- Open Access
The finite continuous nonsymmetric Jacobi transform and applications
© Bouzaffour; licensee Springer. 2012
- Received: 27 December 2011
- Accepted: 9 May 2012
- Published: 9 May 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.
- Special functions
- Integral transform
- Sampling theory.
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 ).
Then the operator Y α, 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.
2.1 The nonsymmetric Jacobi polynomials
Then the solution (14) is now immediate. ■
2.2 Orthogonality relations
2.3 Extension of the nonsymmetric Jacobi polynomials
In the next theorem we give a new integral representation for the Jacobi function.
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. ■
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.
where is given by relation (25).
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.
This establish (34). ■
In the next theorem we derive an inversion formula for the finite continuous nonsymmetric Jacobi transform.
uniformly on compact subsets of (-π, π).
The result follows from Lemma 3.1 in . ■
The following theorem gives a series representation for the finite continuous nonsymmetric Jacobi transform.
with absolute convergence, uniform on strips of finite width in ℂ around ℝ.
Thus, we can expand the function with the aid of Whittaker-Shannon-Kotel'nikov theorem.
This research is supported by NPST Program of King Saud University, project number 10-MAT1293-02.
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