**2. Acceleration of convergence of Fourier series**

### **2.1 Basic definitions**

Known the jumps (1), one can construct, e.g. piecewise-polynomial *g*. As a result, 2-periodic extension of the function *F* ¼ ð Þ *f* � *g* is *q* times continuously differen-

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

*fs* � *gs* � �*e*

Therefore, taking into account only first 2ð Þ *n* þ 1 Fourier coefficients and trun-

*g x*ð Þþ <sup>X</sup>*<sup>n</sup>*

The "spectral" method proposed by Kurt Eckhoff in [4] (1993) turned out to be

easy to obtain using the integration in parts the following asymptotic representation

*Ap*,*<sup>k</sup>* ð Þ *iπs*

However, the computing of the jumps f g *Ask*ð Þ*f* directly by the function *f* sharply limits the scope of practical application of the method of A. N. Krylov.

*k*¼�*n*

*<sup>i</sup>πsx* <sup>þ</sup> *rs*ð Þ *<sup>x</sup>* ,

*fs* � *gs* � �*e*

*<sup>i</sup>πsx* (2)

� � (see also [5]). It is

(4)

*<sup>k</sup>*þ<sup>1</sup> <sup>þ</sup> *rs*,*rn* <sup>¼</sup> *o s*�*<sup>q</sup>* ð Þ, *<sup>s</sup>* ! <sup>∞</sup>*:* (3)

*s*¼�*n*

cating the remainder term *rs*, it is possible to approximate *f* in the form

*def*

*f x*ð Þ� *g x*ð Þ¼ <sup>X</sup>*<sup>n</sup>*

*f x*ð Þ≃ *f <sup>n</sup>*ð Þ¼ *x*

more practical, as it is based only on the use of coefficients *fs*

X *q*�1

*k*¼0

ð Þ �<sup>1</sup> *<sup>s</sup>*þ<sup>1</sup> 2ð Þ *iπs*

As a function *g* from (2) K. Eckhoff used those Bernoulli polynomials f g *Bk*ð Þ *x* , *x*∈ ½ � �1, 1 , *k*≥0, which Fourier coefficients f g *bk*,*<sup>s</sup>* have the following sim-

0, *s* ¼ 0,

Denoting *B*0ð Þ¼ *x* 1, polynomials f g *Bk*ð Þ *x* , *k* ¼ 0, 1, … , *n*, *x*∈½ � �1, 1 compose a basis on the space of polynomials of degree *n*. Bernoulli polynomials extended to the

where the quantities f g *ω<sup>s</sup>* are given explicitly, converges to *f* with the rate

the following system of linear equations with the Vandermonde matrix, obtained by principal part (1) choosing the indexes *s* ¼ *sk*, *k* ¼ 1, 2, … , *m q*ð Þ þ 1 , *θn*≤ ∣*sk*∣ ≤*n*,

> *A*~ *<sup>p</sup>*,*<sup>k</sup>* ð Þ *iπs*

Further, the problem is solved by the Krylov method. It is natural to call this acceleration scheme the Krylov-Eckhoff method **(KE-method)**. Over the past two

K. Eckhoff suggests to find approximate values of jumps *A*~ *<sup>p</sup>*,*<sup>k</sup>*≈*Ap*,*<sup>k</sup>*

X *q*

*k*¼0

*Ap*,*kBk <sup>x</sup>* � *ap* � <sup>1</sup> � � <sup>þ</sup> <sup>X</sup>*<sup>n</sup>*

*s*¼�*n*

*ωse*

*<sup>k</sup>*þ<sup>1</sup> , *<sup>s</sup>* <sup>¼</sup> *<sup>s</sup>*1, *<sup>s</sup>*2, … , *sm q*ð Þ <sup>þ</sup><sup>1</sup> (6)

*<sup>i</sup>πsx*, (5)

� � by solving

*<sup>k</sup>*þ<sup>1</sup> , *<sup>s</sup>* 6¼ 0, *<sup>k</sup>* <sup>¼</sup> 1, 2, *:* …

where *rs*ð Þ¼ *<sup>x</sup> o s*�*<sup>q</sup>* ð Þ, *<sup>s</sup>* ! <sup>∞</sup>, *<sup>x</sup>*<sup>∈</sup> ½ � �1, 1 .

tiable on whole axis, and then

of the Fourier coefficients:

ple form

*o n*�*<sup>q</sup>* ð Þ, *<sup>n</sup>* ! <sup>∞</sup>.

0<*θ* ¼ *const* <1

**30**

*fs* ¼ � <sup>1</sup> 2 X*m p*¼1 *e* �*iπs ap*

*bk*,*<sup>s</sup>* ¼

8 ><

>:

real axis with period 2 as piecewise-smooth functions. According to Krylov's scheme, the sequence

*def* <sup>X</sup>*<sup>m</sup>*

*p*¼1

X *q*

*k*¼0

*Fn*ð Þ¼ *x*

*fs* ¼ � <sup>1</sup> 2 X*m p*¼1 *e* �*i πs ap*

The classical definition of the partial sums of the Fourier series is based on a gradual increase in the frequencies of the Fourier system. Below we use a more general notations from the work [17].

**Definition 1.** *Let us call the truncated Fourier series any sum of the form*

$$S\_n(\boldsymbol{\kappa}) \stackrel{def}{=} \sum\_{k \in D\_n} f\_k \exp\left(i\,\pi k \boldsymbol{\kappa}\right), \ \boldsymbol{\kappa} \in [-1, 1],\tag{7}$$

where

$$f\_s = \frac{1}{2} \int\_{-1}^{1} f(\varkappa) e^{-i\pi s \varkappa} d\varkappa, \quad s = 0, \pm 1, \pm 2, \ldots \tag{8}$$

are Fourier coefficients of *f*, and *Dn* ¼ f g *dk* , *k* ¼ 1, … , *n*, is a set of *n* different integers (*n*≥1). We will assume that *D*<sup>0</sup> ¼ Ø.

**Definition 2.** *Let n*≥1 *be a fixed integer. Consider a system of functions Un* ¼ f g exp ð Þ *iπ λ<sup>k</sup> x* , *λ<sup>k</sup>* ∈ , *x*∈½ � �1, 1 , *k* ¼ 1, 2, … , *n, where* f g *λ<sup>k</sup> are arbitrary parameters. Consider the linear span Qn* ¼ *span U*f g*<sup>n</sup> . We call a function q*∈ *Qn as quasipolynomial of degree at most n.*

It is easy to see that *<sup>q</sup>*<sup>∈</sup> *Qn* <sup>P</sup> , if and only if when either *q x*ð Þ� 0 or *q x*ð Þ¼ *<sup>k</sup>P<sup>β</sup><sup>k</sup>* ð Þ *x* exp ð Þ *iπ λ<sup>k</sup> x* , where polynomials *P<sup>β</sup><sup>k</sup>* ð Þ *x* ≢ 0 have the exact degree *β<sup>k</sup>* and *<sup>m</sup>* <sup>¼</sup> <sup>P</sup> *<sup>k</sup>* 1 þ *β<sup>k</sup>* ð Þ≤ *n*. The number *m* will be considered below as the degree of the quasi-polynomial *q*.

P **Definition 3.** *We call the system* f g *λ<sup>k</sup>* ⊂ *as parameters of the quasi-polynomial q x*ð Þ¼ *<sup>k</sup>P<sup>β</sup><sup>k</sup>* ð Þ *x* exp ð Þ *iπ λ<sup>k</sup> x* ∈ *Qn and the number* 1 þ *β<sup>k</sup> as multiplicity of the parameter λk.*

**Definition 4.** *Let parameters* f g *<sup>λ</sup><sup>k</sup>* , *<sup>k</sup>* <sup>¼</sup> 1, … , *n are* <sup>1</sup> *fixed. We denote by Qn*ð Þ f g *<sup>λ</sup><sup>k</sup> the set of all corresponding q x*ð Þ∈ *Qn.*

**Remark 1.** *The set of quasi-polynomials q x*ð Þ∈ *Qn is invariant with respect to a linear change of the variable x. The set of quasi-polynomials q x*ð Þ∈ *Qn*ð Þ f g *λ<sup>k</sup> is invariant with respect to a shift of the variable x.*

For a fixed integer *n* ≥1, consider a set of non-integer parameters f g *λ<sup>k</sup>* ⊂ , *k*∈ *Dn*, and the following infinite sequence

<sup>1</sup> Unless otherwise stated, we will not exclude the repetition of values of *<sup>λ</sup><sup>k</sup>* in a set f g *<sup>λ</sup><sup>k</sup>* .

$$t\_{r,s} \stackrel{\text{def}}{=} (-\mathbf{1})^{s-r} \left( \prod\_{\substack{p \in D\_n \\ p \neq r}} \frac{s-p}{r-p} \right) \prod\_{k \in D\_n} \frac{r-\lambda\_k}{s-\lambda\_k}, \quad r \in D\_n, \ s = 0, \ \pm 1, \ldots \tag{9}$$

Since the Bernoulli polynomial *Bk*ð Þ *x* corresponds to one *k* -multiple parameter *λ <sup>j</sup>* ¼ 0 in the KE - method, formula (14) can be considered as a generalization of

**Remark 3.** *Like the Bernoulli polynomials in the KE- method, quasi-polynomials*

If in the sequence (9) parameters f g *λ<sup>k</sup>* and its corresponding multiplicities f g *nk*

1

CCCA Y *j*∈ *Dm*

*r* � *λ <sup>j</sup> s* � *λ <sup>j</sup>* � �*<sup>n</sup> <sup>j</sup>*

� �, *j* ∈ *Dm, be given and the subset* Λ *consists*

*Tr*ð Þ¼ *x* exp ð Þ *iπ rx* ,*r*∈Λ1*:* (16)

� � are corresponding positive integers, and

, 9<sup>∗</sup> ð Þ (15)

� �, *k*≥1 *can be decomposed into a Fourier series with coefficients (8), for which*

*s* � *p r* � *p*

*<sup>j</sup>* <sup>∈</sup> *Dm n <sup>j</sup>* ¼ *n*, *λ<sup>p</sup>* 6¼ *λ<sup>q</sup>* if *p* 6¼ *q*. However, this sequence is still defined only for

Consider now the possibility of including in the consideration of some integer parameters in (10). First of all, note that if for some *j* ∉ *Dn*, *λ <sup>j</sup>* is integer, then

Finally, we note that in the latter case, the sequence *tr*,*<sup>s</sup>* can be represented as a sum of simple fractions with respect to *s*∈ . Bearing in mind *Lemma 1*, it is not

i. *If there is an λ<sup>k</sup>* ∈Λ, *λ<sup>k</sup>* ∉ *Dn, then the system T*f g*<sup>r</sup> from (10) does not exist.*

ii. *If* Λ ¼ ð Þ Λ<sup>1</sup> ∪Λ<sup>2</sup> ⊂ *Dn, where* Λ<sup>1</sup> ⊂ *Dm contains only different integers and*

*With a λ<sup>r</sup>* ∈Λ<sup>2</sup> *we can apply Lemma 1 and also find the explicit form of*

The following result is a generalization of Theorem 1 in [17] to the case that in

**Theorem 1.** *Suppose the sequence (15) is given and the possible integer parameters in*

iii. *If* Λ 6¼ Ø *and* Λ 6¼ *Dn then the system T*f g *<sup>r</sup>*ð Þ *x* ,*r*∈ *Dm*nΛ *is determined based on the sequence, similar to (15), in which n <sup>j</sup>* ¼ 1, ∀*j, and n is replaced by n*ð Þ � *p*

If *λ <sup>j</sup>* are integers for *j*, *j*∈ *Dn*, then, firstly, it is natural to accept *tr*,*<sup>r</sup>* ¼ 1, and secondly, we notice that for *s* 6¼ *r* the number of the products in *tr*,*<sup>s</sup>* are

� �*. Then.*

Bernoulli polynomials to the case of any integer *λ <sup>j</sup>* .

*A Fast Method for Numerical Realization of Fourier Tools*

*DOI: http://dx.doi.org/10.5772/intechopen.94186*

*tr*,*<sup>s</sup>* ¼ �ð Þ<sup>1</sup> *<sup>s</sup>*�*<sup>r</sup>* <sup>Y</sup>

� �.

**Lemma 2.** *Let in (*9<sup>∗</sup> *) the parameters λ <sup>j</sup>*

*tr*,*<sup>λ</sup> <sup>j</sup>* ¼ ∞. So here the system f g *Tr* from (10) does not exist.

Λ<sup>2</sup> ⊂ *Dm contains only multiple integers, then*

*where p is the number of elements* Λ*.*

*quasi-polynomials and have the following explicit form*

formula (15) there are integer values among the parameters f g *λ<sup>k</sup>* .

f g *λ<sup>k</sup> satisfy the condition λ<sup>k</sup>* ∈ *Dm. Then the corresponding functions T*f g*<sup>r</sup> are*

are known, then we will use the representation ð Þ *s* ¼ 0, �1, …

0

BBB@

*p*∈ *Dn p*6¼*r*

Λ *<sup>j</sup>*,*<sup>k</sup>*

P

reduced.

**33**

*fs* <sup>¼</sup> *O s*j j�*<sup>k</sup>* � �, *<sup>s</sup>* ! <sup>∞</sup>*.*

where *r*∈ *Dn*, *Dm* ⊂ *Dn*, *nq*

non-integer numbers in *λ<sup>p</sup>*

difficult to proof, that.

*of all integer parameters from λ <sup>j</sup>*

*functions T*f g *<sup>r</sup>*ð Þ *x .*

**2.3 Explicit form of the system** f g *Tr*

**Remark 2.** *The last product in (3) is invariant with respect to the numbering of the parameters* f g *λ<sup>k</sup> . Here this numbering is tied to the set k*∈ *Dn.*

*In the general case, the parameters* f g *λ<sup>k</sup> (i.e., and tr*,*s) may depend on n. In order not to complicate the notation, we will indicate this dependence as needed.*

Further we denote (see [17] for details)

$$\begin{aligned} T\_r(\mathbf{x}) & \stackrel{\text{def}}{=} \exp\left(i\pi rx\right) + \sum\_{s \notin D\_n} t\_{r,s} \exp\left(i\pi sx\right), r \in D\_n, \ \mathbf{x} \in [-1, 1], \\\ f(\mathbf{x}) & \simeq F\_n(\mathbf{x}) \stackrel{\text{def}}{=} \sum\_{r \in D\_n} f\_r T\_r(\mathbf{x}), R\_n(\mathbf{x}) \stackrel{\text{def}}{=} f(\mathbf{x}) - F\_n(\mathbf{x}) \end{aligned} \tag{10}$$

The system f g *Tr*ð Þ *x* , 1*=*2 exp ð Þ *iπ rx* ,*r*∈ *Dn*, is biorthogonal on the segment *x*∈½ � �1, 1 and *L*2-error of approximation *f x*ð Þ≃*Fn*ð Þ *x* can be found from the formula

$$\left\| \left| R\_{\mathfrak{n}} \right| \right\|^2 = \sum\_{s \in D\_{\mathfrak{n}}} \left\| f\_s - \sum\_{r \in D\_{\mathfrak{n}}} f\_r t\_{r,s} \right\|^2 \tag{11}$$

Here we confine ourselves to the one-dimensional case. A similar universal algorithm for multivariate truncated Fourier series was proposed and numerically implemented in [1].

The following obvious formula (*x*∈½ � �1, 1 )

$$\exp\left(i\pi\lambda\mathbf{x}\right) = \sum\_{\iota=-\infty}^{\infty} \text{sinc}(\pi(\iota-\lambda)) \exp\left(i\pi\kappa\mathbf{x}\right), \ \lambda \in \mathbb{C},\tag{12}$$

where sincð Þ¼ *z* sin ð Þ*z =z*, sinc 0ð Þ¼ 1, *z*∈ , plays a key role in the future. This Fourier series can be repeatedly differentiated by the parameter *λ*.

### **2.2 The case of integer or multiple parameters**

To include the case of a quasi-polynomial *q*∈ *Qn* containing some integer parameters from f g *λ<sup>k</sup>* , consider the following quasi-polynomials associated with formula (12) and sequence (9)

$$\begin{split} \Lambda\_{j,k}(\boldsymbol{x}) & \stackrel{\text{def}}{=} \\ \frac{i^{2-k}}{2\pi^{k-1}(k-1)!} \frac{d^{k-1}}{d\lambda\_j^{k-1}} \left(\csc(\pi\lambda\_j) \exp\left(i\pi\lambda\_j \boldsymbol{x}\right)\right) &= \\ \sum\_{s=-\infty}^{\infty} \frac{(-1)^{s} \exp\left(i\pi\kappa\boldsymbol{x}\right)}{2\left(i\pi(s-\lambda\_p)\right)^k}, \\ \lambda\_j &\in \mathbb{C}, \quad j \in D\_n, \quad k \ge 1, \ \boldsymbol{x} \in [-1, 1]. \end{split} \tag{13}$$

**Lemma 1.** *If λ <sup>j</sup> is an integer parameter, then*

$$\Lambda\_{j,k}(\mathbf{x}) = B\_k(\mathbf{x}) \exp\left(i\pi\lambda\_j\mathbf{x}\right) \tag{14}$$

where *Bk*ð Þ *x* is a Bernoulli polynomial (see (4)).

*A Fast Method for Numerical Realization of Fourier Tools DOI: http://dx.doi.org/10.5772/intechopen.94186*

*tr*,*<sup>s</sup>* ¼ *def*

> *Tr*ð Þ¼ *x def*

implemented in [1].

ð Þ �<sup>1</sup> *<sup>s</sup>*�*<sup>r</sup>* <sup>Y</sup>

0

BBB@

*p*∈ *Dn p*6¼*r*

Further we denote (see [17] for details)

*f x*ð Þ≃*Fn*ð Þ¼ *x*

exp ð Þþ *<sup>i</sup><sup>π</sup> rx* <sup>X</sup>

The following obvious formula (*x*∈½ � �1, 1 )

exp ð Þ¼ *<sup>i</sup>π λ<sup>x</sup>* <sup>X</sup><sup>∞</sup>

**2.2 The case of integer or multiple parameters**

Λ *<sup>j</sup>*,*<sup>k</sup>*ð Þ¼ *x def*

*i* 2�*k* 2*π<sup>k</sup>*�<sup>1</sup>ð Þ *k* � 1 !

**Lemma 1.** *If λ <sup>j</sup> is an integer parameter, then*

where *Bk*ð Þ *x* is a Bernoulli polynomial (see (4)).

X∞ *s*¼�∞

formula (12) and sequence (9)

**32**

*def* X *r* ∈ *Dn*

*s* � *p r* � *p*

*parameters* f g *λ<sup>k</sup> . Here this numbering is tied to the set k*∈ *Dn.*

1

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

CCCA Y *k*∈ *Dn*

*to complicate the notation, we will indicate this dependence as needed.*

*s*∈ �*Dn*

k k *Rn* <sup>2</sup> <sup>¼</sup> <sup>X</sup> *s*∈ �*Dn*

*s*¼�∞

Fourier series can be repeatedly differentiated by the parameter *λ*.

*r* � *λ<sup>k</sup> s* � *λ<sup>k</sup>*

**Remark 2.** *The last product in (3) is invariant with respect to the numbering of the*

*In the general case, the parameters* f g *λ<sup>k</sup> (i.e., and tr*,*s) may depend on n. In order not*

*frTr*ð Þ *x* , *Rn*ð Þ¼ *x*

*fs* � <sup>X</sup> *r* ∈ *Dn*

The system f g *Tr*ð Þ *x* , 1*=*2 exp ð Þ *iπ rx* ,*r*∈ *Dn*, is biorthogonal on the segment *x*∈½ � �1, 1 and *L*2-error of approximation *f x*ð Þ≃*Fn*ð Þ *x* can be found from the formula

> � � � � �

Here we confine ourselves to the one-dimensional case. A similar universal algorithm for multivariate truncated Fourier series was proposed and numerically

where sincð Þ¼ *z* sin ð Þ*z =z*, sinc 0ð Þ¼ 1, *z*∈ , plays a key role in the future. This

csc *πλ <sup>j</sup>*

� � exp *<sup>i</sup>π λ <sup>j</sup> <sup>x</sup>* � � � � <sup>¼</sup>

<sup>Λ</sup> *<sup>j</sup>*,*<sup>k</sup>*ð Þ¼ *<sup>x</sup> Bk*ð Þ *<sup>x</sup>* exp *<sup>i</sup>π λ <sup>j</sup> <sup>x</sup>* � � (14)

To include the case of a quasi-polynomial *q*∈ *Qn* containing some integer parameters from f g *λ<sup>k</sup>* , consider the following quasi-polynomials associated with

> *d<sup>k</sup>*�<sup>1</sup> *dλ<sup>k</sup>*�<sup>1</sup> *j*

*λ <sup>j</sup>* ∈ , *j*∈ *Dn*, *k*≥ 1, *x*∈½ � �1, 1 *:*

ð Þ �<sup>1</sup> *<sup>s</sup>* exp ð Þ *<sup>i</sup><sup>π</sup> sx* 2 *iπ s* � *λ<sup>p</sup>* � � � � *<sup>k</sup>* ,

*tr*,*<sup>s</sup>* exp ð Þ *iπ sx* ,*r*∈ *Dn*, *x*∈½ � �1, 1 ,

*def f x*ð Þ� *Fn*ð Þ *<sup>x</sup>*

*frtr*,*<sup>s</sup>*

� � � � �

2

sincð Þ *π* ð Þ *s* � *λ* exp ð Þ *iπ sx* , *λ*∈ , (12)

, *r*∈ *Dn*, *s* ¼ 0, � 1, … *:* (9)

(10)

(11)

(13)

Since the Bernoulli polynomial *Bk*ð Þ *x* corresponds to one *k* -multiple parameter *λ <sup>j</sup>* ¼ 0 in the KE - method, formula (14) can be considered as a generalization of Bernoulli polynomials to the case of any integer *λ <sup>j</sup>* .

**Remark 3.** *Like the Bernoulli polynomials in the KE- method, quasi-polynomials* Λ *<sup>j</sup>*,*<sup>k</sup>* � �, *k*≥1 *can be decomposed into a Fourier series with coefficients (8), for which fs* <sup>¼</sup> *O s*j j�*<sup>k</sup>* � �, *<sup>s</sup>* ! <sup>∞</sup>*.*

If in the sequence (9) parameters f g *λ<sup>k</sup>* and its corresponding multiplicities f g *nk* are known, then we will use the representation ð Þ *s* ¼ 0, �1, …

$$t\_{r,s} = (-1)^{s-r} \left( \prod\_{\substack{p \in D\_n \\ p \neq r}} \frac{s-p}{r-p} \right) \prod\_{j \in D\_m} \left( \frac{r-\lambda\_j}{s-\lambda\_j} \right)^{n\_j}, \tag{15}$$

where *r*∈ *Dn*, *Dm* ⊂ *Dn*, *nq* � � are corresponding positive integers, and P *<sup>j</sup>* <sup>∈</sup> *Dm n <sup>j</sup>* ¼ *n*, *λ<sup>p</sup>* 6¼ *λ<sup>q</sup>* if *p* 6¼ *q*. However, this sequence is still defined only for non-integer numbers in *λ<sup>p</sup>* � �.

Consider now the possibility of including in the consideration of some integer parameters in (10). First of all, note that if for some *j* ∉ *Dn*, *λ <sup>j</sup>* is integer, then *tr*,*<sup>λ</sup> <sup>j</sup>* ¼ ∞. So here the system f g *Tr* from (10) does not exist.

If *λ <sup>j</sup>* are integers for *j*, *j*∈ *Dn*, then, firstly, it is natural to accept *tr*,*<sup>r</sup>* ¼ 1, and secondly, we notice that for *s* 6¼ *r* the number of the products in *tr*,*<sup>s</sup>* are reduced.

Finally, we note that in the latter case, the sequence *tr*,*<sup>s</sup>* can be represented as a sum of simple fractions with respect to *s*∈ . Bearing in mind *Lemma 1*, it is not difficult to proof, that.

**Lemma 2.** *Let in (*9<sup>∗</sup> *) the parameters λ <sup>j</sup>* � �, *j* ∈ *Dm, be given and the subset* Λ *consists of all integer parameters from λ <sup>j</sup>* � �*. Then.*


$$T\_r(\boldsymbol{\kappa}) = \exp\left(i\boldsymbol{\pi}r\boldsymbol{\kappa}\right), r \in \Lambda\_1. \tag{16}$$

*With a λ<sup>r</sup>* ∈Λ<sup>2</sup> *we can apply Lemma 1 and also find the explicit form of functions T*f g *<sup>r</sup>*ð Þ *x .*

iii. *If* Λ 6¼ Ø *and* Λ 6¼ *Dn then the system T*f g *<sup>r</sup>*ð Þ *x* ,*r*∈ *Dm*nΛ *is determined based on the sequence, similar to (15), in which n <sup>j</sup>* ¼ 1, ∀*j, and n is replaced by n*ð Þ � *p where p is the number of elements* Λ*.*
