Data Mining and Business Intelligence - Artis Consulting

22 Pages·2009·0.34 MB·English

A SPLINE PRODUCT QUASI-INTERPOLATION METHOD FOR
WEAKLY SINGULAR FREDHOLM INTEGRAL EQUATIONS∗
EERO VAINIKKO AND GENNADI VAINIKKO†
Abstract. A discrete method of accuracy O(hm) is constructed and justi(cid:28)ed for a class of
Fredholm integral equations of the second kind with kernels that may have weak diagonal and
boundarysingularities. Themethodisbasedonimprovingtheboundarybehaviourofthekernel
withthehelpofachangeofvariables,andontheproductintegrationusingquasi-interpolationby
smoothsplinesoforderm.
Keywords. Weaklysingularintegralequations,boundarysingularities,splinequasi-interpo-
lation,splineinterpolation,productintegrationmethods,Nystr(cid:246)mtypemethods
AMS subject classi(cid:28)cations. 65R20,65D07,65D30
1. Introduction. In the present paper we call attention to some high order
fully discrete methods for a class of Fredholm integral equations of the second kind
withdi(cid:27)erentweaksingularities. Theideaofthemethodisclosetothosein[2],[4],
[9]. Consider the weakly singular integral equation
Z 1
(1.1) u(x)= (cid:0)a(x,y)|x−y|−ν +b(x,y)(cid:1)u(y)dy+f(x), 0≤x≤1,
0
dependingontheparameterν,0<ν <1. Atraditionalproductintegrationmethod
(see [2], [4] and the literature quoted there) is based on the approximation of (1.1)
by the equation
Z 1
u(x)= (cid:0)|x−y|−νP (a(x,y)u(y))+P (b(x,y)u(y))(cid:1)dy+f(x), 0≤x≤1,
n,m n,m
0
where P is an interpolation projector onto the space of piecewise polynomial
n,m
functions of degree m−1 associated with a subdivision of [0,1] into n subinter-
vals by some knots 0 = x < x < ... < x = 1; P is applied to the
0,n 1,n n,n n,m
products a(x,y)u(y) and b(x,y)u(y) as functions of y treating x as a parameter.
This results to some Nystr(cid:246)m type methods of accuracy O(n−m) provided that the
grid {x , x ,..., x } is properly graded to compensate the generic boundary
0,n 1,n n,n
singularities of the derivatives of the exact solution to (1.1); even the accuracy
O(n−m−1+ν) can be achieved for a skilled choice of the interpolation points, see
[13]. The method of work [9] is di(cid:27)erent: the authors (cid:28)rst perform a change of
variables which improves the boundary behaviour of the solution and of the coef-
(cid:28)cient functions a(x,y), b(x,y), and after that, using a polynomial approximation
of the solution, they apply a Gauss type quadrature formula for the integral in the
transformed equation.
In our method we follow [9], [10] performing a smoothing change of variables
and then, instead of discontinuous piecewise polynomial interpolation by P , we
n,m
useaproductquasi-interpolation(oreventherealinterpolation)bysmoothsplines
ofdegreem−1ontheuniformgridofthestepsizeh=1/n. Thisleadstosomere-
ductionofdegreesoffreedomcomparedwiththeuseofpiecewisepolynomials. Our
algorithm still has features of the Nystr(cid:246)m method. In Nystr(cid:246)m type methods for
weakly singular integral equations, the computation of the quadrature coe(cid:30)cients
∗ThisworkwassupportedbytheEstonianInformationTechnologyFoundation,grantNo07-
03-01-01,andbytheEstonianScienceFoundation,grantNo7353.
†Faculty of Mathematics and Computer Science, University of Tartu, Liivi 2, Tartu 50409,
Estonia([email protected]@ut.ee)
1
2 E.VAINIKKOANDG.VAINIKKO
is a laborious and delicate part of the algorithms. The use of quasi-interpolation
(orrealinterpolation)bysmoothsplinesonuniformgridsenablesustoobtainsim-
ple but still somewhat delicate formulae for the quadrature coe(cid:30)cients, and so we
obtain a fully discrete method of accuracy O(hm). This is the main message of the
present paper.
In our considerations, the coe(cid:30)cient functions a(x,y) and b(x,y) may have
boundary singularities with respect to y. Denote by T the integral operator of
equation (1.1),
Z 1
(Tu)(x)= (cid:0)a(x,y)|x−y|−ν +b(x,y)(cid:1)u(y)dy.
0
Lemma 1.1. Assume that a,b∈C([0,1]×(0,1)) satisfy the inequalities
|a(x,y)|≤cy−λ0(1−y)−λ1, |b(x,y)|≤cy−µ0(1−y)−µ1, (x,y)∈[0,1]×(0,1),
where 0<ν <1, λ ,λ ,µ ,µ ∈R, λ +ν <1, λ +ν <1, µ <1, µ <1. Then
0 1 0 1 0 1 0 1
T maps C[0,1] into C[0,1], and T : C[0,1]→C[0,1] is compact.
The proof is standard, cf. [5], [7]; a detailed argument can be found in [11].
For m ∈ N, θ0,θ1 ∈ R, θ0,θ1 < 1, denote by C?m(0,1) and Cm,θ0,θ1(0,1) the
weighted spaces of functions u∈C[0,1]∩Cm(0,1) such that, respectively,
m
X
kuk := sup xk(1−x)k|u(k)(x)|<∞,
Cm(0,1)
? 0<x<1
k=0
m
X
kuk := sup w (x)w (1−x)|u(k)(x)|<∞
Cm,θ0,θ1(0,1) k−1+θ0 k−1+θ1
0<x<1
k=0
where

1, ρ<0,

w (r)= 1/(1+|logr|), ρ=0, r,ρ∈R, r >0.
ρ
 rρ, ρ>0,
Clearly, Cm[0,1]⊂Cm,θ0,θ1(0,1)⊂Cm(0,1).
?
Lemma 1.2. ([11]). Let a,b∈Cm([0,1]×(0,1)) and
(1.2)|∂k∂la(x,y)|≤cy−λ0−l(1−y)−λ1−l, |∂k∂lb(x,y)|≤cy−µ0−l(1−y)−µ1−l,
x y x y
(x,y)∈[0,1]×(0,1), k+l≤m,
where m ∈ N, 0 < ν < 1, λ +ν < 1, λ +ν < 1, µ < 1, µ < 1. Then T maps
0 1 0 1
Cm(0,1) into Cm(0,1), T : Cm(0,1) → Cm(0,1) is bounded and T2 : Cm(0,1) →
? ? ? ? ?
C?m(0,1) is compact. Further, T maps Cm,θ0,θ1(0,1) with θ0 = max{λ0 +ν,µ0},
θ1 = max{λ1+ν,µ1} into Cm,θ0,θ1(0,1) and T : Cm,θ0,θ1(0,1) → Cm,θ0,θ1(0,1) is
compact.
Here ∂k∂l = (∂/∂x)k(∂/∂y)l. An example of a satisfying (1.2) is given by
x y
a(x,y)=y−λ0(1−y)−λ1a(x,y) where a∈Cm([0,1]×[0,1]).
Denote N(I − T) = {u ∈ C[0,1] : u = Tu}. The following theorem is a
consequence of Lemmas 1.1 and 1.2.
Theorem 1.3. Assume the conditions of Lemma 1.2 and N(I −T) = {0}.
Then for f ∈Cm(0,1), equation (1.1) has a solution u∈Cm(0,1) which is unique
? ?
in C[0,1], and k u k ≤ c k f k where the constant c is independent
Cm(0,1) Cm(0,1)
? ?
of f. Further, if f ∈ Cm,θ0,θ1(0,1), θ0 = max{λ0 +ν,µ0}, θ1 = max{λ1 +ν,µ1}
PRODUCTQUASI-INTERPOLATIONMETHOD 3
then u ∈ Cm,θ0,θ1(0,1), and k u kCm,θ0,θ1(0,1)≤ cm,θ0,θ1 k f kCm,θ0,θ1(0,1) where the
constant c is independent of f.
m,θ0,θ1
The main results of the paper are established under assumptions of Theorem
1.3. Using the approach of [12], our treatment can be extended to the case where
f has a singularity at a point x = ξ, ξ ∈ (0,1), and/or a and b have certain
singularities on the lines x=ξ and y =ξ or on a system of such inner points/lines
(cid:21) we can build the approximate solution on [0,ξ] and [ξ,1] in a similar manner as
this is done on [0,1] in the Section 4.
The rest of the paper is organised as follows. In Section 2 we recall some
results about the interpolation and quasi-interpolation of functions by splines with
auniformknotsetand, inparticular, abouttheoptimalityofsuchapproximations.
InSection3wetransformequation(1.1)intoanewformbythechangeofvariables.
Section 4 is central in the paper (cid:21) we introduce the product quasi-interpolation
method for the transformed equation, examine its accuracy and present the matrix
form of the method. In Section 5 we discuss di(cid:27)erent computational details of the
method and in Section 6 we test the algorithms on a numerical example.
Actually the method can be applied to somewhat more general class of equa-
tions rather than (1.1). The case of integral equation with logarithmic diagonal
singularity of the kernel will be treated in a separate work with more attention to
the computational environment. In [17] the method is modi(cid:28)ed for the Volterra
integral equation corresponding to (1.1).
2. Approximation tools.
2.1. The father B-spline. Thefather B-spline B of order minthetermi-
m
nology of [3], [15], or of degree m−1 in the terminology of [6], [16], can be de(cid:28)ned
by the formula
m
(2.1) B (x)= 1 X(−1)iC(m)(x−i)m−1, x∈R, m∈N.
m (m−1)! i +
i=0
Here, as usual, 0!=1, 00 :=lim xx =1,
x↓0
(cid:18) m (cid:19) m! (cid:26) (x−i)m−1, x−i≥0,
C(m) = = , (x−i)m−1 =
i i i!(m−i)! + 0, x−i<0.
Let us recall some properties of B : B ∈C(m−2)(R),
m m
(2.2) suppB =[0,m], B (x)=B (m−x)>0 for 0<x<m,
m m m
(2.3) B(m−1)(x)=(−1)lC(m−1) for l<x<l+1, l=0,...,m−1,
m l
Z
X
(2.4) B (x−j)=1 for x∈R, B (x)dx=1.
m m
j∈Z R
2.2. Spline interpolation on the uniform grid in R. Introduce in R the
uniform grid hZ={ih: i∈Z} of the step size h>0. Denote by S , m∈N, the
h,m
space of splines of order m (of degree m−1) and defect 1 with the knot set hZ.
The family B-spline B (h−1x−j), j ∈ Z, belongs to S , and the same is true
m h,m
forP d B (h−1x−j)witharbitrarycoe(cid:30)cientsd ; therearenoproblemswith
j∈Z j m j
the convergence of the series since it is locally (cid:28)nite: it follows from (2.2) that
4 E.VAINIKKOANDG.VAINIKKO
i
X X
d B (h−1x−j)= d B (h−1x−j) for x∈[ih,(i+1)h), i∈Z.
j m j m
j∈Z j=i−m+1
Given a function f ∈ C(R), bounded or of at most polynomial growth as
|x|→∞, we determine the interpolant Q f∈S by the conditions
h,m h,m
X
(2.5) (Q f)(x)= d B (h−1x−j), x∈R,
h,m j m
j∈Z
m m
(2.6) (Q f)((k+ )h)=f((k+ )h), k ∈Z.
h,m 2 2
Form=1andm=2,Q f istheusualpiecewiseconstant,respectively,piecewise
h,m
linearinterpolantwhichcanbedeterminedoneverysubinterval[ih,(i+1)h),i∈Z,
independentlyofothersubintervals. Form≥3,thevalueofQ f atagivenpoint
h,m
x ∈ R depends on the values of f at all interpolation knots (k+ m)h, k ∈ Z. It
2
occurs (see [14], [16]) that for m ≥ 3 conditions (2.5)(cid:21)(2.6) really determine d ,
j
j ∈Z, uniquely in the space of bounded or polynomially growing bisequences (d ),
j
namely,
X m X m
(2.7) d = α f((k+ )h)= α f((j−k+ )h), j ∈Z,
j j−k,m 2 k,m 2
k∈Z k∈Z
where
(2.8) α =Xm0 zlm,m0−1 z|k|, k ∈Z, m =(cid:26) (m−2)/2 if m is even,
k,m P0 (z ) l,m 0 (m−1)/2 if m is odd,
l=1 m l,m
and z ∈(−1,0), l=1,...,m , are the roots of the characteristic polynomial
l,m 0
X m
P (z)= B (k+ )zk+m0
m m 2
|k|≤m0
(it is a polynomial of degree 2m ); it occurs that P has exactly m simple roots
0 m 0
z , l = 1,...,m , in the interval (−1,0), and the remaining m roots are of the
l,m 0 0
form z =1/z ∈(−∞,−1), l=1,...,m .
l+m0,m l,m 0
Denote by BC(R) the space of bounded continuous functions on R equipped
with the norm k f k∞= supx∈R|f(x)|, by Vm,∞(R) the space of functions having
bounded mth (distributional) derivative in R, by Wm,∞(R) the standard Sobolev
spaceoffunctionsonRhavingboundedderivativesoforder≤m,andbyWm,∞(R)
(0,1)
the space of functions f ∈Wm,∞(R) with supports in (0,1).
Lemma 2.1. ([6,21,22]). For f ∈Vm,∞(R), it holds f−Q f ∈BC(R) and
h,m
(2.9) kf −Q f k ≤ Φ π−mhm kf(m) k
h,m ∞ m+1 ∞
where Φ = 4 P∞ (−1)km , m∈N, is the Favard constant,
m π k=0 (2k+1)m
Φ <Φ <Φ <...<4/π <...<Φ <Φ <Φ , lim Φ =4/π.
1 3 5 6 4 2 m
m→∞
Thefollowinglemmatellsthat,insomesense,thesplineinterpolationyieldsthe
bestapproximationofthefunctionclassesWm,∞(R)andVm,∞(R),asymptotically
also of Wm,∞(R), compared with other methods that use the same information as
(0,1)
PRODUCTQUASI-INTERPOLATIONMETHOD 5
Qh,mf (cid:21) the values f|Zh,m of f on the grid Zh,m ={(j+ m2)h: j ∈Z}. Denote by
C(Z ) the vector space of all (grid) functions de(cid:28)ned on Z .
h,m h,m
Lemma 2.2. ([21,22]). For given γ >0, we have by Lemma 2.1
sup kf −Q f k ≤Φ π−mhmγ,
h,m ∞ m+1
f∈Vm,∞(R),kf(m)k∞≤γ
whereas for any mapping M : C(Z ) → C(R) (linear or nonlinear, continuous
h h,m
or discontinuous), it holds
sup kf −Mh(f|Zh,m)k∞≥Φm+1π−mhmγ,
f∈Wm,∞(R),kf(m)k∞≤γ
lihm→i0nff∈W(m0,,1∞)(Rs)u,pkf(m)k∞≤γ kf −Mh(f|Zh,m)k∞ /(Φm+1π−mhmγ)≥1.
2.3. Quasi-interpolation by splines. Let m ≥ 3. In a quasi-interpolant
Q(p) f, the in(cid:28)nite sum (2.7) de(cid:28)ning the coe(cid:30)cients d of the spline interpolant
h,m j
(2.5)(cid:21)(2.6) is replaced by a (cid:28)nite sum:
(2.10) (Q(p) f)(x)=Xd(p)B (h−1x−j), x∈R,
h,m j m
j∈Z
(2.11) d(p) = X α(p) f((j−k+ m)h), p∈N.
j k,m 2
|k|≤p−1
A simple truncation of the series in (2.7) does not give acceptable results. Using a
special di(cid:27)erence calculus for fast decaying bisequences, the following formulae for
α(p) have been proposed in [8]:
k,m
p−1 (cid:18) (cid:19)
(2.12) α(p) = X(−1)k+q 2q γ , |k|≤p−1,
k,m k+q q,m
q=|k|
(2.13) γ =1, γ =Xm0 (1+zl,m)zlm,m0+q−1 , q ≥1.
0,m q,m (1−z )2q+1P0 (z )
l=1 l,m m l,m
Then it occurs that
(2.14) kQ f −Q(p) f k ≤c h2p kf(2p) k for f ∈V2p,∞(R)
h,m h,m ∞ m,p ∞
with a constant c that can be described. A consequence of (2.14) is that for
m,p
f ∈ Vm,∞(R) with uniformly continuous f(m) and 2p > m, it holds k Q f −
h,m
Q(p) f k h−m → 0 as h → 0, i.e., the quasi-interpolant Q(p) f is asymptotically
h,m ∞ h,m
ofthesameaccuracyastheinterpolantQ f. Itisreasonabletotakethesmallest
h,m
p∈N for which 2p>m; denote it by m ,
1
(cid:26) m +1, m even (cid:27)
(2.15) m = 2 =m−m .
1 m+1, m odd 0
2
Denote also Q0 :=Q(m1), α0 :=α(m1), |k|<m . Note that (Q0 f)(x) is well
h,m h,m k,m k,m 1 h,m
de(cid:28)nedforx∈[ih,(i+1)h]withani∈Ziff isgivenon[(i−m+1)h,(i+m)h]∩Z .
h,m
6 E.VAINIKKOANDG.VAINIKKO
Lemma 2.3. ([8]). For i∈Z, f ∈Cm[(i−m+1)h, (i+m)h], it holds
(2.16) max |f(x)−(Q0 f)(x))|
h,m
ih≤x≤(i+1)h
≤(Φ π−m+q c0 )hm sup |f(m)(x)|.
m+1 m m
(i−m+1)h≤x≤(i+m)h
For a relatively compact set M in C[−δ,1+δ], δ >0, it holds
(2.17) sup max |f(x)−(Q0 f)(x))|→0 as h→0.
h,m
f∈M 0≤x≤1
Formulae for qm :=k Qh,m kBC(R)→BC(R), qm0 :=k Q0h,m kBC(R)→BC(R) and c0m
from (2.16) are presented in [8] with the following numerical values:
m 3 4 5 6 7 8 9 10 20
c0 0.016 0.019 0.015 0.0085 0.0060 0.0030 0.0022 0.0010 6.5e-6
m
q 1.414 1.549 1.706 1.816 1.916 2.000 2.075 2.142 2.583
m
q c0 0.023 0.029 0.026 0.015 0.012 0.006 0.005 0.002 1.7e-5
m m
q0 1.250 1.354 1.329 1.403 1.356 1.413 1.378 1.419 1.514
m
Relatively small values of the norms q and q0 tell us about good stability prop-
m m
erties of interpolation and quasi-interpolation processes.
We assumed that m≥3. For m=1 and m=2, we may put Q0 =Q .
h,m h,m
3. A smoothing change of variables. In the integral equation (1), we per-
form the change of variables
(3.1) x=ϕ(t), y =ϕ(s), 0≤t≤1, 0≤s≤1,
where ϕ: [0,1]→[0,1] is de(cid:28)ned by the formula
1 Z t
(3.2) ϕ(t)= σr0−1(1−σ)r1−1dσ,
c
? 0
Z 1 Γ(r )Γ(r )
c = σr0−1(1−σ)r1−1dσ = 0 1 ,
? Γ(r +r )
0 0 1
Γ is the Euler gamma function. We assume that r ,r ∈R, r ≥1, r ≥1, but we
0 1 0 1
keep in mind that practicable algorithms correspond to the case where at least one
of parameters r and r is natural. If r ∈N, (3.2) takes the form
0 1 1
ϕ(t)= 1 tr0rX1−1 (−1)kC(r1−1)tk, 0≤t≤1, c = (r1−1)! ,
c r +k k ? r (r +1)...(r +r −1)
? 0 0 0 0 1
k=0
butin ourcomputations, thefollowing expansionof ϕ withpositiveterms occurred
to be preferable (essentially more stable numerically):
(3.3) ϕ(t)=tr0"1+rX1−1r0(r0+1)...(r0+k−1)(1−t)k#, 0≤t≤1.
k!
k=1
If r ,r ∈N, the integral in (3.2) can be alternatively evaluated [9] in a stable way
0 1
by an exact Gauss rule, since the integrand is a polynomial of degree r +r −2.
0 1
PRODUCTQUASI-INTERPOLATIONMETHOD 7
Clearly, ϕ(0) = 0, ϕ(1) = 1 and ϕ(t) is strictly increasing in [0,1]. Hence
ϕ(t)−ϕ(s) > 0, |ϕ(t)−ϕ(s)| = ϕ(t)−ϕ(s)|t−s| for s 6= t, and equation (1.1) takes
t−s t−s
with respect to v(t)=u(ϕ(t)) the form
Z 1
(3.4) v(t)= (cid:0)A(t,s)|t−s|−ν +B(t,s)(cid:1)v(s)ds+g(t), 0≤t≤1,
0
where
(cid:26) ϕ(t)−ϕ(s), t6=s,
(3.5) A(t,s)=a(ϕ(t),ϕ(s))Φ(t,s)−νϕ0(s), Φ(t,s)= t−s
ϕ0(s), t=s,
(3.6) B(t,s)=b(ϕ(t),ϕ(s))ϕ0(s), g(t)=f(ϕ(t)).
Let us characterise the boundary behaviour of functions in equation (3.4). Clearly,
0≤ϕ(t)≤ctr0, 0≤1−ϕ(t)≤c(1−t)r1,
(3.7) |ϕ(k)(t)|≤ctr0−k(1−t)r1−k, 0<t<1, k =1,...m.
Lemma 3.1. For u∈Cm(0,1), v(t)=u(ϕ(t)), it holds v ∈Cm(0,1) and
? ?
(3.8) kv k ≤ckuk
Cm(0,1) Cm(0,1)
? ?
where the constant c is independent of u. Further, if u ∈ Cm,θ0,θ1(0,1), θ0 < 1,
θ <1, then for j =1,...,m, 0<t<1,
1
(3.9) |v(j)(t)|≤ckuk ×
Cm,θ0,θ1(0,1)
 tr0−j, θ0 <0  (1−t)r1−j, θ1 <0 
tr0−j(1+|logt|), θ0 =0 (1−t)r1−j(1+|log(1−t)|), θ1 =0 .
 t(1−θ0)r0−j, θ0 >0  (1−t)(1−θ1)r1−j, θ1 >0 
Proof. Clearly k v k =k u k . The proof of (3.8) and (3.9) is based on (3.7)
∞ ∞
and the formula of Fa(cid:224) di Bruno
(cid:18)d(cid:19)j X j! (cid:18)ϕ0(t)(cid:19)k1 (cid:18)ϕ(j)(t)(cid:19)kj
u(ϕ(t))= u(k1+...+kj)(ϕ(t)) ...
dt k !..k ! 1! j!
1 j
k1+...+jkj=j
where the sum is taken over all k ≥0,...,k ≥0 such that k +2k +...+jk =j.
1 j 1 2 j
Here are the details for (3.9). For 0<t≤ 1, 1≤j ≤m, we have
2
 
1, k +...+k <1−θ
X  1 j 0 
|v(j)(t)|≤c 1+|logt|, k +...+k =1−θ ×
1 j 0
k1+2k2...+jkj=j t(1−θ0−k1−...−kj)r0, k1+...+kj >1−θ0 
×kuk tk1(r0−1)tk2(r0−2)...tkj(r0−j)
Cm,θ0,θ1
X  t(k1+...+kj)r0−j, k1+...+kj <1−θ0 
=c t(1−θ0)r0−j(1+|logt|), k1+...+kj =1−θ0 kukCm,θ0,θ1 .
k1+...+jkj=j t(1−θ0)r0−j, k1+...+kj >1−θ0 
Theconditionk +2k ...+jk =jimpliesthefollowing. (i)Therelationk +...+k <
1 2 j 1 j
1−θ0 is possible only if θ0 < 0, and then tr0−j is the leading term as t → 0 in
8 E.VAINIKKOANDG.VAINIKKO
the last sum (it corresponds to k = ... = k = 0, k = 1). (ii) The relation
1 j−1 j
k +...+k = 1−θ is possible only if θ ≤ 0, θ ∈ Z; in case θ = 0, the leading
1 j 0 0 0 0
term is tr0−j(1+|logt|) whereas in case θ0 ≤ −1, similarly as in (i), the leading
term is tr0−j . (iii) If 0 < θ0 < 1 then all terms in the last sum are t(1−θ0)r0−j.
Theseconsiderationsprove(3.9)for0<t≤ 1. Byasymmetryargumentweobtain
2
(3.9) also for 1 ≤t<1.
2
Lemma 3.2. Let a and b satisfy the conditions of Lemma 1.2. Then for
j =0,...,m, 0≤t≤1, 0<s<1, it holds
(3.10) |∂ja(ϕ(t),ϕ(s))|≤cs−r0λ0−j(1−s)−r1λ1−j,
s
(3.11) |∂jb(ϕ(t),ϕ(s))|≤cs−r0µ0−j(1−s)−r1µ1−j.
s
These inequalities elementarily follow by the formula of Fa(cid:224) di Bruno.
The function Φ(t,s)−ν has singularities at (0,0) and (1,1), the only zeroes of
Φ(t,s) in [0,1]×[0,1]. It is easy to see that
∂kΦ(t,s)(cid:16)(t+s)r0−k−1((1−t)+(1−s))r1−k−1 as t,s→0 or as t,s→1
s
that together with the formula of Fa(cid:224) di Bruno implies the following result.
Lemma 3.3. For j =0,...,m, 0≤t≤1, 0<s<1, it holds
(cid:12) (cid:16) (cid:17)(cid:12)
(3.12) (cid:12)∂j (Φ(t,s))−ν (cid:12)≤c(t+s)−ν(r0−1)−j((1−t)+(1−s))−ν(r1−1)−j.
(cid:12) s (cid:12)
Next we present estimates for functions A(t,s), B(t,s) and ∂m[A(t,s)v(s)],
s
∂m[B(t,s)v(s)] in somewhat speci(cid:28)c form for the needs of Section 4.
s
Corollary 3.4. Let a and b satisfy the conditions of Lemma 1.1. Then the
following holds true: (i) if
(3.13) r ,r ≥1, r >(1−ν)/(1−ν−λ ), r >(1−ν)/(1−ν−λ ),
0 1 0 0 1 1
then with δ :=(1−ν−λ )r −(1−ν)>0, δ :=(1−ν−λ )r −(1−ν)>0, it
0 0 0 1 1 1
holds
(3.14) |A(t,s)|≤csδ0(1−s)δ1, (t,s)∈[0,1]×(0,1);
(ii) if
(3.15) r ,r ≥1, r >1/(1−µ ), r >1/(1−µ ),
0 1 0 0 1 1
then with δ :=(1−µ )r −1>0, δ :=(1−µ )r −1>0, it holds
0 0 0 1 1 1
(3.16) |B(t,s)|≤csδ0(1−s)δ1, (t,s)∈[0,1]×(0,1).
Proof. These estimates are direct consequences of Lemmas 3.2 and 3.3. By
(3.7), (3.10) and (3.12) we have
|A(t,s)|≤cs−r0λ0+r0−1(t+s)−ν(r0−1)(1−s)−r1λ1+r1−1(2−t−s)−ν(r1−1)
that for r , r satisfying (3.13) yields (3.14). Similarly, by (3.7), (3.11) and (3.12),
0 1
|B(t,s)|≤cs−r0µ0+r0−1(1−s)−r1µ1+r1−1
that for r , r satisfying (3.15) yields (3.16).
0 1
PRODUCTQUASI-INTERPOLATIONMETHOD 9
Corollary 3.5. Let a and b satisfy the conditions of Lemma 1.2, and let
u ∈ Cm(0,1), v(t) = u(ϕ(t)). Then the following estimates hold true for (t,s) ∈
?
[0,1]×(0,1): (i) if
(3.17) r ,r ≥1, r >m/(1−ν−λ ), r >m/(1−ν−λ ),
0 1 0 0 1 1
then with δ :=(1−ν−λ )r −m>0, δ :=(1−ν−λ )r −m>0,
0 0 0 1 1 1
(3.18) |A(t,s)|≤csm−(1−ν)+δ0(1−s)m−(1−ν)+δ1,
(3.19) |∂sm[A(t,s)v(s)]|≤cs−(1−ν)+δ0(1−s)−(1−ν)+δ1 kukC?m(0,1);
(ii) if
(3.20) r ,r ≥1, r >m/(1−µ ), r >m/(1−µ ),
0 1 0 0 1 1
then with δ :=(1−µ )r −m>0, δ :=(1−µ )r −m>0,
0 0 0 1 1 1
(3.21) |B(t,s)|≤csm−1+δ0(1−s)m−1+δ1,
(3.22) |∂sm[B(t,s)v(s)]|≤cs−1+δ0(1−s)−1+δ1 kukC?m(0,1);
(iii) if
(3.23) r ,r ≥1, r >(m+ν)/(1−λ ), r >(m+ν)/(1−λ )
0 1 0 0 1 1
then with δ :=(1−λ )r −m−ν >0, δ :=(1−λ )r −m−ν >0,
0 0 0 1 1 1
(3.24) ϕ0(t)ν|A(t,s)|≤csm−(1−ν)+δ0(1−s)m−(1−ν)+δ1,
(3.25) ϕ0(t)ν|∂sm[A(t,s)v(s)]|≤cs−(1−ν)+δ0(1−s)−(1−ν)+δ1 kukC?m(0,1) .
Proof. These estimates are direct consequences of Lemmas 3.1(cid:21)3.3. Observe
that |∂k[ϕ0(t)νΦ(t,s)−ν]|≤c(t+s)−k((1−t)+(1−s))−k.
s
Let us extend A(t,s) and B(t,s) with respect to s outside (0,1) by the zero
value. Underconditions(3.13)and(3.15)weobtaincontinuousfunctionson[0,1]×
R, see (3.14) and (3.16).
4. The product quasi-interpolation method.
4.1. The operator form of the method. Let h = 1/n, n ∈ N, n ≥ m.
For 0 ≤ s ≤ 1, the quasi-interpolant Q0 w of a function w∈ C[−δ,1+δ] has the
h,m
expansion
n−1
X X m
(4.1) (Q0 w)(s)= α0 w((j−p+ )h)B (ns−j).
h,m p,m 2 m
j=−m+1|p|<m1
We approximate equation (3.4) by its discretised version: for 0≤t≤1,
Z 1
(4.2) v (t)= [|t−s|−νQ0 (A(t,s)v (s))+Q0 (B(t,s)v (s))]ds+g(t)
n h,m n h,m n
0
where Q0 is applied to the products A(t,s)v(s) and B(t,s)v(s) as functions of s.
h,m
This can be done for v given on [0,1] since A(t,s) = B(t,s) = 0 for s ≤ 0 and for
s≥1; recall that under condition (3.13) and (3.15), A,B ∈C([0,1]×R).
10 E.VAINIKKOANDG.VAINIKKO
Theorem4.1. (i)LetaandbsatisfytheconditionsofLemma1.1,N(I−T)=
{0}, f ∈ C[0,1], and let r and r satisfy condition (3.13) and (3.15). Then for
0 1
su(cid:30)ciently large n, equation (4.2) has a unique solution v , and
n
(4.3) kv−v k := max |v(t)−v (t)|→0 as n→∞
n ∞ n
0≤t≤1
where v is the solution of equation (3.4).
(ii) Under assumptions of Theorem 1.3, f ∈ Cm(0,1) and conditions (3.17)
?
and (3.20) on r and r , it holds
0 1
(4.4) kv−v k ≤chm kf k .
n ∞ Cm(0,1)
?
(iii) Under assumptions of Theorem 1.3, f ∈ Cm(0,1) and conditions (3.20)
?
and (3.23) on r and r , it holds
0 1
(4.5) max ϕ0(t)ν|v(t)−v (t)|≤chm kf k .
n Cm(0,1)
0≤t≤1 ?
The constant c in (4.4) and (4.5) is independent of n and f.
Proof. Accept the assumptions formulated in (i). Denote by T and T the
n
integral operators of equations (3.4) and (4.2),
Z 1
(Tv)(t)= [|t−s|−νA(t,s)+B(t,s)]v(s)ds,
0
Z 1
(T v)(t)= [|t−s|−νQ0 (A(t,s)v(s))+Q0 (B(t,s)v(s))]ds.
n h,m h,m
0
We claim that T →T compactly in C[0,1] as n→∞, i.e.,
n
(4.6) kT v−Tv k →0 for every v ∈C[0,1],
n ∞
(4.7) (v )⊂C[0,1], kv k ≤1 ⇒ (T v ) is relatively compact in C[0,1].
n n ∞ n n
Indeed,thesets{A(t,·): 0≤t≤1}and{B(t,·): 0≤t≤1}arerelativelycompact
in C[−δ,1+δ], and by Lemma 2.3, for a (cid:28)xed v ∈ C[0,1] extended by v(t) = v(0)
for −δ ≤s≤0, v(t)=v(1) for 1≤s≤1+δ, it holds
sup max |A(t,s)v(s)−Q0 (A(t,s)v(s))|→0 for n→∞,
h,m
0≤t≤1 0≤s≤1
sup max |B(t,s)v(s)−Q0 (B(t,s)v(s))|→0 for n→∞.
h,m
0≤t≤1 0≤s≤1
This together with the equality kQ0 k=q0 implies (4.6). The proof of (4.7) can
h,m m
be built using the Arzela theorem.
Due to the condition N(I −T) = {0}, also N(I −T) = {0}. As well known
(see [1], [2], [7] or [18]), relations (4.6), (4.7) and N(I −T) = {0} imply that, for
su(cid:30)cientlylargen,theoperatorsI−T areinvertibleandtheinversesareuniformly
n
bounded:
(4.8) k(I−T )−1 k ≤c, n≥n .
n C[0,1]→C[0,1] 0
Let v and v be the solutions of equations (3.4) and (4.2), respectively. Then
n
(4.9) v−v =(I−T )−1(Tv−T v),
n n n