Generalization of the Fourier calculus and Wigner function

MYKOLA YAREMENKO

Department of Partial Differential Equations,

The National Technical University of Ukraine,

“Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, UKRAINE

Abstract: - In this paper, we consider

p

l

-periodical functions

 

pcs m



and

 

psn m



, which

are defined on the curve given by the equation:

1, 1

pp

x y p  

on

2

R

as functions of its

length. Considering

 

pcs m



and

 

psn m



as an independent functional system, we

construct the theory similar to Fourier analysis with the proper weights. For these weights, we

establish an analogous of the Riemannian theorem. The adjoint representations are introduced

and dual theory is developed. These Fourier representations can be used for approximation of

the oscillation processes.

Keywords: - General periodic function, Fourier analysis, p-circle, adjoint, p-Laplacian, linear

approximation, spectral theory, oscillation.

Received: August 26, 2021. Revised: April 17, 2022. Accepted: May 18, 2022. Published: July 3, 2022.

Introduction

A curved line given by the equation

1

pp

xy

on

2

R

-plane is called a

p

-curve

and denoted by

Cp

. Let us denote the length of

p

-curve by

p

l

. We introduce a pair of

C1

-

smooth functions

 

pcs



and

 

psn



of the

real argument

0, p

l







defined as

 

for allpcs x R





(1)

and

 

for allpsn y R





, (2)

where coordinates

x

and

y

belongs to

p

-

curve, i.e. bound by the equation

1

pp

xy

,

so that

 

00

4

p

l

psn pcs







and

 

01

4

p

l

pcs psn







, and

   

1 for all

pp

psn pcs R

  

  

. (3)

These functions satisfy the integral identity

   

 



pp

psn pcs

pcs psn d



  







. (4)

p

-Fourier transform

Assume

0,

pp

f L l





and let us write a

Fourier-type series with appropriate weights on

the interval

0, p

l





as

 

   

 

, ,...

,

mm

m

fx

a a pcs mx b psn mx



  



012

(5)

with some real coefficients

, , ,.., , ,...

mm

a a b a b

0 1 1

.

By usual means. integrating the identity

(3) over the period

p

l

, we obtain

   

2

pp

ll

pp

p

l

pcs d psn d

   





00

(6)

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DOI: 10.37394/232021.2022.2.17

Mykola Yaremenko

E-ISSN: 2732-9976

104

Volume 2, 2022

and

 

1p

l

p

a f x dx

l



0

. (7)

Next, we have

     

2p

lp

m

p

a f x pcs mx pcs mx dx

l



2

0

(8)

and

     

2p

lp

m

p

b f x psn mx psn mx dx

l



2

0

. (9)

Thus, we obtain the mapping of the

functions

0,

pp

f L l





in the set of the

infinite series according to the formula

   

       

, ,...

1

2.

p

l

p

lp

l

m

pp

f x f x dx

l

f y pcs my pcs my pcs mx

dy

lf y psn my psn my psn mx

























0

2

0

12 2

0

(10)

Statement (analogous Riemannian

theorem) 1. Assuming

g

is an integrable

function over an arbitrary interval

 

,a b R

then

     

lim 0

bp

ma

g x psn mx psn mx dx



 

2

(11)

and

     

lim 0

bp

ma

g x pcs mx pcs mx dx



 

2

. (12)

Theorem (adjoint) 2. Let

g

be an

integrable function over an arbitrary interval

 

,a b R

then there are

   

lim 0

b

ma

g x psn mx dx

 



(13)

and

   

lim 0

b

ma

g x pcs mx dx

 



. (14)

Adjoint series

Assume

p

fL

then

p

pp

f f L





21

and

we can write

   

, ,...

,

p

m

p

mm

f x f x a

a pcs mx pcs mx

b psn mx psn mx

















2

0

2

12

(15)

where

, , ,..., , ,...

mm

a a b a b

0 1 1

defined as follows

   

1p

lp

p

a f x f x dx

l



2

0

, (16)

     

2p

lp

m

p

a f x f x pcs mx dx

l



2

0

(17)

and

     

2p

lp

m

p

b f x f x psn mx dx

l



2

0

. (18)

The morphism from the real line

to the complex plane

:Epp R Cp

We introduce a function

:Epp R Cp

,

which maps from the real line to the

p

-curve

on the complex plane as follows

     

,Epp i pcs i psn R

   

  

(19)

and dual function

     

,

Epq i pcs i psn

R p q

  







, (20)

assume that

p

is renaming

q

. The function

:Epp R Cp

is a surjective morphism of the

topological groups from the real line

R

to the

p

-curve

Cp

and covering the space of the

p

-

curve

Cp

. In case

2p

, the function

Epp

is a

classical exponent on the complex plane of the

imaginary argument.

From formula (19), we have

     

 

1,

2

pcs Epp i Epp i R

   

   

and

     

 

1,

2

psn Epp i Epp i R

i

   

   

.

We introduce an integral transformation

Tp

of a function

pq

f L L

in the form

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DOI: 10.37394/232021.2022.2.17

Mykola Yaremenko

E-ISSN: 2732-9976

105

Volume 2, 2022

 

    

ˆ

p

f

Epp l i x f x dx Tp f











  



(21)

where

p

l

is a length of the

p

-curve

Cp

.

This integral transformation

Tp

is a

linear mapping relative to the function

f

and in

case

2p

coincides with the Fourier

transformation.

If

2p

then the integral transformation

of function

g

 

    

p

Epp l i x g d Rp g x

  









(22)

coincides with the inverse Fourier transform, in

the general case it is not necessarily true since

the dual structure does not coincide with the

natural complex structure, the inverse transform

is not always given by formula (22).

We define the inverses integral

transformation

Tp1

of a function

 

ˆ

pf



as

 

ˆ

p

f x Tp f x



1

(23)

for all transforms

 

ˆ

pf



.

So, we introduce two types of mappings:

the first is an analog of the Fourier transform

Tp

and its inverse

Tp1

, second is an analog of

the inverse Fourier transform

Rp

and we can

easily define its inverse

Rp1

. These

morphisms do not have the structure of the

group except for

2p

.

Generalization of the Wigner

function

Let functions

 

pn

LR





and

 

qn

LR





then we introduce a general

Wigner function

  

,,W x p



as any quasi-

probability distribution, which satisfies the

following conditions:

1.

      

,,

n

R

W x p dp x x

   





;

2.

  

 

,,

.

n

R

W x p dx

Tp p Tp p









As a consequence of the first condition,

we have

      

,,

nx

R

W x p dpdx x x

   





2

.

For a pair of functions

 

pn

LR





and

 

qn

LR





such that

|0





, we define a

density



in the point

 

,xp

by

      

,,

,, |

W x p

x p x p



   



 



.

The probability density function is a

homogeneous function of degree one so that

   

,,

,,x p x p

   





for all complex

0





.

Let us introduce the generalization of

the Weyl quantization by

    

 

,

n

p

R

Epp l i x x dx



    





,

where



is a symplectic form.

We define an operator

     

 

,,,

p

V Epp l i x Q P

  



,

where

Q

is position operators and

P

is a

momentum.

The Weyl quantization

  

Dp



is

defined by

         

Dp V



   

    

for any test function



.

We estimate

  

p

Dp



   



.

Similarly to the classical case, the new Weyl

quantization is a linear mapping so that

     

Dp Dp Dp

     

  

holds for all complex numbers

,



.

Definition. The Schwartz space is a

space of all functions such that

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DOI: 10.37394/232021.2022.2.17

Mykola Yaremenko

E-ISSN: 2732-9976

106

Volume 2, 2022

   

   

:.

sup , 0

n

nan

x

xR

CR

SR x x a N



















    





Now, let us consider a case when

Epp Exp

. The exponent function satisfies the

characteristic identity

     

Exp a b Exp a Exp b

so the Weyl

product has the property

     

#Dp Dp Dp

   



for some function

,



.

The symbol

#

denotes a non-

commutative product (often called Weyl

product) so that

     

#Dp Dp Dp

   



for some functions.

Let us assume

A

K

and

B

K

are kernels

for the integral operators

A

and

B

respectively. So, we have

 

     

 

     

exp 2

1,

2

exp 2

,

11

,

22

n

RA

n

RA

Dp Dp A x

i z x p

W K x z p z dpdz

i z x y p

K z x y z x y z dpdzdy















  









   



   







2

3

1

we take

 

Dp A





then

 

Dp A





1

and

calculate

 

,,

22

n

A

K x z x z F x z









  





1

,

thus

 

   

, , .Dp Dp x p x p





1

Generally speaking, the product

 

nn

AB

K K S R R

does not commute. So,

we obtain the following lemma.

Lemma 1. Let

A

K

be a kernel of an

operator

   

 

,

nn

A BL L R L R22

. Then the

mapping

Dp1

is an inverse to Weyl

quantization so that

 

nA

Dp A W K





1

and

 

nA

A Dp W K





; the Weyl kernel is given

by

 

1

exp 2 ,

2

1,,

2

n

R

n

K i z x p x z p dp

zx

F x z

  

 



   





  











then

 

   

,

,,

n

Dp Dp x p

W K x p x p









1

holds for

 

n

LR



2

.

Lemma 2. Let

A

K

and

B

K

be integral

kernels of the operators

A

and

B

respectively.

Then the product

      

, , ,

A B A B

K K x z K x K z  

is

correctly defined and is a kernel of the

operator; in other words

     

:n n n n n n

S R R S R R S R R     

.

Proof. Let us denote the multi-indices

by

, , , n

a b N



0

then we estimate

  

   

,

,,

1sup , ,

2maxsup , ,

1 , ,

2m

n

abx z A B

abx z A B L

abx z A B

R

abx z A B

cnR

abx z A B

x z K K x z

x z K x K z

Const x z K x K z

Const







  

    

     

    

     

     

     



1

2

00

   

ax , , .

abx z A B c

cn

x z K x K z



    0

2

Next, we exchange the order of the

supremum and integration and obtain

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DOI: 10.37394/232021.2022.2.17

Mykola Yaremenko

E-ISSN: 2732-9976

107

Volume 2, 2022

   

,

sup , ,

n

abx z A B

x z R

abx z A B

x z R

abx z A B

x z R L

x z K x K z





    

    

    

1

so, we have

   

     

,

sup , ,

1 sup sup , ,

2 max sup sup , , ,

n

nn

abx z A B

x z R L

abx z A B

x z R R

abx z A B

cn

x z R R

x z K x K z

C x z K x K z





 

 

    

     

     

1

2

thus, we obtain

 

nn

AB

K K S R R

.

For the Weyl system, we can formulate

the following Weyl quantization theorem.

Theorem. Let functions

 

,n

SR



2

then the function

 

#n

SR



2

and such

that satisfies the equality

     

#Dp Dp Dp

   



,

where

  

   

 

   

 

    

   

     

 

   

,,

#,

exp 2 , , ,

2

,,

2

exp , , , , .

,

,,

zz

xp

i x p z z

izz

F z F z

i

x p z x p

z

zz

 



   



   

   







   



  

























Proof. Assume

 

,n

SR



2

and

employ the definition of

Dp

, we have

   

 

    

   

 

     

   

,,

, , ,

2

exp 2 ,.

,

,,

,

zz

Dp Dp

F z F z

W z W z

zz

i

z

F z F W z

z











   









   















 













Now, we are going to establish that

 

#n

SR



2

  

 

     

 

   

 

   

 

    

   

 

   

 

,

,,

#,

,,

2

exp 2 ,,

,,

exp 2 , , ,

2

,,

, , ,

2

exp 2 ,

2

exp 2 , , ,

z

zz

xp

z

i

z

F x p

F z F z

i x p z

izz

F z F z

x p z

i

z

izz



 

 









   

  



   

   







   













 































 





    

   

,,

zz

F z F z





   









so

#



belongs

 

n

SR

2

.

Let us denote

K



and

K



kernels,

which belong to

 

n

SR

2

, then we have

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DOI: 10.37394/232021.2022.2.17

Mykola Yaremenko

E-ISSN: 2732-9976

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   

 

   

      

,

, # .

z

Dp Dp x

K K x

K K z z Dp x



  



  



    

  

Next, using the properties of the

exponential function, we have

  

     

 



   

 

   

 

   

 

   



#,

exp 2 , , , ,

exp 2 , , ,

2

exp 2 , , ,

2

exp 2 , , ,

2

,,

n

R

xp

i x p z y

izy

izz

iyy

z y dzd dyd dzd dyd



   



   



   



   

       



  





























8

     

 



     

exp 2 , , , ,

, , , .

2

n

R

i z z x p

z x p z dzd dzd

   



     

  

















4

By changing variables

     

, , , ,

2

y x p z







we are completing

the proof of the theorem.

From semigroup properties of

exponential function follows: let

a

be a symbol

of

 

2n

SR

then the Weyl operator is given by

 

   

 

,

ˆ

1,

2

1,

2exp

n

zp

Ax

a x z p

ip x z z



 















 

 





the kernel of the Weyl operator

A

is

 

   

ˆ,

11

exp ,

22

A

n

p

K x y

ip x y a x y p

 



    

  



    

   

,

and the symbol is written as

 

ˆ

,

11

exp , .

22

A

z

a x p

ip z K x z x z







   





These formulae are circular via to the

semigroup properties.

Since

   

 

   

ˆ,

exp 2 2

R

T x p x

ip x x x x









  





00

0 0 0

(24)

the Weyl operator can be written in the form

 

     

 

,

ˆ

1ˆ

, , .

2

n

Rzp

Ax

a z p T z p x













(25)

Statement. The Weyl operator extends

to the continuous operator

   

ˆ:nn

A S R S R





.

Indeed, Since

 

2n

a S R

the function

   

 

ˆ

,,

2n

xR

a z p x T z p S R

 



for all

functions

 

n

SR





and all multi-indices

n

N





therefore

 

ˆ

x

x A x

 

  

.

Weyl established that correspondence

between symbols

a

and Weyl operators

A

is

one-to-one and linear, unit symbol corresponds

to the identity operator on

 

n

SR



. Thus the

set of all Weyl operators coincides with the set

of all symbols on

 

nn

S R R

. The Weyl

operators are pseudo-differential operators with

rapidly decreasing kernels.

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Since the Weyl operator can be rewritten

as

 

   

 

,

ˆ

1,

2

1,

2exp

n

zp

Ax

a x z p

ip x z z



 















 

 





(26)

so that the kernel of the Weyl operator

A

can

be calculated by the formula

 

   

ˆ,

11

exp ,

22

A

n

p

K x y

ip x y a x y p

 



    

  



    

   

,

(27)

then, the symbol can be represented as

 

ˆ

,

11

exp ,

22

A

z

a x p

ip z K x z x z







   





. (28)

The last three formulae are circular.

Theorem 4. Let

ˆWeyl

Aa

be the Weyl

correspondence then

1. for

 

nn

a S R R

it is necessary

and sufficient

 

ˆ,nn

A

K x y S R R

and

 

   

ˆ

ˆ,

Az

A x K x z z





;

2. the map

ˆ

aA

extends to an

isomorphism

     

 

,

n n n n

S R R L S R S R





,

where

   

 

,

nn

L S R S R



is the

space of continuous linear

operators from

 

n

SR

to

 

n

SR



.

Proof. The theorem follows from the

Schwartz kernel theorem.

Theorem 5. Let the Weyl operator

ˆ

A

corresponds to the symbol

 

, 1 2

r n n

a L R R r   

so

ˆWeyl

Aa

, then

there is a constant

 

Const r

such that the

inequality

 

   

ˆ22

2n r n

nL R L R

LR

A Const r a





(29)

holds for all

 

2n

LR





.

From this theorem follows that for all

symbols

 

2nn

a L R R

corresponding Weyl

operators are

2

L

-bounded. However, there are

examples of the symbols

 

,2

r n n

a L R R r  

on which

2

L

boundness is ruined so that Weyl operators

ˆ

A

are not

2

L

- bounded for these symbols

 

,2

r n n

a L R R r  

.

The complete analysis of

2

L

- regularity

for Weyl operators can be made in terms of the

Calderon -Zygmund theory.

Theorem 6. Let

ˆ

A

be trace-class Weyl

operator on

 

2n

LR

corresponded to symbol

 

, 1 2

r n n

a L R R r   

. Then for

ˆ0A

it

is necessary and sufficient that

 

   

 

,

exp , , , , xp

F a x p

i x p x p a x p





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is continuous and such that the matrix with

entries

 

exp , , ,

2

,,

j j k k

ix p x p

F a x p x p















is positive semidefinite for all possible sets of

     

 

, , , ,...., , 2

1 1 2 2

N

n

NN

x p x p x p R

.

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DOI: 10.37394/232021.2022.2.17

Mykola Yaremenko

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