Nonlocal Operational Calculi for Dunkl Operators

The one-dimensional Dunkl operator $D_k$ with a non-negative parameter $k$, is considered under an arbitrary nonlocal boundary value condition. The right inverse operator of $D_k$, satisfying this condition is studied. An operational calculus of Mikusinski type is developed. In the frames of this operational calculi an extension of the Heaviside algorithm for solution of nonlocal Cauchy boundary value problems for Dunkl functional-differential equations $P(D_k)u=f$ with a given polynomial $P$ is proposed. The solution of these equations in mean-periodic functions reduces to such problems. Necessary and sufficient condition for existence of unique solution in mean-periodic functions is found.

Here the one-dimensional Dunkl x , k ≥ 0, in C 1 (R) under a nonlocal boundary value condition Φ{f } = 0 with an arbitrary non-zero linear functional Φ in C(R) are considered. The right inverse operators L k of D k , defined by D k L k f = f and Φ{L k f } = 0 are studied. To this end, the elements of corresponding operational calculi are developed. A convolution product f * g on C(R), such that L k f = {1} * f , is found. Further, the convolution algebra (C(R), * ) is extended to its ring M k of the multipliers. (C(R), * ) may be conceived as a part of M k due to the embedding f ֒→ f * . The ring M k of multiplier fractions A B , such that A, B ∈ M k and B being non-divisor of zero in the operator multiplication, is constructed.
A Heaviside algorithm for effective solution of nonlocal Cauchy boundary value problems for Dunkl functional-differential equations P (D k )u = f with polynomials P is developed. The solution of these equations in mean-periodic functions reduces to such problems. Necessary and sufficient condition for existence of unique solution in mean-periodic functions is found.
The operational calculus, developed here, is a generalization of the nonlocal operational calculus for D 0 = d dx (see Dimovski [6]). Some background material about the Dunkl operators is taken from our previous paper [7] without proofs. Lemma 1. The right inverse operator Λ k of D k , defined by the initial condition Λ k f (0) = 0 has the form where f e and f o are the even and the odd parts of f , respectively.
The proof is a matter of a simple check (see [7, p. 198]). In the general case, an arbitrary right inverse operator L k of D k has a representation of the form In order L k to be a linear operator, the additive constant C should depend on f and to be a linear functional Ψ{f } in C(R). Hence, an arbitrary linear right inverse operator L k of D k in C(R) has the form with a linear functional Ψ in C(R).
According to the general theory of right invertible operators (Bittner [2], Przeworska-Rolewicz [12]), an important characteristic of L k is its initial projector It maps C 1 (R) onto ker D k = C, i.e. it is a linear functional Φ on C 1 (R). This identity written in the form will be used later. Expressing Φ by Ψ, we obtain Let us note that Φ{1} = 1, which expresses the projector property of F . Considering the right inverse operator L k of D k , it is more convenient to look on L k f = y as the solution of an elementary boundary value problem of the form assuming that Φ is a given linear functional on C(R) with Φ{1} = 1. This restriction of the class of right inverse operators L k of D k is adequate when we are to consider nonlocal Cauchy problems for Dunkl equations.
Theorem 1. Let Φ : C(R) → C be a linear functional, such that Φ{1} = 1. Then the right inverse operator L k of D k , defined by the boundary value condition Φ{L k f } = 0 has the form The proof follows immediately from Lemma 1 and the condition Φ{1} = 1.

Definition 1. The polynomials
are said to be Dunkl-Appell polynomials.
The check is immediate. Similar polynomials are introduced implicitly by M. Rösler and M. Voit [14, p. 346]. where This formula is an analogue of the particular case of the Taylor formula known as the Maclaurin formula.
Proof . Delsarte [4], Bittner [2], and Przeworska-Rolewicz [12] give variants of the Taylor formula for right invertible operators in linear spaces. In our case (5) can be written as where I is the identity operator and F = I − L k D k . In functional form the above identity takes the form where the initial projector F of L k (1) is the linear functional Φ: F projects the space C(R) onto the space C of the constants. Hence which is the Taylor formula (5).
2 Convolutional products for the right inverses L k of D k In Dunkl [8,Theorem 5.1] the similarity operator is found, which transforms the differentiation operator D = d dx into D k : Usually this operator is called intertwining operator. The constant b k is chosen to ensure that The problem of inverting the Dunkl intertwining operator V k is discussed by several authors, see e.g. Trimèche [15], Betankor, Sifi, Trimèche [1], but we will use the explicit formulae from Ben Salem and Kallel [3, p. 159]. Denoting of V k has the following representations: (i) If k = n + r is non-integer with integer part n and r ∈ (0, 1), then (ii) If k is a non-negative integer, then V k transforms C(R) into a proper subspace C k = V k (C(R)) of it. V k is a similarity from a right inverse operator Λ of D 0 = d dx to L k . In order to specify the operator Λ let us define the linear functional in C k . Then define Λ : C k → C k to be the solution y = Λ f of the elementary boundary value problem This solution has the form Lemma 4. The following similarity relation holds Proof . Applying V k to the defining equation D(Λ f ) = f , one obtains In fact, the boundary value condition Φ{Λ f } = 0 can be written as Φ{V k Λ f } = 0. Hence u = V k Λ f is the solution of the boundary value problem D k u = f , Φ{u} = 0, i.e. u = L k f . Therefore The similarity relation (4) allows to introduce a convolution structure * : C(R) × C(R) → C(R), such that L k to be the convolution operator L k = {1} * in C(R).
The operator Λ is defined not only in C k , but in the whole space C(R). This allows to introduce a convolution structure * : C(R) × C(R) → C(R).

Lemma 5. The operation
is a bilinear, commutative and associative operation in It satisfies the boundary value condition Φ{ f * g} = 0 for arbitrary f and g in C(R).
The proof of the assertion that f * g is an inner operation in C k follows directly from the explicit inversion formula for V k (see Xu [16] or Ben Salem and Kallel [3, Theorem 1.1]). In Dimovski [5, p. 52] it is proved that (6) is a bilinear, commutative and associative operation in C(R), and hence in C k = V k (C(R)). The second relation (7) is obvious. The proof of Φ{ f * g} = 0 is also elementary (see Dimovski [5, p. 54]).

Theorem 2. The operation
where n is the integer part of k, is a convolution of L k in C(R) such that and the boundary value condition Φ{f * g} = 0 is satisfied for arbitrary f and g in C(R).
Proof . The assertion of the theorem follows from Lemmas 5 and 4 and a general theorem of Dimovski [5, Theorem 1.3.6, p. 26]. This convolution is introduced in Dimovski, Hristov and Sifi [7].

Remark 1. The convolution (8) reduces to
From (9) and Definition 1 it follows that where A k,N is the Dunkl-Appell polynomial of degree exactly N . This allows also to state the Taylor formula (5) with remainder term in the Cauchy form: 3 The ring of multipliers of the convolutional algebra (C(R), * ) The convolutional algebras (C(R), * ) with convolution product (8), are annihilators-free (or algebras without order in the terminology of Larsen [9, p. 13]). This means that in each of these algebras f * g = 0, ∀ g ∈ C(R), implies f = 0.
for arbitrary f, g ∈ C(R).
As it is shown in Larsen [9], it is not necessary to assume neither that A is a linear operator, nor that it is continuous in C(R). These properties of the multipliers follow automatically from (10). Something more, a general result of Larsen [9, p. 13] implies The simplest multipliers of (C(R), * ) are the numerical operators [α] for α ∈ C, defined by and the convolutional operators f * for f ∈ C(R), defined by Further we need the following characterization result for the multipliers of (C(R), * ):

it admits a representation of the form
where the function m = A{1} is such that m * f ∈ C 1 (R) for all f ∈ C(R).
Proof . Let A : C(R) → C(R) be a multiplier of (C(R), * ). The operator L k f = {1} * f is also a multiplier. Then, according to Theorem 3,

The identity
. It remains to apply D k to (12) in order to obtain (11). Then But m * f = L k D k (m * f ) due to formula (2) since Φ(m * f ) = 0 by Theorem 2. Then Hence A is a multiplier of the convolution algebra (C(R), * ).
The specification of the function m = A{1} is, in general, a nontrivial problem even in the case of the simplest Dunkl operator D 0 = d dx (the usual differentiation). This could be confirmed by the following two examples: , then m is a continuous function of locally bounded variation, i.e. m ∈ BV ∩ C(R) (see Dimovski [5, p. 26]).

Nonlocal operational calculi for D k
Our aim here is to develop a direct operational calculus for solution of the following nonlocal Cauchy problem for the operator D k : Solve the equation P (D k )u = f with a polynomial P and a given f ∈ C(R) under the boundary value conditions Φ{D j k u} = α j , j = 0, 1, 2, . . . , deg P − 1, where α j are given constants and Φ is a nonzero linear functional on C(R).
This is a special case of the problems considered by R. Bittner [2] and D. Przeworska-Rolewicz [12] for an arbitrary right invertible operator D instead of D k .
Our intention here is to propose constructive results and to obtain an explicit solution of the boundary value problems considered. This is done by means of an operational calculus essential part of which is an extension of the Heaviside algorithm.
This operational calculus is developed using a direct algebraic approach based on the convolution (8). Instead of Mikusiński's method [10] of convolutional fractions f g , we follow an alternative approach of multiplier fractions A B , where A and B are multipliers of the convolutional algebra (C(R), * ) and B is a non-divisor of zero in the operator multiplication.
Let us consider the ring M k of the multipliers of the convolutional algebra (C(R), * ). The correspondence α → [α] is an embedding of C into M k . The correspondence f → f * is an embedding of (C(R), * ) in M k . Hence, we may consider C and C(R) as parts of M k .
M k is a commutative ring (Theorem 3). The subset N k of M k , consisting of the non-zero non-divisors of zero with respect to the operator multiplication in M k , is nonempty. Indeed, at least the identity operator I and the right inverse L k of D k belong to N k . In addition, N k is a multiplicative subset, i.e. if A, B ∈ N k , then AB ∈ N k .
Consider the Cartesian product and introduce the equivalence relation M k may be considered both as an extension of the field C of the complex numbers and of the ring (C(R), * ). Formally, this is seen by the embeddings α ֒→ [α] I and f ֒→ f * I .
In the sequel we denote the identity operator I simply by 1. The multiplication operation of the two elements p and q in M k will be denoted simply by pq. Therefore, instead of f * g we will write f g.
For our aims the most important elements of M k are The fraction S k with the identity operator as numerator and with L k as denominator will be called algebraic Dunkl operator. Its relation to the ordinary Dunkl operator D k is given by the following theorem: Then Note that identity (14) should be interpreted as where (D k f ) * and (f * ) are to be understood as convolution operators and [Φ{f }] as the numerical operator determined by the number Φ{f }. S k is neither convolutional nor numerical operator, but an element of M k .
Proof . In Section 1 (equality (2)) we have seen that where Φ{f } is the corresponding constant function {Φ{f }}. Considered as an operator identity, this can be written as ( It remains to multiply by S k to obtain (14).
Relation (14) may be characterized as the basic formula of our operational calculus. Using it repeatedly, we obtain Remark 2. The last formula is equivalent to the Taylor formula (5) in Section 1.
The simplest nonlocal Cauchy problem for D k , determined by a linear functional Φ in C(R) concerns the functional-differential equation with the boundary condition Φ{u} = 0.
It is known that the solution of the homogeneous equation under the initial condition u(0) = 1 is We introduce the Dunkl indicatrix of the functional Φ as the following entire function of exponential type: is the generating function of the Dunkl-Appell polynomials system, i.e.
Here we will skip the simple proof. The linear operator L k,λ defined as the solution u(x) = L k,λ f (x) of the nonlocal Cauchy boundary value problem is said to be the resolvent operator of the Dunkl operator under the boundary value condition Φ{u} = 0.
Theorem 6. The resolvent operator L k,λ admits the convolutional representation Proof . We will use the formula which is true under the assumption f ∈ C 1 (R). It follows from a more general result of Dimovski [5, Theorem 1.38], but in our case it can be verified directly. It gives Hence u = {l k (λ, x) * f (x)} satisfies the equation D k u − λu = f . It remains to verify the boundary value condition Φ{u} = 0. But it follows from the basic property Φ{f * g} = 0 of the convolution (Theorem 2).
It is easy to find the solution of our problem in M k . Using the basic formula of the operational calculus (see Theorem 5), we have D k u = S k u since Φ{u} = 0, and then In order to write the solution we must be sure that S k − λ is non-divisor of zero.
Proof . Let S k − λ be a divisor of zero in M k . Then there exists a multiplier fraction A B such that A = 0 and Multiplying by L k we get Since Φ(L k v) = 0 by the definition of L k (Section 1), then Φ{v} = 0. Applying D k , we get D k v − λv = 0, Φ{v} = 0. According to Ben Salem and Kallel [3], all the non-zero solutions of D k v − λv = 0 are v = C(j k− 1 2 (iλx) + λx 2k+1 j k+ 1 2 (iλx)) with a constant C = 0. The boundary value condition Φ{v} = 0 is equivalent to E k (λ) = 0.
Conversely, if E k (λ) = 0, then there exists a solution v = 0 of the eigenvalue problem D k v − λv = 0, Φ{v} = 0. For this v we have Proof . We have seen that But for the solution u = L k,λ f of the boundary value problem D k u − λu = f , Φ{u} = 0, in the case E k (λ) = 0 we found Since the convolution * is annihilators-free, then (16) follows from the identity

Heaviside algorithm for solving nonlocal Cauchy problems for Dunkl operators
Now we are to apply the elements of the operational calculus developed in the previous section to effective solution of nonlocal Cauchy boundary value problems of the form with given α j ∈ C.
To this end we extend the classical Heaviside algorithm, which is intended for solving initial value problems for ordinary linear differential equations with constant coefficients to the case of Dunkl functional-differential equations.
The extended Heaviside algorithm starts with the algebraization of problem (17). It reduces the problem to a single algebraic equation of the first degree in M k .
2) Represent each of the terms of the equation by the algebraic Dunkl operator S k . This is done by the formulae Thus we obtain the following equation in M k : 3) Verify if P (S k ) is a non-divisor of zero in M k by checking if E k (µ j ) = 0 for all j = 1, 2, . . . , s. 4) If P (S k ) is a non-divisor of zero, then write the solution u in M k : 5) Expand 1 P (S k ) and Q(S k ) P (S k ) into partial fractions: 6) Interpret the partial fractions as convolution operators λ=µ j * , l = 2, 3, . . . .

7) Write the convolutional representation
Example 3. Let P (λ) has only simple zeros µ 1 , µ 2 , . . . , µ m . Then Then the solution u takes the functional form The result of this section can be summarized in the following Remark 3. The term "nonlocal" should not be understood literally. The assertion of Theorem 8 is true also when Φ is a Dirac functional, i.e. Φ{f } = f (a) for a ∈ R. For us the most interesting is the case Φ{f } = f (0). Then E k (λ) ≡ 1 and from the theorem it follows that the initial value problem always has a unique solution. We will use this fact in the following section. The notion of mean-periodic function for the differentiation operator d dt , determined by a linear functional Φ in C(R), is introduced by J. Delsarte [4]: A function f ∈ C(R) is said to be mean-periodic with respect to the functional Φ if it satisfies identically the condition In order to define mean-periodic functions for the Dunkl operator D k we need to recall the definition of the Dunkl translation (shift) operators, introduced by M. Rösler [13] and later studied in M.A. Mourou and K. Trimèche [11]. They are a class of operators M : C(R) → C(R) commuting with D k in C 1 (R).
Definition 5. Let f ∈ C(R) and y ∈ R. Then (T y k f )(x) = u(x, y) ∈ C 1 (R 2 ) is the solution of the boundary value problem T y k is called the translation operator for the Dunkl operator D k . Such a solution exists for arbitrary f ∈ C(R) and it has the following explicit form (see e.g. [13,3]): As usually, the subscripts "e" and "o" denote correspondingly the even and the odd part of a function: As for h e (x, y, t) and h o (x, y, t), they denote respectively h e (x, y, t) = 1 − sign(xy) cos t, Lemma 9. The translation operators satisfy the following basic relations: Proofs can be found in various publications, in particular, in our paper [7]. A natural extension of the notion of mean-periodic function for the Dunkl operator is proposed by Ben Salem and Kallel [3]. Instead of (18) they use the condition to define mean-periodic function f for D k with respect to the functional Φ. Here T y k is the generalized translation operator just defined.
The space of mean-periodic functions for the Dunkl operator D k with respect to a given functional Φ will be denoted by P Φ . We skip the subscript k for sake of simplicity.
k L k f (x)} and use the commutation relation (21) from Lemma 9 Further we will be interested in the solvability of Dunkl differential-difference equations with a polynomial P in the space of the mean-periodic functions P Φ , defined by (22). We intend also to propose an algorithm for obtaining such solutions.
To this end we are to develop an operational calculus for D k in C(R) and to extend the Heaviside algorithm for it. The following result plays a basic role in the application of this algorithm for solution of Dunkl equations in mean-periodic functions.
Theorem 9. The class of mean-periodic functions P Φ is an ideal in the convolutional algebra (C(R), * ), i.e. if f ∈ P Φ and g ∈ C(R), then f * g ∈ P Φ .
From Lemma 10 it follows that L n+1 k f ∈ P Φ for n = 0, 1, 2, . . . , i.e. Since and then we can assert that Φ t T t k (P * f )(x) = 0 for any polynomial P . By an approximation argument it follows that Φ t T t k (g * f )(x) = 0 for arbitrary g ∈ C(R), i.e. that g * f ∈ P Φ .
Corollary 3. Let M : C(R) → C(R) be an arbitrary multiplier of the algebra (C(R), * ). Then M (P Φ ) ⊂ P Φ , i.e. the restriction of M to P Φ is an inner operator in P Φ .
Proof . Let f ∈ P Φ . According to Theorem 4, In the sequel we study the problem of solution of Dunkl equations in mean-periodic functions determined by a linear functional. Proof . The condition f ∈ P Φ is necessary for the existence of a solution u ∈ P Φ . Assume that a function u ∈ P Φ ∩ C (m) (R) is a solution of the Dunkl equation P (D k )u = f . Then mean-periodic are all the functions D j k u, j = 0, 1, 2, . . . , m − 1, i.e.
In order to prove that a solution u of P (D k )u = f with f ∈ P Φ , which satisfies conditions (25), is a mean-periodic function, we consider the function v = Φ y T y k u(x) = Au. Since the operator A commutes with D k , then applying it on the equation P (D k )u = f , we get P (D k )v = 0 due to Af = 0. It remains to find the initial values D j k v(0), j = 0, 1, 2, . . . , m − 1: At the end of the previous section we have seen that the initial value problem P (D k )v = 0, D j k v(0) = 0, j = 0, 1, 2, . . . , m − 1, has only the trivial solution v(x) = 0. Thus we proved that Φ y {T y k u} = 0, i.e. u is mean-periodic. Now we can use operational calculus method for solving nonlocal Cauchy problems for Dunkl equations to find explicitly the mean-periodic solutions of such equations.
To this end, we are to solve the homogeneous nonlocal Cauchy boundary value problem with f ∈ P Φ . In the ring M k of the multiplier fractions it reduces to the single algebraic equation for u P (S k )u = f.
As we have seen in Section 4, P (S k ) is a non-divisor of zero in M k iff none of the zeros of the polynomial P (λ) is a zero of the Dunkl indicatrix E k (λ). If P (S k ) is a divisor of zero, then, in order to ensure the existence of solution of (27) and thus of (26), additional restrictions on f should be imposed. This is the so called resonance case, which we will not treat here. Thus, let P (S k ) be a non-divisor of zero in M k , i.e. {λ : P (λ) = 0} ∩ {λ : E k (λ) = 0} = ∅. Then the formal solution of (27) in M k u = 1 P (S k ) f can be written in explicit functional form. Using the extended Heaviside algorithm of Section 5, we represent 1 P (S k ) as a convolutional operator 1 P (S k ) = {G(x)} * .
Then u = G * f is the desired mean-periodic solution of the Dunkl equation P (D k )u = f . The verification is straightforward. Indeed, G * f ∈ P Φ according to Theorem 9, since f ∈ P Φ . Our considerations of the problem for solving Dunkl equations in mean-periodic functions can be summarized in the following Theorem 11. A Dunkl equation P (D k )u = f with f ∈ P Φ has a unique solution in P Φ iff none of the zeros of the polynomial P (λ) is a zero of the Dunkl indicatrix E k (λ) = Φ j k− 1 2 (iλx) + λx 2k + 1 j k+ 1 2 (iλx) .
In the end, it is possible the Duhamel principle to be extended to the problem for solving Dunkl equations in mean-periodic functions.