Orthogonal polynomials for exponential weights x^2α(1 – x²)^2ρe^–2Q(x) on [0, 1)

Definition 1.1

(see [1, Definition 1.1]). Let I = [0, d). Assume that W = e^–Q where Q : I → [0, ∞) satisfies the following properties:

1. x Q′(x) ∈ C(I) with limit 0 at 0 and Q(0) = 0.

2. Q″ exists in (0, d), while Q^*″ is positive in (0, d ).

3. limx→d−Q(x)=∞.

4. The function

   T(x):=xQ′(x)Q(x),x∈(0,d)

   is quasi-increasing in (0, d), with

   T(x)≥Λ>12,x∈(0,d). (1.3)

5. There exists C₁ > 0 such that

   |Q″(x)|Q′(x)≤C1Q′(x)Q(x),a.e.x∈(0,d). (1.4)

   Then we write W ∈ 𝓛(C²). If also there exists a compact subinterval J of I^*, and C₂ > 0 such that

   Q∗″(x)|Q∗′(x)|≥C2|Q∗′(x)|Q∗(x),a.e.x∈I∗∖{J}, (1.5)

   then we write W ∈ 𝓛(C²+).

Theorem 1.1

Let I = [0, 1) and W ∈ 𝓛(C²) (or W ∈ 𝓛(C²+)). Suppose that there exists λ such that for x ∈ I ∖ {0},

where

and Λ is defined by (1.3). Then we have Ŵ ∈ 𝓛(C²) (or 𝓛(C²+)).

Theorem 1.2

Let α > – 12 , W ∈ 𝓛(C²) and the other assumptions of Theorem 1.1 be valid. Assume that 0 < p ≤ ∞.

1. Let L, ζ ≥ 0. Let β > – 1p if p < ∞ and β ≥ 0 if p = ∞. There exist C₁, n₀ > 0 such that for n ≥ n₀ and P ∈ P_n,

   PW^(x)xβLp(I)≤C1PW^(x)xβLp[La^nn−2,a^n(1−ζη^n)].

   Moreover, given r > 1, there exist C₂, n₀, ν > 0 such that for n ≥ n₀ and P ∈ P_n,

   PW^(x)xβLp(a^rn,1)≤exp⁡(−C2nν)PW^(x)xβLp(Δ^n).

2. Let β > – 1p if p < ∞ and β ≥ 0 if p = ∞. Given 0 < r < 1. Then for n ≥ 1, P ∈ P_n and for some C,

   PW^′(x)φ^n♯(x)xβLp(I)≤CPW^(x)xβLp(I),P′W^(x)xβLp(I)≤Cn2a^nPW^(x)xβLp(I)

   and

   P′W^(x)xβLp[a^rn,1)≤Cna^nT^(a^n)PW^(x)xβLp(I).

3. Let L > 0. Then uniformly for n ≥ 1 and x ∈ [0, â_n(1 + Lη̂_n)],

   λn(Wα,ρ2;x)∼φ^n(x)W^2(x)x+a^nn22α.

   Moreover, there exists C > 0 such that uniformly for n ≥ 1 and x ∈ I,

   λn(Wα,ρ2;x)≥Cφ^n(x)W^2(x)x+a^nn22α.

4. Then uniformly for n ≥ 1,

   supx∈Ipn(Wα,ρ2;x)W^(x)x+a^nn2αx+a^nn2(a^n−x)14∼1.

5. There exist C₃, C₄ > 0 such that for n ≤ 1 and 1 ≤ j ≤ n – 1,

   xjn−xj+1,n≤C3φ^n(xjn),xnn∼a^nn−2

   and

   a^n(1−C4η^n)≤x1n<a^n+α+14.

   Furthermore, for each fixed j and n, x_jn is a non-decreasing function of α.

Theorem 1.3

Let α > – 12 , W ∈ 𝓛(C²+) and the other assumptions of Theorem 1.1 be valid.

1. Let 0 < β < 1, then uniformly for n ≥ 1,

   supx∈I|pn(x)|W^(x)x+a^nn2α∼na^n12,supx∈[a^βn,1)|pn(x)|W^(x)x+a^nn2α∼a^n−12(nT^(a^n))16

   and

   xjn−xj+1,n∼φ^n(xjn),1≤j<n.

   If W ∈ 𝓛(C²), these estimates hold with ∼ replaced by ≤ C.

2. There exists n₀ such that uniformly for n > n₀, 1 ≤ j ≤ n,

   pn′Wα,ρ(xjn)∼φ^n(xjn)−1xjn(a^n−xjn)−14, (1.8)

   pn−1Wα,ρ(xjn)∼a^n−1xjn(a^n−xjn)14,maxx∈Iℓjn(x)W^(x)x+a^nn2α(Wα,ρ)−1(xjn)∼1 (1.9)

   and

   1−x1na^n∼η^n.

   If we assume instead that W ∈ 𝓛(C²), then (1.8) holds with ∼ replaced by ≤ C and (1.9) holds with ∼ replaced by ≥ C.

3. For j ≤ n – 1 and x ∈ [x_j+1,n, x_jn],

   pnWα,ρ(x)∼min|x−xjn|,|x−xj+1,n|φ^n(xjn)−1xjn(a^n−xjn)−14.

Theorem 1.4

Let I = [0, 1), τ > 0, and

1. Then

   T(x)≥1,x∈(0,1). (1.10)

2. If

   τ>4|ρ|,ρ<0,0,ρ≥0, (1.11)

   then we have Ŵ ∈ 𝓛(C²+).

Theorem 1.5

Let I = [0, 1), k ≥ 1, τ > 0, and

1. Then the relation of (1.10) holds.

2. If

   τ>4|ρ|∏j=1kexpj⁡(1),ρ<0,0,ρ≥0, (1.12)

   then we have Ŵ ∈ 𝓛(C²+).

Remark 1.1

For Theorem 1.4 and Theorem 1.5, Levin and Lubinsky in [1, 2] discussed the case when ρ = 0.

Lemma 2.1

1. Let the assumptions of Theorems 1.1 be valid. Then there exist μ > 0 and

   λ=2|ρ|μ,ρ≠0

   such that

   μQ″(x)≥|ρ|q″(x),x∈I∖{0} (2.1)

   and

   μQ∗′′(x)≥|ρ|q∗′′(x),x∈I∗∖{0}. (2.2)

2. Let I = [0, 1) and W ∈ 𝓛(C²). Assume that there exists μ > 0 such that (2.2) is valid. Then (2.1) holds. Moreover, there exists λ such that for x ∈ I ∖ {0}, (1.6) and (1.7) hold.

Proof

The case when ρ = 0 is trivial. Let ρ ≠ 0.

(a) Given any ε > 0, choose

Then

Clearly,

Then inequality (1.6) with the above relations may be written as

so the relation (2.1) follows from (2.5) directly.

If we introduce the notation

then according to (2.1),

With the help of (2.7) for x ∈ (0, 1),

and

Inequalities of (2.7), (2.8) and (2.9) give the estimates

Here according to (1.1) and (1.2) we introduce

and then for s ∈ I^* ∖ {0} = (–1, 1) ∖ {0},

Since s ∈ (–1, 1) ∖ {0} is equivalent to s² ∈ (0, 1), so using (2.12) and (2.10) we obtain

This proves the relation (2.2).

(b) Given (2.2), using (2.12) it is shown that one of the following cases will hols:

Case 1:

and

Case 2:

and

According to (2.7) and (2.8), it is clear that Case 1 gives (2.1).

Here Case 2 means that

and

Using the same arguments as (2.7) and (2.8), we see that Case 2 is contradictory. So we get (2.1).

Further, set λ=2|ρ|μ, where μ is defined by (2.3), then coupling with (2.1) we obtain (1.6) and (1.7).□

Remark 2.1

According to the above Lemma (b), the result implies Q″(x) > 0, x ∈ (0, 1). Moreover, we see that the assumptions of Theorems 1.1 and Lemma 2.1(b) are equivalent.

Lemma 2.2

Let

Then

and

Proof

By a short calculation we gain

□