PORTUGALIAE MATHEMATICA
Vol. 57 Fasc. 2 – 2000
UNIFORM CONVERGENCE RESULTS FOR
CERTAIN TWO-DIMENSIONAL CAUCHY
PRINCIPAL VALUE INTEGRALS *
E. Santi
Abstract: A general uniform convergence theorem for numerical integration of
certain two-dimensional Cauchy principal value integrals is proved. A special instance
of this theorem is given as corollary.
1 – Introduction
In this paper we study the uniform convergence with respect to the parameters
ϑ and ζ, of numerical methods for evaluating the Cauchy principal value (CPV)
integral
(1.1)
Z
J(f ; ϑ, ζ) : = −
1
−1
Z
1
− w1 (x) w2 (x)
−1
ϑ ∈ (−1, 1) ,
e)
f (x, x
e,
dx dx
e − ζ)
(x − ϑ) (x
ζ ∈ (−1, 1) ,
where w1 , w2 are the Jacobi weight functions
(1.2)
w1 (x) : =(1 − x)α1 (1 + x)β1 ,
e) : =(1 − x
e)α2 (1 + x
e ) β2 ,
w2 ( x
αi , βi > −1, i = 1, 2 .
In some recent papers [2, 3, 8], integrals of type (1.1), has been approximated
by cubature rules based on quasi-uniform tensor product spline spaces.
Received : July 29, 1998; Revised : February 18, 1999.
AMS Subject Classification: 65D32, 41A15, 41A63.
Keywords: Cubatures; Cauchy principal value integrals; Splines.
* Work sponsored by M.U.R.S.T. and C.N.R. of Italy.
192
E. SANTI
Supposing that f ∈ Hp (µ, µ), 0 < µ ≤ 1, 0 ≤ p < m−1, where m denotes
the order of the splines, in [2, 3] the authors proved a convergence theorem
for cubature rules obtained by substituting the integrand function f with quasiinterpolating or nodal spline operators; they gave besides, an upper bound for the
remainder term. By subtracting out the singularities, in [8], it has been proved
the uniform convergence of cubature rules based on quasi-interpolating splineprojectors for (ϑ, ζ) belonging to any closed region contained in the integration
domain.
Since in many applications it is necessary to have at one’s disposal rules
uniformly converging for all (ϑ, ζ) ∈ (−1, 1)×(−1, 1), the main object of this
paper is to give general conditions for obtaining such rules. We shall do that in
Section 2, proving the general convergence theorem 1.
In Section 3 we shall apply the obtained results to a particular approximation
operator and we shall give the relative convergence results as corollary of the
above general convergence theorem.
2 – A general uniform convergence theorem
In this section we shall state and prove a general uniform convergence theorem
for two-dimensional CPV integrals (1.1), that can be written in the form
(2.1)
Z
Z
1
e) − f (ϑ, ζ)
f (x, x
e
dx dx
e − ζ)
(x − ϑ) (x
−1 −1
+ f (ϑ, ζ) W1 (ϑ) W2 (ζ) .
where
(2.2)
1
J(f ; ϑ, ζ) = −
Z
e)
− w1 (x) w2 (x
1
w1 (x)
dx ,
x−ϑ
W1 (ϑ) = −
−1
Z
1
W2 (ζ) = −
−1
e)
w2 ( x
e.
dx
e−ζ
x
e with I = [−1, 1], Ie = [−1, 1], and we define some partitions
We denote Ω = I×I,
e
N
e = {x
e
eei }N
XN = {xi }i=1 , X
e
N
e of I and I respectively. Connected with the sei=1
e : N = N1 , N2 , ...; N
e =N
e1 , N
e2 , ...},
quence of tensor product partitions {XN ×X
e
N
we consider a sequence of real positive numbers {∆N Ne } such that
(2.3)
e
e
lim ∆N Ne = 0 .
N →∞
e →∞
N
N
e)}i=1,2,···
Let Φ = {ϕiei (x, x
i=1,2,···N be a set of basis functions real continuous in Ω and
let FN Ne be a linear operator defined by Φ and approximating to f .
TWO-DIMENSIONAL CAUCHY PRINCIPAL VALUE INTEGRALS
We assume
(2.4)
Denoting by
(2.5)
(2.6)
e) = f (x, x
e) − F e (x, x
e) .
rN Ne (x, x
NN
SN Ne (x) =
e) =
TN Ne (x
Z
Z
1
−1
1
−1
e)
w2 ( x
w1 (x)
and considering the cubature rule
(2.7)
193
Z
Z
e) − r e (x, ζ)
rN Ne (x, x
NN
e,
dx
e−ζ
x
e) − r e (ϑ, x
e)
rN Ne (x, x
NN
dx ,
x−ϑ
e) − F e (ϑ, ζ)
FN Ne (x, x
NN
e
dx dx
e − ζ)
(x − ϑ) (x
−1 −1
+ f (ϑ, ζ) W1 (ϑ) W2 (ζ) ,
1
JN Ne (f ; ϑ, ζ) = −
1
e)
− w1 (x) w2 (x
the error term EN Ne (f ; ϑ, ζ) = J(f ; ϑ, ζ) − JN Ne (f ; ϑ, ζ), given by
(2.8)
Z
1
EN Ne (f ; ϑ, ζ) = −
−1
Z
1
e)
− w1 (x) w2 (x
−1
can be written in the form [8],
(2.9)
EN Ne (f ; ϑ, ζ) =
Z
e) − r e (ϑ, ζ)
rN Ne (x, x
NN
e,
dx dx
e − ζ)
(x − ϑ) (x
SN Ne (x) − SN Ne (ϑ)
dx
x−ϑ
−1
+ W1 (ϑ) SN Ne (ϑ) + W2 (ζ) TN Ne (ζ) .
1
w1 (x)
We say that f ∈ H(µ, µ), 0 < µ ≤ 1, if f is a continuous function in Ω
e1 ), (x2 , x
e2 ) ∈ Ω there results |f (x1 , x
e1 ) − f (x2 , x
e2 )| ≤
such that for all (x1 , x
e1 − x
e2 |µ ], with C real constant.
C[|x1 − x2 |µ + |x
For proving theorem 1 below, we need some lemmas [6].
Lemma 1. Let g ∈ H(σ, σ) in Ω, 0 ≤ σ < 1. The function
e) =
ϕ(x, x
e) − g(x, x
e0 )
g(x, x
,
e0 |²̄
|x − x
∀ ²̄, 0 < ²̄ < σ ,
e,
satisfies the H(σ − ²̄) condition for the variable x uniformly with respect to x
e uniformly with respect to x.
and H(σ − ²̄) condition for the variable x
194
E. SANTI
If we define the functions
(2.10)
Z
s(x) =
and
Z
e) =
t(x
(2.11)
we have:
1
−1
1
−1
e)
w2 ( x
w1 (x)
e) − g(x, x
e0 )
g(x, x
e
dx
e−x
e0
x
e) − g(x0 , x
e)
g(x, x
dx ,
x − x0
Lemma 2. Suppose g ∈ H(σ, σ) in Ω, 0 ≤ σ < 1. The functions s and t,
defined in (2.10), (2.11), satisfy a Hölder condition of order σ − ²̄, where ²̄ is an
arbitrary real number such that 0 < ²̄ < σ, i.e.
(2.12)
|s(x) − s(x0 )| ≤ K0 |x − x0 |σ−²̄ ,
(2.13)
e) − t(x
e0 )| ≤ K1 |x
e−x
e0 |σ−²̄ .
|t(x
Theorem 1. Let f ∈ H(µ, µ) in Ω, and assume that the approximation FN Ne
to f is such that
e
e) = 0, ∀ x
e ∈ I,
(i) rN Ne (x, ±1) = 0, ∀ x ∈ I, rN Ne (±1, x
(ii) krN Ne k∞ = O(∆ν e ), 0 < ν ≤ µ,
NN
(iii) rN Ne ∈ H(σ, σ), 0 < σ ≤ µ.
If
(2.14)
ρ + γ − ²̄ > 0
where ρ : = min(σ, ν), γ : = min(α1 , α2 , β1 , β2 , 0) and ²̄ is a positive real number
as small as we like, then
(2.15)
e →∞,
as N → ∞, N
EN Ne (f ; ϑ, ζ) → 0
uniformly for all (ϑ, ζ) ∈ (−1, 1)×(−1, 1) .
Proof: We can write:
EN Ne (f ; ϑ, ζ) =
Z
1
−1
w1 (x)
S
e (x)−SN Ne (ϑ) dx + W1 (ϑ) S
e (ϑ) + W2 (ζ) TN N
e (ζ)
x−ϑ
NN
NN
= T1 + T2 + T3 .
e) =
Taking into account the condition (iii), and assuming in lemma 2 g(x, x
e), x0 = ϑ, x
e0 = ζ, the functions S e (x), T e (x
e) defined in (2.5), (2.6),
rN Ne (x, x
NN
NN
195
TWO-DIMENSIONAL CAUCHY PRINCIPAL VALUE INTEGRALS
satisfy (2.12) and (2.13) respectively. Furthermore, for 0 < ²∗ < σ, by (ii) and
lemma 5 in [6], there results:
∗
ν(1− ²σ )
|SN Ne (x)| ≤ C̄ ∆ e
NN
(2.16)
∀ ζ ∈ (−1, 1) ,
∗
ν(1− ²σ )
(2.17)
∀ ϑ ∈ (−1, 1) .
e)| ≤ C̄1 ∆
|TN Ne (x
e
NN
Consider first T2 . Because (i), in a neighbourhood of x = 1, we have
Z
h
i
h
i
e) − r e (ϑ, ζ) − r e (1, x
e) − r e (1, ζ)
rN Ne (ϑ, x
NN
NN
NN
e)
e
w2 ( x
dx
SN Ne (ϑ) =
e
x−ζ
−1
1
then, by condition (iii) and lemma 1, there results
(2.18)
|SN Ne (ϑ)| ≤ C |ϑ − 1|
σ−²̄
Z
1
−1
e)
w2 ( x
σ−²̄
e = O(1 − ϑ)
dx
.
1−²̄
e − ζ|
|x
Besides, in a neighbourhood of x = 1, by §4.62 of [10],
(2.19)
 ³
´

 O (1 − ϑ)α1 + c
W1 (ϑ) =
³
´

 O | log(1 − ϑ)|
if α1 is not an integer,
if α1 is an integer .
Hence, we can find δ > 0 sufficiently small so that for all ϑ ∈ [1−δ, 1] and ∀ ² > 0,
T2 = O((1− ϑ)σ−²̄+α1 | log(1− ϑ)|) < ² uniformly in ϑ if (2.14) holds.
Similarly, we can find δ̄ > 0 such that for all ϑ ∈ [−1, −1+ δ̄]
³
´
T2 = O (1+ ϑ)σ−²̄+β1 | log(1+ ϑ)| < ²
uniformly in ϑ .
Finally, since W1 (ϑ) = O(1) in [−1+ δ̄, 1−δ] and kSN Ne k∞ = o(1) as N → ∞,
e → ∞, we conclude that
N
T2 = o(1)
e →∞,
as N → ∞, N
uniformly for all (ϑ, ζ) ∈ (−1, 1)×(−1, 1) .
In the same way we can prove that
T3 = o(1)
e →∞,
as N → ∞, N
uniformly for all (ϑ, ζ) ∈ (−1, 1)×(−1, 1) .
196
E. SANTI
Consider now
¯
¯Z
¯
¯ 1
SN Ne (x) − SN Ne (ϑ)
¯
¯
w1 (x)
dx¯
|T1 | = ¯
¯
¯ −1
x−ϑ
¯
¯
¯Z
¯
SN Ne (x) − SN Ne (ϑ)
¯
¯
¯
≤ ¯ w1 (x)
dx¯¯
x−ϑ
¯
¯
U
¯
¯
¯
¯
¯
¯
¯
¯
Z
SN Ne (x) − SN Ne (ϑ)
¯
¯
¯
+¯
dx¯¯
w1 (x)
x−ϑ
¯
¯
¯ |x−ϑ|≥∆
¯
NN
e
¯
¯
x∈U
/
¯
¯
¯
¯
¯
¯
¯
¯
Z
SN Ne (x) − SN Ne (ϑ)
¯
¯
w1 (x)
dx¯¯
+ ¯¯
x−ϑ
¯
¯
¯ |x−ϑ|≤∆
¯
NN
e
¯
¯
x∈U
/
= T11 + T12 + T13
where U : = [−1, −1+ se] ∪ [1−s̄, 1], for some se, s̄ to be determined below.
There results
¯Z
¯ −1+e
s
¯
w1 (x)
¯
¯ −1
S
NN
e (x)−SN Ne (ϑ)
x−ϑ
¯
Z −1+e
¯
s
¯
dx¯ ≤ K0
w1 (x) |x − ϑ|σ−²̄−1 dx
¯
−1
!
ÃZ
−1+e
s
(1 + x)γ+σ+²̄−1 dx
= O
−1
for se sufficiently small. Similarly we have
¯Z
¯
¯ 1
¯
SN Ne (x) − SN Ne (ϑ)
¯
¯
w1 (x)
dx¯ < ²
¯
¯ 1−s̄
¯
x−ϑ
for s̄ sufficiently small, and then
(2.20)
T11 < 2 ² ,
∀² > 0 .
< ²
TWO-DIMENSIONAL CAUCHY PRINCIPAL VALUE INTEGRALS
197
Considering the term T12 , we have
T12
(2.21)
¯
¯
¯
¯
¯
¯
¯
¯
Z
SN Ne (x) − SN Ne (ϑ)
¯
¯
dx¯¯
= ¯¯
w1 (x)
x−ϑ
¯
¯
¯
¯ |x−ϑ|≥∆
N
N
e
¯
¯
x∈U
/
!
Ã
∗
²
ν (1− )
= O ∆ e σ ln(∆N Ne )
NN
= o(1)
e →∞.
as N → ∞, N
Finally, by (2.12), (2.13),
T13
(2.22)
¯
¯
¯
¯
¯
¯
¯
¯
Z
SN Ne (x) − SN Ne (ϑ)
¯
¯
dx¯¯
= ¯¯
w1 (x)
x−ϑ
¯
¯
¯
¯ |x−ϑ|≤∆
N
N
e
¯
¯
x∈U
/
Z
≤ C2
|x − ϑ|σ−²̄−1 dx = o(1)
|x−ϑ|≤∆
NN
e
x∈U
/
e → ∞ uniformly ∀ (ϑ, ζ) ∈ (−1, 1)×(−1, 1).
as N → ∞, N
The thesis follows considering that T1 , T2 , T3 can be made arbitrarily small,
e → ∞.
uniformly in (ϑ, ζ) ∈ (−1, 1)×(−1, 1), as N → ∞, N
We remark that the theorem 1 extends to the two-dimensional case the results
of theorem 1 in [7].
3 – Particular example of theorem 1
In this section we derive uniform convergence results for the below defined
approximation W̄N Ne to f , which we state as corollary.
Consider the sequence of two-dimensional nodal spline operators studied in
[3]:
(3.1)
e)
WN∗ Ne (f ; x, x
=
e) ∈ [τj , τj+1 ]×[τeej , τeej +1 ] ,
(x, x
e
q
qj
ej
X
X
i=pj e
i=p
e
ej
e) f (τj , τeej ) ,
ωmm
e iei (x, x
j = 0, 1, ...N −1 ,
e −1 .
e
j = 0, 1, ..., N
198
E. SANTI
For constructing (3.1), we firstly need to consider the partitions
n
o
XmN : = −1 ≡ x0 < x1 < · · · < x(m−1)N ≡ 1 ,
n
e
e0 < x
e1 < · · · < x
e
X
e : = −1 ≡ x
e≡1
m
eN
(m
e −1)N
of I, Ie respectively. The nodes τi , τeei are defined by
o
τi : = x(m−1)i ,
0 ≤ i ≤ N, with N ≥ m−1 ,
e(m
τeei : = x
e −1)ei ,
e , with N
e ≥ m−1
e
0 ≤ ei ≤ N
,
e) where
and the functions ωmm
e iei are such that ωmm
e iei = ωmi (x) × ωm
eei (x
 Q
m x−τ
k

x ∈ [−1, τi1 −1 ], (i ≤ m−1) ,



τ
−
τ
i

k
 k=0
ωmi (x) : =
















k6=i
si (x)
x ∈ [τi1 −1 , τN −i0 +1 ], (N ≥ m) ,
m
Q
k=0
k6=N −i
x − τN −k
τi − τN −k
x ∈ [τN −i0 +1 , 1], (i ≥ N −(m−1)) ,
m
:
with i0 : = [ m
2 ] + 1, i1 = m − [ 2 ]. For the functions si we have
si (x) : =
+m−1
m −i0X
X
m
αirt B(m−1)(i+t)+r
(x) ,
r=0
i = 0, 1, ..., N ,
t=−i0
(m−1)N −1
where {Bim (x)}i=1−m
We recall that
are the normalized B-splines of order m.
0 < Bim (x) ≤ 1 if x ∈ (xi , xi+m ) ,
Bim (x) = 0 otherwise ,
m (−1) = 1, B
except that B1−m
(m−1)N −1 (1) = 1.
We besides assume
pj : =
qj : =


0






j − i1 + 1
if j = i1 −1, ..., N −i0 ,
N −m+1
if j = N −i0 +1, ..., N −1 ,


m−1







if j = 0, 1, ..., i1 −2 ,
if j = 0, 1, ..., i1 −2 ,
j + i0
if j = i1 −1, ..., N −i0 ,
N
if j = N −i0 +1, ..., N −1 .
TWO-DIMENSIONAL CAUCHY PRINCIPAL VALUE INTEGRALS
199
em
e), peej , qeej . Without loss of generality, we can assume
Likewise we define ω
eei (x
e i.e. we use splines of the same order on both axes.
m = m,
We remark that W ∗ e is a spline operator with the following properties:
NN
e) depends only on the values
(a) W ∗ e is local, in the sense that W ∗ e (f ; x, x
NN
NN
e);
of f in a small neighborhood of (x, x
(b) W ∗ e satisfies the relations: W ∗ e (f ; τi , τeei ) = f (τi , τeei );
NN
NN
(c) W ∗ e has the optimal order polynomial reproduction property, that means
NN
2 , where P 2 is the set of bivariate polynomials
W ∗ e p = p for all p ∈ Pm
m
NN
of total order m.
We say that the collection of product partitions
n
e
e
e e
XmN × X
e , N = N1 , N2 , ..., N = N1 , N2 , ...
m
eN
o
e is quasi-uniform (q.u.) if there exists a positive constant A such
of Ω = I×I,
e , where
that ∆i /δj ≤ A for i and j equal to N or N
∆N = max0≤i≤(m−1)N −1 (xi+1 − xi ) ,
eei+1 − x
eei ) ,
∆Ne = max0≤ei≤(m−1)Ne−1 (x
δN = min0≤i≤(m−1)N −1 (xi+1 − xi ) ,
eei+1 − x
eei ) ,
δNe = min0≤ei≤(m−1)Ne−1 (x
and we shall call a sequence of spline spaces quasi-uniform if they are based on a
sequence of q.u. partitions.
We define ∆N Ne = ∆N + ∆Ne , and suppose that
e →∞.
∆Ne → 0 as N
∆N → 0 as N → ∞ ,
(3.2)
We shall use the results in [3] for deducing the following
Proposition 1. Suppose f ∈ C(Ω), for any sequence of q.u. nodal spline
spaces {W ∗ e }, there results:
NN
(3.3)
kf − WN∗ Ne k∞ ≤ K ω(f ; ∆N Ne ; Ω)
where K is a constant depending only on m and A, and
(3.4)
with
³
ω(WN∗ Ne ; ∆N Ne ; Ω) = O ω(f ; ∆N Ne ; Ω)
ω(φ; ∆; Θ) =
max
´
¯
¯
¯
¯
e − φ(x, x
e + h)
e )¯ .
¯φ(x+h, x
|h|,|e
h|≤∆
(x,e
x),(x+h,e
x+e
h) ∈ Θ
200
E. SANTI
Using the method introduced in [5], we modify the operator (3.1) as
e) = W ∗ e (f ; x, x
e)
W̄N Ne (f ; x, x
NN
h
i
m
e) B1−m
e) − W ∗ e (f ; −1, x
+ f (−1, x
(x)
NN
h
i
h
i
m
e) − W ∗ e (f ; 1, x
e) B(m−1)N
+ f (1, x
−1 (x)
NN
(3.5)
h
i
e m (x
+ f (x, −1) − WN∗ Ne (f ; x, −1) B
1−m e)
and we prove the following
em
e) ,
+ f (x, 1) − WN∗ Ne (f ; x, 1) B
e −1 (x
(m−1)N
Lemma 3. Let f ∈ H(µ, µ) in Ω, and let {W̄N Ne } be a sequence of q.u.
spline operators defined in (3.5). There results:
e r
e) = 0, ∀ x
e ∈ I,
(1) rN Ne (∓1, x
e (x, ∓1) = 0, ∀ x ∈ I;
NN
(2) krN Ne k∞ = O(∆µ e );
NN
(3) rN Ne ∈ H(µ, µ) in Ω
e) = f (x, x
e) − W̄ e (x, x
e).
with rN Ne (x, x
NN
Proof: Property (1) derives by the definition (3.5) and by the interpolating
property of W ∗ e . From the definition (3.5) we have krN Ne k∞ ≤ 3 kf − W ∗ e k
NN
NN
and by (3.3) property (2) holds. Property (3) follows considering that for
e1 ), (x2 , x
e2 ) ∈ Ω we can write
(x1 , x
(3.6)
¯
¯
¯
¯
e
e
(x
,
x
)
−
r
(x
,
x
)
¯rN N
e 1 1
e 2 2 ¯ ≤
NN
¯
¯
¯
¯
¯
¯
¯
¯
e1 ) − f e (x2 , x
e2 )¯ + ¯W̄ e (x1 , x
e1 ) − W̄ e (x2 , x
e 2 )¯ .
≤ ¯fN Ne (x1 , x
NN
NN
NN
Then, by the definition of W̄N Ne , using (3.3) and (3.4), it is possible to prove that
e1 ),
in each of the different cases, according to the positions of the points (x 1 , x
e2 ) in Ω, there results [9]:
(x2 , x
¯
¯
¯
¯
e
e
(f
;
x
,
x
)
−
W̄
(f
;
x
,
x
)
¯W̄N N
¯ ≤ C̄ ω(f ; ∆N N
1
1
2
2
e
e
e ; Ω) .
NN
Therefore the claim follows by (3.6) and the hypothesis on f .
Corollary 1. Let f ∈ H(µ, µ) in Ω and let {FN Ne } be a sequence of q.u.
spline spaces {W̄N Ne } defined in (3.5). Assume that (3.2) holds, then
e →∞
EN Ne (f ; ϑ, ζ) → 0 as N, N
uniformly in (ϑ, ζ) ∈ (−1, 1)×(−1, 1) .
TWO-DIMENSIONAL CAUCHY PRINCIPAL VALUE INTEGRALS
201
Proof: Condition (i) of theorem 1 follows by (1) of lemma 3. By conditions
e , ν = µ)
(2) and (3) of the same lemma, condition (ii) (with ∆N Ne = ∆N + ∆
e
N
and condition (iii) (with σ = µ) hold.
ACKNOWLEDGEMENTS – The author wishes to thank the referee for his insightful
comments.
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Elisabetta Santi,
Dipartimento di Energetica – Università dell’Aquila,
Localita’ Monteluco, 67040 Roio Poggio (AQ) – ITALY
E-mail: esanti@dsiaq1.ing.univaq.it