Luận văn toán học RIGHT INVERTIBLE OPERATORS AND INTERPOLATION PROBLEMS IN LINEAR SPACES

The theory of right invertible operators algebraic analysis started with works of D.. math-This thesis concentrates on the theory of right invertible operators and terpolation problems w

VIETNAM NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS

VIETNAM NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS

Thesis advisor: Prof Dr Hab Nguyen Van Mau

Hanoi - 2012

me to finish this thesis !

I also want to thank Teachers in Horizon School for all the help they havegiven to me during my graduate study I am so lucky to get their support

I wish to thank the other teachers at the Mathematics Department of sity of Science for their teaching, continuous support, tremendous research andstudy environment they have created I also thank to my classmates for theirfriendship and suggestion I will never forget their care and kindness Thankyou for all the help and making the class like a family

Univer-Last, but not least, I would like to express my deepest gratitude to my family.Without their unconditional love and support, I would not be able to do what Ihave accomplished

Hanoi, November 13th, 2012

Student

Azat Yazgulyyev

1.1 Introduction to Algebraic Analysis 4

1.2 D-polynomials 7

2 Interpolation problems 10 2.1 Property (c) 10

2.2 Polynomial Interpolation Problems 14

3 Some Applications in Analysis 16 3.1 The criterion for initial operators to possess the (generalized) c(R) -property 16

3.2 Hermite interpolation problem 20

3.3 Lagrange interpolation problem 20

3.4 Newton interpolation problem 21

3.5 Taylor interpolation problem 22

3.6 Logarithmic and Antilogarithmic mappings 24

3.7 Trigonometric elements and mappings 26

The theory of right invertible operators (algebraic analysis) started with works

of D Przeworska-Rolewicz, and then studied and developed by many other ematicians The algebraic theory of (generalized) invertible operators was stud-ied by Ross Caradus, Zuhair Nashed, Otmar Scherzer, Nguyen Van Mau, NguyenMinh Tuan and by other scientists

math-This thesis concentrates on the theory of right invertible operators and terpolation problems with right invertible operators in linear spaces

in-Algebraic analysis, is an algebraic based theory unifying many different eralizations of derivatives and integrals (not necessarily continuous) The mainconcepts of this algebraic formulation are right invertible linear operators, theirright inverses and associated initial operators Right invertible operators areconsidered to be algebraic counterparts of derivatives and their right inversestogether with initial operators correspond with idea of integration For thereader’s convenience we will try to give a brief & basic concepts and definitionsrelated to the right invertible operators

gen-We use the word ’operator’ in many fields likewise biology, physics, tics, computer programming etc and all of them have a different meanings

linguis-In basic mathematics an operator is a symbol or function representing a matical operation (i.e operator means a function between vector spaces )

mathe-In terms of vector spaces, an operator is a mapping from one vector space (ormodule ) to another As we know that the operators are of critical importance

to both linear algebra and functional analysis Furthermore there are bunch ofapplications in many other fields of pure and applied mathematics

Let’s recall the definition of linear operators:

The most common kind of operators encountered are linear operators Let U

andV be vector spaces over a field𝒦 Operator

A :U→V

is called linear if:

A(αx+βy) = αAx+βAy, ∀x, y∈ U,∀α , β∈ 𝒦).Now, let’s give a definition of right invertible operators:

Let X be a linear space overR and L(X)be the family of all linear operators

in X with the domains being linear spaces of X Then for any A ∈ L(X), let

DA denote the domain of A and let L0 = A∈ L(X): DA = X By the space ofconstants of an operator D∈ L(X)we shall mean the set ZD =ker D

A linear operator D∈ L(X)is said to be right invertible if

DR= Ifor some linear operator R ∈ L0(X) called a right inverse of D and I =idX Thefamily of all right invertible operators in X will be denoted by R(X)

Next we will give a definition of interpolation problem: In the mathematicalfield of numerical analysis interpolation is a method of constructing new datapoints within the range of a discrete set of known data points

One often has a number of data points, obtained by sampling or tation, which represent the values of a function for a limited number of values

experimen-of the independent variable It is experimen-often required to interpolate (i.e estimate)the value of that function for an intermediate value of the independent variable.This maybe achieved by curve fitting (if more than two data points are known)

or regression analysis

Or in other words, simply, we can say estimation of a value between twoknown data points A simple example is calculating the mean of two popula-tion counts made 10 years apart to estimate the population in the fifth year.The interpolation problems are an important part of algebra and calculus It has

a big role in mathematics not only as subjects for studying but also as a powerfultool of continuous and discrete models of the calculus in the theory of equations,approximation theory

This thesis consist of 3 parts:

∙ First chapter introduces the theory of right invertible operators (in ular, examples, definitions and some properties) with the initial operatorsand D-polynomials

partic-∙ The next chapter is about interpolation problems induced by right ible operators We have given the definition of c(R)-property and mentionabout interpolation problems induced by right invertible operators Fur-thermore, because, considering the general problem is complicated, we willmention about some classical interpolation problems (specific cases of thegeneral interpolation problem) Taylor interpolation problem, Lagrange in-terpolation problem, Newton interpolation problem and Hermite interpo-lation problem

invert-∙ In the last chapter, we have given the criterions for the system of initial erators to possess the c(R)-property and generalized c(R)-property You willsee some applications of interpolation problems with right invertible oper-ators in analysis and given some properties of logarithmic and antiloga-rithmic mappings to show how to solve equations with linear combinations

op-of right invertible operators in commutative algebras with the concrete amples by using trigonometric elements

Unfortunately, Prof Danuta Przeworska- Rolewicz has died several months ago

C HAPTER 1

Calculus of right invertible operators

D.Przeworska Rolewicz developed an algebra-based theory around linear, notnecessarily continuous operators D : X → X which admit a right inverse The

”Algebraic analysis” term is used by many authors to indicate an algebraic proach to analytic problems and also used in many different senses In thepresent paper, this term we use in the sense of D Przeworska-Rolewicz sinceour main interest is the calculus of right invertible operators To be more pre-cise below we will state some fundamental facts of algebraic analysis

ap-1.1 Introduction to Algebraic Analysis

Let X be a linear space over R and L(X) be the family of all linear operators

in X with the domains being linear subspaces of X Then for any A ∈ L(X), let

DA denote the domain of A and let L0(X) = {A∈ L(X): DA =X}

A linear operator D ∈ L(X) is said to be right invertible if DR = I, for somelinear operator R ∈ L0(X) called a right inverse of D and I =idX The family ofall right invertible operators in X will be denoted by R(X) If two right inversescommute each other, then they are equal It is known that a power of a rightinvertible operators is again right invertible, as well as a polynomial in a rightinvertible operator under appropriate assumptions However, a linear combina-tion of right invertible operators ( their sum and/or difference) in general is notright invertible

Elements of kerD are said to be constants, since by definition, Dz =0 if andonly if z ∈ kerD The kernel of D is said to be the space of constants By thespace of constants of an operator D ∈ L(X) we shall mean the set ZD = kerD

We should point out that, in general, constants are different than scalars, sincethey are elements of the space X

1.1 Introduction to Algebraic Analysis

Furthermore, by ℛD = {Rγ}γ∈Γ we denote the family of all right inverses of

a given D ∈ R(X) If R ∈ ℛD is a given right inverse of D ∈ R(X), the familyℛD

is characterized by

ℛD = {R+ (I− ℛD)A : A∈ L0(X)} (1.1)Consider a family of right invertible operators Di ∈ R(X)and a correspondingfamily of their right inverses Ri ∈ ℛD for i =1, , n and some n∈ N Then, the

composition D=D1, , Dn is right invertible, i.e D ∈ R(X) and one of its rightinverses R∈ R(X)is given by

R=Rn· · ·R1 (1.2)For any x, y ∈ X we say that y is a primitive element of x whenever Dy = x.Thus, the element Rxis a primitive element of x, for any x ∈ X and R ∈ ℛD Theset

I(x) = {y∈ X : Dy=x} (1.3)

is called the indefinite integral of a given x∈ X One can easily check that

ℛDx= {Rx+ (I− ℛD)Ax : A ∈ L0(X)} = ℛDx+ZD =Rx+ZD (1.4)for any R∈ ℛD and any non-zero element x ∈ X Hence, we obtain

I(x) = ℛDx+ZD =Rx+ZD (1.5)for any x∈ X and R ∈ ℛD

In other words, we can just simply say that, the invertibility of an operator

A∈ L(X)means that the equation Ax=y has a unique solution for every y∈ X.

An element y ∈ dom D is said to be primitive for an x ∈ X if y = Rx for an

R ∈ ℛ𝒟 Indeed, by definition, x = DRx = Dy again, by definition, all x ∈ X

have primitives

Definition 1.1 The operator F ∈ L0(X) is said to be an initial operator of

D ∈ R(X), if the following conditions are satisfied:

1 Im F=ker D, F2= F

2 There exists an R ∈ ℛD e.i FR=0

for every operator D ∈ R(X), denote byℱD the set of all initial operators of D

In other words, any projection operator F∈ L0(X) onto ZD, i.e F2 =F and

Im F = ZD is said to be an initial operator induced by D ∈ R(X) and the family

of all such operators we denote by FD For an initial operator F and x ∈ X, the

1.1 Introduction to Algebraic Analysis

element Fx ∈ ZD is called the initial value of x Additionally, we say that aninitial operator F ∈ ℱD corresponds to R∈ ℛD if FR =0 or equivalently if

By a simple calculation one can verify that FαFβ = Fβ and Fβℛα = ℛα− ℛβ, for

any α, β∈ Γ Hence, for any indices α, β, γ∈ Γ, there is

Fβℛγ−Fαℛγ =Fβℛα, (1.7)

which means that in fact the left side of equation (1.7) is independent of γ The

last property allows one to define the following definite integration operator

Iα β = Fβℛγ−Fαℛγ (1.8)

for any α, β, γ ∈ Γ Amongst many properties of the operator Iβ

α we can mentionthe most intuitive one, namely

Hence for any x ∈ X and its arbitrary primitive element y ∈ X i.e Dy = x, weget

Iα βx= Fβγ−Fαγ (1.10)which is called the definite integral of x

To intuitively demonstrate the basic concepts which we mentioned above, we’llgive an example by taking the usual derivative operator D = d

dx.

Example 1.2 (cf.16) Assume the linear space X= 𝒞0(R) and D= d

dx .Then we recognize the domain DD = 𝒞1(R) and the set (linear subspace) of all constants

of D is ZD = {f ∈ X : f is a constant function} Since ZD is 1-dimensional linear spaceover R, we shall assume the identification ZD ≡ R Thus, the initial operators F in thisexample are projections of X onto R.

𝒞0(R) = all continuous real functions

𝒞1(R) = all real functions having continuous derivative

In the next section we define and analyze, in the sense of algebraic analysis,polynomials corresponding with a given family of right invertible operators, togeneralize the usual polynomials of several variables

1.2 D-polynomials

1.2 D-polynomials

Right invertible operators are considered to be algebraic counterparts of tives and their right inverses together with initial operators correspond with the

deriva-idea of integration These are some examples in terms of algebraic analysis :

usual differential and integral calculus, generalized differential calculus in rings etc

One can associate the concept of D-polynomials with a fixed invertible operator

D, defined in a linear space X However the D-polynomials, definite integrals

as-sociated with a single right invertible operator D constitute the algebraic

coun-terparts corresponding with mathematical analysis for functions of one variable

So, there is a necessity to extend algebraic analysis for functions of many

vari-ables To begin this direction we replace a single operator D by a fixed family D

of right invertible operators and study the corresponding D-polynomials

Preliminaries

Let X be a linear space over a field 𝒦 and L(X) be the family of all linear

mappings

D : U →Vfor any U,V linear subspaces of X

We shall use the notation: dom(D) = U, codom(D) = V and Im D =

Du : u ∈U

for the domain, codomain and image of D, correspondingly

We denote that

N =1, 2, 3 and N0 =0, 1, 2, 3 (1.11)Whenever D1= · · · = Dm =D ∈ L(X), we write Dm = D1 Dm, for m∈ N, and

D0 = I =iddom(D) By the space of constants for D ∈ L(X)we mean the family

For any D∈ L(X)and m ∈ N, we assume the notation

Z0(D) =ker D∖0 and Zm(D) = ker Dm+1

Obviously, for any D ∈ L(X) there exist

Zi(D)\Zj(D) =∅ where i ̸= j (1.14)

Proposition 1.3 Let D ∈ L(X), m ∈ N0 and Zi(D) ̸= (∅) for i = 0, 1 m Then

any element ui ∈ Zi(D), i=0, m are linearly independent

1.2 D-polynomials

Proof: Consider a linear combination u = λ0u0+ · · · +λmum and suppose that

u = 0 for some coefficients λ0 λm ∈ 𝒦 Hence we obtain the sequence ofequations:

Dku = λkDkuk+ · · · +λmDkum = 0, for k = 0, m Step by step, from these

equations we compute λm =0, λ0=0 Let us define

R(X) = D∈ L(X): codom(D) =imD (1.15)i.e each element D ∈ R(X)is considered to be a surjective mapping (onto its do-main) Thus, R(X) consists of all right invertible elements Indeed, it is enough

to know one right inverse in order to determine all right inverses and all tial operators Note that a superposition of a finite number of right invertibleoperators is again a right invertible operator

ini-Definition 1.4 An operator R∈ L(X)is said to be a right inverse of D ∈ R(X) ifdom(R) = im(D) and DR = I ≡ idim(D) ByℛD we denote the family of all rightinverses of D In fact, RD is a nonempty family, since for each y ∈ im(D) we canselect an element x∈ D−1(y)and define R∈ ℛD s.t

R : y →x

Definition 1.5 Any element F ∈ L(X) s.t dom(F) = dom(D), im(F) = Z(D)

and F2 =F is said to be an initial operator induced by D∈ R(X) We say that aninitial operator F corresponds to a right inverse R∈ ℛD, whenever FR=0 or if :

The initial operators play very important role in the calculus of right invertibleoperators If two initial operators commute each other, then they are equal Thefamily of all initial operators induced by D will be denoted by FD

The above formula characterizes initial operators by means of right inverses,whereas formula

by Fx(t) =x(a).

Example 1.7 Let X= RN (the linear space of all sequences ) andD ∈ R(X)- differenceoperator, i.e (Dx)n = xn+ 1−xn for n ∈ N A right inverse R ∈ ℛD is defined by theformulas (Rx)1 = 0 and (Rx)n+1 = ∑n

i = 1

xi while (Fx)n = x1 defines the initial operator

F ∈ ℱD corresponding to R We note that the nontrivial initial operators do exist only foroperator which are right invertible but not invertible The family of all such operators isthen :

R+(X) = {D ∈ R(X) : dim Z(D) >0} (1.20)

Proposition 1.8 Let D ∈ R(X) and R∈ ℛD Then R is not a nilpotent operator.Proof: Suppose that Rn ̸= ∅ and Rn + 1 = 0, for some n ∈ N Then 0 ̸= Rn =

IRn = DRRn = DRn+1 =0, a contradiction

C HAPTER 2

Interpolation problems

It is known that the classical interpolation problems were born very early,starting with the works of Newton, Lagrange But we should mention thatthe construction of the general interpolation problems and algorithms to findits solutions as well as building theories related to interpolation, in general,until now are being researched and developed by mathematicians The generalinterpolation problems induced by a right invertible operators with initial oper-ators was first introduced and considered by D Przeworska-Rolewicz, in 1988

As we mentioned before the initial operators of the right invertible operators has

a huge role in dealing with interpolation problems Moreover, we can say thatinterpolation problems play very important role in establishing the polynomialssatisfying the system of the special conditions

2.1 Property (c)

Here we are going to give a necessary and sufficient condition for the minant induced by a system of initial operators with the property c(R) to bedifferent from zero This will also lead us to a necessary and sufficient conditionfor general interpolation problem to have a unique solution

deter-Definition 2.1 Let D ∈ R(X) An initial operator F0 for D has the property c(R)

for an R ∈ ℛD if there exist scalar ck s.t

F0Rkz = (ckk!)z for all z∈ ker D, k ∈N (2.1)and ck =0 for all k∈ N if F0 =F, where F is initial operator for D corresponding

to R We shall write F0 ∈ c(R) A set FD0 ⊂ FD has the property (c) if for every

F0 ∈ FD0 there exists an R ∈ ℛD s.t F0 ∈ c(R) Put c0 = 1, since F0z = z for all

z∈ ker D

VN =det(dik)i,k=0, ,N−1=0 (2.3)The following question was stated by D.Przeworska-Rolewicz:

Is the determinant VN different from zero for any system{F0· · ·FN− 1}of linearlyindependent initial operators having the property (c(R))?

We can claim that, indeed, the answer is positive for the case N = 1 (by def,

d0 =1) But, in general, the answer for the case N ≥2 is negative (N.V.Mau: see[9]-Chapter 6)

Naturally, the following question arises: Does there exist a subspace X0 ⊂Xsuch that the matrix (dik)(i, k = 0, N−1) has non-zero determinant if andonly if the restrictions of the initial operators{F0, , FN− 1}to X0are linearlyindependent?

We will show that the answer is positive, by the following theorem:

Theorem 2.3 Put

PN(R) = lin{Rkz : z∈ ker D, k =0, , N−1 (2.4)

or more specifically we can describe as:

ker D+R(ker D) + · · · +RN−1(ker D) = PN(R).Suppose F0, , FN − 1 ∈ ℱD have the property (c) Then a necessary and suf-ficient condition for VN ̸= 0, where VN is defined by (2.3), is that F0, FN − 1 arelinearly independent on PN(R)

Proof is based on the below lemma:

Lemma 2.4 Suppose F0, , FN − 1 ∈ ℱD have the property c(R) for an R∈ ℛD.Put

Fi′ = (Fi, FiR, , FiRN−1) for i =0, , N−1, (2.5)

di = (di0, di1, diN− 1) for i=0, , N−1, (2.6)where dik are defined by(2.2) Then the vectors F0′, , FN′ −1 are linearly indepen-dent on ker D if and only if the vectors d0, d1, , dN− 1 are linearly independent

Proof: [cf 9-Chapter 6]

2.1 Property (c)

Corollary 2.5 Let D∈ R(X), R∈ ℛDand F0, FN− 1 ∈ c(R) Then VN defined

by(2.3)is not zero if and only if the operators F0Rk, F1Rk, , FN− 1Rk are linearlyindependent on ker D for each k (0≤k ≤N−1)

Proof: [cf.9-Chapter 6]

Proof of theorem (2.3)

Suppose that VN ̸= 0, Then by corollary, the vectors F0′, F1′· · · , FN′ −1 of the form

(2.5)are linearly independent on ker D This means that the operators F0Rj, , FN − 1Rjare linearly independent on ker D for each j ∈ {0, 1, , N−1 , i.e F0, , FN − 1are linearly independent on the set

ker D+R(ker D) + · · · +RN−1(ker D) = PN(R).Conversely, suppose F0, , FN − 1 ∈ c(R) are linearly independent on PN(R) ByCorollary, we just need to show that the system of vector operators

βiFiRjz=0 for all z∈ ker D

where βi ∈C This means that

Proof: Since dim ker D =1 we have F1Rkz =ckz for all z∈ ker D,

where ck ∈ ℱ (k=1, 2, ) If ck ̸=0 for some k∈ N we conclude that

F1Rk(ker D) = ker D On the other hand, F1Rk(ker D) ⊂ F1RkX ⊂ F1RX ⊂ F1X =

ker D Thus F1RX =ker D

2.1 Property (c)

Every initial operator F of the right invertible operator D possesses(c) −property

if and only if dim ker D = 1 Particularly, if D is the right invertible operator inthe linear spaces X and dimkerD ≥2, then there exists a class of initial operators

of D which doesn’t possesses the (c)-property (In the beginning of the next ter we will give some conditions related to possessing generalized c(R)-property)

chap-Definition 2.7 Let D ∈ R(X), R ∈ ℛD We say that the operator F ∈ ℱD

possesses c(R)-property if for every k ∈N, there exists ck ∈ K such that FRkz =ckzfor all z∈ ker D where R0 =I

Definition 2.8 Let D∈ R(X), R ∈ ℛD, and let Fi ∈ ℱD, i=1, 2, , n The system

of the initial operators{Fi}i=1, ,n is said to possess the generalized c(R)-property

if there are nontrivial subspaces Z1, Z2, , Zpof ker D such that following tions hold

Example 2.9 Put X =c(R), D = dtd22, R =Rt

0

Rs

0 dxThe dim ker D = 2 and e1 = 1, e2 = t are the basic vectors of ker D Consider theoperators Fk given by the following:

c(R)-property, but they do not possess the c(R)-property

2.2 Polynomial Interpolation Problems

Note: Next chapter will include some, definitions, theorems (and their plications) of some classical interpolation problems

ap-Let’s review a little bit about the polynomial interpolations

2.2 Polynomial Interpolation Problems

Briefly, we say that the interpolation problem is nothing but a problem ofsolving a system of linear equations and we define the interpolation problem asfollowing:

Given k+1 distinct points inR,{xj}0≤j≤k and a function f , find a polynomial ofdegree less than or equal to k that agrees with f on these points

The first question we will face is whether our problem is well-posed or not?( In particular, does a solution exist and is it unique?) We usually use the La-grange polynomials to demonstrate the existence

The Lagrange polynomials for{xj}0≤j≤k are defined as below:

Lj(x) = ∏

0 ≤ i ≤ k,i ̸= j

x−xi

xj−xi, 0≤ j≤k (2.7)Since

if P and Q are the same polynomials

By (2.7), we obtain a closed form formula for (2.9) But, it has 2 drawbacks:

1 When we have large number of division and multiplication, the evaluation

of the expression in (2.9) is inefficent

2 If we denote Pk as a value of the interpolating polynomial f on the set ofpoints{xj}0≤j≤k, at some intermediate points ξ, it is impossible to use Pk(ξ)

in order to compute Pk+ 1(ξ) We need to start computation from the ning