DIRICHLET THEOREM FOR JACOBI-DUNKL EXPANSIONS

HAL Id: hal-02126595

https://hal.archives-ouvertes.fr/hal-02126595

Preprint submitted on 12 May 2019

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

DIRICHLET THEOREM FOR JACOBI-DUNKL EXPANSIONS

Frej Chouchene, Iness Haouala

To cite this version:

Frej Chouchene, Iness Haouala. DIRICHLET THEOREM FOR JACOBI-DUNKL EXPANSIONS.

2019. �hal-02126595�

FREJ CHOUCHENE, INESS HAOUALA

DEPARTMENT OF MATHEMATICS,

HIGHER SCHOOL OF SCIENCES AND TECHNOLOGY OF HAMMAM SOUSSE, UNIVERSITY OF SOUSSE, TUNISIA.

Abstract. The purpose of this paper is to study the pointwise convergence of the Jacobi-Dunkl series. Indeed, we recall some properties of the Jacobi-Dunkl coefficients.

Then, we establish a Dirichlet type theorem for expansions in term of Jacobi-Dunkl polynomials.

1. Introduction

The Fourier series is named in honour of Jean-Baptiste Joseph Fourier (1768-1830) who introduced it for the purpose of solving the heat equation in a metal plate, pub- lishing his initial results in [6], and pursuing his study in [7]. The question whether the Fourier series of a periodic function converges to a given function is well researched and an extensive literature exists on this subject. We mention here for example [9, 5, 12].

Indeed, mathematicians studied pointwise, absolute, uniform, quadratic convergences...

It is well known that many problems for partial differential equations are reduced to a power series expansion of the desired solution in terms of special functions or orthogo- nal polynomials. In particular, by using the properties of Jacobi polynomials ([15]), the Fourier-Jacobi series has been studied extensively by many authors and several results concerning the approximation of functions by partial sums of these series are proved (see e.g. [1, 11, 13, 14, 16, 17, 18]).

In this paper, we also discuss this subject. More precisely, we are interested in Jacobi- Dunkl expansions.

In [2], the author defined the Jacobi-Dunkl coefficients associated with Jacobi-Dunkl polynomials given by

ψ^(α,β)_n (θ) :=







R^(α,β)_|n| (cos(2θ)) + iλ^(α,β)n

4(α+ 1)sin(2θ)R^(α+1,β+1)_|n|−1 (cos(2θ)) if n∈Z\ {0},

1 if n= 0,

(1)

2010 Mathematics Subject Classification. 33B10, 33C45, 33C47, 40A30, 42A10, 42A16.

Key words and phrases. Jacobi-Dunkl polynomials, Jacobi-Dunkl coefficients, Jacobi-Dunkl series, quadratic mean convergence, Dirichlet type theorem.

1

where R^(α,β)_m (x), m ∈ N, is the normalized Jacobi polynomial of degree m such that R^(α,β)_m (1) = 1, and λ^(α,β)_n is given by

λ^(α,β)_n := 2 sgn(n)p

|n|(|n|+ρ), n∈Z, with

α≥β ≥ −1

2; α6=−1

2, and ρ:=α+β+ 1 >0.

In the second section, we will give some preliminaries concerning these polynomials.

Then, we will see more properties of the Jacobi-Dunkl coefficients in the third section.

In section 4, we state a theorem about Jacobi-Dunkl convergence in quadratic mean.

Finally, we will focus on pointwise convergence. We establish a Dirichlet type theorem which generalizes the classical one, see [10]. The proof is based on the asymptotic behaviour of Jacobi and Jacobi-Dunkl polynomials studied in [3] and [4].

2. Preliminaries

In this section, we will recall some properties of Jacobi and Jacobi-Dunkl polynomials.

We denote by

(a)_n :=

a(a+ 1)...(a+n−1) if n ∈N\ {0},

1 if n = 0.

(a)_n is called the Pochhammer symbol.

2F₁(a, b;c;z) is the Gauss hypergeometric function, given by

∀a, b∈C, ∀c∈C\Z⁻, ∀z ∈C; |z|<1, ₂F₁(a, b;c;z) :=

+∞

X

n=0

(a)_n(b)_n n!(c)_n zⁿ. The Jacobi polynomialsϕ^(α,β)_m (θ), m∈N, θ∈h

−π 2,π

2 i

, are defined by ϕ^(α,β)_m (θ) := R^(α,β)_m (cos(2θ)) = ₂F₁(−m, m+ρ;α+ 1; (sinθ)²).

The Jacobi operator ∆_α,β defined on C²i 0,π

2 h

is given by

∆_α,β := d²

dθ² + A⁰_α,β A_α,β

d dθ, where

A_α,β(θ) :=

(

2^2ρ(sin|θ|)^2α+1(cosθ)^2β+1 if θ ∈i

−π 2,π

2 h

\ {0},

0 if θ = 0.

For all m ∈ N, ϕ^(α,β)_m is the unique even C^∞-solution on i

−π 2,π

2 h

of the differential equation







∆_α,βu = −λ²_mu, u(0) = 1, u⁰(0) = 0.

The Jacobi-Dunkl operator∧_α,β is defined by

∧_α,βf(θ) := d

dθf(θ) + A⁰_α,β(θ) A_α,β(θ)

f(θ)−f(−θ)

2 , f ∈C¹i

−π 2,π

2 h

, with A⁰_α,β(θ)

A_α,β(θ) = (2α+ 1) cotθ−(2β+ 1) tanθ , θ∈i

−π 2,π

2

h\ {0}.

According to [2], the differential-difference equation

∧_α,βu(θ) = iλ^(α,β)_n u(θ); n ∈Z, u(0) = 1,

admits a unique C^∞-solution on i

−π 2,π

2 h

given by (1), which is related to the Jacobi polynomial and to its derivative by

ψ_n^(α,β)(θ) :=







ϕ^(α,β)_|n| (θ)− i λ^(α,β)n

d

dθϕ^(α,β)_|n| (θ) if n ∈Z\ {0},

1 if n = 0,

(2) and satisfies

∀n∈Z, ∀θ ∈h

−π 2,π

2 i

,

ψ_n^(α,β)(θ) ≤1.

For alln, p ∈Z, we have the following orthogonality property Z ^π₂

−^π

2

ψ_n^(α,β)(θ)ψ^(α,β)p (θ)Aα,β(θ)dθ = h^(α,β)_n ⁻¹

δn,p, (3)

whereh^(α,β)_n = Z ^π₂

−^π₂

ψ^(α,β)_n (θ)

2A_α,β(θ)dθ

!−1

: h^(α,β)₀ = Γ(ρ+ 1) 2^2ρΓ(α+ 1)Γ(β+ 1) and

∀n∈Z\ {0}, h^(α,β)_n = (2|n|+ρ)Γ(α+|n|+ 1)Γ(ρ+|n|) 2^2ρ+1(Γ(α+ 1))²Γ(|n|+ 1)Γ(β+|n|+ 1). Letp∈[1,+∞]. We denote by

• L^p_α,β = L^ph

−π 2,π

2 i

, A_α,β(θ)dθ

: the space of measurable functions f on h

−π 2,π

2 i such that











kfk_p,α,β = Z ^π₂

−^π₂

|f(θ)|^pA_α,β(θ)dθ

!_p¹

<+∞ if 1≤p <+∞, kfk∞,α,β = ess sup

θ∈[^−π2,^π₂]

|f(θ)|<+∞ if p= +∞.

• Le^p_α,β = L^ph 0,π

2 i

, A_α,β(θ)dθ

the space of measurable functions g on h 0,π

2 i

such that









 Z ^π₂

0

|g(θ)|^pA_α,β(θ)dθ

!¹_p

<+∞ if 1≤p <+∞, ess sup

θ∈[^0,π2]

|g(θ)|<+∞ if p= +∞.

The Jacobi coefficients (see [8]) of a functiong ∈Le¹_α,β are defined by

∀m∈N, F_α,β(g)(m) = Z ^π₂

0

g(θ)ϕ^(α,β)_m (θ)A_α,β(θ)dθ.

The Jacobi-Dunkl coefficients (see [2]) of a functionf ∈L¹_α,β are defined by

∀n ∈Z, Ff(n) :=

Z ^π₂

−^π

2

f(θ)ψn^(α,β)(θ)A_α,β(θ)dθ, and satisfy

∀n ∈Z, |Ff(n)| ≤ kfk_1,α,β. Now, we consider the analog of the Fourier series given by

+∞

X

n=−∞

Ff(n)ψ_n^(α,β)(θ)h^(α,β)_n , θ ∈h

−π 2,π

2 i

. Forn∈N, we denote its partial sum by

S_n^f(θ) :=

n

X

k=−n

Ff(k)ψ_k^(α,β)(θ)h^(α,β)_k , θ∈h

−π 2,π

2 i

. 3. Jacobi-Dunkl coefficients

Let f ∈L¹_α,β. We put for all k ∈N,

ak(f) :=Ff(k) +Ff(−k), and

b_k(f) :=







− i

λ^(α,β)_k [Ff(k)− Ff(−k)] if k ∈N\ {0},

0 if k = 0.

Hence, by (2) we can writeS_n^f(θ), forn ∈N\ {0} and θ ∈h

−π 2,π

2 i

, as S_n^f(θ) = a₀(f)

2 h^(α,β)₀ +

n

X

k=1

ak(f)ϕ^(α,β)_k (θ) +bk(f) d

dθϕ^(α,β)_k (θ)

h^(α,β)_k . Remark 3.1.

For allk ∈N, we have these relations:

(1) Ff(k) = a_k(f) +iλ^(α,β)_k b_k(f)

2 .

(2) Ff(−k) = a_k(f)−iλ^(α,β)_k b_k(f)

2 .

Proposition 3.2.

For allk ∈N, we have the following integral representations:

(1) a_k(f) = 2 Z ^π₂

−^π₂

f(θ)ϕ^(α,β)_k (θ)A_α,β(θ)dθ.

(2) bk(f) = 2

λ^(α,β)_k 2

Z ^π₂

−^π

2

f(θ) d

dθϕ^(α,β)_k (θ)Aα,β(θ)dθ, k 6= 0.

Proof.

(1) a_k(f) = Ff(k) +Ff(−k) = Z ^π₂

−^π₂

f(θ)h

ψ_k^(α,β)(θ) +ψ_−k^(α,β)(θ)i

A_α,β(θ)dθ.

Since we know that

ψ_k^(α,β)(θ) +ψ^(α,β)_−k (θ) = 2<

ψ_k^(α,β)(θ)

= 2ϕ^(α,β)_k (θ), then, we obtain the result.

(2) b_k(f) = i λ^(α,β)_k

Z ^π₂

−^π₂

f(θ)h

ψ^(α,β)_−k (θ)−ψ_k^(α,β)(θ)i

A_α,β(θ)dθ, k6= 0.

As we have

ψ_−k^(α,β)(θ)−ψ_k^(α,β)(θ) = 2i=

ψ_k^(α,β)(θ)

=− 2i λ^(α,β)_k

d

dθϕ^(α,β)_|k| (θ), then, we get the equality.

Remarks 3.3.

Let k ∈N.

(1) If the function f is even, then b_k(f) = 0 and a_k(f) = 4

Z ^π₂

0

f(θ)ϕ^(α,β)_k (θ)A_α,β(θ)dθ.

(2) If the function f is odd, then ak(f) = 0 and bk(f) = 4

λ^(α,β)_k 2

Z ^π₂

0

f(θ) d

dθϕ^(α,β)_k (θ)Aα,β(θ)dθ, k6= 0.

Proposition 3.4.

Let f be in L¹_α,β, a real-valued function. For all k∈N, we have these properties:

(1) Ff(−k) =Ff(k).

(2) a_k(f) = 2<(Ff(k))∈R. (3) b_k(f) = 2

λ^(α,β)_k =(Ff(k))∈R, k 6= 0.

Proof.

(1) Ff(−k) = Z ^π₂

−^π

2

f(θ)ψ^(α,β)_−k (θ)Aα,β(θ)dθ = Z ^π₂

−^π

2

f(θ)ψ_k^(α,β)(θ)Aα,β(θ)dθ =Ff(k).

(2) a_k(f) = Ff(k) +Ff(−k) =Ff(k) +Ff(k) = 2<(Ff(k)).

(3) For all k ∈Z\ {0}, we have b_k(f) = i

λ^(α,β)_k [Ff(−k)− Ff(k)] = i λ^(α,β)_k

hFf(k)− Ff(k)i

= 2

λ^(α,β)_k =(Ff(k)).

In the following parts, we will study for a suitable given function f, the convergence of the series

+∞

X

n=−∞

Ff(n)ψ_n^(α,β)(θ)h^(α,β)_n .

4. Convergence in quadratic mean Theorem 4.1. For all f ∈L²_α,β, we have

n→+∞lim kS_n^f −fk2,α,β = 0.

Proof. Let f ∈L²_α,β and n ∈N. Z ^π₂

−^π₂

S_n^f(θ)−f(θ)

2A_α,β(θ)dθ = Z ^π₂

−^π₂

S_n^f(θ)−f(θ)

Sn^f(θ)−f(θ)

A_α,β(θ)dθ

= Z ^π₂

−^π

2

S_n^f(θ)

2A_α,β(θ)dθ− Z ^π₂

−^π

2

S_n^f(θ)f(θ)A_α,β(θ)dθ

− Z ^π₂

−^π

2

f(θ)Sn^f(θ)A_α,β(θ)dθ+ Z ^π₂

−^π

2

|f(θ)|²A_α,β(θ)dθ := I₁+I₂+I₃+I₄.

We have by the orthogonality property (3), I₁ =

Z ^π₂

−^π₂ n

X

k=−n

Ff(k)ψ_k^(α,β)(θ)h^(α,β)_k (θ)

! _n X

p=−n

Ff(p)ψp^(α,β)(θ)h^(α,β)p (θ)

!

A_α,β(θ)dθ

=

n

X

k=−n n

X

p=−n

Ff(k)Ff(p)h^(α,β)_p h^(α,β)_k Z ^π₂

−^π₂

ψ_k^(α,β)(θ)ψ^(α,β)p (θ)A_α,β(θ)dθ

!

=

n

X

k=−n n

X

p=−n

Ff(k)Ff(p)h^(α,β)_p h^(α,β)_k

h^(α,β)_k −1

δ_k,p

=

n

X

k=−n

Ff(k)Ff(k)h^(α,β)_k

=

n

X

k=−n

|Ff(k)|²h^(α,β)_k .

Furthermore

I₂ = − Z ^π₂

−^π

2

n

X

k=−n

Ff(k)ψ^(α,β)_k (θ)h^(α,β)_k

!

f(θ)A_α,β(θ)dθ

= −

n

X

k=−n

Ff(k)h^(α,β)_k Z ^π₂

−^π

2

f(θ)ψ^(α,β)_k (θ)Aα,β(θ)dθ

!

= −

n

X

k=−n

Ff(k)Ff(k)h^(α,β)_k

= −

n

X

k=−n

|Ff(k)|²h^(α,β)_k

= −I1. We also have

I₃ =I₂ =I₂ =−I₁. Then

Z ^π₂

−^π₂

S_n^f(θ)−f(θ)

2A_α,β(θ)dθ =kfk_2,α,β−

n

X

k=−n

|Ff(k)|²h^(α,β)_k . By the Plancherel formula [2, Theorem 3.4], we obtain

n→+∞lim Z ^π₂

−^π₂

S_n^f(θ)−f(θ)

2A_α,β(θ)dθ = 0.

5. Dirichlet type convergence Notation 5.1.

For alln ∈N, θ, φ∈h

−π 2,π

2 i

. We denote by

D^(α,β)_n (θ, φ) :=

n

X

k=−n

ψ_k^(α,β)(θ)ψ^(α,β)_k (φ)h^(α,β)_k .

D_n^(α,β)(θ, φ) is the analog of the Dirichlet kernel associated with the Fourier series.

Proposition 5.2.

Let f ∈L¹_α,β, n∈N and θ ∈h

−π 2,π

2 i

. We have

S_n^f(θ) = Z ^π₂

−^π₂

f(φ)D^(α,β)_n (θ, φ)A_α,β(φ)dφ.

Proof.

S_n^f(θ) =

n

X

k=−n

Z ^π₂

−^π

2

f(φ)ψ^(α,β)_k (φ)A_α,β(φ)dφ

!

ψ^(α,β)_k (θ)h^(α,β)_k

= Z ^π₂

−^π₂

f(φ)

n

X

k=−n

ψ^(α,β)_k (φ)ψ^(α,β)_k (θ)h^(α,β)_k

!

A_α,β(φ)dφ

= Z ^π₂

−^π

2

f(φ)D_n^(α,β)(θ, φ)Aα,β(φ)dφ.

Proposition 5.3.

Let n∈N and θ∈h

−π 2,π

2 i

. We have Z ^π₂

−^π₂

D_n^(α,β)(θ, φ)A_α,β(φ)dφ= 1.

Proof.

Z ^π₂

−^π

2

D^(α,β)_n (θ, φ)A_α,β(φ)dφ=

n

X

k=−n

Z ^π₂

−^π

2

ψ_k^(α,β)(φ)A_α,β(φ)dφ

!

ψ^(α,β)_k (θ)h^(α,β)_k . As we know, by the orthogonality property (3), that

Z ^π₂

−^π₂

ψ^(α,β)_k (φ)A_α,β(φ)dφ=

h^(α,β)₀ −1

δ_0,k, then, we get

Z ^π₂

−^π

2

D_n^(α,β)(θ, φ)A_α,β(φ)dφ=ψ₀^(α,β)(θ) = 1.

Proposition 5.4.

(1) ∀n ∈N, ∀θ, φ∈h

−π 2,π

2 i

; θ6=±φ, we have

D^(α,β)_n (θ, φ) = Γ(α+n+ 2)Γ(ρ+n+ 1)

2^2ρ−1(Γ(α+ 1))²(2n+ρ+ 1)n! Γ(β+n+ 1) × 1

cos(2θ)−cos(2φ)

×

ϕ^(α,β)_n+1 (θ)ϕ^(α,β)_n (φ)−ϕ^(α,β)_n (θ)ϕ^(α,β)_n+1 (φ) + λ^(α,β)n λ^(α,β)_n+1 4(n+ 1)(n+ρ)

×

=ψ^(α,β)_n+1 (θ)=ψ_n^(α,β)(φ)− =ψ_n^(α,β)(θ)=ψ^(α,β)_n+1 (φ) , with =ψ_n^(α,β)(θ) = ψ_n(θ)−ψ_n(−θ)

2i .

(2) ∀n ∈N, ∀θ, φ∈h

−π 2,π

2 i

, we have (a) D^(α,β)_n (θ, φ)∈R.

(b) D^(α,β)_n (θ, θ)>0.

(c) D^(α,β)_n (φ, θ) =D^(α,β)_n (θ, φ).

Proof.

(1) The case n = 0 is obvious, and we have the result in [4, theorem 3.1], for n ∈N\ {0}.

(2) (a) We deduce the result from (1), for θ 6=±φ. We also have D^(α,β)_n (θ, θ) =

n

X

k=−n

ψ^(α,β)_k (θ)

2

h^(α,β)_k ∈R,

and

D^(α,β)_n (θ,−θ) =

n

X

k=−n

ψ_k^(α,β)(θ) 2

h^(α,β)_k

= h^(α,β)₀ +

n

X

k=1

ψ_k^(α,β)(θ)2

+

(ψ_k^(α,β)(θ)² h^(α,β)_k

= h^(α,β)₀ + 2<

n

X

k=1

ψ_k^(α,β)(θ)2!

∈R.

(b) D^(α,β)_n (θ, θ) = h^(α,β)₀ +

n

X

k=−n,k6=0

ψ^(α,β)_k (θ)

2

h^(α,β)_k >0.

(c) D^(α,β)_n (φ, θ) =D^(α,β)n (θ, φ) = D_n^(α,β)(θ, φ).

Theorem 5.5.

Let f be a piecewise continuous function on h

−π 2,π

2 i

and θ∈h

−π 2,π

2

i\ {0} such that i) f(−θ) = f(θ),

ii) f is differentiable on θ and −θ.

Then we have

n→+∞lim S_n^f(θ) =f(θ).

Proof.

Letn∈N and θ ∈h

−π 2,π

2

i\ {0}. By Proposition 5.3, we can write

f(θ)−S_n^f(θ) = Z ^π₂

−^π₂

[f(θ)−f(φ)]D^(α,β)_n (θ, φ)A_α,β(φ)dφ.

From [4, Theorem 3.1], we have for all θ6=±φ f(θ)−S_n^f(θ) = l^(α,β)_n

Z ^π₂

−^π₂

f(θ)−f(φ) cos(2θ)−cos(2φ)

ϕ^(α,β)_n+1 (θ)ϕ^(α,β)_n (φ)−ϕ^(α,β)_n (θ)ϕ^(α,β)_n+1 (φ) + λ^(α,β)n λ^(α,β)_n+1

4(n+ 1)(n+ρ)

× =ψ_n+1^(α,β)(θ)=ψ^(α,β)_n (φ)− =ψ^(α,β)_n (θ)=ψ_n+1^(α,β)(φ)

A_α,β(φ)dφ, where

l^(α,β)_n := Γ(α+n+ 2)Γ(ρ+n+ 1)

2^2ρ−1(Γ(α+ 1))²(2n+ρ+ 1)n! Γ(β+n+ 1). For allφ ∈h

−π 2,π

2

i\ {±θ}, we put

g_θ(φ) := f(θ)−f(φ) cos(2θ)−cos(2φ).

Since we have supposed that f is a piecewise continuous function onh

−π 2,π

2 i

, theng_θ is also piecewise continuous on h

−π 2,π

2

i\ {±θ}.

Furthermore, we have

φ→θlimg_θ(φ) =−1 2

1

sin(2θ)f⁰(θ).

And from hypothese i) of our theorem, we deduce that

φ→−θlim g_θ(φ) = 1 2

1

sin(2θ)f⁰(−θ).

Under the assumption ii) of the theorem, these limits exist and are finite.

We still callg_θ the extension ofg_θ onh

−π 2,π

2 i

. Thus, g_θ ∈L²_α,β. In the following, we denote by

g∨θ(φ) := gθ(−φ), φ∈h

−π 2,π

2 i

, g_θ¹ := (g_θ)_|

[0, π2], g_θ² := (gθ)_|[^−π2,0],

∨ _

g_θ²(φ) := g²_θ(−φ), φ∈h 0,π

2 i

. Now, we write

f(θ)−S_n^f(θ) = I₁+I₂+I₃+I₄,

where

I₁ := l_n^(α,β)ϕ^(α,β)_n+1 (θ) Z ^π₂

−^π₂

g_θ(φ)ϕ^(α,β)_n (φ)A_α,β(φ)dφ, I₂ := −l_n^(α,β)ϕ^(α,β)_n (θ)

Z ^π₂

−^π₂

g_θ(φ)ϕ^(α,β)_n+1 (φ)A_α,β(φ)dφ, I₃ := l_n^(α,β) λ^(α,β)n λ^(α,β)_n+1

4(n+ 1)(n+ρ)=ψ^(α,β)_n+1 (θ) Z ^π₂

−^π₂

g_θ(φ)=ψ^(α,β)_n (φ)A_α,β(φ)dφ, I4 := −l_n^(α,β) λ^(α,β)n λ^(α,β)_n+1

4(n+ 1)(n+ρ)=ψ^(α,β)_n (θ) Z ^π₂

−^π

2

gθ(φ)=ψ^(α,β)_n+1 (φ)Aα,β(φ)dφ.

Combining the fact that

l^(α,β)_n ∼

+∞

1

2^2ρ(Γ(α+ 1))²n^2α+1, and the result (35) of [3], we get

l^(α,β)_n ϕ^(α,β)_n+1 (θ) ∼

+∞n^α+12cos

(2n+ 2 +ρ)|θ| −(2α+ 1)^π₄

√πΓ(α+ 1)A^2α−1

4 ,^2β−1₄ (θ) . Moreover, we have

Z ^π₂

−^π

2

g_θ(φ)ϕ^(α,β)_n (φ)A_α,β(φ)dφ=F_α,β



g¹_θ+

_∨

g²_θ



(n).

From the Parseval formula for the Jacobi coefficients (see [2]), we obtain F_α,β



g¹_θ+

_∨

g_θ²



(n) = o

n^−(α+12) . Thus, lim

n→+∞I1 = 0.

We use the same proof as for I₁ to show that

n→+∞lim I₂ = lim

n→+∞−



n^α+12cos

(2n+ρ)|θ| −(2α+ 1)^π₄

√πΓ(α+ 1)A2α−1

4 ,^2β−1₄ (θ) F_α,β



g¹_θ +

_∨

g²_θ



(n+ 1)





= 0.

Otherwise, we have Z ^π₂

−^π₂

g_θ(φ)=ψ^(α,β)_n (φ)A_α,β(φ)dφ= 1 2iF_∨

g_θ−g_θ (n).

By [2, corollary 3.5], we have F_∨

g_θ−g_θ

(n) = o

n^−(α+12) .

Furthermore, we get, from [3, Theorem 4.7], that

=ψ^(α,β)_n+1 (|θ|) ∼

+∞

2^2ρΓ(α+ 1)

√π

n⁻(^α+12) A2α−1

4 ,^2β−1₄ (θ)sinh

(2n+ 2 +ρ)|θ| −(2α+ 1)π 4 i

. Since we have

n→+∞lim

λ^(α,β)n λ^(α,β)_n+1

4(n+ 1)(n+ρ) = 1, then, we get

n→+∞lim I₃ = 0.

We use the same reasons as forI₃ to show that

n→+∞lim I₄ = lim

n→+∞−l_n^(α,β) λ^(α,β)n λ^(α,β)_n+1

4(n+ 1)(n+ρ)=ψ^(α,β)_n (θ) Z ^π₂

−^π₂

g_θ(φ)=ψ^(α,β)_n+1 (φ)A_α,β(φ)dφ

= 0.

Hence, we obtain

n→+∞lim

f(θ)−S_n^f(θ)

= lim

n→+∞I₁+I₂+I₃+I₄ = 0.

Which achieves the proof.

References

[1] H. Bavinck, Approximation processes for Fourier-Jacobi expansions, Appl. Anal., 5(1976), 293- 312.

[2] F. Chouchene, Harmonic analysis associated with the Jacobi-Dunkl operator on i

−π 2,π

2 h

, J.

Comput. Appl. Math.178(2005), 75-89.

[3] F. Chouchene, Bounds, asymptotic behavior and recurrence relations for the Jacobi-Dunkl poly- nomials, Int. J. Open Problems Complex Analysis,6(1)(2014), 49-77.

[4] F. Chouchene, Recurrence and Christoffel-Darboux formulas for the Jacobi-Dunkl polynomials and applications, Comm. Math. Anal., 16(1)(2014), 123-142.

[5] G.B. Folland, Fourier analysis and its applications, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, 1992.

[6] J.-B. J. Fourier, M´emoire sur la propagation de la chaleur dans les corps solides, (nepublikov´ano) pro Institute de France, Paris, podno 21. prosince, 1807.

[7] J.-B. J. Fourier, Th´eorie analytique de la chaleur. Didot, Paris, 1822.

[8] G. Gasper, Positivity and the convolution structure for Jacobi series, Ann. Math. 93 (1971), 112-118.

[9] L. Grafakos, Classical Fourier Analysis, 3^d edition, Springer, New York, 2014.

[10] J.-P. Kahane, P.G. Lemari´e-Rieusset, S´eries de Fourier et Ondelettes, Cassini, 1998.

[11] G. Kavernadze, Uniform convergence of Fourier-Jacobi series, J. Approx. Theory, 117 (2002), 207-228.

[12] A. Lesfari, Distributions, Analyse de Fourier et transformation de Laplace, Ellipses dition Mar- keting, 2012.

[13] Z. Li, Y. Xu, Summability of the product Jacobi series, J. Approx. Theory,104(2000), 287-301.

[14] H.N. Hhaskar, S. Tikhonov, Wiener type theorems for Jacobi series with nonnegative coefficients, Proc. Amer. Math. Soc.,140(3)(2012), 977-986.

[15] G. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloq. Pub. Vol.23, Amer. Math. Soc.

Providence, R. I., 1975.

[16] Y. Xu, Mean convergence of generalized Jacobi series and interpolating polynomials, I, J. Approx.

Theory,72(1993), 237-251.

[17] Y. Xu, Mean convergence of generalized Jacobi series and interpolating polynomials, II, J. Approx.

Theory,76(1994), 77-92.

[18] W. zu Castell, F. Filbir, and Y. Xu, Ces`aro means of Jacobi expansions on the parabolic biangle, J. Approx. Theory159(2009), 167-179.

E-mail address: [email protected] E-mail address: [email protected]