PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 96 (110) (2014), 227–231 DOI: 10.2298/PIM1410227N

ON A CONJECTURE OF NEVAI

Incoronata Notarangelo

Abstract. It is shown that a conjecture concerning the derivatives of orthogonal polynomials, proved by Nevai in 1990 for generalized Jacobi weights, holds for doubling weights as well.

1. Introduction and main result

In 1990, an international conference on approximation theory was held at the University of South Florida, bringing together for the first time a large number of approximation theorists from the United States and Soviet countries. In the proceedings of this conference Nevai conjectured that the following theorem could be extended to the case in which the orthogonal polynomials p_{n}(α) are replaced by their derivatives.

Theorem 1.1 (Máté, Nevai, Totik [6]). Let 0 < p ⩽ ∞. Then there is a constant C with the property that for every measure α supported in [−1, 1] such that α′ > 0 almost everywhere there, the inequality

( ∫_{−1}^{1} | \frac{f(t)}{\sqrt{α′(t)}(1 − t^{2})^{1/4}} |^{p}dt) ^{1/p} ⩽ C lim inf_{n→∞} ( ∫_{−1}^{1} |f(t)p_{n}(α, t)|^{p}dt) ^{1/p}

holds for every measurable function f in [−1, 1].

We emphasize that Theorem 1.1 is useful in several contexts (e.g., Fourier series in orthogonal systems, related interpolation and quadrature processes).

Conjecture (Nevai [7]). Let 0 < p ⩽ ∞. Then there is a constant C with the property that for every measure α supported in [−1, 1] such that α′ > 0 almost everywhere there, the inequality

2010 Mathematics Subject Classification: 42C05.

Key words and phrases: Orthogonal polynomials, derivatives of orthogonal polynomials, doubling weights.

Dedicated to Professor Giuseppe Mastroianni on the occasion of his retirement.

This work was supported by University of Basilicata (local funds) and by National Group of Computing Science (GNCS–INDAM): Progetto Giovani Ricercatori 2013 “Weighted polynomial approximation and its application to the numerical treatment of integral equations”.

( ∫_{−1}^{1} | \frac{f(t)}{\sqrt{α′(t)}(1 − t^{2})^{3/4}} |^{p}dt) ^{1/p} ⩽ C lim inf_{n→∞} \frac{1}{n}( ∫_{−1}^{1} |f(t)p′_{n}(α, t)|^{p}dt) ^{1/p}

holds for every measurable function f in [−1, 1].

In the same paper, Nevai proved that the conjecture is true in the case of sufficiently smooth generalized Jacobi weight functions, concluding that “Nevertheless, I do not know how to prove similar results for general weights or measures”.

Theorem 1.2 (Nevai [7]). Let

U(x) = ∏_{k=1}^{m} |x − y_{k}|^{A_{k}} and W (x) = g(x) ∏_{k=1}^{m} |x − y_{k}|^{B_{k}}

where y_{k} ∈ [−1, 1], A_{k} > −1, B_{k} > −1, and g is a nonnegative function. If r ⩾ 0, p ⩾ 1 and 0 < c_{1} ⩽ g(x) ⩽ c_{2} < ∞ for x ∈ [−1, 1], then there is a positive constant C such that

∫_{−1}^{1} \frac{U(x)}{W (x)^{\frac{r+p}{2}} (1 − x^{2})^{\frac{r+3p}{4}}} dx ⩽ C lim inf_{n→∞} \frac{1}{n^{p}} ∫_{−1}^{1} |p_{n}(W, x)|^{r}|p′_{n}(W, x)|^{p}U(x) dx.

Subsequently, Nevai and Xu proved Theorem 1.2 with the interval of integration [−1, 1] replaced by an arbitrary subinterval Δ ⊆ [−1, 1], in order to show the convergence of the Hermite interpolating polynomial to a continuous and differentiable function in weighted L^{p}-norm (see [8]).

The aim of this paper is to prove that Theorem 1.2 holds with the generalized Jacobi weights replaced by doubling weights. To be more precise, we will prove the following equivalent statement.

Theorem 1.3. Let u, w be doubling weights, with w > 0 a.e. in [−1, 1], and φ(x) = \sqrt{1 − x^{2}} . If r ⩾ 0, 1 ⩽ p < ∞, then there exists a positive constant C such that

∫_{−1}^{1} \frac{u(x)}{w(x)^{\frac{r+p}{2}} φ(x)^{\frac{r+p}{2}}} dx ⩽ C lim inf_{m→∞} \frac{1}{m^{p}} ∫_{−1}^{1} |p_{m}(w, x)|^{r}|p′_{m}(w, x) φ(x)|^{p}u(x) dx.

We note that, if uφ^{−p} is a doubling weight, the statement can be rewritten as

∫_{−1}^{1} \frac{u(x)}{w(x)^{\frac{r+p}{2}} (1 − x^{2})^{\frac{r+3p}{4}}} dx ⩽ C lim inf_{m→∞} \frac{1}{m^{p}} ∫_{−1}^{1} |p_{m}(w, x)|^{r}|p′_{m}(w, x)|^{p}u(x) dx.

Before proving Theorem 1.3, in the next Section we recall some preliminary results.

2. Basic facts

In the sequel C will stand for a positive constant that can assume different values in each formula. Moreover, letting A and B be positive quantities depending on some parameters, we will write A ∼ B if there exists a positive constant C (independent of these parameters) such that (A/B)^{±1} ⩽ C. Finally, we will denote by ℙ_{m} the set of all algebraic polynomials of degree at most m.

The definition and the main properties of the doubling weights can be found in Stein’s book [9] (see also [1, 2, 3, 4, 5]). Here, we recall only the results used in the proof of Theorem 1.3. We say that a weight u satisfies the doubling property if, for some constant L

∫_{2I} u(t) dt ⩽ L ∫_{I} u(t) dt

for all intervals I ⊂ [−1, 1], where 2I denotes the twice enlarged I (enlarged from its center). For instance, the above generalized Jacobi weights satisfy the doubling property. We emphasize that a doubling weight can vanish on a set of positive measure (see [9, Chap. I, Sect. 8.8]).

Let u be a doubling weight. Setting

Δ_{m}(x) = \frac{φ(x)}{m} + \frac{1}{m^{2}} = \frac{\sqrt{1 − x^{2}}}{m} + \frac{1}{m^{2}} ,

we define

u_{m}(x) = \frac{1}{Δ_{m}(x)} ∫_{x−Δ_{m} (x)}^{x+Δ_{m} (x)} u(t) dt.

From the inequality

u_{m}(x) ⩽ C( 1 + m|x − t| + m|\sqrt{1 − x^{2}} − \sqrt{1 − t^{2}} |)^{s}u_{m}(t),

where x, t ∈ [−1, 1] and s is a constant, it follows that (see [4])

(1)  u_{m}(x) ∼ u_{m}(t),  |x − t| ⩽ CΔ_{m}(x).

Moreover, in [4] Mastroianni and Totik proved that

u_{bm}(x) ∼ u_{m}(x), b ⩾ 1, (2)

∫_{−1}^{1} |P_{m}(x)|^{p}u_{m}(x) dx ∼ ∫_{−1}^{1} |P_{m}(x)|^{p}u(x) dx (3)

for any P_{m} ∈ ℙ_{m} and for p > 0. Note that (3) holds also with P_{m} replaced by a generalized polynomial of the form T_{m}(x) = ∏ _{i} |x − x_{i}|^{A_{i}} , m = ∑ _{i} A_{i}, A_{i} ⩾ 1(see [4]).

In all the previous relations C and the constants in “∼” are independent of m and P_{m}.

Let w be a doubling weight and {p_{m}(w)}_{m∈ℕ} be the corresponding sequence of orthonormal polynomials with positive leading coefficients. We denote by x_{k} := x_{m,k} (w), k = 1, …, m, the zeros of p_{m}(w) located as −1 < x_{1} < x_{2} < ⋯< x_{m} < 1 . Furthermore, we set x_{0} = −1 and x_{m+1} = 1. Then the equivalence

(4)  Δx_{k} = x_{k+1} − x_{k} ∼ Δ_{m}(x_{k}) = \frac{\sqrt{1 − x_{k}^{2}}}{m} + \frac{1}{m^{2}} , k = 0, …, m,

holds with the constants in “∼” independent of m, as shown by Mastroianni and Totik in [5].

3. Proof of Theorem 1.3

We use arguments similar to those in [7, pp. 87–90].

Let x ∈ [x_{k}, x_{k+1} ], k = 1, …, m. Since p_{m}(w, x_{k}) = 0, we have

|p_{m}(w, x)|^{\frac{r}{p} +1} ⩽ ( \frac{r}{p} + 1) ∫_{x_{k}}^{x} |p_{m}(w, t)|^{\frac{r}{p}} |p′_{m}(w, t)| dt,

⩽ ( \frac{r}{p} + 1) ∫_{x_{k}}^{x_{k+1}} |p_{m}(w, t)|^{\frac{r}{p}} |p′_{m}(w, t)| dt,

and

|p_{m}(w, x)|^{r+p} ⩽ ( \frac{r}{p} + 1)^{p}(Δx_{k})^{p−1} ∫_{x_{k}}^{x_{k+1}} |p_{m}(w, t)|^{r}|p′_{m}(w, t)|^{p}dt,

having used the Hölder inequality in the case p > 1.

Then, multiplying both sides by u_{m}(x) and using (1), we obtain

|p_{m}(w, x)|^{r+p} u_{m}(x) ⩽ C(Δx_{k})^{p−1} ∫_{x_{k}}^{x_{k+1}} |p_{m}(w, t)|^{r}|p′_{m}(w, t)|^{p}u_{m}(t) dt,

and, integrating over [x_{k}, x_{k+1} ],

∫_{x_{k}}^{x_{k+1}} |p_{m}(w, x)|^{r+p} u_{m}(x) dx ⩽ C(Δx_{k})^{p} ∫_{x_{k}}^{x_{k+1}} |p_{m}(w, t)|^{r}|p′_{m}(w, t)|^{p}u_{m}(t) dt.

By (4) and since Δ_{m}(x_{k}) ∼ Δ_{m}(t), it follows that

∫_{x_{k}}^{x_{k+1}} |p_{m}(w, x)|^{r+p} u_{m}(x) dx ⩽ C ∫_{x_{k}}^{x_{k+1}} |p_{m}(w, t)|^{r}|p′_{m}(w, t)Δ_{m}(t)|^{p}u_{m}(t) dt

whence, summing on k = 1, …, m, we get

∫_{−1}^{1} |p_{m}(w, x)|^{r+p} u_{m}(x) dx ⩽ C ∫_{−1}^{1} |p_{m}(w, t)|^{r}|p′_{m}(w, t)Δ_{m}(t)|^{p}u_{m}(t) dt.

Now, since u is doubling, also the weight φ^{p}u satisfies the doubling property and (φ^{p}u)_{m}(x)∼ m^{p}Δ_{m}(x)^{p}u_{m}(x). So, by (2) and (3), we get

∫_{−1}^{1} |p_{m}(w, x)|^{r+p} u(x) dx ⩽ \frac{C}{m^{p}} ∫_{−1}^{1} |p_{m}(w, t)|^{r}|p′_{m}(w, t)φ(t)|^{p}u(t) dt,

where C is independent of m.

Then, using Theorem 1.1, with f and p replaced by u^{\frac{1}{r+p}} and r+p, respectively, we obtain

∫_{−1}^{1} \frac{u(x)}{w(x)^{\frac{r+p}{2}} φ(x)^{\frac{r+p}{2}}} dx ⩽ C lim inf_{m→∞} \frac{1}{m^{p}} ∫_{−1}^{1} |p_{m}(w, t)|^{r}|p′_{m}(w, t)φ(t)|^{p}u(t) dt,

which completes the proof.

Acknowledgement. The author wishes to thank Professors Giuseppe Mastroianni, Paul Nevai, and Vilmos Totik for their useful remarks.

References

1. G. Mastroianni, V. Totik, Jackson type inequalities for doubling and A_{p} weights, Proc. Third Internat. Conf. on Functional Analysis and Approximation Theory, Vol. I (Acquafredda di Maratea, 1996); Rend. Circ. Mat. Palermo (2) Suppl. No. 52, Vol. I (1998), 83–99.

2. , Jackson type inequalities for doubling weights. II, East J. Approx. 5 (1999), 101–116.

3. , Best approximation and moduli of smoothness for doubling weights, J. Approx Theory 110 (2001), 180–199.

4. , Weighted polynomial inequalities with doubling and A_{∞} weights, Constr. Approx. 16(1) (2000), 37–71.

5. , Uniform spacing of zeros of orthogonal polynomials, Constr. Approx. 32(2) (2010), 181–192.

6. A. Máté, P. Nevai, V. Totik, Necessary conditions for weighted mean convergence of Fourier series in orthogonal polynomials, J. Approx. Theory 46(3) (1986), 314–322.

7. P. Nevai, Orthogonal polynomials, recurrences, Jacobi matrices, and measures, in: A. A. Gonchar and E. B. Saff (ed.), Progress in Approximation Theory, Springer Ser. Comput. Math. 19, Springer, New York, 1992, 79–104.

8. P. Nevai, Y. Xu, Mean convergence of Hermite interpolation, J. Approx. Theory 77 (1994), 282–304.

9. E. M. Stein, Harmonic Analysis. Princeton University, Princeton, NJ, 1993.

Department of Mathematics, Computer Sciences and Economics University of Basilicata Potenza Italy

    incoronata.notarangelo@unibas.it