
15
-
6
i
5
2
2
i
3
64
15
-
2
i
5
164
35
-
2
i
3
58
15
2
i
5
174
35
-
2
i
7
2
6
i
5
164
35
2
i
7
368
63
,and, for the reciprocal metrics,(38)
γ
⋆
Q
=
9.66896
-
6.48244
i
-
7.7305
2.88875
i
4.08105
6.48244
i
5.05372
-
4.87307
i
-
2.51961
2.60449
i
-
7.7305
4.87307
i
8.6524
-
1.75798
i
-
5.20898
-
2.88875
i
-
2.51961
1.75798
i
1.58252
-
0.861328
i
4.08105
-
2.60449
i
-
5.20898
0.861328
i
3.44531
.Expanding the given function forn
=
5,(39)
f
5
∞
=
γ
⃡
⋆
f
∞
=
∑
r
=
1
5
γ
k
r
μ
r
Q
k
∞
=
∑
r
=
1
5
μ
k
Q
k
∞
,where μ
k
=
∑
r
=
1
5
γ
k
r
Q
r
∣
f
∞ or(40)
f
5
=
μ
1
μ
2
μ
3
μ
4
μ
5
1
2
x
+
i
3
x
2
+
i
x
4
x
3
+
i
x
2
5
x
4
+
i
x
3
≡
μ
∣
Q
,with(41)
μ
=
∫
-
1
1
Q
1
∗
ξ
Q
2
∗
ξ
Q
3
∗
ξ
Q
4
∗
ξ
Q
5
∗
ξ
f
ξ
d
ξ
γ
⋆
Q
T
.Considering γ
⋆
[
Q
] formerly calculated, the expansion becomes(42)
f
5
x
=
1.38287
-
8.50196
×
10
-
16
i
Q
1
x
+
8.46164
×
10
-
16
+
0.502949
i
Q
2
x
-
1.0059
-
2.95875
×
10
-
16
i
Q
3
x
-
6.04446
×
10
-
16
+
0.110628
i
Q
4
x
+
0.442512
-
9.03615
×
10
-
16
i
Q
5
x
.Figure2 shows the superposition of the function f
(
x
) and its expansion f
5
(
x
). It can be noticed that the five-term approximation is not good.Figure 2
Superposition of the graphics off
(
x
) and f
5
(
x
).From the explicit expression (26), the metrical matrix for n
=
10 can be determined:(43)
γ
Q
=
2
2
i
2
2
i
3
2
2
i
5
2
2
i
7
2
2
i
9
-
2
i
14
3
-
2
i
3
58
15
-
6
i
5
134
35
-
10
i
7
242
63
-
14
i
9
382
99
2
2
i
3
64
15
-
2
i
5
164
35
-
6
i
7
104
21
-
10
i
9
508
99
-
14
i
11
-
2
i
3
58
15
2
i
5
174
35
-
2
i
7
118
21
-
2
i
3
598
99
-
10
i
11
906
143
2
6
i
5
164
35
2
i
7
368
63
-
2
i
9
652
99
-
6
i
11
1016
143
-
10
i
13
-
2
i
5
134
35
6
i
7
118
21
2
i
9
670
99
-
2
i
11
1082
143
-
6
i
13
106
13
2
10
i
7
104
21
2
i
3
652
99
2
i
11
1104
143
-
2
i
13
556
65
-
2
i
5
-
2
i
7
242
63
10
i
9
598
99
6
i
11
1082
143
2
i
13
1694
195
-
2
i
15
2434
255
2
14
i
9
508
99
10
i
11
1016
143
6
i
13
556
65
2
i
15
2464
255
-
2
i
17
-
2
i
9
382
99
14
i
11
906
143
10
i
13
106
13
2
i
5
2434
255
2
i
17
3438
323
,with the reciprocal matrix given by(44)
γ
⋆
Q
=
194.628
182.705
i
-
245.619
449.976
i
431.287
837.612
i
-
402.514
768.318
i
142.611
264.765
i
182.705
i
173.81
227.829
i
-
441.019
395.46
i
831.682
366.933
i
-
766.695
129.629
i
264.765
-
245.619
227.829
i
338.212
529.699
i
-
651.753
967.53
i
641.618
887.652
i
-
234.589
307.523
i
449.976
i
-
441.019
529.699
i
1306.76
825.923
i
-
2699.58
706.803
i
2604.53
236.383
i
-
922.568
431.287
395.46
i
-
651.753
825.923
i
1425.36
1406.66
i
-
1518.75
1258.88
i
583.739
434.151
i
837.612
i
831.682
967.53
i
-
2699.58
1406.66
i
5953.4
1114.41
i
-
5973.08
348.073
i
2170.75
-
402.514
366.933
i
641.618
706.803
i
-
1518.75
1114.41
i
1703.66
958.733
i
-
677.455
325.651
i
768.318
i
-
766.695
887.652
i
2604.53
1258.88
i
-
5973.08
958.733
i
6154.17
286.437
i
-
2279.55
142.611
129.629
i
-
234.589
236.383
i
583.739
348.073
i
-
677.455
286.437
i
275.799
95.1856
i
264.765
i
264.765
307.523
i
-
922.568
434.151
i
2170.75
325.651
i
-
2279.55
95.1856
i
856.67
.Consequently, the ten-term expansion is given by(45)
f
10
∞
=
γ
⃡
⋆
f
∞
=
∑
r
=
1
10
γ
k
r
μ
r
Q
k
∞
=
∑
r
=
1
10
μ
k
Q
k
∞
,where μ
k
=
∑
r
=
1
10
γ
k
r
Q
r
∣
f
∞ or, in matrix form,(46)
f
10
=
μ
1
μ
2
μ
3
μ
4
μ
5
μ
6
μ
7
μ
8
μ
9
μ
10
1
2
x
+
i
3
x
2
+
i
x
4
x
3
+
i
x
2
5
x
4
+
i
x
3
6
x
5
+
i
x
4
7
x
6
+
i
x
5
8
x
7
+
i
x
6
9
x
8
+
i
x
7
10
x
9
+
i
x
8
≡
μ
∣
Q
,with(47)
μ
=
∫
-
1
1
Q
1
∗
Q
2
∗
Q
3
∗
Q
4
∗
Q
5
∗
Q
6
∗
Q
7
∗
Q
8
∗
Q
9
∗
Q
10
∗
ξ
f
ξ
d
ξ
γ
⋆
Q
T
,where γ
⋆
[
Q
] is given by the new matrix Q or, analogously, calculating the coefficients by the equation(48)
μ
k
=
∑
r
=
1
n
∫
-
1
1
Q
r
⋆
ξ
f
ξ
d
ξ
γ
r
k
′
.Consequently, the new approximation is given by(49)
f
10
x
=
2.03777
-
1.8553
×
10
-
13
i
Q
1
x
+
1.82597
×
10
-
13
+
1.04832
i
Q
2
x
-
2.09663
-
2.15423
×
10
-
13
i
Q
3
x
-
5.49712
×
10
-
13
+
0.661302
i
Q
4
x
+
2.64521
-
3.28239
×
10
-
13
i
Q
5
x
+
1.1432
×
10
-
12
+
0.320826
i
Q
6
x
-
1.92496
-
2.75674
×
10
-
13
i
Q
7
x
-
1.10567
×
10
-
12
+
0.071728
i
Q
8
x
+
0.573824
-
9.09373
×
10
-
14
i
Q
9
x
+
3.92054
-
3.2685
i
×
10
-
13
Q
10
x
.Figure3(a) shows the almost perfect superposition of the function f
(
x
) and the real part of its expansion f
10
(
x
). Figure 3(b) shows the local rest for the expansion.Figure 3
Approximation off
(
x
).
(a)
Superposition of the graphics off
(
x
) and f
10
x
(b)
Local error of the expansionf
10
(
x
) of f
(
x
)


## 4. Conclusion

Considering that using nonorthogonal bases is being increased in spectral numerical methods requiring Gram-Schmidt procedures that are generally difficult from the operational point of view, this paper presents a simpler method for expanding functions, based onbracket formalism and calledcoadjoint method.The mathematical ideas of thecoadjoint method were presented and the examples have shown its practicability.It can be added thatcoadjoint method is an efficient and concise tool for nonorthogonal bases with low computational costs. As it is not necessary to have orthogonal bases, more general function can be used in the numerical methods, increasing the quality of the whole process.



---

*Source: 9718962-2016-07-27.xml*