The Stability of the Gauss-Laguerre Rule for Cauchy P.V. Integrals on the Half Line ()
1. Introduction
A careful analysis of Gauss-Laguerre formulas for ordinary integrals can be found in [1] . The present paper is instead aimed at the use of the Gauss-Laguerre formulas for the approximation of the Cauchy principal value integrals. The main results on the topic can be found in [2] (see also the references therein). The aim of the present paper is to give the upper bound of the stability factor when this kind of quadrature rule is used.
We consider the class of singular principal value integrals
(1)
where f satisfies the smoothness conditions
(2)
and
(3)
where
is the weighted Ditzian-Lubinsky modulus of smoothness [3] .
We propose a Gauss-Laguerre type quadrature formula to evaluate the singular principal value integral
defined by (1) assuming that the function f has good integration property at the superior limit of integration interval; this assumption is the same that assures the boundedness of
. In this first part we study the stability of the proposed procedure with respect to the distance of the singularity t from the quadrature knots. The proposed method to compute (1) is well known in the case of bounded intervals. Even though the fundamental idea is not new, a thorough investigation of this algorithm is of interest since there are significant differences between the case of a bounded interval and the case of an unbounded interval from the point of view of approximation theory.
2. The Stability of Gauss-Laguerre Quadrature Rule
The GL type formula to evaluate
is constructed by interpolating the function f on
, and on the singularity t assuming that
. Taking into account that such formula can be written
(4)
we have
Therefore, the formula (4) has degree of exactness 2n, i.e.
whenever f is a polynomial of degree ≤ 2n.
Obviously, from a theoretical point of view, this formula turns out to be convergent if the function f is sufficiently smooth. Furthermore, it has the advantage of simplicity in the computation of the coefficients, but unfortunately it may exhibit numerical cancellation and generally it cannot converge when a knot
is very close to t. In order to establish a bound of the amplification factor of (4) which depends on the position of t with respect to the points
, we need some some notations and preliminary lemmas.
If A and B are two expressions depending on some variables, then we write
if and only if
uniformly for the variables under consideration.
We bring here some properties of the knots
and of the Christoffel constants
of the GL formula (4). These properties can be found in [4] and [5] where are proved for a more general class of weight functions.
Let
be the zeros of the n-th Laguerre orthogonal polynomial ordered in increasing order. We have
(5)
with some constants
and
independent of
and
.
The Christoffel constants
admit the following bounds
(6)
uniformly for
and where
.
Let
and
be such that
.
For any
,
we denote by
the knot closest to t, defined by
Lemma 1. We have
Proof. It easy seen that
and
Thus
Hence performing the integrations we get the lemma. □
Lemma 2. For any
such that
, and with some constant C independent of f, n and t,
where
.
Proof. We write
(7)
First, we obtain from (6)
(8)
Here we have used Lemma 1.
Now, if
then the last sum of (7) is equal to the first sum in (8). On the other hand, if
, we have
; then
So, in all cases,
(9)
Finally, again by using (6),
(10)
taking into account the definition of
. Combining (7), (9) and (10) the assertion follows. □
Lemma 3. For any
such that
, and with some constant C independent of f, n and t,
where
.
Proof. The lemma has been proved [2] with respect to the Hermite weight. Following the same steps of the proof of the Lemma 3.3 in [2] , it is possible to derive the assertion. Here we omit the details. □
In order to estimate the stability of the GL formula (4), we define
Theorem 4. For any
such that
, and with some constant C independent of n and t,
where
.
Proof. Following the same steps of the proofs of the Lemmas 2 and 3, we have
(11)
(12)
respectively. Then, the assertion follows by (11) and (12) taking into account the definition of
. □
Corollary 1. Assume that
, where the space
is defined in (2)
and f satisfies the condition (3). If
and n are such that
, then
with some constant C independent of n and t.
Proof. Taking into account that the assumptions on t and n give
, we have
and
. Thus, the assertion follows from Theorem 4. □
Now assume that
is fixed. In order to specify the bound of the amplification factor
, we need to use a suitable subsequence of
. To derive it, the following lemma turns out helpful.
Lemma 5. Let
. There exists
such that uniformly for
,
,
(13)
(14)
where
is the Mhaskar-Rakhmanov-Saff number [6] , and
are positive constants.
Proof. At first we remark that
, (cfr. [5] ).
Let us denote
(15)
the fundamental Lagrange polynomials with respect to the points
, zeros of the n-th Laguerre polynomial
.
By using the relation
(see Theorem 1.3(c) in [7] ), we obtain from (15)
(16)
By using
(see Theorem 1.3(a), (b) and Theorem 1.4 in [7] ) and taking into account that
and
, we obtain (13) from (16). Similarly we can prove (14). □
Assuming that
is fixed, we define the set
where the parameter
is chosen a priori. Being t fixed we have that
. Thus, in view of Lemma 5, we deduce that
is an infinite set. Indeed, if
, then
.
Finally, we remark that to construct the formula we have assumed
. To derive a rule and the related error bound for all t, it would be of interest to investigate the limit case
. Of course, this would require additional assumptions on the function f. However, the restriction
, does not influence the effective approximation of
. Indeed, for any fixed
it is possible to construct the subsequence
and the related error can be derived.
Corollary 2. If
is fixed, then
with some constant C independent of
.
Proof. The assertion follows from Theorem 4 taking into account the definition of the set
. □
3. Conclusion
Theorem 4 provides the upper bound of the amplification factor of the Gauss-Laguerre quadrature formula in the case of Cauchy principal value integrals. Corollary 2 instead provides a useful definition of a subsequence of the Gauss-Laguerre formula with respect to which the amplification factor is particularly useful in applications.