aa r X i v : . [ m a t h . QA ] D ec An algebraic analysis frameworkfor quantum calculus
Piotr Multarzy´nski
Faculty of Mathematics and Information Science
Warsaw University of Technology e-mail: [email protected]
ABSTRACT
An algebraic analysis framework for quantum calculus is proposed. The quan-tum derivative operator D τ,σ is based on two commuting bijections τ and σ de-fined on an arbitrary set M equipped with a tension structure determined by asingle tension function θ , i.e. a 1-dimensional case is analyzed here. The wellknown cases, i.e. h - and q -calculi together with their symmetric versions, can beobtained owing to special choice of mappings τ and σ . Keywords:
Quantum calculus; difference quotient operator; right invertible operator
MSC 2000: 12H10, 39A12, 39A70
The term ”algebraic analysis” is used by many authors to indicate an algebraic approachto analytic problems and, in fact, it is used in many different senses. In the presentpaper this term we use in the sense of D. Przeworska-Rolewicz [8] since our main interesthere is the calculus of right invertible operators. The examples of such operators can bethe usual derivative ddx as well as the divided difference operators studied in quantumcalculus [5]. In Section 4 we interpret the quantum h - and q -definite integrals within thealgebraic analysis framework. Then, in Section 6 some more general setting is proposed.Namely, we analyze linear operators defined on function (commutative) algebras andsatisfying certain product rules being the modified versions of the Leibniz rule. Suchoperators have many properties which are quite analogous with the corresponding onesfor differential operators. In parallel, there is a natural possibility to define some kind ofalgebraic integration associated with right invertible operators. The algebraic conceptof definite integration, with respect to a given right invertible linear operator, has beendefined by using initial operators within the algebraic analysis framework [8].1or the reader’s convenience, below we present some basics of algebraic analysis andquantum calculus. Quantum calculus is in fact a sort of a discrete calculus, in which some discrete versionsof differentiation and integration are studied. In the present paper we are going tocompare the proposal of quantum calculus integration with the corresponding generalidea of integration based on the calculus of right invertible operators. For the reader’sconvenience we give a short survey of the basic concepts concerning the right invertibleoperators but the comprehensive treatment of the topic one can find in Reference [8].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 D A denotethe domain of A and let L ( X ) = { A ∈ L ( X ) : D A = X } . By the space of constants ofan operator D ∈ L ( X ) we shall mean the set Z D = kerD . A linear operator D ∈ L ( X )is said to be right invertible if DR = I , for some linear operator R ∈ L ( X ) called aright inverse of D and I = id X . The family of all right invertible operators in X willbe denoted by R ( X ). In turn, by R D = { R γ } γ ∈ Γ we shall denote the family of all rightinverses of a given D ∈ R ( X ). If R ∈ R D is a given right inverse of D ∈ R ( X ), thefamily R D is characterized by R D = { R + ( I − RD ) A : A ∈ L ( X ) } . (2.1)Consider a family of right invertible operators D i ∈ R ( X ) and a corresponding familyof their right inverses R i ∈ R D , for i = 1 , . . . , n and some n ∈ N . Then, the composition D = D . . . D n is right invertible, i.e. D ∈ R ( X ), and one of its right inverses R ∈ R D is given by R = R n . . . R . (2.2)For any x, y ∈ X , we say that y is a primitive element of x whenever Dy = x . Thus,the element Rx is a primitive element of x , for any x ∈ X and R ∈ R D . The set I ( x ) = { y ∈ X : Dy = x } (2.3)is called the indefinite integral of a given x ∈ X . One can easily check, that R D x = { Rx + ( I − RD ) Ax : A ∈ L ( X ) } = R D x + Z D = Rx + Z D , (2.4)for any R ∈ R D and any non-zero element x ∈ X . Hence, we obtain I ( x ) = R D x + Z D = Rx + Z D , (2.5)for any x ∈ X and R ∈ R D .Any projection operator F ∈ L ( X ) onto Z D , i.e. F = F and ImF = Z D , is saidto be an initial operator induced by D ∈ R ( X ) and the family of all such operators we2enote by F D . For an initial operator F and x ∈ X , the element F x ∈ Z D is called theinitial value of x . Additionally, we say that an initial operator F ∈ F D corresponds to R ∈ R D if F R = 0 or equivalently if F = I − RD, (2.6)on the domain of D . The two families R D and F D uniquely determine each other.Indeed, for any R ∈ R D we define F = I − RD ∈ F D . On the other hand, for any F ∈ F D , we define R = R − F R , where R ∈ R D can be any since the result isindependent of the choice of R . Thus, for any γ ∈ Γ we have F γ = I − R γ D andconsequently F D = { F γ } γ ∈ Γ .By a simple calculation one can verify that F α F β = F β and F β R α = R α − R β , forany α, β ∈ Γ. Hence, for any indices α, β, γ ∈ Γ, there is F β R γ − F α R γ = F β R α , (2.7)which means that in fact the left side of equation (2.7) is independent of γ . The lastproperty allows one to define the following definite integration operator I βα = F β R γ − F α R γ , (2.8)for any α, β, γ ∈ Γ. Amongst many properties of the operator I βα we can mention themost intuitive one, namely I βα D = F β − F α . (2.9)Hence, for any x ∈ X and its arbitrary primitive element y ∈ X , i.e. Dy = x , we obtain I βα x = F β y − F α y , (2.10)which is called the definite integral of x .To intuitively demonstrate the basic concepts of algebraic analysis, we end thissection with two important examples. In the first example we take the usual derivativeoperator D = ddx while in the second example we consider D = D h being the followingdifference operator defined by D h f ( x ) = f ( x + h ) − f ( x ) h , (2.11)and giving rise to h-quantum calculus. Example 1.1
Assume the linear space X = C ( R ) (all continuous real functions)and D = ddx . Then we recognize the domain D D = C ( R ) (all real functions havingcontinuous derivative) and the set (linear subspace) of all constants of D is Z D = { f ∈ X : f is a constant function } . Since Z D is a 1-dimensional linear space over R , weshall assume the identification Z D ≡ R . Thus, the initial operators F in this exampleare projections of X onto R . To see why the name ”initial operator” is intuitivelyconsonant, it is enough to notice that F a : X ∋ f f ( a ) ∈ Z D ≡ R , a ∈ R , is3 projection operator associating with any f its value at a . Obviously, { F a : a ∈ R } ⊂ F D but { F a : a ∈ R } 6 = F D . The reason is that any convex combination ofinitial operators is again an initial operator [8]. For example, one can easily check that F ab = ( F a + F b ) ∈ F D and F ab = F c , if a = b , for any a, b, c ∈ R . Therefore, although F D can be viewed as an indexed family, i.e. F D = { F γ } γ ∈ Γ , we cannot naturally identifyΓ with R . As an example of a right inverse of D = ddx we can take R : X → C ( R ),such that R a f ( x ) = x R a f ( t ) dt , for a fixed a ∈ R . Let us notice that F a is the initialoperator corresponding to R a , for any a ∈ R . Example 1.2
Let X = R R be the linear space of all real functions and consider D defined by formula (2.11), for a fixed h >
0. Evidently, the linear space Z D consists ofall h -periodic functions. Then, the operator R defined by Rf ( x ) = − h − ⌊ xh ⌋ − P m =0 f ( x + mh ) x ∈ ( −∞ , x ∈ [0 , h ) h ⌊ xh ⌋ P m =1 f ( x − mh ) x ∈ [ h, ∞ ) (2.12)fulfils the condition DR = I and therefore it is a right inverse of D . In the above formulathe floor brackets ⌊·⌋ stand for the integer value function of its argument. Then, let usdefine the operator F by formula F f ( x ) = f ( x − j xh k h ) , (2.13)for any x ∈ R . Since F f ( x + h ) = F f ( x ), for any x ∈ R , the function F f is h -periodic,i.e. F f ∈ Z D . On the other hand, for any function f ∈ Z D , there is F f = f . Hence, theoperator F is a projection of X onto Z D and therefore it is an initial operator inducedby D = D h . Moreover, one can check the property (2.6) which means that F is theinitial operator corresponding to R given by (2.12). By using formula (2.1), the family R D is fully determined by the above operator R , then with the help of (2.6) we obtainthe family F D . In this section we briefly recall the main elements of quantum calculus but more detailedstudy of the topic, motivation and many properties reflecting the analogies with theusual differential calculus the reader will find in [5]. For the history of q-calculus, itsrelation to other mathematical and physical areas and the imposing list of referenceswe recommend [1]. 4et f : R → R be an arbitrary function and consider the well known differencequotient f ( x ) − f ( x ) x − x , (3.1)for some x = x . The limit of the last expression when x → x , if it exists, defines thederivative of f at x . Now, if we take x = x + h for a fixed h = 0 or x = qx for afixed q = 1 and do not take the corresponding limit, we enter the so-called quantum h -or q -calculus. For any f : R → R one defines its h -differential d h fd h f ( x ) = f ( x + h ) − f ( x ) , (3.2)and its q -differential δ q f δ q f ( x ) = f ( qx ) − f ( x ) . (3.3)In particular, for the identity mapping e defined on R , i.e. e ( x ) ≡ x , we have d h e ( x ) = h and δ q e ( x ) = ( q − x . Quite commonly the simplified notation is used, i.e. d h e ( x ) ≡ d h x and δ q e ( x ) = δ q x . In applications, the two versions of quantum calculus (i.e. h - or q -calculus) are considered separately, which allows one to denote both differentials by thesame symbol, i.e. one can write d h or d q (instead δ q ) and recognize them from context.The above two symbols d h , δ q can be viewed as the linear operators d h : f d h f and δ q : f δ q f defined on some R -algebra A of real functions. However, the algebra A should be invariant w.r.t. the h- or q-shifts, i.e. functions x f ( x + h ) or x f ( qx )should be the elements of A , for any f ∈ A .One can easily verify the following Leibniz product rules d h ( f g )( x ) = d h ( f )( x ) g ( x + h ) + f ( x ) d h ( g )( x ) , (3.4)and similarly δ q ( f g )( x ) = δ q ( f )( x ) g ( qx ) + f ( x ) δ q ( g )( x ) . (3.5)The above Leibniz formulae define the corresponding classes ( A -modules) of lineardifference-like operators, defined on some R -algebra A of functions.Evidently, the above product rules are also fulfilled by operators D h and ∆ q , calledthe quantum derivatives and defined as D h ( f )( x ) = d h f ( x ) d h e ( x ) ≡ d h f ( x ) d h x (3.6)and similarly ∆ q ( f )( x ) = δ q f ( x ) δ q e ( x ) ≡ δ q f ( x ) δ q x . (3.7) Remark:
Since δ q e (0) = 0, the expression ∆ q ( f )( x ) is not defined at x = 0 unless f ′ (0) does exist. Therefore, the q -calculus can be developed in algebras A of functionsdefined on R \ { } or in algebras of functions defined on R and differentiable at x = 0.5n h -calculus, an h -antiderivative of a function f : R → R is defined to be anyfunction g : R → R such that D h g ( x ) = f ( x ), for any x ∈ R . The family of all h -antiderivatives of a given function f is called the indefinite h -integral and is denotedby Z f ( x ) d h x . (3.8)Then, the definite h -integral is defined by formula b Z a f ( x ) d h x = h ( f ( a ) + f ( a + h ) + . . . + f ( b − h )) if a < b if a = b , − h ( f ( b ) + f ( b + h ) + . . . + f ( a − h )) if a > b (3.9)for any a, b ∈ R , such that a and b differ by an integer multiple of h .Concerning q -calculus, in this paper we shall assume q ∈ (0 , ∪ (1 , ∞ ). Thisrestriction follows from the physical motivation that the two quantum parameters areusually related by q = e h . The last exponential relation transforms the real line R onto R + = (0 , ∞ ). Consequently, the h -calculus for functions defined on R and the q -calculus for functions defined on R + can be unified within a more general framework(generalized quantum calculus). A q -antiderivative of a function f is said to be anyfunction g such that ∆ q g ( x ) = f ( x ). A special q -antiderivative, the so-called Jacksonintegral of f , is formally derived in [5] as the geometric series expansion g ( x ) = (1 − q ) x ∞ X m =0 q m f ( xq m ) . (3.10)Then, formula (3.10) is used to define b Z f ( x ) d q x = (1 − q ) b ∞ X m =0 q m f ( bq m ) , (3.11)and finally define the definite q -integral [4] b Z a f ( x ) d q x = b Z f ( x ) d q x − a Z f ( x ) d q x , (3.12)for 0 < a < b . Since formula (3.10) has been derived formally, one needs to examinethe conditions when it converges to a q -antiderivative. Within the algebraic analysisframework, used in this paper, we construct many q -antiderivatives which are finitesums and no condition has to be examined to justify their convergency. However, theabove Jackson integral can be recovered in this approach provided the correspondinginfinite expansion is convergent.At the end of this section let us briefly discuss the lack of symmetry one can noticeconcerning the product rules (3.4), (3.5). 6n the strength of formulae (3.4), (3.5), for any a, b ∈ R such that a + b = 1, onecan write the following combinations d h ( f g )( x ) = d h ( f )( x )( ag ( x ) + bg ( x + h )) + ( bf ( x ) + af ( x + h )) d h ( g )( x ) , (3.13)and analogously δ q ( f g )( x ) = δ q ( f )( x )( ag ( x ) + bg ( xq )) + ( bf ( x ) + af ( xq )) δ q ( g )( x ) . (3.14)If a = b , the above combined formulae (3.13), (3.14) are equivalent with (3.4) and (3.5),respectively, i.e. the corresponding classes of operators defined coincide. On the otherhand, for a = b = the corresponding symmetric product rule defines a larger class ofoperators, in general. However, for some algebras the symmetric rule can be equivalentwith its all non-symmetric counterparts. A non-trivial example of an algebra, for whichthe symmetric product rule implies all the other ones, is the algebra of polynomials A = R [ x ]. Indeed, assume h = 1, a = b = and consider the symmetric product rule D ( f g )( x ) = D ( f )( x ) · g ( x ) + g ( x + 1)2 + f ( x ) + f ( x + 1)2 · D ( g )( x ) , (3.15)for any f, g ∈ R [ x ]. One can prove that D = D (1 A ) · d , where d f ( x ) = f ( x + 1) − f ( x ). Hence, any operator D is proportional to the usual difference operator d andconsequently it fulfills (non-symmetric) formula (3.4). In turn, an algebra for which thesymmetric product rule (3.15) is weaker than any non-symmetric one is for examplethe R -algebra (of real functions) A = gen ( { e, z } ) generated by the identity e ( x ) = x and the integer valued function z ( x ) = ⌊ x ⌋ . In this section we present an approach to quantum integration within the algebraicanalysis framework [8].Let us construct the following right inverses R hs , s ∈ R . Namely, for h < R hs f ( x ) = −⌊ x − s − h ⌋ P m =1 hf ( x − mh ) x ∈ ( −∞ , s )0 for x ∈ [ s, s − h ) − ⌊ x − s − h ⌋− P m =0 hf ( x + mh ) x ∈ [ s − h, ∞ ) (4.1)and for h > R hs f ( x ) = − −⌊ x − sh ⌋− P m =0 hf ( x + mh ) x ∈ ( −∞ , s )0 for x ∈ [ s, s + h ) ⌊ x − sh ⌋ P m =1 hf ( x − mh ) x ∈ [ s + h, ∞ ) (4.2)7y a straightforward calculation one can show that D h R hs = I , for any h = 0 and s ∈ R .Then, on the strength of formula (2.5), the indefinite h -integral of a function f ∈ R R can be written as Z f ( x ) d h x = R hs f ( x ) + Z D h ≡ R h f ( x ) + Z D h , (4.3)where s ∈ R is an arbitrarily fixed index, e.g. s = 0, and the notation (3.8) was used.Define the operators F hs by F hs f ( x ) = f ( x − (cid:22) x − s | h | (cid:23) · | h | ) , (4.4)for any s ∈ R .One can verify that F hs are the initial operators induced by D h , for any s ∈ R .Indeed, the function F hs f is h -periodic, since F hs f ( x + h ) = f ( x + h − (cid:22) x + h − s | h | (cid:23) · | h | ) == f ( x + h − (cid:18)(cid:22) x − s | h | (cid:23) + h | h | (cid:19) · | h | ) = F hs f ( x ) . Moreover, for any h -periodic function f , we have the evident identity F hs f = f , whichproves that F hs is a surjective projection onto Z D h . Therefore F hs is an initial operatorinduced by D h , for any s ∈ R . One can also verify that the initial operators F hs correspond to (4.1) and (4.2), respectively.In turn, the definite h -integrals are defined in a general manner by using formula(2.8). Within this approach we obtain a large class of definite h -integrals, with theintegration limits being arbitrary (indices of) initial operators. Below we considerdefinite h -integrals determined by the (particular) initial operators F hs , for s ∈ R . As weshall see, these integrals can be used to obtain the ordinary h -definite integrals definedby formula (3.9). Namely, by formula (2.8), for any a, b ∈ R and the corresponding(particular) initial operators F ha , F hb we obtain I ba = F hb R h − F ha R h . (4.5)The concrete right inverse R h , for s = 0, is used above only for simplicity since theresult is independent of this choice. Assume h > f at x , i.e. I ba f ( x ) = R h f ( x − (cid:22) x − bh (cid:23) · h ) − R h f ( x − (cid:22) x − ah (cid:23) · h ) . (4.6)Then, the ordinary h -definite integrals, defined intuitively in [5], are obtained here asthe value of I ba f ( x ) at any point x = a + kh , k ∈ Z . Indeed, assume 0 < k ∈ Z and b = a + kh . Then, for x = a we obtain I ba f ( a ) = R h f ( a + kh ) − R h f ( a ) =8 k − X j =0 ( R h f ( a + ( j + 1) h ) − R h f ( a + jh )) = k − X j =0 hD h R h f ( a + jh ) == k − X j =0 hf ( a + jh ) = h ( f ( a ) + f ( a + h ) + . . . + f ( b − h )) . If a = b , the result is obviously I ba f ( a ) = 0. In turn, for a > b and a = b + kh , forsome 0 < k ∈ Z , we have I ba f ( b ) = R h f ( b ) − R h f ( b + kh ) == k − X j =0 ( R h f ( b + jh ) − R h f ( b + ( j + 1) h )) = − k − X j =0 hD h R h f ( b + jh ) = − k − X j =0 hf ( b + jh ) = − h ( f ( b ) + f ( b + h ) + . . . + f ( a − h )) . The above calculation demonstrates how the ordinary h -definite integrals, defined by(3.9), emerge from the algebraic analysis approach used here.Directly from definition of the initial operator concept and from (4.5), we concludethat I ba f ∈ Z D , i.e. it is an h -periodic function and I ba f ( x ) = I ba f ( a ), for any x ∈ a + h Z .Let us emphasize the conceptual difference between definitions (3.9) and (4.5). In h -calculus, by formula (3.9) one defines the definite integral to be a scalar-valued linearfunctional, while in the algebraic analysis approach the corresponding definite integralvalue is an h -periodic function (non-constant, in general). The above two formulationsof definite integrals are equivalent for functions defined on the domain a + h Z , for somefixed a, h ∈ R . Remark:
Imagine that an action functional of a physical system is defined as an h -integral of some lagrangian. Consequently, such an action is h -periodic and its h -periodicity can be viewed as a physical symmetry giving rise to a corresponding con-servation law.Concerning q -calculus, we shall work here with functions f defined on the domain(0 , + ∞ ) and q ∈ (0 , ∪ (1 , + ∞ ). By analogy with the above right inverse operators R hs we first construct the operators ρ qs , where s ∈ (0 , + ∞ ), being the (particular) rightinverses of δ q . Then, we define the corresponding (particular) right inverses P qs of ∆ q .Namely, for q ∈ (0 ,
1) we have ρ qs f ( x ) = −⌊ log q sx ⌋ P m =1 f ( xq − m ) x ∈ (0 , s )0 for x ∈ [ s, sq − ) , − ⌊ log q sx ⌋− P m =0 f ( xq m ) x ∈ [ sq − , ∞ ) (4.7)9nd for q ∈ (1 , ∞ ) we have ρ qs f ( x ) = − −⌊ log q xs ⌋− P m =0 f ( xq m ) x ∈ (0 , s )0 for x ∈ [ s, sq ) . ⌊ log q xs ⌋ P m =1 f ( xq − m ) x ∈ [ sq, ∞ ) (4.8)One can easily verify that δ q ρ qs = I , for any s ∈ (0 , + ∞ ). Now, to find the rightinverses P qs of the divided difference operator ∆ q , defined by (3.7), we can write∆ q = T − q ◦ δ q , (4.9)where T q is the invertible operator defined as T q f ( x ) = ( q − xf ( x ) , (4.10)and apply formula (2.2), i.e. P qs = ρ qs ◦ T q . For q ∈ (0 ,
1) the result is P qs f ( x ) = −⌊ log q sx ⌋ P m =1 ( q − xq − m f ( xq − m ) x ∈ (0 , s )0 for x ∈ [ s, sq − ) , − ⌊ log q sx ⌋− P m =0 ( q − xq m f ( xq m ) x ∈ [ sq − , ∞ ) (4.11)and for q ∈ (1 , ∞ ) there is P qs f ( x ) = − −⌊ log q xs ⌋− P m =0 ( q − xq m f ( xq m ) x ∈ (0 , s )0 for x ∈ [ s, sq ) . ⌊ log q xs ⌋ P m =1 ( q − xq − m f ( xq − m ) x ∈ [ sq, ∞ ) (4.12)Although a single right inverse operator can generate all the other ones by formula(2.1), the right inverses P qs can be used to reach certain q -antiderivative, the so-calledJackson integral, being an infinite series, derived formally in [5]. From this approachit becomes clear that Jackson integral is not the only q -antiderivative existing andeven though it is divergent for certain function f , we can still work with other q -antiderivatives of f , well defined by the finite sums, which are never threatened by adivergency problem.Namely, in the lower part of formula (4.11) we put s → Z f ( x ) d q x = (1 − q ) x ∞ X m =0 q m f ( xq m ) , (4.13)10or x ∈ (0 , + ∞ ).As a next step we formulate definite integrals in terms of algebraic analysis andcompare them with definite q -integrals originally defined in q -calculus. In analogy toformula (4.4) let us consider the operators G a defined by G a f ( x ) = f ( xq −⌊ log q xa ⌋ ) , (4.14)for any function f : (0 , ∞ ) → R and a ∈ (0 , ∞ ). Evidently, for a ∈ (0 , ∞ ), operators G a are surjective onto the family of all q-periodic functions defined on (0 , ∞ ). One canalso verify the property G a = G a , for any a ∈ (0 , ∞ ). Therefore the operators G a arethe initial operators induced by the operator ∆ q , for any a ∈ (0 , ∞ ).Now, according to formula (2.8), we obtain a q -definite integral determined by theinitial operators G a and G b I ba = G b P qs − G a P qs , (4.15)for any a, b, s ∈ (0 , + ∞ ) (the above result is independent of s ).In order to interpret formula (3.12) within this framework, for any a, b ∈ (0 , + ∞ ),we should take q ∈ (0 ,
1) and sufficiently big positive s for which a, b ∈ [ q − s +1 , + ∞ ),since the last interval corresponds with (0 , + ∞ ) when s → + ∞ . Assume a < b = aq k ,for some 0 > k ∈ Z and calculate I ba f ( a ) = G b P qs f ( a ) − G a P qs f ( a ) = P qs f ( aq −⌊ log q ab ⌋ ) − P qs f ( aq −⌊ log q aa ⌋ ) == P qs f ( aq k ) − P qs f ( a ) = (1 − q ) aq k ⌊ log q aqk ⌋− s X m =0 q m f ( aq k q m ) −− (1 − q ) aq k ⌊ log q a ⌋− s X m =0 q m f ( aq m ) = (1 − q ) a − X m = k q m f ( aq m ) . On the other hand, from formula (3.12) we obtain b Z a f ( x ) d q x = b Z f ( x ) d q x − a Z f ( x ) d q x == (1 − q ) b ∞ X m =0 q m f ( bq m ) − (1 − q ) a ∞ X m =0 q m f ( aq m ) == (1 − q ) a − X m = k q m f ( aq m ) , which coincides with the previous result. Let us notice that formula (3.12) defines q -definite integral provided the Jackson q -antiderivative is a convergent series. A simpleexample of a function f , for which such a formulation of a definite q -integral cannot11e applied is f ( x ) = x , for which Jackson q -antiderivative is evidently divergent. Butfortunately, according to formula (2.8), a definite integral depends only on the ini-tial operators and is completely independent of a particular choice of a right inverseused in the calculation. Therefore, divergency of Jackson integral merely means thatthis particular q -antiderivative cannot be used in the calculation of a given q -definiteintegral.Let us end this section with the example of a definite q -integral for the above men-tioned function f ( x ) = x , where we assume q ∈ (0 ,
1) and 0 < a < b = aq k , for somenegative k ∈ Z . We obtain b Z a x d q x = (1 − q ) a − X m = k q m aq m = (1 − q ) · ( − k ) = (1 − q ) log q ab . An interesting observation is that the above definite q -integral depends only on theratio of its limits a and b . The usual quantum calculus, i.e. h - or q -calculus [5], is based on very special differenceand divided difference operators. As one can easily notice, formulae (3.2), (3.3) can berealized for functions defined on an arbitrary set M while there arises a problem withformulae (3.6), (3.7) since the differences appeared in the corresponding denominatorsare undefined unless M is equipped with the usual algebraic structure. In order to avoidthat problem we propose here to study more general formulation of quantum calculusin a tension space ( M, θ ).Let M = ∅ and assume the following definition. Definition 5.1.
By a tension function on M we understand any function θ : M × M → R such that θ ( p , p ) + θ ( p , p ) = θ ( p , p ) , (5.1) for any p , p , p ∈ M . Directly from the above definition, we can prove that any tension function is skewsymmetric, i.e. for any p , p ∈ M there is θ ( p , p ) = − θ ( p , p ) . (5.2) Definition 5.2.
By a tension space we shall mean a pair ( M, θ ) , where M = ∅ and θ is a tension function on M . In this paper we shall assume that (
M, θ ) is a nontrivial tension space, i.e. thereexist points p, q ∈ M for which θ ( p, q ) = 0 . (5.3)12 emark: One can easily check that a linear combination of tension functions on M is a tension function again. Consequently, any family { θ j } j ∈ J of tension functions on M generates the linear space L = Lin ( { θ j } j ∈ J ), the so-called tension structure on M . Then, by a (multidimensional) tension space we can understand the pair ( M, L ).However, in this paper we consider only a tension space (
M, θ ) defined by a singletension function θ .With a tension function θ we shall associate the equivalence relation in M definedby the formula p ∼ q iff θ ( p, q ) = 0 . (5.4)Then the equivalence classes of this relation are the following[ p ] = { q ∈ M : θ ( p, q ) = 0 } . (5.5)One can easily check that the function ˆ θ given byˆ θ ([ p ] , [ q ]) = θ ( p, q ) , (5.6)for p, q ∈ M , is a well defined tension function on the quotient set ˆ M ≡ M/ ∼ . Thuswe have constructed the ”effective” tension space ( ˆ M , ˆ θ ) . On the quotient set ˆ M = M/ ∼ we have the natural linear ordering relation[ p ] (cid:22) [ q ] iff ˆ θ ([ p ] , [ q ]) ≤ . (5.7)We shall also write [ p ] ≺ [ q ] whenever [ p ] (cid:22) [ q ] and simultaneously [ p ] = [ q ].Then, there is a natural metric g θ defined on ˆ M by g θ ([ p ] , [ q ]) = | ˆ θ ([ p ] , [ q ]) | , (5.8)for any p, q ∈ M .In the sequel we will often use mappings θ q : M → R defined by θ q ( p ) = θ ( p, q ) , (5.9)for any p, q ∈ M . One can easily verify that θ q = θ q , whenever q ∼ q . Intuitively,the mapping θ q we can interpret as a potential function defined on M , associating ascalar potential θ q ( p ) with any point p ∈ M and such that θ q ( q ) = 0 at q ∈ M . Definition 5.3.
A mapping τ : M → M is said to be rightward θ -directed if [ p ] ≺ [ τ ( p ) ] , (5.10) and it is said to be leftward θ -directed if [ τ ( p ) ] ≺ [ p ] , (5.11) for any p ∈ M . We say that τ is a θ -directed mapping if it is either rightward orleftward θ -directed mapping. τ = id M and τ n = τ ◦ τ n − , for any n ∈ N . Proposition 5.4.
For any θ -directed mapping τ : M → M and any n ∈ N , thecomposition τ n has no fixed points, i.e. τ n ( p ) = p , (5.12) for p ∈ M . Proof:
Let τ be a rightward θ -directed mapping. Then we have inequalities θ ( τ ( p ) , p ) >
0, ... , θ ( τ n ( p ) , τ n − ( p )) >
0, for any n ∈ N and p ∈ M . Consequently, θ ( τ n ( p ) , p ) = θ ( τ n ( p ) , τ n − ( p )) + . . . + θ ( τ ( p ) , p ) > . Analogously, for a leftward θ -directed mapping we show that θ ( τ n ( p ) , p ) < n ∈ N and p ∈ M . (cid:3) Let us notice that condition (5.12) is not a consequence of the weaker assumptionthat θ ( τ ( p ) , p ) = 0, for any p ∈ M . In that case there would be τ ( p ) = p but notnecessarily τ n ( p ) = p , for any n ∈ N and p ∈ M . Definition 5.5.
We say that θ is homogeneous with respect to τ (shortly, τ -homogeneous)if there exists t ∈ R , the so-called τ -homogeneity coefficient, such that θ ( τ ( p ) , τ ( p )) = t · θ ( p , p ) , (5.13) for any p , p ∈ M . Proposition 5.6.
Let τ : M → M be a θ -directed mapping and θ be a τ -homogeneoustension function. Then, for the τ -homogeneity coefficient we get t > . Proof:
Suppose that t is a τ -homogeneity coefficient for some τ -homogeneous ten-sion function θ and assume that τ is a θ -directed mapping. Then θ ( τ ( p ) , τ ( p )) and θ ( τ ( p ) , p ) are of common sign and θ ( τ ( p ) , τ ( p )) = t · θ ( τ ( p ) , p ). Directly from Definition(5.3) we get t = 0. Hence we conclude that t > (cid:3) Proposition 5.7.
Let θ be τ -homogeneous and ∼ be the equivalence relation defined by(5.4). Then, we have the implication p ∼ q ⇒ τ ( p ) ∼ τ ( q ) (5.14) for any p, q ∈ M , or equivalently τ ([ p ]) ⊂ [ τ ( p ) ] , (5.15) for any p ∈ M . roof: Suppose that p ∼ q , i.e. θ ( p, q ) = 0. Then we have θ ( τ ( p ) , τ ( q )) == t · θ ( p, q ) = 0. (cid:3) In general, the inclusion (5.15) cannot be inverted, which can be confirmed by thefollowing
Example:
Assume M = R × [0 , + ∞ ), θ (( x , y ) , ( x , y )) = x − x and τ ( x, y ) = ( x +1 , y +1). Then we obtain τ ([( x, y )]) = { x +1 }× [1 , + ∞ ) and [ τ ( x, y )] = { x +1 }× [0 , + ∞ ),i.e. τ ([( x, y )]) [ τ ( x, y )]. Proposition 5.8.
Let θ be τ -homogeneous with the τ -homogeneity coefficient t = 0 .Then we have p ≁ q ⇒ τ ( p ) ≁ τ ( q ) , (5.16) for any p, q ∈ M . Proof: θ ( τ ( p ) , τ ( q )) = t · θ ( p, q ) = 0, whenever p ≁ q . (cid:3) Corollary 5.9.
Let θ be τ -homogeneous with a τ -homogeneity coefficient t . Assumethat p ≁ q and τ ( p ) ∼ τ ( q ) , for some p , q ∈ M . Then t = 0 and consequently τ ( p ) ∼ τ ( q ) , or equivalently [ τ ( p ) ] = τ ( M ) , for any p, q ∈ M . τ, σ )-calculus Let σ, τ : M → M be two commuting bijections and assume A ⊂ R M to be a σ ∗ , τ ∗ -invariant R -algebra, i.e. σ ∗ A , τ ∗ A ⊂ A . Definition 6.1.
By the ( τ, σ ) -quantum differential we mean the mapping d τ,σ : A → A given by d τ,σ f ( p ) = f ( τ ( p )) − f ( σ ( p )) , (6.1) for p ∈ M . One can easily check that the quantum differential d τ,σ is a linear operator and itfulfills the following Leibniz product rule d τ,σ ( f · g )( p ) = d τ,σ f ( p ) · g ( τ ( p )) + f ( σ ( p )) · d τ,σ g ( p ) , (6.2)for any functions f, g ∈ A and p ∈ M . Definition 6.2.
By a ( τ, σ ) -quantum derivation we shall mean any linear operator δ : A → A that fulfills formula (6.2).
Since the elements f, g ∈ A commute, the following combinations are also fulfilled δ ( f · g )( p ) = [ af ( σ ( p )) + bf ( τ ( p ))] · δg ( p ) + δf ( p ) · [ bg ( σ ( p )) + ag ( τ ( p ))] , (6.3)where a, b ∈ R are coefficients such that a + b = 1. If a = b , formula (6.3) is equivalentwith (6.2). In turn, when a = b = , formula (6.3) becomes symmetric δ ( f · g )( p ) = H ( f )( p ) · δg ( p ) + δf ( p ) · H ( g )( p ) , (6.4)15here H ( f )( p ) = f ( σ ( p ))+ f ( τ ( p ))2 . In general, formula (6.4) is weaker than (6.2) butthere exist algebras A for which both formulae are equivalent, i.e. they define the same A -module of linear operators (e.g. A = R [ x ], compare the corresponding comment inSection 3). Remark:
The mapping H : A → A , defined above, is linear and preserving the unity1 A but in general it is not an algebra homomorphism. The last defect is precisely thereason why operators defined by (6.4) are not differential operators.Now, we assume [ σ ( p )] τ,σ ≺ [ τ ( p )] τ,σ , (6.5)for any p ∈ M , and define the quantum ( τ, σ )-derivative operator in a tension space( M, θ ). Definition 6.3.
By the ( τ, σ ) -quantum derivative we shall mean the mapping D τ,σ : A → A given by D τ,σ f ( p ) = d τ,σ f ( p ) θ ( τ ( p ) , σ ( p )) ≡ d τ,σ f ( p ) d τ,σ θ q ( p ) , (6.6) for any f ∈ A , independently of q ∈ M . The assumption (6.5) prevents formula (6.6) from zero-valued denominator. How-ever, owing to the evident symmetry D τ,σ = D σ,τ , all properties associated with theoperator D τ,σ remain unchanged if the direction of (6.5) is reversed. Equivalently,relation (6.5) can be formulated as[ p ] τ,σ ≺ [ τ σ − ( p )] τ,σ , (6.7)for any p ∈ M . By Definition (5.3) it means that τ σ − is a rightward θ -directedbijection. Indeed, it is enough to replace p by σ − ( p ) in formula (6.5) and obtain (6.7).Evidently, the quantum derivative D τ,σ fulfills the product rule (6.2).In order to formulate the idea of quantum integration (or the Taylor interpolationpolynomial) we shall need the right inverse operators defined for the above quantumdifferential (6.1) and quantum derivative (6.6).The following definition will play an important role in our further analysis. Definition 6.4.
We say that a family of subsets M k ⊂ M , k ∈ Z , is a ( τ, σ ) -partitionof M = ∅ if1) S k ∈ Z M k = M ,2) M k ∩ M k = ∅ , for any k = k ,3) τ σ − : M k → M k +1 is a bijective mapping, for any k ∈ Z . To shorten our notation, the circle symbol ” ◦ ” is omitted for the composition ofmappings above and later on. 16 roposition 6.5. If M k ⊂ M , k ∈ Z , is a ( τ, σ ) -partition of M = ∅ , then M = ∅ andthe composed mapping ( τ σ − ) m , for any m ∈ Z , has no fixed points. Proof:
Suppose M = ∅ . Then, by condition (3) we get M k = ∅ , for all k ∈ Z , whichcontradicts condition (1). In turn, let ( τ σ − ) m ( p ) = p for some p ∈ M k and m = 0.Then by condition (3) we obtain p = ( τ σ − ) m ( p ) ∈ M k ∩ M k + m which contradictscondition (2). (cid:3) With a given ( τ, σ )-partition of M we associate the following integer-valued function ⌊·⌋ τ,σ : M → Z , defined by ⌊ p ⌋ τ,σ = k iff p ∈ M k , (6.8)for any k ∈ Z . We shall omit the indices and write ⌊·⌋ whenever τ and σ are fixed.Automatically, for any p ∈ M , from the above formula we conclude p ∈ M ⌊ p ⌋ . (6.9) Proposition 6.6.
For any p ∈ M there is ⌊ τ σ − ( p ) ⌋ = ⌊ p ⌋ + 1 . (6.10) Proof.
Let ⌊ p ⌋ = k , i.e. p ∈ M k for some k ∈ Z . Then, τ σ − ( p ) ∈ M k +1 andconsequently ⌊ τ σ − ( p ) ⌋ = ⌊ p ⌋ + 1. (cid:3) Remark:
Since σ is a bijection, we can always replace p by σ ( p ) and repeat formula(6.10) in the following equivalent form ⌊ τ ( p ) ⌋ = ⌊ σ ( p ) ⌋ + 1 . (6.11) Definition 6.7.
By a ( τ, σ ) -partition function (partition function, for short) of M wemean any integer valued function λ : M → Z such that λ ( τ σ − ( p )) = λ ( p ) + 1 , (6.12) for any p ∈ M . One can easily prove the following
Proposition 6.8.
For any ( τ, σ ) -partition function λ of M , the family of sets M k = λ − ( k ) ⊂ M , (6.13) where k ∈ Z , is a ( τ, σ ) -partition of M . In the sequel, we say that the ( τ, σ )-partition of M given by formula (6.13) isdetermined by λ . Naturally, for a given ( τ, σ )-partition of M determined by λ we have ⌊ p ⌋ = λ ( p ) , (6.14)for any p ∈ M . With any ( τ, σ )-partition of M we associate the following17 roposition 6.9. A right inverse of the ( τ, σ ) -differential d τ,σ is given by the formula r τ,σ f ( p ) = − −⌊ p ⌋− P m =0 f ( τ m σ − m − ( p )) if ⌊ p ⌋ ≤ − if ⌊ p ⌋ = 0 ⌊ p ⌋ P m =1 f ( τ − m σ m − ( p )) if ⌊ p ⌋ ≥ . (6.15) Proof:
For ⌊ σ ( p ) ⌋ = k ≤ − ⌊ τ ( p ) ⌋ = k + 1 ≤ −
1. Then d τ,σ r τ,σ f ( p ) = r τ,σ f ( τ ( p )) − r τ,σ f ( σ ( p )) = − − k − X m =0 f ( τ m +1 σ − ( m +1) ( p ))++ − k − X m =0 f ( τ m σ − m ( p )) = − − k − X m =1 f ( τ m σ − m ( p )) + − k − X m =0 f ( τ m σ − m ( p )) = f ( p ) . For ⌊ σ ( p ) ⌋ = − ⌊ τ ( p ) ⌋ = 0. Then d τ,σ r τ,σ f ( p ) = 0 − r τ,σ f ( σ ( p )) = − ( − − X m =0 f ( τ m σ − m ( p )) = f ( p ) . For [ σ ( p ) ] τ,σ = k ≥ τ ( p ) ] τ,σ = k + 1 ≥
1. Then d τ,σ r τ,σ f ( p ) = r τ,σ f ( τ ( p )) − r τ,σ f ( σ ( p )) = k +1 X m =1 f ( τ − ( m − σ m − ( p )) −− k X m =1 f ( τ − m σ m ( p )) = k X m =0 f ( τ − m σ m ( p )) − k X m =1 f ( τ − m σ m ( p )) = f ( p ) . (cid:3) Next, by using formula (2.2) we can find the right inverse R τ,σ of the ( τ, σ )-derivative D τ,σ . Proposition 6.10.
A right inverse R τ,σ of the ( τ, σ ) -derivative D τ,σ is given by R τ,σ f ( p ) = − −⌊ p ⌋− P m =0 θ ( τ m +1 σ − m − ( p ) , τ m σ − m ( p )) f ( τ m σ − m − ( p )) if ⌊ p ⌋ ≤ −
10 if ⌊ p ⌋ = 0 ⌊ p ⌋ P m =1 θ ( τ − m +1 σ m − ( p ) , τ − m σ m ( p )) f ( τ − m σ m − ( p )) if ⌊ p ⌋ ≥ . (6.16) Proof:
Let us define the operator T τ,σ by formula T τ,σ f ( p ) = θ ( τ ( p ) , σ ( p )) · f ( p ) . (6.17)18hus we write D τ,σ = T − τ,σ ◦ d τ,σ and using formula (2.2) we obtain R τ,σ = r τ,σ ◦ T τ,σ . (6.18)Finally, we apply (6.15) and after some calculations obtain formula (6.16). (cid:3) Remark:
Let us notice that the tension function θ makes no explicit contribution onthe construction of the right inverse r τ,σ . The only connection between r τ,σ and θ isthrough formula (6.5) which means that τ σ − is a θ -directed mapping. On the otherhand, by formula (6.18), the right inverse R τ,σ depends on θ explicitly.Now, let us determine the initial operator F τ,σ induced by D τ,σ and correspondingwith R τ,σ . Since F τ,σ = I − R τ,σ D τ,σ = I − r τ,σ d τ,σ , (6.19)it becomes simultaneously the initial operator for d τ,σ corresponding with r τ,σ . Proposition 6.11.
The initial operator F τ,σ induced by D τ,σ and corresponding with R τ,σ is given by the formula F τ,σ f ( p ) = f (( τ σ − ) −⌊ p ⌋ ( p )) . (6.20) Proof:
For ⌊ p ⌋ ≤ −
1, we have r τ,σ d τ,σ f ( p ) = − −⌊ p ⌋− X m =0 f ( τ τ m σ − m − ( p )) + −⌊ p ⌋− X m =0 f ( στ m σ − m − ( p )) == − −⌊ p ⌋ X m =1 f ( τ m σ − m ( p )) + −⌊ p ⌋− X m =0 f ( τ m σ − m ( p )) = f ( p ) − f (( τ σ − ) −⌊ p ⌋ ( p ))If ⌊ p ⌋ = 0, there is r τ,σ d τ,σ f ( p ) = 0. For ⌊ p ⌋ ≥
1, we have r τ,σ d τ,σ f ( p ) = ⌊ p ⌋ X m =1 f ( τ τ − m σ m − ( p )) − ⌊ p ⌋ X m =1 f ( στ − m σ m − ( p )) == ⌊ p ⌋ X m =0 f ( τ − m σ m ( p )) − ⌊ p ⌋ X m =1 f ( τ − m σ m ( p )) = f ( p ) − f (( τ σ − ) −⌊ p ⌋ ( p )) . (cid:3) If a ( τ, σ )-partition of M is determined by a partition function λ , we shall index theright inverses or initial operators by λ , i.e. we shall write r λ ≡ r τ,σ , R λ ≡ R τ,σ and F λ ≡ F τ,σ .If λ and λ are two ( τ, σ )-partition functions of M and R is an arbitrary right inverseof the ( τ, σ )-quantum derivative D τ,σ , according to formula (2.8) the correspondingdefinite ( τ, σ )-integrals are given by I λ λ = F λ R − F λ R . (6.21)19 xample:
Let (
M, θ ) be a tension space, D τ,σ be a quantum ( τ, σ )-derivation ofan algebra A ⊂ R M and η be another tension function on M such that the bijectivemapping τ σ − is η -directed. Additionally, assume that η is τ - and σ -homogeneous withboth homogeneity coefficients equal 1. Then, for any point s ∈ M , the function λ s defined by λ s ( p ) = (cid:22) η ( p, s ) η ( τ σ − ( s ) , s ) (cid:23) (6.22)is a ( τ, σ )-partition function. In particular, when M = R , τ ( x ) = x + h , σ ( x ) = x , η ( x, y ) = x − y , for x, y, h, s ∈ R , h >
0, we get the partition function λ s ( x ) = ⌊ x − sh ⌋ used in h -calculus (see Section 3). Hence we obtain the right inverse operators as wellas the initial operators F λ s corresponding with λ s . Consequently, the ( τ, σ )-definiteintegral, for a, b ∈ R , is given as I ba = F λ b R − F λ a R , (6.23)where R is an arbitrary right inverse of D τ,σ .At the end, let us make a comment about higher order ( τ, σ )-difference-like oper-ators. Let M = ∅ and A n ⊂ R M n be a sequence of R -algebras, for n ∈ N , and let A = A . Assume ( p , . . . , p n ) ∈ M n and define µ p ,...,p n = { f ∈ A n : f ( p , . . . , p n ) = 0 } ,the ideal of A n , for any n ∈ N . Definition 6.12.
A linear mapping
Λ : A → A n , for a fixed n ∈ N , is said to be ofpre-order n if Λ( µ p · . . . · µ p n ) ⊂ µ p ,...,p n . For example, let us explicitely formulate the rule fulfilled by an operator Λ of pre-order n = 1. From the above definition we obtainΛ(( f − f ( p ))(( f − f ( p )) ∈ µ p ,p , (6.24)which means that Λ(( f − f ( p ))(( f − f ( p ))( p , p ) = 0 . (6.25)Formula (6.25) can be written equivalently asΛ( f f )( p , p ) − f ( p )Λ( f )( p , p ) − f ( p )Λ( f )( p , p )++ f ( p ) f ( p )Λ(1)( p , p ) = 0 . (6.26)Now, let us define δ : A → A by formula δ ( f )( p ) = Λ( f )( τ ( p ) , σ ( p )) , (6.27)for any p ∈ M . Directly from formula (6.26) we obtain δ ( f f )( p ) − f ( τ ( p )) δ ( f )( p ) − f ( σ ( p )) δ ( f )( p )++ f ( σ ( p )) f ( τ ( p )) δ (1)( p ) = 0 . (6.28) Definition 6.13.
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