Asymptotic behavior of the partial derivatives of Laguerre kernels and some applications

J. Math. Anal. Appl. 421 (2015) 314–328

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Asymptotic behavior of the partial derivatives of Laguerre kernels and some applications Francisco Marcellán a,b , M. Francisca Pérez-Valero b,∗ , Yamilet Quintana c a

Instituto de Ciencias Matemáticas (ICMAT), Campus de Cantoblanco, Universidad Autónoma de Madrid, Calle Nicolás Cabrera, no. 13-15, 28049 Madrid, Spain b Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain c Departamento de Matemáticas Puras y Aplicadas, Universidad Simón Bolívar, Edificio Matemáticas y Sistemas (MYS), Apartado Postal: 89000, Caracas 1080 A, Venezuela

a r t i c l e

i n f o

Article history: Received 21 March 2014 Available online 11 July 2014 Submitted by P. Nevai Keywords: Orthogonal polynomials Laguerre polynomials Asymptotics Diagonal Christoffel–Darboux kernels Sobolev-type orthogonal polynomials

a b s t r a c t The aim of this paper is to present some new results about the asymptotic behavior of the partial derivatives of the kernel polynomials associated with the Gamma distribution. We also show how these results can be used in order to obtain the inner relative asymptotics for certain Laguerre–Sobolev type polynomials. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Let μ be a finite positive Borel measure supported on an infinite subset of R. It is well-known that the polynomial kernels (also called reproducing, Christoffel–Darboux or Dirichlet kernels) associated with the sequences of orthogonal polynomials corresponding to μ are frequently used as a basic tool in spectral analysis, convergence of orthogonal expansions [2,23,27], and other aspects of mathematical analysis (see [26] and the references therein). In the setting of orthogonal polynomial theory these kernels have been especially used by Freud and Nevai [4,21,22] and, more recently, the remarkable Lubinsky’s works [9,10] have caused heightened interest in this topic. Also, other interesting and related results corresponding to Fourier–Sobolev expansions may be found in [11–15,17–19,25]. * Corresponding author. E-mail addresses: [email protected] (F. Marcellán), [email protected] (M.F. Pérez-Valero), [email protected] (Y. Quintana). http://dx.doi.org/10.1016/j.jmaa.2014.07.012 0022-247X/© 2014 Elsevier Inc. All rights reserved.

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Our goal here will be to analyze the asymptotic behavior of the partial derivatives of the diagonal Christoffel–Darboux kernels corresponding to classical Laguerre orthogonal polynomials (in short, the diagonal Laguerre kernels), i.e., we will consider the n-th Christoffel–Darboux kernel Kn (x, y), given by Kn (x, y) =

n α   α (y) L (x)L k

k=0

k

α , L  α α L k k

,

and its partial derivatives Kn(j,k) (x, y) :=

∂ j+k Kn (x, y) , ∂xj ∂y k

0 ≤ i, j ≤ n,

 α (x)}n≥0 is the sequence of monic polynomials orthogonal with respect to the inner where, as it is usual, {L n product ∞ f, gα =

f (x)g(x)xα e−x dx,

α > −1, f, g ∈ P,

0

and P denotes the linear space of polynomials with real coefficients. Then, for c > 0 we will study the (j,k) asymptotic behavior of Kn (c, c), 0 ≤ j, k ≤ n. From this starting point, we will focus our attention on the study of asymptotic properties of the sequences of polynomials orthogonal with respect to the following Sobolev-type inner product on the linear space of polynomials with real coefficients P: f, gS = f, gα +

N 

Mk f (k) (c)g (k) (c),

(1)

k=0

where c > 0, Mk ≥ 0, for k = 0, . . . , N − 1, and MN > 0. (j,k) To the best of our knowledge, asymptotic properties of the diagonal Laguerre kernels Kn (c, c), 0 ≤ j, k ≤ n, are not available in the literature up to those cases where the following situations have been considered. • Case 1: c ≥ 0 and either j = k = 0 or 0 ≤ j, k ≤ 1 (cf. [6,8]). • Case 2: c = 0 and either 0 ≤ j, k ≤ 1 or 0 ≤ j, k ≤ n (cf. [3,24]). The outline of the paper is as follows. Section 2 provides some basic background about structural and asymptotic properties of the classical Laguerre polynomials. The estimates of the partial derivatives of the (j,k) diagonal Christoffel–Darboux kernels Kn (c, c) (Theorem 2) are deduced. In Section 3 we prove our main result (Theorem 3), where an estimate in the Laguerre weighted L2 -norm for the difference between Laguerre  α,M (x), orthogonal with respect to orthonormal polynomials and the Laguerre–Sobolev type polynomials L n (1) with c > 0, is obtained. Let consider the multi-indexes M = (M0 , . . . , MN ) of nonnegative real numbers, MN > 0. The notation un ∼n vn will always mean that the sequence un /vn converges to 1 when n tends to infinity. Positive constants will be denoted by C, C1 , Ci,j , . . . and they may vary at every occurrence. Any other standard notation will be properly introduced whenever needed. 2. Asymptotics for the partial derivatives of the diagonal Laguerre kernels  α (x)}n≥0 , {L ˜ α (x)}n≥0 , and {L(α) For α > −1, let {L n (x)}n≥0 be the sequences of monic, orthonormal n n n and normalized Laguerre polynomials with leading coefficient equal to (−1) n! , respectively. The following

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proposition summarizes some structural and asymptotic properties of the classical Laguerre polynomials (see [6,7,19] and the references therein). α Proposition 2.1. Let {L n (x)}n≥0 be the sequence of monic Laguerre orthogonal polynomials. Then the following statements hold. 1. For every n ∈ N, 

α α L n , Ln

 α

 α 2 Ln α = Γ (n + 1)Γ (n + α + 1). = 

(2)

2. Hahn’s condition. For every n ∈ N, d α  α+1 (x). L (x) = nL n−1 dx n

(3)

3. The n-th Laguerre kernel Kn (x, y) satisfies the Christoffel–Darboux formula (cf. [27, Theorem 3.2.2]): α  α (y) − L  α (x)L  α (y)

Ln+1 (x)L n n n+1 , Kn (x, y) =  α 2 x − y L n α 1

x = y, n ≥ 0.

(4)

4. The so-called confluent form of the above kernel is Kn (x, x) =

α   1   α   α  (x)L  α (x) , Ln+1 (x)Ln (x) − L n n+1 α 2  Ln α

n ≥ 0.

5. (See [27, Theorem 8.22.2].) Perron generalization of Fejér formula on R+ . Let α ∈ R. Then for x > 0 we have   −1/2 x/2 −α/2−1/4 α/2−1/4 L(α) e x n cos 2(nx)1/2 − απ/2 − π/4 n (x) = π p−1     · Ak (x)n−k/2 + O n−p/2 k=0

  + π −1/2 ex/2 x−α/2−1/4 nα/2−1/4 sin 2(nx)1/2 − απ/2 − π/4 p−1     · Bk (x)n−k/2 + O n−p/2 ,

(5)

k=0

where Ak (x) and Bk (x) are certain functions of x independent of n and regular for x > 0. The bound for the remainder holds uniformly in [, ω]. For k = 0 we have A0 (x) = 1 and B0 (x) = 0. In the next result, we show a confluent form for the partial derivatives of the kernel polynomial Kn−1(x, y) for x = y = c. Proposition 2.2. For every n ∈ N and 0 ≤ j, k ≤ n − 1, we have (k,j) Kn−1 (c, c)

=

j!k!  α 2 (j + k + 1)!L n−1 α



j  j+k+1 l=0

l

 α (l) α (j+k+1−l)   α (l) α (j+k+1−l) n  n (c) L  n−1  n−1 (c) L (c) − L (c) . × L

(6)

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Proof. For k = 0 and 0 ≤ j ≤ n − 1, it is enough to follow a standard technique in literature (see, for instance [1, p. 269]) by taking derivatives in (4) with respect to the variable y and then to evaluate at y = c. Thus we obtain (0,j)

Kn−1 (x, c) =



j!  α 2 (x L n−1 α

− c)j+1

  α    α  α α Tj x, c; L n−1 Ln (x) − Tj x, c; Ln Ln−1 (x) ,

(7)

where Tj (x, c; f ) is the j-th Taylor polynomial of f around y = c. α α Using the Taylor expansion of L n (x) and Ln−1 (x) in (7), we only need to look for the coefficients of j+k+1 (x − c) therein in order to find (6). 2  (α) Taking p = 1 in (5), it is difficult to analyze the behavior of L n (x), x ∈ R+ , for n large enough, i.e.  α (x) = (−1)n Γ (n + 1)π −1/2 ex/2 x−α/2−1/4 nα/2−1/4 L n     · cos 2(nx)1/2 − απ/2 − π/4 1 + O n−1/2 . So, we can rewrite the above expression as follows     α (x) = (−1)n Γ (n + 1)n α2 − 14 σ α (x) cos ϕα (x) 1 + O n−1/2 , L n n

(8)

where 1/2 ϕα − n (x) = 2(nx)

απ π − , 2 4

and σ α (x) = π −1/2 ex/2 x−α/2−1/4 , is a function independent of n. Now, our task is to find the asymptotic behavior of the diagonal Laguerre kernels. In order to do this we must estimate expressions like α+n0 α+n0 α cos ϕα n−n1 (c) cos ϕn−n2 (c) − cos ϕn−n3 (c) cos ϕn (c).

Under some conditions on the parameters n0 , n1 , n2 , and n3 , we will prove that the above expression goes to zero when n tends to infinity and, moreover, we can compute its speed of convergence. The result reads as follows. Lemma 1. Let n, n0 ∈ N, α > −1, and c ∈ R+ . For fixed n1 , n2 , n3 ∈ N with n3 = n1 + n2 and n ≥ n3 , let us consider α+n0 α+n0 α F α,c (n) := cos ϕα n−n1 (c) cos ϕn−n2 (c) − cos ϕn−n3 (c) cos ϕn (c).

Then, ⎧ −1 −1 ⎪ ⎪ 4 (n2 − n1 − n3 )(n2 − n1 + n3 )cn ⎪ √ ⎪ ⎨ −1 (n2 − n1 − n3 ) cn−1/2 2 F α,c (n) ∼n 1 −1 ⎪ ⎪ 4 (n2 − n1 − n3 )(n2 − n1 + n3 )cn ⎪ ⎪ √ −1/2 ⎩1 2 (n2 − n1 − n3 ) cn

if n0 ≡ 0 mod 4, if n0 ≡ 1 mod 4, if n0 ≡ 2 mod 4, if n0 ≡ 3 mod 4.

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Proof. By using trigonometric identities, a straightforward computation yields F α,c (n) = f α,c (n) + g c (n), where

f

α,c

   (n0 + 1)π (n) = − sin cn + c(n − n1 ) + c(n − n2 ) + c(n − n3 ) − απ − 2    √  × sin c(n − n2 ) + c(n − n1 ) − c(n − n3 ) − cn , √



and g c (n) = − sin



× sin

c(n − n1 ) −



c(n − n2 ) +

 √  c(n − n3 ) − cn

   √ n0 π . c(n − n1 ) − c(n − n3 ) + cn − c(n − n2 ) − 2

Our first technical step will be to show that lim n3/2 sin



n→∞

c(n − n2 ) +



c(n − n1 ) −



c(n − n3 ) −

√

√  n1 n2 c . cn = 4

(9)

√ √ √ √ Since h(n) = ( n − n − n2 ) − ( n − n1 − n − n3 ) can be written as h(n) = k(n) − k(n − n1 ), with √ √ k(n) = n − n − n2 , then, according to the mean value theorem, we obtain h(n) = n1 k (ξn ) = Let l(n) =

√1 . n

n1 2



1 1 √ −√ , ξn ξn − n2

where n − n1 ≤ ξn ≤ n.

By using the mean value theorem we get h(n) =

 n1 n1  −n1 n2 −3/2 l(ξn ) − l(ξn − n2 ) = n2 l (δn ) = δn , 2 2 2

where n − n1 − n2 ≤ ξn − n2 ≤ δn ≤ ξn ≤ n. Taking into account that limn→∞ the first factor in f α,c (n) is bounded, we obtain

δn n

= 1, we get (9). Since

lim n1/2 f α,c (n) = 0 = lim nf α,c (n).

n→∞

n→∞

(10)

In our second technical step will show that the speed of convergence of the first factor in g c (n) is n−1/2 . In a similar way to the previous situation, we have lim

n→∞

=

√

n sin

    √ c(n − n1 ) − c(n − n3 ) + cn − c(n − n2 )

1√ c(n3 − n1 + n2 ). 2

Then, the speed of convergence of g c (n) can be determined by discussing the following four cases (i) If n0 ≡ 0 mod 4, using that sin(x − 2π) = sin(x), then lim ng c (n) =

n→∞

−1 c(n2 − n1 − n3 )(n2 − n1 + n3 ) = 0. 4

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(ii) If n0 ≡ 1 mod 4, using that sin(x − π2 ) = cos(x), then lim n1/2 g c (n) =

n→∞

−1 √ c(n2 − n1 − n3 ) = 0. 2

(iii) If n0 ≡ 2 mod 4, using that sin(x − π) = − sin(x), then 1 c(n2 − n1 − n3 )(n2 − n1 + n3 ) = 0. 4

lim ng c (n) =

n→∞

(iv) If n0 ≡ 3 mod 4, using that sin(x −

= − cos(x), then

3π 2 )

lim n1/2 g c (n) =

n→∞

1√ c(n2 − n1 − n3 ) = 0. 2

Finally, from the above analysis and (10) the statement of the lemma follows. 2 Theorem 2. For c > 0, we get the behavior of the partial derivatives of the diagonal Laguerre kernels  (k,j) Kn−1 (c, c)

∼n

C0,k,j n C1,k,j n

j+k+1 2

if j + k ≡ 0 mod 2,

j+k 2

if j + k ≡ 1 mod 2,

where 0 ≤ j, k ≤ n − 1 and √ (−1) 2 +j α σ (c)σ α+j+k+1 (c) c, k+j+1 j+k+1 k−j α σ (c)σ α+j+k+1 (c)c. = (−1) 2 +j k+j j+k

C0,k,j = C1,k,j

Proof. Without loss of generality, we can assume that j ≤ k. From (3) and (6), we obtain (k,j)

Kn−1 (c, c) =

j!k!nj+k+1 α (j + k + 1)!L

2 n−1 α

×

j  j+k+1  l

l=0

  α+j+k+1−l (c) − L  α+l (c)L  α+j+k+1−l (c) .  α+l (c)L L n−1−l n−l n−j−k−1+l n−j−k−2+l

(11)

Now, using (2) and (8), we get (k,j) Kn−1 (c, c)

∼n

j  j+k+1 l=0

×

l

(−1)j+k

j+k j!k! σ α (c)σ α+j+k+1 (c)n 2 +1 F α+l,c (n − l). (j + k + 1)!

On the other hand, for all l = 0, . . . , j, Lemma 1 with n0 = j + k + 1 − 2l, n1 = 1, n2 = j + k + 1 − 2l, and n3 = j + k + 2 − 2l implies that ⎧ (−1)l (j + k + 1 − 2l)cn−1 ⎪ ⎪ ⎪ ⎪ ⎨ (−1)l √cn−1/2 F α+l,c (n − l) ∼n ⎪ (−1)l+1 (j + k + 1 − 2l)cn−1 ⎪ ⎪ ⎪ √ ⎩ (−1)l+1 cn−1/2

if j + k + 1 ≡ 0 mod 4, if j + k + 1 ≡ 1 mod 4, if j + k + 1 ≡ 2 mod 4, if j + k + 1 ≡ 3 mod 4.

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Since the above relation can be reduced to  j+k √ (−1)l+ 2 cn−1/2 α+l,c F (n − l) ∼n j+k−1 (−1)l+ 2 (j + k + 1 − 2l)cn−1

if j + k ≡ 0 mod 2, if j + k ≡ 1 mod 2,

we obtain the statement of the theorem with C0,k,j = (−1)

j+k 2

j √  j+k+1 j!k! σ α (c)σ α+j+k+1 (c) c (−1)l , (k + j + 1)! l l=0

C1,k,j = (−1)

j+k−1 2

j  j!k! j+k+1 α α+j+k+1 σ (c)σ (c)c (j + k + 1 − 2l)(−1)l+1 . (k + j + 1)! l l=0

Using the identities



j  j+k+1 j+k (−1)l = (−1)j , l j l=0

j  l=0

j+k+1 (j + k − 1)! (k − j), (j + k + 1 − 2l)(−1)l+1 = (−1)j+1 (j + k + 1) j!k! l

the above expressions read √ (−1) 2 +j α σ (c)σ α+j+k+1 (c) c, = k+j+1 j+k+1 k−j α σ (c)σ α+j+k+1 (c)c. = (−1) 2 +j k+j j+k

C0,k,j C1,k,j

Thus, we conclude the proof of the theorem. 2 Remark 1. Notice that Theorem 2 generalizes the asymptotic behavior of the diagonal Laguerre kernels given in [6] (where only the case 0 ≤ j, k ≤ 1 has been analyzed). The interested reader can find an analogous result of Theorem 2 when c = 0, 0 ≤ j, k ≤ 1, and c = 0, 0 ≤ j, k ≤ n − 1, in [3,24], respectively. Also, it is worthwhile to point out that with a different approach the authors of [8] obtained a lower bound for the Christoffel functions when c ≥ 0. 3. Inner relative asymptotics As an application of Theorem 2, we will study the inner relative asymptotics for a certain family of Laguerre–Sobolev type orthogonal polynomials. More precisely, we compare the behavior of the Sobolev and standard Laguerre polynomials on (0, ∞) for n large enough. The main result in this section guarantees the norm convergence of the Laguerre–Sobolev polynomials to the Laguerre ones in the Laguerre L2-norm. Before to deal with the general case, we are going to analyze a more simple framework. For example, let us consider the Sobolev type inner product f, gS = f, gα + M f  (c)g  (c),

(12)

where α > −1, c > 0 and M > 0. Notice that this is just a particular case of the family of inner products  M,α defined in [16]. Let {L (x)}n≥0 be the monic Laguerre–Sobolev polynomials orthogonal with respect n to (12). We also consider the normalization

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321

 M,α  M,α (x) = Ln (x) , L n α L n α i.e., the normalized Laguerre–Sobolev type orthogonal polynomials with the same leading coefficient as the classical orthonormal Laguerre polynomial of degree n. Then (see [16, Eq. (2.8)]) ˜ α (x) =  M,α (x) − L L n n

˜ α ) (c) M (L n (1,1)

1 + M Kn−1 (c, c)

(0,1)

Kn−1 (x, c).

Let consider the standard L2 -Laguerre norm of the previous expression, i.e.  M,α 2 ˜α   Ln − L n α =

˜ α ) (c)]2 M 2 [(L n (1,1)

(1 + M Kn−1 (c, c))2

(1,1)

Kn−1 (c, c) ≤

˜ α ) (c)]2 [(L n (1,1)

.

Kn−1 (c, c)

Now, from Proposition 2 we obtain (1,1)

3

Kn−1 (c, c) ∼n Cn 2 , and, on the other hand, 

 α+1  n!(−1)n−1 (α+1) ˜ α  (c) = nLn−1 (c) = L L (c), n 1/2 n−1 α  (Γ (n + 1)Γ (n + α + 1)) Ln α

from which it follows that  α  2 ˜ n (c) = L

 (α+1) 2 n! L (c) ≤ Cn1/2 . Γ (n + α + 1) n−1

As a consequence,  M,α  ˜ α 2 ≤ Cn−1 ,  Ln − L n α so, we have proved the norm convergence of the n-th Laguerre–Sobolev type orthogonal polynomial to the n-th Laguerre one:  M,α  ˜ α  = 0. Ln − L lim  n α

n→∞

3.1. The multi-index case  α,M (x) the corresponding monic orthogonal Let us consider the Sobolev type inner product (1) and L n polynomial of degree n. Also, we consider the normalization  α,M  α,M (x) = Ln (x) , L n α L n α i.e., the Laguerre–Sobolev polynomials with the same leading coefficient as the orthonormal Laguerre ones. From now on, we will denote by j1 < · · · < jq the indexes such that Mj1 −1 = · · · = Mjq −1 = 0. Theorem 3. With the above notation, the inner relative asymptotics for the Laguerre–Sobolev polynomials orthogonal with respect to (1) reads  α,M  ˜ ˜ α  = 0. lim L −L n n α

n→∞

(13)

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Proof. Following a standard technique we can expand the Laguerre–Sobolev type orthogonal polynomials in terms of the Laguerre classical ones to obtain ˜ α (x) −  α,M (x) = L L n n

N n−1 

 α,M (j)  α (j)  ˜ ˜ α (x) Mj L (c) L (c)L n k k

k=0 j=0

˜ α (x) − =L n

N 

 α,M (j) (j,0)  Mj L (c)Kn−1 (c, x). n

(14)

j=0

 α,M )(j) (c) when j = 0, . . . , N , j = j1 − 1, . . . , jq − 1, are needed. At this point, estimations for (L n In order to do that, we can write (14) evaluated at x = c in a matrix form as follows, ALα,M = Lα , where ⎛

(1,0)

1 + M0 Kn−1 (c, c)

⎜ (0,1) ⎜ M0 Kn−1 (c, c) ⎜ ⎜ (0,2) M0 Kn−1 (c, c) A=⎜ ⎜ ⎜ .. ⎜ . ⎝

M1 Kn−1 (c, c) (1,1)

M2 Kn−1 (c, c)

(2,0)

...

MN Kn−1 (c, c)

(2,1)

...

MN Kn−1 (c, c)

...

MN Kn−1 (c, c) .. .

1 + M1 Kn−1 (c, c)

M2 Kn−1 (c, c)

M1 Kn−1 (c, c) .. .

(1,2)

1 + M2 Kn−1 (c, c) .. .

(1,N ) M1 Kn−1 (c, c)

(2,N ) M2 Kn−1 (c, c)

(0,N ) M0 Kn−1 (c, c)

(2,2)

..

.

(N,0) (N,1) (N,2)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

(N,N )

. . . 1 + MN Kn−1 (c, c)

 α  α   α (N ) T ˜ (c), L ˜ (c), . . . , L ˜ Lα = L (c) , n n n and  α,M  α,M   α,M (N ) T    Lα,M = L (c), L (c), . . . , L (c) . n n n Here, v T denotes the transpose of the vector v. Then, applying Cramer’s rule we get 

 α,M L n

(m−1)

(c) =

det(Am ) , det(A)

for m = 1, . . . , N + 1,

where Am is the matrix obtained by replacing the m-th column in the matrix A by the column vector Lα . Thus, by using Lemmas 5 and 6, for n large enough we obtain  α,M (m−1)  −2m−1 ˜n  L (c) ≤ Cn 4 ,

(15)

where C is a positive constant which does not depend on n. Finally, in order to obtain (13) we take norm in (14). Thus  N 2    α,M   α,M (j)   (j,0) α 2 ˜   Ln − Ln α ≤  Mj Ln (c)Kn−1 (c, x)   j=0

≤ (N + 1)

α

N  j=0

 α,M (j) 2 (j,j)  Mj2 L (c) Kn−1 (c, c). n

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From Theorem 2 and (15) we get N   α,M  −4(j+1)−2 2j+1 ˜ α 2 ≤ (N + 1)  4 Ln − L Cj Mj2 n n 2 n α j=0 −1

≤ Cn

2

.

Remark 2. Notice that in [5] estimates in the weighted L2 -norm for the difference between continuous Sobolev orthogonal polynomials associated with a vector of measures (ψW, W ) and standard orthogonal polynomials associated with W , where W is an exponential weight W (x) = e−2Q(x) and ψ is a measurable and positive function on a set of positive measure, such that the moments of the Sobolev product are finite, have been obtained in terms of the Mhaskar–Rakhmanov–Saff number. The authors assume that Q is an even and convex function on the real line such that Q is continuous in (0, ∞) and Q > 0 in (0, ∞), as well  (x) as for some 0 < α < β, α ≤ xQ Q (x) ≤ β, x ∈ (0, ∞) holds. The study of analogue estimates as above for general exponential weights constitutes an interesting problem in which we are working. Acknowledgments The work of the first author (FM) has been supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, grant MTM 2012-36732-C03-01. The work of the second author (MFPV) has been supported by the Research Fellowship Program, Ministerio de Ciencia e Innovación (MTM 2009-12740-C03-01) and Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, grant MTM 2012-36732-C03-01. The work of the third author (YQ) has been partially supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, grant MTM 2012-36732-C03-01. Appendix A. Estimates for det(A) and det(Am ) First of all we will need the following well-known result, see for instance, [20, vol. III, p. 311]. Lemma 4 (Cauchy’s double alternant). Let x1 , . . . , xn , y1 , . . . , yn be real numbers. Then, !

1 det xi + yj

#

" =

1≤i
#

1≤i,j≤n

− xj )(yi − yj )

1≤i,j≤n (xi

+ yj )

.

Let us denote M :=

N +1 $

Ml−1 ,

N +1 

Q :=

l=1 l=j1 ,...,jq

l.

l=1 l=j1 ,...,jq

Lemma 5. With the notation introduced in Section 3, we have det(A) ∼n C1 n

2Q−(N +1)+q 2

,

(A.1)

where C1 is a positive constant independent of n. In particular, there exists a constant C2 > 0 such that det(A) > C2 n for n large enough.

2Q−(N +1)+q 2

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324

Proof. We denote by aij , 1 ≤ i, j ≤ N + 1, the (i, j) entry of the matrix A. Notice that these entries verify ⎧ (j−1,i−1) ⎪ (c, c) ⎨ Mj−1 Kn−1 aij ∼n 1 ⎪ ⎩ 0 Then, from Theorem 2, we obtain ⎧ i+j−1 ⎪ Mj−1 C0,j−1,i−1 n 2 , if ⎪ ⎪ ⎪ i+j−2 ⎨M C j−1 1,j−1,i−1 n 2 , if aij ∼n ⎪ 1 if ⎪ ⎪ ⎪ ⎩ 0 if

for j such that Mj−1 > 0, if i = j and Mj−1 = 0, if i = j and Mj−1 = 0.

i + j ≡ 0 mod 2 and Mj−1 > 0, i + j ≡ 1 mod 2 and Mj−1 > 0, i = j and Mj−1 = 0,

(A.2)

i = j and Mj−1 = 0.

Using the definition of determinant and (A.2), we get 

det(A) =

sgn(δ)a1,δ(1) · · · aN +1,δ(N +1)

δ∈SN +1



∼n

sgn(δ)Cδ n

−p1 (δ) 2

n−p2 (δ)

δ∈SN +1

N +1 $

n

l+δ(l) 2

,

(A.3)

l=1 l=j1 ,...,jq

where SN +1 is the group of permutations of the set {1, . . . , N + 1}, p1 (δ) (resp. p2 (δ)) is the number of indexes l in {1, . . . , N + 1}\{j1 , . . . , jq } such that l + σ(l) is even (resp. odd) and N +1 $

Cδ =

Mδ(l)−1 C0,δ(l)−1,l−1 .

l=1 l=j1 ,...,jq

Let us define the set % δ ∈ SN +1

Δ=

:

l + δ(l) is even for all l = 1, . . . , N + 1, and δ(l) = l, for l = j1 , . . . , jq

& .

Notice that p12(δ) + p2 (δ) attains a minimum when p2 (δ) = 0. Then, the asymptotic behavior of (A.3) will be given by the terms corresponding to permutations in Δ, if they do not vanish. Thus, we have to check ' that δ∈Δ sgn(δ)Cδ is not zero. 

sgn(δ)Cδ =

δ∈Δ



sgn(δ)

 δ∈Δ

Mδ(l)−1 C0,δ(l)−1,l−1

l=1 l=j1 ,...,jq

δ∈Δ

=

N +1 $

sgn(δ)

N +1 $

Mδ(l)−1 (−1)

l=1 l=j1 ,...,jq

√ σ α (c)σ α+l+σ(l)−1 (c) c

l=1 l=j1 ,...,jq

Recalling that σ α+l+δ(l)−1 (c) = π −1/2 ec/2 c N +1 $

l+σ(l) +l 2

σ α+l+δ(l)−1 (c) = π −

−α+l+δ(l)−1 2

N +1−q 2

e

1 . l + δ(l) − 1

c−1/4 , we get

(N +1−q)c 2

c−

α(N +1−q)+2Q−(N +1−q) 2

 N +1−q −Q  α N +1−q N +1−q −Q = σ α−1 (c) c = σ (c) c 2 c .

(A.4)

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325

After some computations, (A.4) becomes 

2(N +1−q) N +1−q−Q   sgn(δ)Cδ = M (−1)2Q σ α (c) c sgn(δ)

δ∈Δ

N +1 $ l=1 l=j1 ,...,jq

δ∈Δ

1 . l + δ(l) − 1

Now, let us consider {1, 2, . . . , N + 1}\{j1 , j2 , . . . , jq } = {r1 , r2 , . . . , rK1 } ∪ {s1 , s2 , . . . , sK2 } where ri is odd for i = 1, 2, . . . , K1 and si is even for i = 1, 2, . . . , K2 . Notice that K1 + K2 = N + 1 − q. Then, we have  δ∈Σ

sgn(δ)

N +1 $ l=1 l=j1 ,...,jq

 1 = l + δ(l) − 1



sgn(δ) sgn(ξ)

δ∈SK1 ξ∈SK2



=

sgn(δ)

δ∈SK1

#

=#

K1 $

K1 $

1 r + r − 1 s + s −1 i j δ(i) ξ(j) i=1 j=1 

1

r + rδ(i) − 1 i=1 i

1≤i
1≤i
#

− rj )

2

+ rj − 1)

K2 $

1

#

sgn(ξ)

ξ∈SK2

1≤i
1≤i
K2 $

1 s + s ξ(j) − 1 j=1 j

− sj )2

+ sj − 1)

,

where we have used Lemma 4 in the sense ! det

ri −

1 2

1 + rj −

" 1 2



= 1≤i,j≤K1

sgn(δ)

δ∈SK1

K1 $

1

i=1

ri + rδ(i) − 1

.

Finally, (A.4) becomes  δ∈Δ

# # 2 (si − sj )2  α 2(N +1−q) N +1−q−Q 1≤i
which is, as desired, different from zero. Then, we can state that det(A) ∼n Cn

2Q−(N +1)+q 2

,

(A.5)

where C is a positive constant independent of n. This concludes the proof. 2 Lemma 6. For n large enough, there exists a constant C > 0 such that   +q − 34 det(Am ) ≤ Cn 2Q−m−N 2 . Proof. Notice that for i = m, the entries of the matrix Am are the same as those of the matrix A. Their asymptotic behavior was given in (A.2). Let us denote by a ˆim the (i, m) entry of the matrix Am . According to (8), we have  α (i−1) i 3 ˜n a ˆim = L (c) ∼n (−1)n−i+1 σ α+i−1 (c)n 2 − 4 cos ϕn+i−1 n−i+1 (c), for i = 1, . . . , N + 1.

(A.6)

F. Marcellán et al. / J. Math. Anal. Appl. 421 (2015) 314–328

326

We expand det(Am ) along the m-th column: N +1 

(−1)i+m a ˆim det Bim ,

det(Am ) =

(A.7)

i=1

where Bim is the N × N matrix obtained by deleting of A the i-th row and the m-th column. Using (A.6) in (A.7), we obtain det(Am ) ∼n

N +1 

(−1)n+m+1 σ α+i−1 (c)n 2 − 4 cos ϕn+i−1 n−i+1 (c) det Bim , i

3

i=1

where det Bim can be computed as 

det Bim =

sgn(σ)

σ∈SN

N $

bl,σ(l) =

l=1



sgn(ψ)

ψ∈Ψ

N +1 $

al,ψ(l) ,

(A.8)

l=1 l=i,j1 ,...,jq

with % Ψ=

ψ ∈ SN +1

ψ(l) = l, for l = j1 , . . . , jq and ψ(i) = m

:

& .

Now, we will discuss two cases: 1. Case i + m even. The highest power of n that can be reached in the sum (A.8) appears when l + ψ(l) is even for all l = 1, . . . , N + 1, l = i, j1 , . . . , jq . This means that det Bim ∼n C



2Q−i−m−N +q 2 sgn(γ)Cγ n ,

γ∈Γ

with ⎧ ⎨ Γ = γ ∈ SN +1 ⎩

:

⎫ l + γ(l) is even for all l = 1, . . . , N + 1, ⎬ , γ(l) = l, for l = j1 , . . . , jq ⎭ and γ(i) = m

whenever 

sgn(γ)Cγ = 0.

(A.9)

γ∈Γ

2. Case i + m odd. +q−1 In this case, the highest power of n in the sum (A.8) could be at most 2Q−i−m−N − 1, when the 2 permutation ψ satisfies that l + ψ(l) is odd for one l ∈ {1, . . . , N + 1}\{i, j1 , . . . , jq }, and it is even for the remainder indexes. We obtain the highest power of n for the first case, and after checking (A.9), we conclude det(Am ) ∼n

N +1  i=1

(−1)n+m+1 σ α+i−1 (c)n

2Q−m−N +q − 34 2

cos ϕn+i−1 n−i+1 (c).

F. Marcellán et al. / J. Math. Anal. Appl. 421 (2015) 314–328

327

Then, for n large enough, there exists a constant C > 0 such that   +q − 34 det(Am ) ≤ Cn 2Q−m−N 2 . In order to conclude the proof we must check that (A.9) holds. Indeed, 

sgn(γ)

γ∈Γ

=

N +1 $

Mγ(l)−1 C0,γ(l)−1,l−1

l=1 l=i,j1 ,...,jq

 2(N −q) 2N +i+m−2q−2Q  M 2 (−1)Q−i σ α (c) c sgn(γ) Mm−1 γ∈Γ

N +1 $ l=1 l=i,j1 ,...,jq

1 . l + γ(l) − 1

Let suppose now that m is even. Let {1, 2, . . . , N + 1}\{i, j1 , j2 , . . . , jq } = {r1 , r2 , . . . , rK1 } ∪ {m, s1 , s2 , . . . , sK2 }, where ri is odd for i = 1, 2, . . . , K1 , and si is even for i = 1, 2, . . . , K2 . Notice that K1 + K2 = N − q. We can write + , K1 N +1  $ $  1 1 = det B, sgn(γ) sgn(δ) l + γ(l) − 1 r + rδ(i) − 1 i=1 i γ∈Γ

l=1 l=i,j1 ,...,jq

where

δ∈SK1

⎛

1 m+i−1 1 s1 +i−1

1 m+s1 −1 1 s1 +s1 −1

.. .

.. .

1 sK2 +i−1

1 sK2 +s1 −1

⎜ ⎜ B=⎜ ⎜ ⎝

... ... .. . ...

1 m+sK2 −1 1 s1 +sK2 −1

.. .

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

1 sK2 +sK2 −1

Using Lemma 4, 

N +1 $

sgn(γ)

γ∈Γ

l=1 l=i,j1 ,...,jq

1 l + γ(l) − 1

# # − sl )(i − sl ) 1≤i
#K2

l=1 (m

=

In an analogue way, if m is odd, let {1, 2, . . . , N + 1}\{i, j1 , j2 , . . . , jq } = {m, r1 , r2 , . . . , rK1 } ∪ {s1 , s2 , . . . , sK2 }, where ri is odd for i = 1, 2, . . . , K1 , and si is even for i = 1, 2, . . . , K2 , and 

N +1 $

sgn(γ)

γ∈Γ

# # − rl )(i − rl ) 1≤i
#K1 =

l=1 l=i,j1 ,...,jq

1 l + γ(l) − 1

l=1 (m

This is different from zero and we get our statement. 2

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