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

The following result is a generalization of Theorem 1 in [17] to the case that in formula (15) there are integer values among the parameters f g *λ<sup>k</sup>* .

**Theorem 1.** *Suppose the sequence (15) is given and the possible integer parameters in* 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 quasi-polynomials and have the following explicit form*

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

$$T\_r(\boldsymbol{\kappa}) = \sum\_{j \in D\_n} \sum\_{k=1}^{n\_j} c\_{rj,k} \ \Lambda\_{j,k}(\boldsymbol{\kappa}), r \in D\_n, \ \boldsymbol{\kappa} \in [-1, 1] \tag{17}$$

**Step 3.** Solve resulting equation by the least square method.

**Step 5.** Realize the approximation *f* ≃ *Fn* (see (10) and (17)). **Remark 4.** *Least square method provides only one solution in Step 3.* The following theorem is the main result of the paper [17].

*is f x*ð Þ� *Fn*ð Þ *<sup>x</sup>* , *<sup>x</sup>*∈½ � �1, 1 *), it is necessary and sufficient that* <sup>Λ</sup><sup>⊆</sup> *Dn* <sup>∪</sup> *<sup>D</sup>*<sup>~</sup> *<sup>n</sup> .*

� �, *s* ∈ *Dn* ∪ *D*~ *n, of the f be given. Applying Algorithm 1, we get parameters* f g *λ<sup>k</sup>* , *k*∈ *Dn. The accelerating the convergence of truncated Fourier series*

*tion 2) and the sets Dn, <sup>D</sup>*<sup>~</sup> *<sup>n</sup> and the Fourier coefficients fs*

*<sup>S</sup>*2*<sup>n</sup>*ð Þ¼ *<sup>x</sup>* <sup>X</sup>

**Remark 5.** *There are no Fourier coefficients fr*

**3. An numerical algorithm for Fourier transform**

*<sup>F</sup>*ð Þ¼ *<sup>ω</sup>* <sup>1</sup>

ffiffiffiffiffi <sup>2</sup>*<sup>π</sup>* <sup>p</sup>

*<sup>F</sup>*ð Þ¼ *<sup>ω</sup>* <sup>1</sup>

ffiffiffiffiffi <sup>2</sup>*<sup>π</sup>* <sup>p</sup> <sup>X</sup><sup>∞</sup> *k*¼�∞

on each segment ð Þ *k* � *h*, *k* þ *h* , ∀*k*, using Algorithm A and formula (17).

if *k* ¼ 0, then for functions exp ð Þ *iπ λ*<sup>1</sup> *x* , we have (see (12), here *k* ¼ 0)

exp ð Þ *iπ λ*<sup>1</sup> *t e*

ð<sup>∞</sup> �∞ *f t*ð Þ*e*

*k*∈ *Dn* ∪ *D*~ *<sup>n</sup>*

of the corresponding polynomial *P z*ð Þ¼ <sup>P</sup>*<sup>n</sup>*

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

*A Fast Method for Numerical Realization of Fourier Tools*

convergence of a truncated Fourier series.

we define by formula (17).

**3.1 Algorithm construction scheme**

ð*h* �*h*

**35**

*given. Denote by*

P

*fs*

**Step 4.** Again, according to the Vieta's formulae find all rootsf g *zk* , *k* ¼ 1, 2, … , *n*

**Theorem 2.** *{The phenomenon of the over-convergence}. Let f* ∈ *Qn (see Defini-*

Λ *the set of integer parameters in the representation f* ¼ *<sup>k</sup>Pmk* ð Þ *x* exp ð Þ *iπ μ<sup>k</sup> x . In order for the approximation by Algorithm* A *to be exact (that*

It is now natural to formulate the following definition of the acceleration of the

**Definition 5.** *Let f* <sup>∈</sup> *<sup>L</sup>*2½ � �1, 1 *and the sets Dn, <sup>D</sup>*<sup>~</sup> *<sup>n</sup> and the Fourier coefficients*

*coefficients are used in Algorithm* A *to an optimal choice of parameters* f g *λ<sup>k</sup>* , *k*∈ *Dm.*

Consider the Fourier transform *F* of a function *f* ∈*L*2ð Þ �∞, ∞ the form

The method of the item **2** can be applied by linear change of variable to function *f* defined on any finite segment. For a fixed *h*> 0 formula (21) can be rewritten as

On the other hand, our method allows one to approximate function *<sup>f</sup>* <sup>≃</sup> <sup>P</sup> *frTr*

Our idea is as follows. First, based on the required 1 error, you can leave a fnite number of intervals. Second, on the remaining intervals, the integrals are calculated explicitly. Really, each function *Tr*ð Þ *x* is a linear combination of functions of the form exp ð Þ *iπ λ*<sup>1</sup> *x* or its derivative with respect to *λ* of a certain order. For example,

*i t<sup>ω</sup> dt* <sup>¼</sup> <sup>2</sup> *<sup>h</sup>* sincð Þ *<sup>h</sup>*ð Þ *πλ*<sup>1</sup> <sup>þ</sup> *<sup>ω</sup>*

ð*<sup>k</sup>*þ*<sup>h</sup> k*�*h f t*ð Þ*e*

*<sup>p</sup>*¼<sup>0</sup>*ep <sup>z</sup>n*�*p*, *<sup>e</sup>*<sup>0</sup> <sup>¼</sup> 1, and put f g *<sup>λ</sup><sup>k</sup>* <sup>¼</sup> f g *zk* .

*f <sup>k</sup>* exp ð Þ *iπ kx* , *x*∈½ � �1, 1 , (20)

� �,*r*∈ *D*~ *<sup>m</sup> in formula (17). These*

*i t<sup>ω</sup> dt*, *<sup>ω</sup>*<sup>∈</sup> ð Þ �∞, <sup>∞</sup> (21)

*i t<sup>ω</sup> dt* (22)

� �, *s*∈ *Dn* ∪ *D*~ *n, of the f be*

where

$$\mathcal{L}\_{rj,k} = \frac{(-1)^{r+1} (i\pi)^k \prod\_{p \in D\_n} (r - \lambda\_p)^{n\_p}}{2 \left(n\_j - k\right)! \prod\_{p \in D\_n} (r - p)}$$

$$p \neq r$$

$$\frac{d^{n\_j - k}}{d\lambda\_j^{n\_j - k}} \left(\frac{\prod\_{p \in D\_n} (\lambda\_j - p)}{\prod\_{p \in D\_n} (\lambda\_j - \lambda\_p)^{n\_p}}\right)$$

$$p \neq j$$

### **2.4 The adaptive algorithm and over-convergence phenomenon**

In the main Theorem of paper [17], the phenomenon of over-convergence was theoretically substantiated. The basis of this result was the following **Algorithm** A.

Let *Sn*ð Þ *x* be a truncated Fourier series (see (7)). Consider formulas (9)–(10) with symbolic (not numerical) parameters *λ<sup>q</sup>* � �, *q*∈ *Dn*. Let us now choose a new set of integer numbers *<sup>D</sup>*<sup>~</sup> *<sup>n</sup>* <sup>¼</sup> <sup>~</sup> *dk* n o, *<sup>k</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>*, *<sup>D</sup>*<sup>~</sup> *<sup>n</sup>* <sup>∩</sup> *Dn* <sup>¼</sup> Ø. To determine the parameters *λ<sup>q</sup>* � �, *<sup>q</sup>*<sup>∈</sup> *Dn*, we additionally use Fourier coefficients *fs* � �, *s*∈ *D*~ *<sup>n</sup>*, and solve the following system of equations (compare with (11))

$$f\_s - \sum\_{r \in D\_n} f\_r t\_{r,s} = 0, s \in \tilde{D}\_n \tag{18}$$

regarding parameters *λ<sup>q</sup>* � �. Note that Eq. (18) is essentially nonlinear. If the solution exists, then the system f g *Tr* from (4) is used to approximate *f* by *Fn* (see (10)).

Here we show how Algorithm A can be step-by-step realized using given 2*n* Fourier coefficients *f dk* n o <sup>⋃</sup> *<sup>f</sup>* <sup>~</sup> *dk* n o n o (see (7)).

**Step 1.** Using the representation (9), we formally bring the left side of (18) to a common denominator, and fix the conditions *<sup>λ</sup><sup>q</sup>* 6¼ <sup>~</sup> *dk*, *k* ¼ 1, 2, … , *n*. Then Eq. (18) will take the form (*s*∈ *D*~ *<sup>n</sup>*)

$$\begin{split} \left( f\_s \prod\_{q \in D\_v} (s - \lambda\_q) \right) &= \\ \sum\_{r \in D\_v} \frac{(-1)^{s - d\_r} f\_r}{\text{sinc}(d\_r - \lambda\_r)} \left( \prod\_{p \nmid \, d\_r} \frac{s - d\_p}{d\_r - d\_p} \right) \prod\_{q \in D\_v} \left( d\_r - \lambda\_q \right) . \end{split} \tag{19}$$

**Step 2.** Using the Vieta's formula decompose the products in (18) containing the parameters f g *λ<sup>k</sup>* and denote

$$e\_k = (-\mathbf{1})^k \sum\_{i\_1 < i\_2 < \dots < i\_k} \lambda\_{i\_1} \lambda\_{i\_2} \dots \lambda\_{i\_k}, \ k = \mathbf{1}, \mathbf{2}, \dots, n.$$

It is not difficult to see that, as a result, Eq.(18) will take the form of a linear system of equations with respect to the variables f g *ek* .

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

*Tr*ð Þ¼ *<sup>x</sup>* <sup>X</sup>

with symbolic (not numerical) parameters *λ<sup>q</sup>*

n o

common denominator, and fix the conditions *<sup>λ</sup><sup>q</sup>* 6¼ <sup>~</sup>

*s* � *λ<sup>q</sup>* � � <sup>¼</sup>

ð Þ �<sup>1</sup> *<sup>s</sup>*�*dr*

sincð Þ *dr* � *λ<sup>r</sup>*

*ek* ¼ �ð Þ<sup>1</sup> *<sup>k</sup>* <sup>X</sup>

system of equations with respect to the variables f g *ek* .

*fr*

*i*<sup>1</sup> <*i*<sup>2</sup> < … <*ik*

set of integer numbers *<sup>D</sup>*<sup>~</sup> *<sup>n</sup>* <sup>¼</sup> <sup>~</sup>

regarding parameters *λ<sup>q</sup>*

Fourier coefficients *f dk*

will take the form (*s*∈ *D*~ *<sup>n</sup>*)

parameters f g *λ<sup>k</sup>* and denote

**34**

*fs* Y *q*∈ *Dn*

X *r*∈ *Dn*

parameters *λ<sup>q</sup>*

where

*j*∈ *Dm*

X*n j*

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

*cr*,*j*,*<sup>k</sup>* Λ *<sup>j</sup>*,*k*ð Þ *x* ,*r*∈ *Dn*, *x*∈½ � �1, 1 (17)

*<sup>p</sup>*<sup>∈</sup> *Dm r* � *λ<sup>p</sup>* � �*np*

ð Þ *r* � *p*

1

CCCCA

, *<sup>k</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>*, *<sup>D</sup>*<sup>~</sup> *<sup>n</sup>* <sup>∩</sup> *Dn* <sup>¼</sup> Ø. To determine the

� �. Note that Eq. (18) is essentially nonlinear. If the solution

*q*∈ *Dn*

*λ<sup>i</sup>*<sup>1</sup> *λ<sup>i</sup>*<sup>2</sup> , … , *λik* , *k* ¼ 1, 2, … , *n:*

*dr* � *λ<sup>q</sup>* � �*:*

� �, *q*∈ *Dn*. Let us now choose a new

*frtr*,*<sup>s</sup>* <sup>¼</sup> 0, *<sup>s</sup>* <sup>∈</sup> *<sup>D</sup>*<sup>~</sup> *<sup>n</sup>* (18)

*dk*, *k* ¼ 1, 2, … , *n*. Then Eq. (18)

(19)

� �, *s*∈ *D*~ *<sup>n</sup>*, and

*p* ∈ *Dn p* 6¼ *r*

*<sup>λ</sup> <sup>j</sup>* � *<sup>p</sup>* � �

*λ <sup>j</sup>* � *λ<sup>p</sup>* � �*np*

*k*¼1

ð Þ *<sup>i</sup><sup>π</sup> <sup>k</sup>*<sup>Q</sup>

In the main Theorem of paper [17], the phenomenon of over-convergence was theoretically substantiated. The basis of this result was the following **Algorithm** A. Let *Sn*ð Þ *x* be a truncated Fourier series (see (7)). Consider formulas (9)–(10)

� �, *<sup>q</sup>*<sup>∈</sup> *Dn*, we additionally use Fourier coefficients *fs*

exists, then the system f g *Tr* from (4) is used to approximate *f* by *Fn* (see (10)). Here we show how Algorithm A can be step-by-step realized using given 2*n*

> Y *p* ∈ *Dn p* 6¼ *dr*

(see (7)). **Step 1.** Using the representation (9), we formally bring the left side of (18) to a

> *s* � *dp dr* � *dp* ! <sup>Y</sup>

**Step 2.** Using the Vieta's formula decompose the products in (18) containing the

It is not difficult to see that, as a result, Eq.(18) will take the form of a linear

<sup>2</sup> *<sup>n</sup> <sup>j</sup>* � *<sup>k</sup>* � �! <sup>Q</sup>

Q *p* ∈ *Dn p* 6¼ *r*

0

BBBB@

**2.4 The adaptive algorithm and over-convergence phenomenon**

*dk* n o

solve the following system of equations (compare with (11))

*fs* � <sup>X</sup> *r*∈ *Dn*

⋃ *f* <sup>~</sup> *dk* n o n o

Q *p*∈ *Dn p* 6¼ *j*

*cr*,*j*,*<sup>k</sup>* <sup>¼</sup> ð Þ �<sup>1</sup> *<sup>r</sup>*þ<sup>1</sup>

*dn <sup>j</sup>*�*<sup>k</sup> dλ n <sup>j</sup>*�*k j*

**Step 3.** Solve resulting equation by the least square method.

**Step 4.** Again, according to the Vieta's formulae find all rootsf g *zk* , *k* ¼ 1, 2, … , *n* of the corresponding polynomial *P z*ð Þ¼ <sup>P</sup>*<sup>n</sup> <sup>p</sup>*¼<sup>0</sup>*ep <sup>z</sup>n*�*p*, *<sup>e</sup>*<sup>0</sup> <sup>¼</sup> 1, and put f g *<sup>λ</sup><sup>k</sup>* <sup>¼</sup> f g *zk* .

**Step 5.** Realize the approximation *f* ≃ *Fn* (see (10) and (17)).

**Remark 4.** *Least square method provides only one solution in Step 3.*

The following theorem is the main result of the paper [17].

**Theorem 2.** *{The phenomenon of the over-convergence}. Let f* ∈ *Qn (see Definition 2) and the sets Dn, <sup>D</sup>*<sup>~</sup> *<sup>n</sup> and the Fourier coefficients fs* � �, *s*∈ *Dn* ∪ *D*~ *n, of the f be given. Denote by* P Λ *the set of integer parameters in the representation f* ¼ *<sup>k</sup>Pmk* ð Þ *x* exp ð Þ *iπ μ<sup>k</sup> x . In order for the approximation by Algorithm* A *to be exact (that is f x*ð Þ� *Fn*ð Þ *<sup>x</sup>* , *<sup>x</sup>*∈½ � �1, 1 *), it is necessary and sufficient that* <sup>Λ</sup><sup>⊆</sup> *Dn* <sup>∪</sup> *<sup>D</sup>*<sup>~</sup> *<sup>n</sup> .*

It is now natural to formulate the following definition of the acceleration of the convergence of a truncated Fourier series.

**Definition 5.** *Let f* <sup>∈</sup> *<sup>L</sup>*2½ � �1, 1 *and the sets Dn, <sup>D</sup>*<sup>~</sup> *<sup>n</sup> and the Fourier coefficients fs* � �, *s* ∈ *Dn* ∪ *D*~ *n, of the f be given. Applying Algorithm 1, we get parameters* f g *λ<sup>k</sup>* , *k*∈ *Dn. The accelerating the convergence of truncated Fourier series*

$$S\_{2n}(\mathbf{x}) = \sum\_{k \in D\_n \cup \tilde{D}\_n} f\_k \exp\left(i\pi k \mathbf{x}\right), \ \mathbf{x} \in [-1, 1], \tag{20}$$

we define by formula (17).

**Remark 5.** *There are no Fourier coefficients fr* � �,*r*∈ *D*~ *<sup>m</sup> in formula (17). These coefficients are used in Algorithm* A *to an optimal choice of parameters* f g *λ<sup>k</sup>* , *k*∈ *Dm.*
