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POLYNOMIAL APPROXIMATION WITH POLLACZECK-LAGUERRE WEIGHTS ON THE REAL SEMIAXIS. A SURVEY.

1. Introduction. This paper is a short survey on weighted polynomial approximations of functions defined on the real semiaxis. The function may grow exponentially both at 0 and at +[infinity]. As far as we know, this topic has received attention in the literature only recently (see [12, 13, 14, 15, 16]). We consider weight functions of the form

(1.1) [mathematical expression not reproducible]

Even though w can be seen as a combination of a Pollaczeck-type weight [mathematical expression not reproducible] and a Laguerre-type weight [mathematical expression not reproducible], one cannot investigate the problem by reducing it to a combination of a Pollaczeck-type case (on, say, [0,1]) and a Laguerre-type case (on [1, +00)).

We are going to present the main results concerning orthogonal polynomials, polynomial inequalities, function spaces with new moduli of smoothness, and estimates for the best polynomial approximation with respect to the weight w. We also pay due attention to Gaussian rules and Lagrange interpolation in weighted [L.sup.2] -norms. The behaviour of the related Fourier sums and their discrete versions, the Lagrange polynomials, in the [L.sup.p]-norms remains an open problem.

In the sequel c, C will stand for positive constants that may assume different values in each formula, and we shall write C [not equal to] C(a, b,...) when C is independent of a, b,... Furthermore, A ~ B means that if A and B are positive quantities depending on some parameters, then there exists a positive constant C independent of these parameters such that [(A/B).sup.+1] < C. Finally, we denote by [P.sub.m] the set of all algebraic polynomials of degree at most m. As usual N, Z, R, will stand for the sets of all natural, integer, and real numbers, while Z+ and [R.sup.+] denote the sets of positive integer and positive real numbers, respectively.

2. Orthogonal polynomials. First of all we note that the weight w defined by (1.1) can be reduced to a weight belonging to the class F([C.sup.2]+), introduced by Levin and Lubinsky in [7, pp. 7-8], by a linear transformation. Let us recall the definition of this class for the reader's convenience.

Let I = (c, d) be an interval with -[infinity] < c < 0 < d < +oo, and g : I G R be a weight function with g = [e.sup.-Q], Q : I [member of] [0, +oo) satisfying the following properties: (i) Q' is continuous in I and Q(0) = 0;

(ii) Q" exists and is positive in I \ {0};

(iii) li[m.sub.a - c]+ Q(x) = li[m.sub.x-d]- Q(x) = oo;

(iv) the function

is quasi-decreasing in (c, 0) and quasi-increasing in (0, d) with

T(x) > [LAMBDA] > 1, x [member of] E I \ {0} ;

(v) there exist [C.sub.1], [C.sub.2] > 0 and a compact subinterval JCI such that

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

Then we say that g [member of] D([C.sup.2]+).

With the previous notation, we can state the following lemma.

LEMMA 2.1 (see [16, pp. 817-818]). Letting w be the weight in (1.1), there exists a [lambda] > 0 such that the weight w defined as

[mathematical expression not reproducible]

with

[mathematical expression not reproducible]

belongs to the class F([C.sup.2] +).

Therefore, we have that w(y) = Cw(y + [lambda]), where [lambda] is the unique positive zero of

[mathematical expression not reproducible]

We can deduce the properties of the orthogonal polynomials with respect to our weight w from the results obtained by Levin and Lubinsky using the inverse transformation. The Mhaskar-Rakhmanov-Saff (MRS) numbers, [[epsilon].sub.[tau]] = [[epsilon].sub.[tau]](w) and [a.sub.[tau]] = [a.sub.[tau]](w), related to

[mathematical expression not reproducible]

are defined by

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

PROPOSITION 2.2 (see [16, pp. 820] and [7, p. 13]). For [tau] > 0, [[epsilon].sub.T] is a decreasing function, and [a.sub.[tau]] is an increasing function of[tau] , and

[mathematical expression not reproducible]

with

(2.1) [mathematical expression not reproducible]

and

(2.2) [mathematical expression not reproducible]

Let us denote by [{p.sub.m](w)}.sub.me]N the sequence of orthonormal polynomials defined by

[mathematical expression not reproducible] + lower degree terms, [[gamma].sub.m] = [[gamma].sub.m](w) > 0,

and

[mathematical expression not reproducible]

The zeros of [p.sub.m](w) lie in the MRS interval associated with Aw. Here and for the rest of the paper, we use the notation [[epsilon].sub.[tau]] = [[epsilon].sub.[tau]] (y/w) and [a.sub.[tau]] = [a.sub.[tau]] ([mathematical expression not reproducible]), taking into account that, by definition, [mathematical expression not reproducible]. The next proposition provides further information concerning the distribution of these zeros.

PROPOSITION 2.3 (see [14, pp. 1656-1657] and [7, pp. 312-324]). The zeros of[p.sub.m](w) are located as

[[epsilon].sub.m] < [x.sub.1] < [x.sub.2] < * * * < [x.sub.m] < [a.sub.m] ,

with

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

where the constants in "~" are independent of m.

The distance between two consecutive zeros [DELTA][x.sub.k] = [x.sub.k+1] - [x.sub.k]. can be estimated by

[DELTA][x.sub.k] ~ [psi].sub.m]([x.sub.k]), k = 1,..., m - 1,

where

[mathematical expression not reproducible]

and the constants in "~" are independent ofk and m.

Now, letting [theta] [member of] (0,1) be fixed, we define two indexes j1 = j1(m) and j2 = j2(m) as

(2.3) [mathematical expression not reproducible]

and

For the sake of completeness, if {[x.sub.k] : [x.sub.k] < [epsilon][[theta].sub.m]} or {xk : [x.sub.k] > a[[theta].sub.m]} are empty, then we set [x[j.sub.1] = x1 or [x.sub.j2] = [x.sub.m], respectively.

From Proposition 2.3, it follows that

[mathematical expression not reproducible]

Let

[mathematical expression not reproducible]

be the mth Christoffel function and

[[lambda].sub.k](w) = [[lambda].sub.m](w, [x.sub.k]) , k = 1,. .., m ,

be the Christoffel numbers related to w.

PROPOSITION 2.4 (see [7, p. 257]). We have

[mathematical expression not reproducible]

where [psi].sub.m] is given by

[mathematical expression not reproducible]

and the constants in "~" are independent of m. In particular, for [theta] [member of] (0,1), we get

[mathematical expression not reproducible]

From the numerical point of view, in order to compute the zeros of [p.sub.m] (w) and the Christoffel numbers, we use a procedure given in [14] (see also [18, ([section]) 4.2]) and the MATHEMATICA package OrthogonalPolynomials (cf. [3] and [19]), which is freely downloadable from the website: http://www.mi.sanu.ac.rs/~gvm/.

For the sake of brevity we omit the description of the numerical procedures for the computation of the zeros of [p.sub.m](w), the Christoffel numbers, and the Mhaskar-Rahmanov-Saff numbers [[epsilon].sub.m] and [a.sub.m]. The interested reader can find all the details about these procedures in [14, pp. 1676-1680] (cf. [15]).

The following estimates are crucial tools in order to study the convergence of several approximation processes.

PROPOSITION 2.5 (see [7, pp. 325 and 360]). We have

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

where the constants in "~" are independent of m.

PROPOSITION 2.6 (cf. [7, p. 25]). For the leading coefficient of[p.sub.m](w), we have

[mathematical expression not reproducible]

where q(x) = 1 ([x.sup.[alpha]]+ [x.sup.[beta]] - [gamma] log x).

3. Polynomial inequalities. Letting w be given by (1.1), x [member of] [R.sub.+], we introduce the weight function

(3.1) [mathematical expression not reproducible]

[mathematical expression not reproducible]

In the sequel, by a slight abuse of notation, we denote by || * ||.sub.p] the quasinorm of the Lp -spaces for 0 < p < 1 defined in the usual way.

LEMMA 3.1 (see [16, p. 809]). Let [delta] G R and n = m + \\[delta]\.. For any [P.sub.m] [euro] [P.sub.m] with 0 < p < 00, we have

||Pmu||.sub.p] <C||[P.sub.m]u||.sub.Lp [[epsilon]nan]],

where C = C(m, [P.sub.m]) and [[epsilon].sub.n], [a.sub.n] are defined by (2.1) and (2.2). On the other hand, for any s > 1, we have

||[P.sub.m]u||.sub.L]p([R.sub.+]\[[[epsilon].sub.sm],[a.sub.sm]]) < [Ce-.sup.cmv] ||[P.sub.m]u||.sub.p],

where

(3.2) [mathematical expression not reproducible]

and C and c are independent of m and [P.sub.m].

For the rest of the paper, let

[mathematical expression not reproducible]

The following lemma is of independent interest and gives rise to a useful procedure for verifying polynomial inequalities.

LEMMA 3.2 (see [16, p. 809]). For a sufficiently large m (say m > [m.sub.0]), there exists a polynomial R[e.sub.m] [member of] [P.sub.m], with [pounds sterling] a fixed integer, such that

[R.sub.lm](x) ~ w(x)

and

[mathematical expression not reproducible]

for x [member of] [[[epsilon].sub.m], [a.sub.m]], where [[epsilon].sub.m] = [[epsilon].sub.m](w) and [a.sub.m] = [a.sub.m](w) are defined by (2.1) and (2.2). The constants in "~" andC are independent of m.

By Lemmas 3.1 and 3.2 we reduce the problem for the polynomial inequalities related to the weight u on (0, +oo) to analogous inequalities on bounded intervals with Jacobi weights. In fact, we get:

THEOREM 3.3 (see [16, p. 810]). Let 0 <p < [infinity]. Then, for any [P.sub.m] [euro] [P.sub.m], we have

(3.3) [mathematical expression not reproducible]

and

(3.4) [mathematical expression not reproducible]

where C [not equal to] C(m, [P.sub.m]).

We want to emphasize that the presence of the algebraic factor [x.sup.[delta]] in the definition of u allows us to iterate the Bernstein inequality (3.3) as follows:

[mathematical expression not reproducible]

for 1 [less than or equal to] r [member of] Z. Also, the factor

[mathematical expression not reproducible]

in the Markoff inequality (3.4) is smaller than the one appearing in the analogous inequality (see [17])

[mathematical expression not reproducible]

with the generalized Laguerre weight [mathematical expression not reproducible] on (0, +[infinity]), whereas the factors of the Bernstein inequalities for the weights u and w[beta] are the same.

Using standard arguments, the Markoff inequality (3.4) can be deduced from the Bernstein inequality (3.3) and the Schur inequality stated in the following theorem.

THEOREM 3.4 (see [16, p. 810]). Let 0 <p [less than or equal to] oo. Then, for any [P.sub.m] [euro] [P.sub.m], we have

[mathematical expression not reproducible]

where v[delta][98(x).sup.[delta]] = x andC [not equal to] C(m, [P.sub.m]).

In analogy with the Bernstein and Markoff inequalities, we give two versions of the Nikolskii inequality.

THEOREM 3.5 (see [16, p. 810]). Let 0 < p < q [less than or equal to] oo. Then, for any [P.sub.m] [euro] [P.sub.m], we have

(3.5) [mathematical expression not reproducible]

and

(3.6) [mathematical expression not reproducible]

where C = C(m, [P.sub.m]).

In analogy with different weighted polynomial inequalities, the factor [mathematical expression not reproducible] in the second Nikolskii inequality is the same as the one appearing in the Markoff inequality.

4. Function spaces, K--functionals, and moduli of smoothness. Let us now define some function spaces related to the weight u (see [13, pp. 168-172]). By [L.sup.p.sub.u], 1 < p < [infinity], we denote the set of all measurable functions f such that

[mathematical expression not reproducible]

while, for p = oo, by a slight abuse of notation, we set

[mathematical expression not reproducible]

with the norm

[mathematical expression not reproducible]

For smoother functions we introduce the Sobolev-type spaces

[mathematical expression not reproducible]

where 1 < p < oo, 1 < r [member of] Z + , [phi](x) := x, and AC(0, +oo) denotes the set of all absolutely continuous functions on (0, +oo). We equip these spaces with the norm

[mathematical expression not reproducible]

To characterize some subspaces of [L.sup.p.sub.u], we introduce the following moduli of smoothness. Let us consider the intervals

[mathematical expression not reproducible]

with [alpha] and [beta] in (3.1), h > 0 sufficiently small, and c > 1 an arbitrary but fixed constant. For

any f G [L.sup.p.sub.u], 1 < p < [infinity], r > 1, and t > 0 sufficiently small (say t < t0), we set

[mathematical expression not reproducible]

where

[mathematical expression not reproducible]

Moreover, we introduce the following K-functional

[mathematical expression not reproducible]

and its main part

[mathematical expression not reproducible]

The main part of the K-functional is equivalent to the main part of the previous modulus of smoothness as the following lemma shows.

LEMMA 4.1 (see [13, p. 171]). Letr [greater than or equal to] 1 and0 < t < t0 for some t0 < 1. Then, for any f [member of] [L.sup.p.sub.u], 1 < p < [infinity], we have

where the constants in "~" are independent of f and t.

Then we define the complete rth modulus of smoothness by

[mathematical expression not reproducible]

with c > 1 a fixed constant. We emphasize that the behaviour of [[omega].sup.[tau].sub.[phi]](f, t).sub.up] is independent of the constant c. Moreover, the following lemma shows that this modulus of smoothness is equivalent to the K-functional.

LEMMA 4.2 (see [13, p. 172]). Letr [greater than or equal to] 1 and0 <t < t0 for some t0 < 1. Then, for any f [member of] [L.sup.p.sub.u], 1 [less than or equal to] p [less than or equal to] oo, we have

[mathematical expression not reproducible]

where the constants in "~" are independent of f andt.

By means of the main part of the modulus of smoothness, for 1 < p < oo, we can define the Zygmund-type spaces

[mathematical expression not reproducible]

s [member of] R+, with the norm

[mathematical expression not reproducible]

We remark that, in the definition of [Z.sup.p.sub.u](u), the main part of the rth modulus of smoothness [[ohm].sup.r] (f,t).sub.up] can be replaced by the complete modulus [mathematical expression not reproducible] as can be deduced from Theorem 5.1 in next section.

5. Weighted approximation and embedding theorems.

5.1. Estimates for the best weighted approximation. Let us denote by

[mathematical expression not reproducible]

the error of the best polynomial approximation of a function f [member of] [L.sup.p.sub.u], 1 [less than or equal to] p [less than or equal to] oo. The following Jackson, weak Jackson, and Stechkin inequalities hold true.

THEOREM 5.1 (see [13, p. 173]). For any f [member of] [L.sup.P.sub.U], 1 [less than or equal to] p [less than or equal to] oo, and m > r [greater than or equal to] 1, we have

(5.1) [mathematical expression not reproducible]

[mathematical expression not reproducible]

and, assuming [mathematical expression not reproducible],

[mathematical expression not reproducible]

Finally, for any f [member of] [L.sup.p.sub.u], 1 < p < oo, we get

(5.2) [mathematical expression not reproducible]

In any case C is independent of m and f.

In particular, for any f [member of] [W.sup.P.sub.T](u), 1 [less than or equal to] p [less than or equal to] +[infinity], we obtain

(5.3) [mathematical expression not reproducible]

[mathematical expression not reproducible]

whereas, for any f [member of] [Z.sup.p.sub.s](u), 1 [less than or equal to] p [less than or equal to] +[infinity], we get

(5.4) [mathematical expression not reproducible]

[mathematical expression not reproducible]

From (5.1), (5.2), and (5.4) we deduce the following equivalences

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

for 1 < p < 00 and r > s.

5.2. Embedding theorems. Now, using Theorem 5.1, the dyadic decomposition, the Nikolskii inequalities (3.5) and (3.6), we can show some embedding theorems, connecting different subspaces of [L.sup.p.sub.u].

THEOREM 5.2 (see [12, p. 159]). For any f [member of] [L.sup.P.sub.U], 1 [less than or equal to] p [less than or equal to] [infinity], such that

[mathematical expression not reproducible]

where [eta] = (2[alpha] + 2)/(2[alpha] + 1), we have

[mathematical expression not reproducible]

where C depends only on r.

In the following theorem, we replace [eta]/p by 1/p.

THEOREM 5.3 (see [12, pp. 159-160]). For any f [member of] [L.sup.p.sub.u], 1 < p < oo, such that

[mathematical expression not reproducible]

we have

[mathematical expression not reproducible]

where C depends only on r.

From Theorem 5.3 we can easily deduce the following corollary, which is useful in several contexts.

COROLLARY 5.4 (see [12, p. 160]). If f [euro] [mathematical expression not reproducible], 1 < p [less than or equal to] [infinity], is such that

then f is continuous on (0, +[infinity]).

6. Quadrature rules and Lagrange interpolation. Here we are going to show a slight extension of the results proved in [14] for [gamma] = 0.

6.1. Gaussian formulas. The Gaussian rule related to the weight [mathematical expression not reproducible] can be defined by the equality

(6.1) [mathematical expression not reproducible]

[mathematical expression not reproducible]

where [x.sub.k] are the zeros of [p.sub.m](w), [[lambda].sub.k](w) are the Christoffel numbers, and (6.1) has to hold for any polynomial P2m-1 [member of] [P.sub.2m-1]. Thus the error of the Gaussian rule for any continuous function f is given by

[mathematical expression not reproducible]

Let us consider the weight

(6.2) [sigma](x) = [(1 + x).sup.[delta]][w.sup.[alpha]](x), [delta] [greater than or equal to] 0,0 < a [less than or equal to] 1.

Naturally, taking also into account Lemma 3.1, the results of Sections 3 and 4 hold with u replaced by [sigma]. If we assume that f [member of] [C.sub.[sigma]], then we can write

[mathematical expression not reproducible]

and the next proposition easily follows.

PROPOSITION 6.1 (cf. [14, p. 1660]). Ifw/[sigma] [member of] [L.sup.1], then, for any f [member of] [C.sub.a], we have

(6.3) [mathematical expression not reproducible]

where C = C(m, f).

This proposition generalizes a result due to Uspensky [20], who first proved convergence of Gaussian rules on unbounded intervals related to Laguerre and Hermite weights (see also [9, pp. 341-345] and [11]).

Notice that the assumption w/[sigma] [member of] [L.sup.1] in Proposition 6.1 is fulfilled if a = 1 and [delta] > 1 or if a < 1 and [delta] is arbitrary. The error estimate (6.3) implies convergence of the Gaussian rules for any f [member of] [C.sub.a]. For a sm[infinity]ther function, for instance f [member of] W ([sigma]), by (6.3) and (5.3), we obtain

[mathematical expression not reproducible]

where C [not equal to] C(m, f) and [a.sub.m] ~ [m.sup.1][1.sup.[beta]].

Thus, a natural question is how to establish the degree of convergence of [e.sub.m] (f) if the function f is infinitely differentiable, i.e., f [member of] C[infinity]([R.sub.+]). We recall that Aljarrah [1, 2] showed estimates of [e.sub.m] (f) related to Hermite or Freud weights for analytic functions in some domains of the complex plane containing the quadrature nodes. For precise estimates, considering the same class of functions and different weights, we refer to [8]. Here we consider the case of infinitely differentiable functions on [R.sub.+] with the condition that ([[f.sup.m])[sigma])(x) is uniformly bounded with respect to m and x. We note that the derivatives of the function can increase exponentially for x[right arrow]0 and x [right arrow] +[infinity].

THEOREM 6.2 (cf. [14, p. 1660]). Let [sigma] be the weight in (6.2) with 0 < a < 1 and [delta] arbitrary. For any infinitely differentiable function f, if K (f) := su[p.sub.m] [f.sup.(m)][sigma] < +[infinity], then we have

[mathematical expression not reproducible]

In order to study the behaviour of the Gaussian rule in the Sobolev spaces [W.sup.1.sub.1](w), it is natural to investigate whether estimates of the form

(6.4) [mathematical expression not reproducible]

hold true. We recall that, as shown in the previous section, for the error of the best approximation,

[mathematical expression not reproducible]

On the other hand, inequality (6.4) holds, mutatis mutandis, for Gaussian rules on bounded intervals related to Jacobi weights. But, as for many exponential weights (see, e.g., [4, 5, 10, ]), inequality (6.4) is false in the sense of the following theorem.

THEOREM 6.3 (cf. [14, p. 1661]). [mathematical expression not reproducible], and [gamma] > 0. Then, for any f [member of] [W.sup.1.sub.1](w), we have

[mathematical expression not reproducible]

where C is independent of m and f. Moreover, for a sufficiently large m (say m [greater than or equal to] [m.sub.0]), there exists a function [f.sub.m], with [greater than or equal to], and a constant C [not equal to] C(m, [f.sub.m]) such that

[mathematical expression not reproducible]

Nevertheless, estimates of the form (6.4) are required in different contexts. To obtain this kind of error estimates, using also an idea from [10], we are going to modify the Gaussian rule.

With [theta] [member of] (0,1) fixed, we define two indexes [j.sub.1] = j1(m) and [j.sub.2] = [j.sub.2](m) as in (2.3). Then, for a sufficiently large N, let [[P.sup.*.sub.N] denote the following subset of all polynomials of degree at most N,

[mathematical expression not reproducible]

Naturally, [p.sub.m](w) [member of] [[P.sup.*.sub.N], for N > [GREATER THAN OR EQUAL TO], and [theta] [member of] (0,1) arbitrary. Now, in analogy with (6.1), we define the new Gaussian rule by means of the equality

[mathematical expression not reproducible]g

which holds for every [Q.sub.2m]-1 [euro] P[I.sub.m-1].

Then, for any continuous function f, the "truncated" Gaussian rule is defined as

(6.5) [mathematical expression not reproducible]

whose error [e.sup.*.sub.m] (f) is the difference between the integral and the quadrature sum.

Compared to the Gaussian rule (6.1), in the formula (6.5), the terms of the quadrature sum corresponding to the zeros that are "close" to the MRS numbers are dropped. From the numerical point of view, this fact has two consequences. First, it avoids overflow phenomena (taking into account that, in general, the function f is exponentially increasing at the endpoints of R+). Moreover, it produces a computational saving, which is evident in the numerical treatment of linear functional equations (see [15]).

We are now going to study the behaviour [e.sup.*.sub.m] (f) in [C.sub.a] and [W.sup.1.sub.T] (w). We will see that the errors [e.sub.m] (f) and [e.sup.*.sub.m] (f) have essentially the same behaviour in [C.sub.a] but not in [W.sup.*.sub.T]. (w) since em (f) satisfies (6.4), while [e.sub.m] (f) does not. The behaviour of [e.sup.*.sub.m] (f) in [C.sub.a] is given by the following proposition.

PROPOSITION 6.4 (cf. [14, p. 1662]). Assume that w/[sigma] [member of] L . Then, for any f [member of] [C.sub.a], we get

(6.6) [mathematical expression not reproducible]

[mathematical expression not reproducible]

where [mathematical expression not reproducible], v is given by (3.2), C = C(m, f), and c = c(m, f). In particular, if f [member of] [mathematical expression not reproducible], then the inequality (6.6) becomes

[mathematical expression not reproducible]

For sm[infinity]ther functions, the analogue of Theorem 6.2 is given by the following statement. THEOREM 6.5 (cf. [14,p. 1662]). Ifthe weight [sigma] and the function f satisfy the assumption of Theorem 6.2, then,for any 0 < [mu] < [alpha](1 - 1/(2[beta]))/([alpha] + 1/2) fixed, we get

[mathematical expression not reproducible]

where [mathematical expression not reproducible] andC [not equal to] C(m, f).

For functions f [member of] [W.sup.1.sub.1](w) or f [member of] Zj(w), 1 < s [member of] R+, the following theorem states the required estimates.

THEOREM 6.6 (cf. [14, pp. 1662-1663]). For any f [member of] [W.sup.1.sub.1](w), we have

(6.7) [mathematical expression not reproducible]

Moreover, for any f [member of] [Z.sup.1.sub.s](w), with s > 1, we get

(6.8) [mathematical expression not reproducible]

[mathematical expression not reproducible]

where r > s > 1. In both cases C and c do not depend on m and f, and v is given by (3.2).

In conclusion, inequality (6.7) is the required estimate and, by (6.8), it can be generalized as

[mathematical expression not reproducible]

for f [member of] [Z.sup.1.sub.s](w), s > 1. In particular, if s is an integer, recalling (6.7), the Zygmund norm can be replaced by the Sobolev norm.

Finally, we emphasize that the previous estimate cannot be improved since, in these function spaces, [e.sup.*.sub.m] (f) converges to 0 with the order of the best polynomial approximation.

6.2. Lagrange interpolation in [mathematical expression not reproducible] Here we want to apply the results in Section 6.1 to estimate the error of the Lagrange interpolation process based on the zeros of [p.sub.m](w). If f [member of] [C.sup.0]([R.sub.+]), then the Lagrange polynomial interpolating f at the zeros of [p.sub.m](w) is defined by

[mathematical expression not reproducible]

and we are going to study the error [mathematical expression not reproducible] for different function classes. Since

(6.9) [mathematical expression not reproducible]

[mathematical expression not reproducible]

and we are dealing with an unbounded interval, we cannot expect an analogue of the theorem

by Erdo's and Turan [6]. On the other hand, if f [member of] [C.sub.u], with [mathematical expression not reproducible], then it is easily seen that

[mathematical expression not reproducible]

Nevertheless, as for the Gaussian formula, if [mathematical expression not reproducible], then [L.sub.m](w, f) has not an optimal behaviour, i.e., an estimate of the form

[mathematical expression not reproducible]

does not hold. In order to overcome this gap, for any f [member of] [C.sup.0]([R.sub.+]), we introduce the following "truncated" Lagrange polynomial

[mathematical expression not reproducible]

where [j.sub.1] and [j.sub.2] are given by (2.3).

Naturally, in general [L.sup.*.sub.m](w, P) = P for arbitrary polynomials P [member of] [P.sub.m-1] (for example, [L.sub.m](w, 1) = 1). But [L.sup.*.sub.m](w, Q) = Q for any Q [member of] [P.sub.m-1] and [L.sup.*.sub.m](w, f) [member of] [P.sub.m-1] for any f [member of] [C.sup.0]([R.sub.+]). Therefore, the operator [L.sup.*.sub.m](w) is a projector from C[degrees]([R.sub.+]) into [P.sub.m-1].

Moreover, considering the weight

(6.10) [mathematical expression not reproducible]

we can show that every function f [member of] [L.sup.p.sub.u] can be approximated by polynomials of [P.sup.*.sub.m]. To this aim we define

[mathematical expression not reproducible]

LEMMA 6.7 (cf. [14, p. 1664]). For any f [member of] [L.sup.p.sub.u], where u is given by (6.10) and 1 < p < +[infinity], we have

[mathematical expression not reproducible]

where [mathematical expression not reproducible], v is given by (3.2), C = C(m, f), and c = c(m, f).

As an immediate consequence of the previous lemma and equality (6.9), we get the following:

PROPOSITION 6.8 (cf. [14, p. 1664]). For any f [member of] [C.sub.u], with u as (6.10), [delta] > 1/2, we have

[mathematical expression not reproducible]

where [mathematical expression not reproducible], [theta] [member of] (0,1), v is given by (3.2), C [not equal to] C(m, f), and c [not equal to] c(m, f).

We are going to study the behaviour of the sequence [{L.sup.*.sub.m](w)}.sub.m] in the Sobolev spaces

[mathematical expression not reproducible], which is interesting in different contexts. We observe that, since no results concerning the sequence of the Fourier sum {[S.sub.m](w)}.sub.m] are known, we cannot deduce the behaviour of {[L.sup.*.sub.m](w)}.sub.m] from that of {[S.sub.m](w)}.sub.m]. Therefore, we need a different approach. The following theorem describes the behaviour of the operator [L.sup.*.sub.m](w) in different function spaces. THEOREM 6.9 (cf. [14, p. 1664]). Assume f [member of [mathematical expression not reproducible] and

(6.11) [mathematical expression not reproducible]

Then we have

(6.12) [mathematical expression not reproducible]

[mathematical expression not reproducible]

where v is given by (3.2) and the constants C, c are independent of m and f.

Note that, by Corollary 5.4, assumption (6.11) implies f [member of] [C.sup.0]([R.sub.+]). The error estimate (6.12) has interesting consequences. Firstly, if

[mathematical expression not reproducible]

i.e., [mathematical expression not reproducible], then the order of convergence of the process is [mathematical expression not reproducible]. If, instead, [mathematical expression not reproducible], with r [greater than or equal to] 1 an integer, then we have

[mathematical expression not reproducible]

This means that the process converges with the error of the best approximation for the considered classes of functions.

Secondly, we are now able to show the uniform boundedness of the sequence {[L.sup.*.sub.m](w)} in the Sobolev spaces.

THEOREM 6.10 (cf. [14, p. 1665]). With the previous notation, for any f [member of] [W.sup.1.sub.1][mathematical expression not reproducible], r > 1, we have

[mathematical expression not reproducible]

Moreover, for any [mathematical expression not reproducible], s > r, we have

[mathematical expression not reproducible]

REMARK 6.11. In all the estimates for [e.sup.*.sub.m](f) and (f - [L.sup.*.sub.m](w,f)), a constant C = C(m, f) appears. We have not indicated the dependence on the parameter [theta] [member of] (0,1)

since [theta] is fixed. Nevertheless, it is useful to observe that C = C([theta]) = O I ([theta].sup.(1-[theta])2]). So, it is clear that the parameter [theta] cannot assume the value 0 or 1, and the "truncation" is necessary in this sense (see [14] for more details).

REFERENCES

[1] R. ALJARRAH, Error estimates for Gauss-Jacobi quadrature formulae with weights having the whole real line as their support, J. Approx. Theory, 30 (1980), pp. 309-314.

[2] , An error estimate for Gauss-Jacobi quadrature formula with the Hermite weight w(x) = exp(-[x.sup.2]), Publ. Inst. Math. (Beograd) (N.S.), 33 (1983), pp. 17-22.

[3] A. S. CVETKOVI[C AND G. V. MILOVANOVI[C, The Mathematica Package "OrthogonalPolynomials", Facta Univ. Ser. Math. Inform., 9 (2004), pp. 17-36.

[4] M. C. DE BONIS, G. MASTROIANNI, AND I. NOTARANGELO, Gaussian quadrature rules with exponential weights on (-1, 1), Numer. Math., 120 (2012), pp. 433-464.

[5] B. DELLA VECCHIA AND G. MASTROIANNI, Gaussian rules on unbounded intervals, J. Complexity, 19 (2003), pp. 247-258.

[6] P. ERDOS AND P. TURAN, On interpolation. I. Quadrature- and mean-convergence in the Lagrange-interpolation,

Ann. Math., 38 (1937), pp. 142-155.

[7] A. L. LEVIN AND D. S. LUBINSKY, Orthogonal Polynomials for Exponential Weights, Springer, New York, 2001.

[8] D.S. LUBINSKY, Geometric convergence of Lagrangian interpolation and numerical integration rules over unbounded contours and intervals, J. Approx. Theory, 39 (1983), pp. 338-360.

[9] G. MASTROIANNI AND G.V. MILOVANOVI[C, Interpolation Processes. Basic Theory and Applications, Springer, Berlin, 2008.

[10] G. MASTROIANNI AND G. MONEGATO, Truncated quadrature rules over (0, [infinity]) and Nystrom type methods, SIAM J. Numer. Anal., 41 (2003), pp. 1870-1892.

[11] G. MASTROIANNI AND I. NOTARANGELO, A Lagrange-type projector on the real line, Math. Comp., 79 (2010), pp. 327-352.

[12] __, Embedding theorems with an exponential weight on the real semiaxis, Electron. Notes Disc. Math., 43 (2013), pp. 15-160.

[13] __, Polynomial approximation with an exponential weight on the real semiaxis, Acta Math. Hungar., 142 (2014), pp. 167-198.

[14] G. MASTROIANNI, I. NOTARANGELO, AND G.V. MILOVANOVIC, Gaussian quadrature rules with an exponential weight on the real semiaxis, IMA J. Numer. Anal., 34 (2014), pp. 1654-1685.

[15] __, A Nystrom method for a class of Fredholm integral equations on the real semiaxis, Calcolo, 54 (2017), pp. 567-585.

[16] G. MASTROIANNI, I. NOTARANGELO AND J. SZABADOS, Polynomial inequalities with an exponential weight on (0, +[infinity]), Mediterr. J. Math., 10 (2013), pp. 807-821.

[17] G. MASTROIANNI AND J. SZABADOS, Polynomial approximation on the real semiaxis with generalized Laguerre weights, Stud. Univ. Babes [section]Bolyai Math., 52 (2007), pp. 105-128.

[18] G. V. MILOVANOVIC, Construction and applications of Gaussian quadratures with nonclassical and exotic weight function, Stud. Univ. Babes-Bolyai Math., 60, (2015), pp. 211-233.

[19] G. V. MILOVANOVIC AND A. S. CVETKOVIC, Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type, Math. Balkanica, 26 (2013), pp. 169-184.

[20] J.V. USPENSKY, On the convergence of quadrature formulas related to an infinite interval, Trans. Amer. Math. Soc., 30 (1928), pp. 542-559.

GIUSEPPE MASTROIANNI ([dagger]), GRADIMIR V. MILOVANOVIC ([double dagger]), AND INCORONATA NOTARANGELO ([section])

Dedicated to Walter Gautschi on the occasion of his 90th birthday

(*) Received May 7, 2018. Accepted August 20, 2018. Published online on November 15, 2018. Recommended by L. Reichel. The first author was supported in part by the University of Basilicata (local funds), the second author by the Serbian Academy of Sciences and Arts (No. [PHI]-96), and the third author by the University of Basilicata (local funds) and by INdAM-GNCS.

([dagger]) Department of Mathematics, Computer Sciences and Economics, University of Basilicata, Via dell'Ateneo Lucano 10, 85100 Potenza, Italy (giuseppe.mastroianni@unibas.it).

([double dagger]) Serbian Academy of Sciences and Arts, Belgrade, Serbia & University of Nis, Faculty of Sciences and Mathematics, 18000 Nis, Serbia (gvm@mi.sanu.ac.rs).

([section]) Department of Mathematics, Computer Sciences and Economics, University of Basilicata, Via dell'Ateneo Lucano 10, 85100 Potenza, Italy (incoronata.notarangelo@unibas.it).

DOI: 10.1553/etna_vol50s36
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Author:Mastroianni, Giuseppe; Milovanovic, Gradimir V.; Notarangelo, Incoronata
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