Jacobi-Sobolev Orthogonal Polynomials: Asymptotics for N-Coherence of Measures

Let us introduce the Sobolev-type inner product 〈f, g〉 〈f, g〉1 λ〈f ′, g 〉2, where λ > 0 and 〈f, g〉1 ∫1 −1 f x g x 1 − x α 1 x dx, 〈f, g〉2 ∫1 −1 f x g x 1 − x α 1 1 x β 1 / ∏M k 1|x − ξk |Nk 1 dx ∑M k 1 ∑Nk i 0 Mk,if i ξk g i ξk , with α, β > −1, |ξk | > 1, and Mk,i > 0, for all k, i. A Mehler-Heine-type formula and the inner strong asymptotics on −1, 1 as well as some estimates for the polynomials orthogonal with respect to the above Sobolev inner product are obtained. Necessary conditions for the norm convergence of Fourier expansions in terms of such Sobolev orthogonal polynomials are given.


Introduction
For a nontrivial probability measure σ, supported on −1, 1 , we define the linear space L p dσ of all measurable functions f on −1, 1 such that f L p dσ < ∞, where
Using the standard Gram-Schmidt method for the canonical basis x n n≥0 in the linear space of polynomials, we obtain a unique sequence up to a constant factor of polynomials Q α,β,N n n≥0 orthogonal with respect to the above inner product. In the sequel, they will called Jacobi-Sobolev orthogonal polynomials.
For M 1 and N 1 0, the pair of measures dμ α,β , dν α,β M 1,0 δ ξ is a 0-coherent pair, studied in 2-4 see also 5 in a more general framework . In 6 , the authors established the distribution of the zeros of the polynomials orthogonal with respect to the above Sobolev inner product 1.3 when M 1 and N 1 0. Some results concerning interlacing and separation properties of their zeros with respect to the zeros of Jacobi polynomials are also obtained assuming we are working in a coherent case. More recently, for a noncoherent pair of measures, when α β, M 2, N 1 N 2 0, and ξ 1 −ξ 2 , the distribution of zeros of the corresponding Sobolev orthogonal polynomials as well as some asymptotic results more precisely, inner strong asympttics, outer relative asymptotics, and Mehler-Heine formulas for these sequences of polynomials are deduced in 7-9 . In the Jacobi case, some analog problems have been considered in 10, 11 .
The aim of this contribution is to study necessary conditions for W N,p -norm convergence of the Fourier expansion in terms of Jacobi-Sobolev orthogonal polynomials. In order to prove it, we need some estimates and strong asymptotics for the polynomials Q α,β,N n x as well as for their derivatives Q α,β,N n x . A Mehler-Heine-type formula, inner strong asymptotics, upper bounds in −1, 1 , and W N,p norms of Jacobi-Sobolev orthonormal Journal of Inequalities and Applications 3 polynomials are obtained. Thus, we extend the results of 10 for generalized N-coherent pairs of measures.
The structure of the manuscript is as follows. In Section 2, we give some basic properties of Jacobi polynomials that we will use in the sequel. In Section 3, an algebraic relation between the sequences of polynomials Q α,β,N n n≥0 and Jacobi orthonormal polynomials is stated. It involves N 1 where N M k 1 N k 1 consecutive terms of such sequences in such a way that we obtain a generalization of the relations satisfied in the coherent case. Upper bounds for the polynomials Q α,β,N n x and their derivatives in −1, 1 are deduced. The inner strong asymptotics as well as a Mehler-Heine-type formula are obtained. Finally, the asymptotic behavior of these polynomials with respect to the W N,p norm is studied. In Section 4, necessary conditions for the convergence of the Fourier expansions in terms of the sequence of Jacobi-Sobolev orthogonal polynomials are presented.
Throughout this paper, positive constants are denoted by c, c 1 , . . . and they may vary at every occurrence. The notation u n ∼ v n means that the sequence u n /v n converges to 1 and notation u n ∼ v n means c 1 u n ≤ v n ≤ c 2 u n for sufficiently large n.

Preliminaries
For α, β > −1, we denote by p α,β n n≥0 the sequence of Jacobi polynomials which are orthonormal on −1, 1 with respect to the inner product We will denote by k π n the leading coefficient of any polynomial π n x , and π n x k π n −1 π n x . Now, we list some properties of the Jacobi orthonormal polynomials which we will use in the sequel.
e Mehler-Heine formula (see [12,Theorem 8 where α, β are real numbers and J α z is the Bessel function of the first kind. This formula holds locally uniformly, that is, on every compact subset of the complex plane.

2.8
Let {s n x } ∞ n 0 be the sequence of orthonormal polynomials with respect to the inner product 1.5 , and let and Next, we will consider the polynomials where a n,i A n,i n Taking into account that the zeros of the polynomial Π N x orthogonal with respect to where a 1 if q α and a −1 if q β.
b When x ∈ −1, 1 and α, β ≥ −1/2, we get the following estimate for the polynomials v n : c Mehler-Heine type formula. We get where α, β are real numbers, and J α z is the Bessel function of the first kind. This formula holds locally uniformly, that is, on every compact subset of the complex plane.
d Inner strong asymptotics. When

Asymptotics of Jacobi-Sobolev Orthogonal Polynomials
Let {Q α,β,N n x } ∞ n 0 denote the sequence of polynomials orthogonal with respect to 1.3 normalized by the condition that they have the same leading coefficient as v n x , that is, k Q α,β,N n a n−1,0 k p α,β n .
The following relation between Q α,β,N n and v n x holds.

3.6
As a conclusion,

Journal of Inequalities and Applications 9
We will prove 3.16 by induction. When l 0, it is a trivial result. On the other hand, applying 3.1 in a recursive way, we get

3.17
Taking into account 3.7 , we have b 1 N 1,n 0. Thus, 3.16 follows for l 1. Now, we assume 3.16 holds for l ≥ 1. Again, from 3.1 ,  N, N 1, . . . , n. First, the following inequality holds:  Next, we will give some properties of the Jacobi-Sobolev orthogonal polynomials.
Proof. a Using Proposition 3.2, we have Proof. Here, we will prove only the case when α ≥ β. 3.38 Proof. The inequality holds for θ ∈ 0, c/n , as well as for θ ∈ π − c/n, π . Therefore, from Propositions 3.3 and 3.4, the statement follows immediately.
Next, we deduce a Mehler-Heine-type formula for Q where α, β are real numbers, and J α z is the Bessel function of the first kind.
Proof. To prove the proposition, we use the same technique as in 17 . a Multiplying in 3.1 by n 1 −α−1/2 , we obtain 3.45 On the other hand, from Proposition 2.3 c , V n n≥0 is uniformly bounded on compact subsets of C. Thus, for a fixed compact set K ⊂ C, there exists a constant C, depending only on K, such that when z ∈ K, Thus, the sequence Y n n≥0 is uniformly bounded on K ⊂ C. As a conclusion, Notice that the upper estimate in 3.54 and 3.55 can also be proved using the bounds for Jacobi-Sobolev polynomials given in Corollary 3.5. In order to prove the lower bound in 3.51 we will need the following. 3.57