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The finite continuous nonsymmetric Jacobi transform and applications
Advances in Difference Equations volume 2012, Article number: 55 (2012)
In this paper we consider the differential difference operator
where (τf)[z] = f[z-1] The eigenfunction of this operator equal to 1 at 1 is called nonsymmetric Jacobi function. We define the finite continuous nonsymmetric Jacobi transform as an extension of the nonsymmetric Fourier Jacobi series. The basic properties including the inversion formula for this transform are studied. We also derive a sampling expansion associated to Y α, β.
2010 Mathematics Subject Classification: 33C45, 3352, 94A20.
The differential-difference operators play a prominent role in the theory of special functions and harmonic analysis. Dunkl, Heckman and in particular Cherednik it was next shown that there are related orthogonal systems of special functions which are not Weyl group invariant (see [1–3]), but are in a sense more simple, and from which the earlier Weyl group invariant special functions can be obtained by symmetrization. The specialization of these operators to the case of rank one has its own interest, because everything can be done there in a much more explicit way, and new results for special functions in one variable can be obtained. In the rank one case the Weyl group has order 2, and Weyl group symmetry turns down to a symmetry under the map t → -t or z → z-1 (see ).
In this work we consider the differential-difference operator
After substitution of z = e-t, k1 = α - β , , the operator Yα, βbecomes
Then the operator Yα, β, coincides with the Heckman's operator  (see also , (1,12)) in the case of root system BC1 and the operator coincides with Opdam operator corresponding to the root system A1.
The outline of this paper is as follow: In section 2, we show that the unique polynomials eigenfunction of the operator Y α, β equal to 1 at 1 is related to Jacobi polynomials and to their derivatives. We call these the nonsymmetric Jacobi polynomials. We establish also a new orthogonality relations for these polynomials and we give integral representation for the nonsymmetric Jacobi function. In section 3, we study the finite continuous nonsymmetric Jacobi transform and we give for this transform an inversion formula. In section 4, sampling theorem associated with the differential-difference operator is investigated.
2 The nonsymmetric Jacobi function
2.1 The nonsymmetric Jacobi polynomials
We note the Laurent polynomials f in z by f[z]. Symmetric Laurent polynomials (where c k = c -k ) are related to ordinary polynomials f (x) in by . Write
are symmetric Laurent polynomials. Consider Jacobi polynomials as normalized symmetric Laurent polynomials:
where , is the Jacobi polynomials given by (see ))
For α, β > -1 these polynomials satisfy the orthogonality relations
Theorem 1. For n ∈ ℤ the differential-difference equation
has a unique solution given by
Proof. From (6), we can write
Then the following equation
is equivalent to the system
Then the solution (14) is now immediate. ■
The Laurent polynomials are called nonsymmetric Jacobi polynomials. For |z| = 1, we have
The following formula
In particular, if , we obtain
2.2 Orthogonality relations
Consider < .,. > α, β , defined in (8), as a symmetric bilinear form on the space of symmetric Laurent polynomials. With the identification f ↔ (f1, f2) between a Laurent polynomial f and a pair of symmetric Laurent polynomials (see (6)) we look for a symmetric bilinear form on the space of Laurent polynomials of the form
such that the nonsymmetric Jacobi polynomials given by(16) are orthogonal with respect to this form, i.e.
By (8) and (16) the orthogonality certainly holds if |n| ≠ |m|. Thus we have to determine C in (17) such that for n = 1, 2,...,
By (17) this turns down to
A priori, it is not clear that C is independent of n. However, from(10) we compute
Thus C is independent of n and the form (17) becomes more explicitly
Here <.,. > α, β is defined by in (8). This form is positive definite if α > -1 and β > -1. The inner product (17) can also be written in terms of an integral with positive weight function. First observe that (6) implies that
Hence, for Laurent polynomials g, h we obtain from (17) and (20) that
where A α, β (θ), is defined in (9). So the orthogonality of the Laurent polynomials with respect to the inner product (22) can be rewritten as the following orthogonality for the Laurent polynomials given by (16):
2.3 Extension of the nonsymmetric Jacobi polynomials
The mapping , extends to a holomorphic function on ℂ defined by the same formula with n replaced by μ - ρ (see ). The function is an symmetric entire function in μ, satisfying the relation
In the next theorem we give a new integral representation for the Jacobi function.
Theorem 2. For and λ, μ ∈ ℂ such that μ2 = λ2 +ρ2, the function has the Laplace integral representation
Proof. We start with the integral representation in 
and from , we have the integral representation
where J0 is the Bessel function of first kind and index zero. If we substitute the expression (26) in (27) and a simple calculation show the result. ■
Lemma 1. The operator Y α, βsatisfies
We can use a similar argument to that used in Theorem 2.1 to show that: For λ, μ ∈ ℂ, such that μ2 = λ2 + ρ2, the differential-difference problem
has a unique solution given by
we call the function , the nonsymmetric Jacobi function. For the nonsymmetric Jacobi function reduces to the nonsymmetric Jacobi polynomials . In the next theorem we give an integral representation for the nonsymmetric Jacobi function.
Theorem 3. For and λ ∈ ℂ, the function has the Laplace integral representation
where K (θ, ϕ) is a function on (-π, π), continuous on (-|θ|, |θ|), supported in [-|θ|, |θ|] and for |ϕ | < |θ|, we have
where is given by relation (25).
3 The finite continuous nonsymmetric Jacobi transform
The finite continuous nonsymmetric Jacobi transform is defined by 
Inversion formula for the finite continuous nonsymmetric Jacobi transform (1.1) has been studied in many papers (see, [11, 12, 9, 13]). In particular, it was inverted in the special Gegenbauer cases α = β = integer ≥ 0 by MacRobert. Butzer, Stens and Wehrens  considered the Legendre case and pointed the relationship with sampling theory. Subsequent Walter and Zayed  found similar results for values such that α + β is a non-negative integer. The interest in this transform was revived by the work of T.H. Koornwinder and G. G. Walter , who remove the restriction on α and β , requiring only that α > -1 and β > -1 In this section we define the finite continuous nonsymmetric Jacobi transform and study some of its basic properties. The idea is to replace the nonsymmetric Jacobi polynomials in the Fourier series by the nonsymmetric Jacobi function. Let f be a function on (-π, π) such that
is well-defined for all λ ∈ ℝ. Then is called the finite continuous nonsymmetric Jacobi transform of f . Recall that each function f defined in (-π, π) may be decomposed uniquely into the sum f = f e + f o , where the even part f e is defined by f e (x) = (f(x) + f(-x))/2 and the odd part f o by f o = (f(x) - f (-x))/2.
Proposition 1. For all λ, μ ∈ ℂ such that μ2 = λ2 + ρ2, we have
where the operator J is given by
Proof. Let f = f e + f o , for λ ∈ ℂ, we have
By integration by parts, we obtain
We deduce (33) from the fact that
From the formula (15) we have
This establish (34). ■
In the next theorem we derive an inversion formula for the finite continuous nonsymmetric Jacobi transform.
Theorem 4. Let f ∈ C2pwith , we have
Proof. Let f = f e + f o , then we have
In the other hand it is easy to see that f e , J f o ∈ C2pwith and by Theorem 4.1 of , we get
4 Sampling theorem
The well-known Whittaker-Shannon-Kotelnikov theorem says that if
and if g ∈ L2 ((-π, π)), then
Proof. From the relation
and by formula (2.9) in the second member of the last equality becomes
and after simplification we obtain
We have ■
Proposition 2. Let f ∈ L2 ((-π, π), A α, β (|θ|)) and let f be once continuously differentiable on (-π, π). Then
uniformly on compact subsets of (-π, π).
Proof. Let f = f e + f o . The Proposition 3: 1 give the following equality
Using the relation
The result follows from Lemma 3.1 in . ■
The following theorem gives a series representation for the finite continuous nonsymmetric Jacobi transform.
Theorem 5. Let f ∈ C2p, where 2p > 2 + max . Then
with absolute convergence, uniform on strips of finite width in ℂ around ℝ.
If we substitute the expression (30) in (3: 2) we obtain
Thus, we can expand the function with the aid of Whittaker-Shannon-Kotel'nikov theorem.
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This research is supported by NPST Program of King Saud University, project number 10-MAT1293-02.
The author declares that they have no competing interests.
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Cite this article
Bouzaffour, F. The finite continuous nonsymmetric Jacobi transform and applications. Adv Differ Equ 2012, 55 (2012). https://doi.org/10.1186/1687-1847-2012-55
- Special functions
- Integral transform
- Sampling theory.