On the denseness of Jacobi polynomials

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PII. S0161171204305314 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON THE DENSENESS OF JACOBI POLYNOMIALS

SARJOO PRASAD YADAV

Received 29 May 2003

LetXrepresent either a spaceC[−1,1]orLα,βp (w), 1≤p <∞, of functions on[−1,1]. It is well known thatXare Banach spaces under the sup and thep-norms, respectively. We prove that there exist the best possible normalized Banach subspacesXk_α,βofXsuch that the system of Jacobi polynomials is densely spread on these, or, in other words, eachf∈Xk_α,β can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Explicit representation forf∈Xkα,βhas been given.

2000 Mathematics Subject Classiﬁcation: 41A10, 42C10, 46B25.

1. Introduction. Let 1≤p≤ ∞,w(x)=(1−x)α₍₁₊_x)β_,_{α, β >}₋_1,_x_∈_[₋₁_,₁_]_{, and} letLpα,β(w)denote the Banach space of functionsf:[−1,1]→ withf xjwp<∞,

j =0,1,2, . . . . Ifp = ∞, assume f is continuous. LetXkα,β be the linear manifolds of Lpα,β(w)with parametersα,β, andksuch that for allf∈X

k

α,β, the following endpoint, called “thepole” condition, is satisﬁed:

t

0

f (cosϕ)−Aϕ2α+1_dϕ₌_o_t2α+2_. _(1.1)

Also, for somek, the following additional condition, called “theantipole” condition, is assumed to be satisﬁed:

h

0

_{f (}₋_cos_ϕ)_ϕβ−α+k_dϕ₌_o(₁_). _(1.2)

It may be noted that the ﬁrst condition would have been satisﬁed byA=f (1), where

f is continuous atx=1. Also, “the antipole” condition (1.2) is weaker than that in [15, Theorem 9.1.4]. Let{gn}be a c-sequence inXkα,β and let its limit beg. Theng will satisfy (1.1) and (1.2), proving thatg∈Xk

α,β. Thus, the spacesXkα,βare normalized Banach subspaces ofX≡Lpα,β(w)with norm

fXk_α,β=c1fX

c1≥1

. (1.3)

In all, we are going to prove that Xk

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We associate a Fourier-Jacobi expansion for allf∈Xk α,βas

f (x)∼

∞

n=0 ˆ

f (n)ω(α,β)n Rn(α,β)(x), (1.4)

where

ω(α,β)n = 1

−1

Rn(α,β)(x) 2

w(x)dx

−1

=(2n+α+β+1)Γ(n+α+β+1)Γ(n+α+1)

2α+β+1_Γ_(n₊_β₊₁₎_Γ_(n₊₁₎_Γ_(α₊₁₎_Γ_(α₊₁₎

n2α+1_L(n)

L(n)≡1+O ₁

n

,

(1.5)

ˆ

f (n)is thenth Fourier-Jacobi transform off given by

ˆ

f (n)= 1

−1

f (x)R(α,β)n (x)w(x)dx, (1.6)

Rn(α,β)(x)is the normalized Jacobi polynomial such that

Rn(α,β)(x)=P (α,β) n (x)

Pn(α,β)(1)

, (1.7)

wherePn(α,β)(x)is thenth Jacobi polynomial of degreenand order(α, β)(see [15]).

2. Preliminaries. We choose a linear combinationσk

n of Jacobi polynomials as

σnk

f ,cosϑ,Xkα,β

=cn+cn−1R (α,β)

1 (cosϑ)+cn−2R (α,β) 2 (cosϑ)

+···+c0R (α,β) n (cosϑ)

≡

n

ν=0

cn−νR(α,β)ν (cosϑ),

(2.1)

where coeﬃcientsciare given by

cn−ν≡ _Ak

n−ν

Ak n

ˆ

f (ν)ω(α,β)ν , (2.2)

ν=0,1, . . . , nandn=0,1,2, . . . . Ak

i (i=0,1,2, . . .)are binomial coeﬃcients ofxiin the expansion of(1−x)−k−1_{. Thus,}_σk

n(f ,cosϑ,Xkα,β)is thenth Cesàro mean of orderkof the Fourier-Jacobi expansion given in (1.4) (see [15, page 244]). Recently in [22], we have proved the following result:

σ1 n

f ,cosϑ,X1 α,β

−f (cosϑ)_X1

α,β →0 asn → ∞, (2.3)

for the casek=1 under only the saturation-type condition (1.1) at the polex=1 for

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has also been solved in [18]. The best possible cases of general orderkhave not been handled so far. We settle the problem by proving

σnk

f ,cosϑ,Xkα,β

−f (cosϑ)_Xk

α,β →0 (2.4)

asn→ ∞, by deciding the best possible span of k forα≥β≥ −1/2. We settle the complete problem in four steps given as Theorems3.1, 3.2,3.3, and3.4. The subject

approximation of functionsin terms of polynomials which indicate itsdensenessis an important part ofanalysiswith many notable contributions, such as Riesz [14], Pollard [11,12,13], Newman and Rudin [10], Askey [1,2], Badkov [3], Nevai [9], Máté et al. [7], Xu [16,17], Lasser and Obermaier [5], Li [6], Mhaskar [8], Yadav [18,19,20,21,22]. The central ideas used in our proofs are the convolution structure for Jacobi series [2], and endpoint convergence of Fourier-Jacobi expansions [15].

3. Main results. We prove the following with prior assumption thatα≥β≥ −1/2.

Theorem_3.1. _{Statement (2.4) holds true for}_k≥_α+β+₁_{only if the pole condition}

(1.1) is satisfied.

Theorem3.2. Forβ >−1/2,α+1/2< k < α+β+1, statement (2.4) is true if “the

pole” condition (1.1) and “the antipole” condition (1.2) are satisfied.

Theorem3.3. Fork < α+1/2, statement (2.4) is not true, that is, there exist functions

f∈Xkα,βso that the norm in (2.4) tends to infinity.

Theorem_3.4. _For_α+₁_/₂_{< k < α}+_β+₁_{, but without “the antipole” condition (1.2),}

statement (2.4) is not true. That is, there exist functions not satisfying (1.2), having their kth Cesàro transform divergent.

4. Results to be used. The following formulae and lemmas have been used to com-plete the proofs of our theorems:

Pn(α,β)(1)=

n+α n

(4.1)

(see [15, (4.1.1)]);

Pn(α,β)(−x)=(−1)nPn(β,α)(x) (4.2)

(see [15, (4.1.3)]). Forα,βarbitrary reals,ca ﬁxed positive number, andn→ ∞,

Pn(α,β)(cosθ)=       

θ−α−1/2_O_n−1/2_, c

n≤θ≤ π

2,

Onα_, ₀_≤_θ_≤ c

n

(4.3)

(cf. [15, (7.32.5)]), and

Pn(α,β)(cosθ)=On−1/2 (4.4)

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Forα >−1,β >−1, andc/n≤θ≤π−c/n,

Pn(α,β)(cosθ)=n−1/2K(θ)

cos(Nθ+γ)+Onsinθ−1 (4.5)

(see [15, Theorem 8.21.13]) and

Pn(α,β)(cosθ)=n−1/2K(θ)cos(Nθ+γ)+On−3/2 (4.6)

forα,βarbitrary and 0< θ < π(see [15, Theorem 8.21.8]), where

K(θ)=π−1/2 _sin_θ

2

−α−1/2_cos_θ

2

−β−1/2

, N=n+α+β₂+1, (4.7)

andγ= −(α+1/2)π /2. The error term in (4.6) holds uniformly in the interval[, π−], for >0. These formulae have been indirectly used with division by

Pn(α,β)(1)=

n+α n

(4.8)

to substitute the value ofR(α,β)n (cosϕ)in the following lemmas which are crucial in the proofs of Theorems3.1,3.2,3.3, and3.4.

Lemma4.1. Letf∈Xk_α,β. Then, asn→ ∞,

π

0

F (ϕ)Rn(α+k+1,β)(cosϕ)dϕ=               

on−2α−2_, _for_{k > α}₊1 2,

on−α−k−3/2_log_n_, _for_k₌_α₊1 2,

on−α−k−3/2_, _for_{k < α}₊1 2,

(4.9)

where

F (ϕ)=f (cosϕ)−Asinϕ

2

2α+1_cos_ϕ

2 2β+1

, (4.10)

provided that, forβ >−1/2andα+1/2< k < α+β+1, the antipole condition (1.2) is satisfied. For−1< β≤ −1/2(orβ >−1/2butk≥α+β+1), no antipole condition is necessary.

Proof ofLemma4.1. Letnbe large enough and

π

0 F (ϕ)R

(α+k+1,β)

n (cosϕ)dϕ= c/n

0 +

δ

c/n+ π−δ´

δ + π−c/n

π−δ´ + π

π−c/n= 5

i=1

Ti, (4.11)

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We denote thenth Cesàro mean of orderkof the series (1.4) byσk

n(f ,cosϑ,Xkα,β) deﬁned by (2.1).Lemma 4.2, which is basic in the proofs of Theorems3.1,3.2,3.3, and 3.4, provides new convergence criteria for Jacobi series at the endpoints of the interval

[−1,1](see [21]).

Lemma4.2. Iff∈Xk_α,β(α >−1,β >−1), then the series (1.4) is(C, k)-summable to

Aatx=1, fork > α+1/2, or

lim n→∞σ

k n

f ,cosϑ,Xkα,β

−A=0 (4.12)

atϑ=0, fork > α+1/2, provided that in the case

β >−1

2, α+ 1

2< k < α+β+1, (4.13)

the antipole condition (1.2) is satisfied. (For−1< β≤ −1/2or for k≥α+β+1, no antipole condition is necessary.) For k≤α+1/2or for k > α+1/2, but without the antipole condition (1.2), statement (4.12) is not true.

Proof ofLemma4.2. We have

σnk(x)=

Akn −1n

ν=0

Akn−νf (ν)ωˆ (α,β)

ν R(α,β)ν (x).

IfAis any constant, then

σk

n(x)−A=

Ak n

−1n

ν=0

Ak−1 n−ν

ν

i=0 ˆ

f (ν)ω(α,β)_i R(α,β)_i (x)−A

.

Using (1.5), we get, atx=1,

σnk(1)−A=

Akn −1n

ν=0

Akn−−1ν ν

i=0 ˆ

f (ν)ω(α,β)i R (α,β) i (1)−A

= π

0

f (cosϕ)−AAkn −1n

ν=0

α+1 2ν+α+β+1

. Akn−−1νω (α+1,β)

ν R(αν +1,β)(cosϕ)ρ(α,β)(ϕ)dϕ,

(4.14)

where

ρ(α,β)_(ϕ)_≡sinϕ 2

2α+1_cos_ϕ

2 2β+1

(4.15)

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5. Proofs

Proof ofTheorem3.1. We substitute∆λ_n,ν=Ak_n−₋1_ν/Ak_nforν≤nand zero other-wise in [19, equation (2.11)] to get

σnk

f , (cosϑ),Xkα,β

−f (cosϑ)

= π

0

Tψf (cosϑ)−f (cosϑ) n

ν=0

α+1 2ν+α+β+2

Ak−1 n−ν

Akn

ω(αν+1,β)Rν(α+1,β)(cosψ)ρ(α,β)(ψ)dψ,

(5.1)

whereTψf (cosϑ)is the generalized translate off (cosϑ)in the interval[0, π ](see [2]). Also,

TψfX≤ fX (5.2)

forα≥β≥ −1/2, and

lim

ψ→0Tψf=f (5.3)

in the sense of strong limit. (See Bavinck [4, page 770].) But

_T_ψ_f₋_f_X_≤_T_ψ_f₋_A_X₊_f₋_A_X_≤_A₆_f₋_A_X _(5.4)

for some nonnegative constantA6. The constantA6 will be independent off by the Banach-Steinhaus theorem. Also,

Tψf−fX≥Tψf−AX−f−AX≥A7f−AX. (5.5)

Again,A7will be independent off. Thus, we have the asymptotic equality

Tψf−fX f−AX. (5.6)

Thus, we compare (5.1) and (4.14) to get _σk

n

f ,cosϑ,Xkα,β

−f (cosϑ)_Xk α,β≤A8

_σk n

f ,1,Xkα,β

−A_Xk

α,β (5.7)

forα≥β≥ −1/2 and an absolute constantA8. But the right-hand side of this inequality tends to zero asntends to∞byLemma 4.2, fork≥α+β+1. Thus, statement (2.4) follows. This completes the proof ofTheorem 3.1.

Proof ofTheorem3.2. Forβ >−1/2,α+1/2< k < α+β+1, statement (4.12)

holds if the antipole condition (1.2) is satisﬁed, that is, inTheorem 3.2, for allf∈Xkα,β satisfying both linear conditions (1.1) and (1.2). Thus, the arguments of the proof of Theorem 3.1apply to the proof ofTheorem 3.2and statement (2.4) holds in this case also. This completes the proof ofTheorem 3.2.

Proof ofTheorem_3.3. _{It is well known that (see Szegö [15, equation (9.41.17),}

page 262]) there exist continuous functions such that fork=α+1/2,

σk n

f ,1,Xkα,β

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This is suﬃcient to conclude, because of the regularity that there exist functionsf∈

Xkα,βsuch that

σk n

f ,cosϑ,Xkα,β

−f (cosϑ)_Xk

α,β → ∞ (5.9)

fork≤α+1/2. Thus, in this case, statement (2.4) is not true. This completes the proof ofTheorem 3.3.

Proof ofTheorem_3.4. _{Here, by an example, we show that there exist functions}

such that the sequence{σk

n(1)}given in (4.14) diverges showing that statement (2.4) is not true. We consider a functionf (x)=(1+x)µ_{given in [15, page 265]. Its Jacobi} series at the endpointx=1 is

∞

n=0

ω(α,β)n 1

−1(1−x)

α₍₁₊_x)β+µ_R(α,β)

n (x)R(α,β)n (1)dx

≡

∞

n=0

(−1)nω(α,β)n 1

−1(1−x)

α+µ₍₁₊_x)β_R(β,α) n (x)dx.

(5.10)

The principal part of the general term of the series (5.10) is approximately

(−1)n_nα−β−2µ−1 _or ₍₋₁₎n_Aα−β−2µ−1

n (5.11)

asR(α,β)n (1)=1. The expansion in (5.10) holds good forµ+β >−1 or−β−1< µ(see [15, page 265]), but the antipole condition (1.2) is not satisﬁed ifµ≤1/2(α−β−k−1). So, for

−β−1< µ≤1/2(α−β−k−1), the Fourier-Jacobi series exists for the functionf (x)= (1+x)µ_{, but it does not satisfy the antipole condition (1.2). Also, for}_k_≤_(α₋_β₋₂_µ₋₁₎_, the expansion in (5.10) is not(C, k)-summable forβ >−1/2,α+1/2< k < α+β+1 (see [15, page 265]). Thus, the sequence given in (4.14) diverges, which, by (5.1), leads to the conclusion that statement (2.4) is not true. This provesTheorem 3.4.

Acknowledgment. _{The author expresses his gratitude to both referees of the}

pa-per for their kind pa-perusal.

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Sarjoo Prasad Yadav: Department of Mathematics/ Computer Applications, Government Model Science College, APS University, 486001 Rewa, Madhya, India