Error Bounds for Numerical Integration of Oscillatory Bessel Transforms with Algebraic or Logarithmic Singularities

0.14 MB·

by  Hongchao Kang

Error Bounds for Numerical Integration of Oscillatory
Bessel Transforms with Algebraic or Logarithmic
Singularities
1.
HongchaoKang2
DepartmentofMathematics,SchoolofScience,HangzhouDianziUniversity,Hangzhou,Zhejiang
310018,PRChina
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CongpeiAn3
n
a
J DepartmentofMathematics,JinanUniversity,Guangzhou510632,China

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Abstract In this paper, we present and analyze the Clenshaw-Curtis-Filon methods for comput-

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A ingtwoclasses ofoscillatory Besseltransforms withalgebraic orlogarithmicsingularities. More
N
importantly, forthesequadrature ruleswederivenewcomputational sharperrorboundsbyrigor-
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h
ous proof. These new error bounds share the advantageous property that some error bounds are
t
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m optimalonωforfixedN,whileothererrorboundsareoptimalonN forfixedω. Furthermore,we
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provefromthepresented errorbounds ininversepowersofωthattheaccuracy improvesgreatly,

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forfixedN,asωincreases.
v
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4 Keywords: oscillatory, Besseltransforms singularities, Clenshaw-Curtis-Filon methods, quadra-
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2 tureruleserrorbounds.
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1 Introduction
r
a
Highly oscillatory Bessel transforms arise widely in mathematical and numerical modeling of
oscillatoryphenomenainmanyareasofsciencesandengineeringsuchasastronomy,electromag-
1This work is supported by National Natural Science Foundation of China (Grant Nos.11301125, 11071260,
11226305,11301222), Scientific Research Startup Foundation of Hangzhou Dianzi University (KYS075613017),
the Fundamental Research Funds for the Central Universities (No. 21612336) and NSF of Guangdong (No.
S2012040007860).
2E-mailaddress:[e-mail]
3E-mailaddress:[e-mail],[e-mail]
1
netics,acoustics, scattering problems, physicaloptics,electrodynamics, andapplied mathematics
[2,3,10,16]. Inthispaper, wefocus onnewcomputational sharp errorbounds ofthequadrature
rulesforsingular oscillatory Besseltransforms oftheforms
b
I [f] = xαf(x)J (ωx)dx, (1.1)
1 m
Z
0
b
I [f] = xαln(x)f(x)J (ωx)dx, (1.2)
2 m
Z
0
where f(x) is suitably smooth in [0,b], J (z) is the Bessel function [1] of the first kind and of
m
order mwithRe(m) > −1,ωisalarge parameter, barerealand finite, andα > −1. Inparticular,
it should be noticed that transforms (1.1) and (1.2) are integrals with algebraic and logarithmic
singularities, respectively.
In most of the cases, such integrals cannot be calculated analytically and one has to re-
sort to numerical methods [9]. The numerical evaluation can be difficult when the parameter ω
is large, because in that case the integrand is highly oscillatory. The singularities of algebraic
or logarithmic type and possible high oscillations of the integrands in (1.1) and (1.2) make the
above integrals very difficult to approximate accurately using standard methods, e.g., Gaussian
quadrature rules. It is well known [9] that a prohibitively large number of quadrature points is
needed if one uses a classic rule such as Gaussian quadratures, or any quadrature method based
on(piecewise) polynomial interpolation oftheintegrands.
In the last few decades, much progress has been made in developing numerical schemes for
b
generalized Bessel transform f(x)J (ωg(x))dx without singularity. For example, the modi-
a m
R 1
fiedClenshaw-Curtis method[28]wasintroduced forefficiently computing f(x)J (ωx)dxfor
0 m
R
m being an integer in 1983; the Levin method [21], Levin-type method [27], and generalized
b
quadraturerules[11,32]werealsoavailableforapproximating f(x)J (ωx)dxforRe(m)> −1.
a m
R
However,theLevinmethod,Levin-typemethodsandgeneralized quadraturerulescannotbeused
if0 < [a,b]. In addition, based onadiffeomorphism transformation, the reference [33]extended
1
the Filon-type method to the efficient computation of the integrals f(x)J (ωg(x))dx with the
0 m
R
exotic oscillator g(x) satisfying that for r ≥ 0, and g(0) = g′(0) = ··· = g(r)(0) = 0,g(r+1)(0) ,
0,g′(x) , 0 for x ∈ (0,1], where Re(m) > 1/(r+1). In many situations the accuracy of the
Filon-typemethodproposedin[33]issignificantlyhigherthanthatofothermethods. Asamatter
of fact, it requires the solution of a linear system that becomes more ill-conditioned as the num-
berofinterpolation nodes increases, andone has toadopt higher-order digit arithmetic togetthe
required accuracy. Furthermore, to avoid the Runge phenomenon, the Clenshaw-Curtis-Filon-
type method [34] based on Clenshaw-Curtis points is designed for computing Bessel transform
2
b
f(x)J (ωx)dx without singularity. Here, it should be also mentioned that the homotopy per-
a m
R b
turbation method in [4, 5] was presented to compute f(x)J (ωx)dx. Recently, Chen [6][7]
a m
R b
also proposed two different complex integration methods for approximating f(x)J (ωx)dx if
a m
R
0 < [a,b]. To the best of our knowledge, so far little research has been done on the numerical
computation oftheintegrals (1.1)and(1.2)withanalgebraicorlogarithmicsingularity.
Consequently, our aim is to demonstrate high efficiency of the proposed quadrature rules for
such integrals (1.1) and (1.2) by constructing error bounds. In the next section, we propose the
Clenshaw-Curtis-Filon methods for computing the integrals (1.1) and (1.2). Here, the required
modified moments can be efficiently calculated by a recurrence relation. Section 3 sets up new
and computational sharp error bounds ofthese quadrature rules by theory analysis. In Section 4,
wedesign ahigher order method and derive itserror estimate in inverse power ofω. Fromthese
new error bounds, it can be seen that for fixed ω, the error bounds are optimal on N, while for
fixed N theerror bounds areoptimal onω. Moreover, for fixed N, thelarger the values of ω,the
highertheaccuracy.
2 Clenshaw-Curtis-Filon methods for computing (1.1) and (1.2)
Chebyshev interpolation has precisely the same effect as taking partial sum of an approximation
Chebyshevseriesexpansion[24]. Supposethat f(x)isabsolutelycontinuouson[0,b]. LetP f(x)
N
denoteaninterpolant of f(x)ofdegree N intheClenshaw-Curtispoints
b b kπ
x = + cos , k = 0,1,...,N. (2.3)
k
2 2 N !
Then,thepolynomial P f(x)canbeexpressed by(see[24,Eq. 6.27,6.28])
N
N N
2
P f(x) = ′′a T∗∗(x), where a = ′′f(x )T∗∗(x ), (2.4)
N j j j N k j k
Xj=0 Xk=0
where the double primes indicate that the first and last terms of the sum are to be halved, T∗∗(x)
j
denotes theshifted Chebyshev polynomial of thefirstkind ofdegree jon[0,b]. Thecoefficients
a canbecomputedefficiently byFFT[8,30].
j
The Clenshaw-Curtis-Filon (CCF) methods for (1.1) and (1.2) are defined, respectively, as
follows,
b
ICCF[f] = xαP f(x)J (ωx)dx
1 Z N m
0
N
= bα+1 ′′a M , (2.5)
j j
Xj=0
3
and
b
ICCF[f] = xαln(x)P f(x)J (ωx)dx
2 Z N m
0
N
= bα+1 ′′a [ln(b)M + M ], (2.6)
j j j
Xj=0
e
where,forr = bω,
1
M = xαT∗(x)J (rx)dx, (2.7)
j j m
Z
0
1
M = xαln(x)T∗(x)J (rx)dx, (2.8)
j j m
Z
0
e
arecalledthemodifiedmoments,whereT∗(x)denotestheshiftedChebyshevpolynomialon[0,1],
j
andwhichcanbecomputedefficiently, asdescribed below.
Fastcomputationsofthemodifiedmoments:
ThehomogeneousrecurrencerelationofthemodifiedmomentsM ,wasprovidedbyPiessens
j
[20][29],asfollows:
r2 r2
M +[(j+3)(j+3+2α)+α2−m2− ]M
j+4 j+2
16 4
+[4(m2−α2)−2(j+2)(2α−1)]M
j+1
3r2
−[2(j2−4)+6(m2−α2)−2(2α−1)− ]M
j
8
+[4(m2−α2)−2(j−2)(2α−1)]M
j−1
r2 r2
+[(j−3)(j−3−2α)+α2−m2− ]M + M = 0, (2.9)
j−2 j−4
4 16
Itisworthtonoticethat
∂
M = M .
j j
∂α
Therefore, by differentiating the above recurrence erelation (2.9) with respect to α, we find M
j
satisfying thefollowingrecurrence relation: e
r2 r2
M +[(j+3)(j+3+2α)+α2−m2− ]M
j+4 j+2
16 4
+[4e(m2−α2)−2(j+2)(2α−1)]M e
j+1
3r2
−[2(j2−4)+6(m2−α2)−2(2α−e1)− ]M
j
8
+[4(m2−α2)−2(j−2)(2α−1)]M e
j−1
r2 r2
+[(j−3)(j−3−2α)+α2−m2−e ]M + M
j−2 j−4
4 16
= −2(α+ j+3)M +4(2α+ j+2)Me +4(3αe+1)M
j+2 j+1 j
+4(2α− j+2)M +2(j−α−3)M . (2.10)
j−1 j−2
4
Because of the symmetry of the recurrence relation of the Chebyshev polynomials T (x), it
j
is convenient to get T (x) = T (x), j = 1,2,..., and, consequently T∗ (x) = T∗(x), M = M
−j j −j j −j j
and M = M . It can be verified easily that both (2.9) and (2.10) are valid, not only for j ≥ 5,
−j j
but foer all inetegers of j. Unfortunately, for (2.9) and (2.10) both the forward recursion and the
backward recursion are asymptotically unstable [20, 29]. Nevertheless, in practical applications
the instability is less pronounced if ω ≥ 2j. Practical experiments demonstrate that M and M
j j
can be computed accurately using the forward recursion as long as ω ≥ 2j. But for ω < 2j tehe
lossofsignificantfiguresincreasesandrecursionintheforwarddirection isnolongerapplicable.
In this case Lozier’s algorithm [22] or Oliver’s algorithm [25] has to be used. This means that
both (2.9) and (2.10) have to be solved as a boundary value problem with six initial values and
two end values. The solution of this boundary value problem requires the solution of a linear
system of equations having a band structure. The end value can be estimated by the asymptotic
expansionsin[20]orcanbesetequaltozero. TheLozier’salgorithmincorporatesanumericaltest
for determining the optimum location of the endpoint, when the end value is set to be zero. The
advantage is that a user-required accuracy is automatically obtained, without computation of the
asymptoticexpansion. Fordetailsonecanreferto[20,22,25,29]. Tostarttherecurrencerelation
with k = 0,1,2,3,..., we only need M ,M ,M , and M . By plugging the shifted Chebyshev
0 1 2 3
polynomials T∗(x) = 1,T∗(x) = 2x−1,T∗(x) = 8x2−8x+1andT∗(x) = 32x3−48x2+18x−1
0 1 2 3
on[0,1]into(2.7),weobtain
M = G(ω,m,α),
0
M = 2G(ω,m,α+1)− M ,
1 0
M = 8G(ω,m,α+2)−4M −3M ,
2 1 0
M = 32G(ω,m,α+3)−6M −15M −10M .
3 2 1 0
Then,itisapparentfromtheaboveequalities that
∂
M = G(ω,m,α), (2.11)
0
∂α
e ∂
M = 2 G(ω,m,α+1)− M , (2.12)
1 0
∂α
e ∂ e
M = 8 G(ω,m,α+2)−4M −3M , (2.13)
2 1 0
∂α
e ∂ e e
M = 32 G(ω,m,α+3)−6M −15M −10M , (2.14)
3 2 1 0
∂α
e e e e
where,from [1,p.480], [15,p.676] and[23, p.44], wefindseveralmomentsformulae asfollows,
5
forℜ(m+α) > −1,
1
G(ω,m,α) = xαJ (rx)dx
m
Z
0
2αΓ(m+α+1) 1
= 2 + [(α+m−1)J (r)S (r)− J (r)S (r)], (2.15)
rα+1Γ(m−α+1) rα m α−1,m−1 m−1 α,m
2
rm α+m+1 α+m+3 r2
G(ω,m,α) = F ( ; ,m+1;− ), (2.16)
2m(α+m+1)Γ(m+1)1 2 2 2 4
Γ(m+α+1) ∞ (m+2j+1)Γ(m−α+1 + j)
G(ω,m,α) = 2 2 J (r), (2.17)
rΓ(m−α+1) Γ(m+α+3 + j) m+2j+1
2 Xj=0 2
whereS (z),Γ(z), F (µ;ν,λ;z)denoteaLommelfunction ofthesecondkind,thegammafunc-
µ,ν 1 2
tion, a class of generalized hypergeometric function, respectively. Moreover, F (µ;ν,λ;z) con-
1 2
vergesforall|z|. From[32,p.346],S (z)canbeexpressedintermsof F (µ;ν,λ;z),namely,
µ,ν 1 2
zµ+1 µ−ν+3 µ+ν+3 z2
S (z) = F (1; , ;− )
µ,ν (µ+ν+1)(µ−ν+1)1 2 2 2 4
2µ−1Γ(µ+ν+1)
− 2 (J (z)−cos(π(µ−ν)/2)Y (z)), (2.18)
πΓ(ν−µ) ν ν
2
whereY (z)isaBesselfunctionofthesecondkindoforderν. Theright-handsidesof(2.11-2.14)
ν
involvethederivativesofthegeneralizedhypergeometricfunctionwithrespecttotheparameterα,
whichhavebeenshownin[19]. Therequiredderivativesofthegammafunctionarealsodescribed
intermsofthePsi(Digamma)functionψ (z),suchas[1]
0
Γ′(z) = Γ(z)ψ (z). (2.19)
0
Theefficientimplementationofthemodifiedmomentsisbasedonthefastcomputationofthe
Lommel functions S (z) and the hypergeometric function F (µ;ν,λ;z). Excellent references
µ,ν 1 2
in this area are [14, 31]. Obviously, when programming the proposed algorithm in a language
like Matlab, we can calculate the values of Γ(z),J (z) and F (µ;ν,λ;z) by invoking the biult-in
m 1 2
functions ‘ gamma(z)’, ‘ besselj(m,z)’ and calling mfun(‘ hypergeom’, [µ],[ν,λ],z) from Maple,
respectively.
Thecomputationof S (z):
µ,ν
(1)Forlarge|z|and|argz|< π,wecancalculateefficientlyS (z)bytruncatingthefollowing
µ,ν
asymptotic expansion (see[31,pp. 351-352]) ininversepowersofz:
(µ−1)2−ν2 [(µ−1)2 −ν2][(µ−3)2−ν2]
S (z) = zµ−1 1− + −...
µ,ν
( z2 z4
[(µ−1)2−ν2]...[(µ−2p+1)2−ν2]
+(−1)p +O(zµ−2p−2),
z2p )
6
(2)Forsmall|z|,weprefertocomputeS (z)using(2.18).
µ,ν
So, when r = bω is large, such as r ≥ 50, we prefer to compute the moments using (2.15).
when r = bω is small, for example r < 50, the moments (2.17) are available. This may be due
to the property that J (r) is a fast decreasing function of m when m > r. Practical experiments
m
also demonstrate that J (r) can decrease to zero quite rapidly when m is a little larger than r.
m
Fortunately, themoments(2.16)isavailable forallr.
3 Error bounds of the CCF methods (2.5) and (2.6)
To obtain results that are absolutely reliable for numerical computations, it is necessary to con-
struct a upper bound for the corresponding error. In the following, we will consider new and
computational error bounds. Thesenewerror bounds share that forfixed N, theerror bounds are
optimalonω,whileforfixedωtheerrorboundsareoptimalon N.
In the following, in order to derive these new error bounds in inverse powers of ω, we first
giveLemmas3.1and3.2.
Lemma3.1 Foreveryt ∈ [0,b](b > 0)andα > −1,itistruethat,forω ≥ 1,
t O( 1 ), if −1< α < 0,
xαJm(ωx)dx =  ωα+1 (3.20)
Z0  O(ω1), if α ≥ 0,
t  O(1+ln(ω)), if −1< α ≤ 0,
xαln(x)Jm(ωx)dx =  ωα+1 (3.21)
Z0  O(ω1), if α > 0.

Proof: Wedivideourproofinthreesteps.
(1)For−1 < α < 0,settingy = ωxyieldsthat
t 1 ωt
xαJ (ωx)dx = yαJ (y)dy,
Z m ωα+1 Z m
0 0
t 1 ωt ωt
xαln(x)J (ωx)dx = yαln(y)J (y)dy−ln(ω) yαJ (y)dy .
Z m ωα+1 "Z m Z m #
0 0 0
Obviously,whethertheintegralupperlimitωtintheright-sideoftheabovetwoformulaeisfinite
or not, by convergence tests for improper integrals (Cauchy’s test or Dirichelet’s test), we know
thattheresulting defectorinfiniteintegrals areconvergent. Itleadstothefirstidentities in(3.20)
and(3.21).
∞
(2)Forα = 0,combining J (t)dt = 1[1,p.486]andthemomentsformula(2.15),wehave
0 m
R
t 1 ωt 1
J (ωx)dx = J (y)dy = O .
m m
Z0 ω Z0 (cid:18)ω(cid:19)
7
If0 < ωt ≤ 1,from[1,p.362]and[26],wehave
|J (x)| ≤ 1,m ≥ 0,x ∈ ℜ. (3.22)
m
So,thefirstidentity in(3.21)followsthat
t 1 ωt y
ln(x)J (ωx)dx = ln( )J (y)dy
(cid:12)(cid:12)Z0 m (cid:12)(cid:12) ω (cid:12)(cid:12)Z0 ω m (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) 1 (cid:12)(cid:12) ωt (cid:12)(cid:12)
(cid:12) (cid:12) ≤ (cid:12) |ln(y)−ln(ω)||(cid:12)J (y)|dy
m
ω Z
0
1 1
≤ −ln(y)+ln(ω) dy
ω Z
0
(cid:0) (cid:1)
1+ln(ω)
= . (3.23)
ω
Ifωt > 1,fromtheproofof (3.23),wethenobtain
t 1 ωt y
ln(x)J (ωx)dx = ln( )J (y)dy
(cid:12)(cid:12)Z0 m (cid:12)(cid:12) ω (cid:12)(cid:12)Z0 ω m (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) 1 (cid:12)(cid:12) 1 (cid:12)(cid:12) 1 ωt
(cid:12) (cid:12) ≤ (cid:12) (ln(y)−ln(ω))J(cid:12) (y)dy + (ln(y)−ln(ω))J (y)dy
ω (cid:12)(cid:12)Z0 m (cid:12)(cid:12) ω (cid:12)(cid:12)Z1 m (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
1+(cid:12)(cid:12) ln(ω) 1 ωt (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
≤ (cid:12) + (ln(y)−l(cid:12)n(ω))J(cid:12) (y)dy . (3(cid:12).24)
ω ω (cid:12)(cid:12)Z1 m (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Usingthemeanvaluetheoremforintegrals, w(cid:12) ehave (cid:12)
(cid:12) (cid:12)
ωt ξ ωt
(ln(y)−ln(ω))J (y)dy = −ln(ω) J (y)dy+ln(t) J (y)dy, for1 ≤ ξ ≤ ωt.
m m m
Z Z Z
1 1 ξ
Then,itfollowsthat
ωt ξ ωt
(ln(y)−ln(ω))J (y)dy ≤ ln(ω) J (y)dy +|ln(t)| J (y)dy
(cid:12)(cid:12)Z1 m (cid:12)(cid:12) (cid:12)(cid:12)Z1 m (cid:12)(cid:12) (cid:12)(cid:12)Zξ m (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) = O(1+(cid:12)ln(ω)), (cid:12) (cid:12) (cid:12) (3.25)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
ξ ωt
whichisduetothefactthatboth J (y)dyand J (y)dyconvergebyreferringtotheidentity
1 m ξ m
∞ R R
J (t)dt = 1 and the moments formula (2.15). Thus, combining (3.24) and (3.25) yields the
0 m
R
firstidentity in(3.21).
(3)Forα > 0,byintegrating bypartsandnoting thedifferential relation[1,pp.361,439]
d[xm+1J (ωx)] = ωxm+1J (ωx)dx, (3.26)
m+1 m
andtogetherwiththefirstidentities in(3.20)and(3.21),wefind
t 1 t
xαJ (ωx)dx = xα−m−1d[xm+1J (ωx)]
m m+1
Z ω Z
0 0
1 t
= xαJ (ωx)t −(α−m−1) xα−1J (ωx)dx
ω " m+1 0 Z m+1 #
(cid:12) 0
(cid:12)
1 (cid:12)
= O .
ω
(cid:18) (cid:19)
8
Similarly,weobtain
t 1
xαln(x)J (ωx)dx = O .
m
Z0 (cid:18)ω(cid:19)
Thiscompletes theproof.
FromLemma3.1,weproveLemma3.2.
Lemma3.2 For f ∈C[0,b],α > −1andω ≥ 1,itistruethat,
b b
tαf(t)J (ωt)dt ≤ C (ω)(|f(b)|+ |f′(t)|dt), (3.27)
(cid:12)(cid:12)Z0 m (cid:12)(cid:12) 1 Z0
(cid:12) (cid:12)
b(cid:12)(cid:12) (cid:12)(cid:12) b
t(cid:12)αln(t)f(t)J (ωt)dt(cid:12) ≤ C (ω)(|f(b)|+ |f′(t)|dt), (3.28)
(cid:12)(cid:12)Z0 m (cid:12)(cid:12) 2 Z0
(cid:12) (cid:12)
(cid:12) (cid:12)
where (cid:12)(cid:12) (cid:12)(cid:12)
c1 , if −1 < α < 0,
C1(ω) =  ωα+1
 cω2, if α ≥ 0,
 c3(1+ln(ω)), if −1 < α < 0,
C2(ω) =  ωα+1
 cω4, if α ≥ 0,
andc (k = 1,2,3,4)arefourconstantsindependent ofωand f.
k
Proof: Setting F(t) = t xαJ (ωt)dt,t ∈ [0,b],togetherwithLemma3.1,wethenhave
0 m
R
c1 , if −1 < α < 0,
C1(ω) = ||F(t)||∞ =  ωα+1
 cω2, if α ≥ 0,
Thesetogetherimpliesthat,byintegratingbyparts,
b b
tαf(t)J (ωr)dt = f(t)dF(t)
(cid:12)(cid:12)Z0 m (cid:12)(cid:12) (cid:12)(cid:12)Z0 (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)f(t)F(t)b− (cid:12)(cid:12)(cid:12) bF(t)f′(t)dt
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)0 Z0 b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
≤ |(cid:12)f(b)||F(b)|+ |F(t)||f′(t)(cid:12)|dt
Z
0
b
≤ |f(b)|||F(t)|| + |f′(t)|dt||F(t)||
∞ ∞
Z
0
b
= C (ω)(|f(b)|+ |f′(t)|dt).
1
Z
0
It is now obvious that the assertion (3.27) holds. The proof of (3.28) can be completed by the
methodanalogous tothatusedabove.
Meanwhile,itshouldalsobenotedthatthefollowingLemma3.3alsoplaysanimportantrole
intheconstruction oferrorbounds.
9
Lemma3.3 (see [34]) Let n be a nonnegative integer. If f is analytic with |f(z)| ≤ M in the
regionE boundedbytheellipsewithfoci±1andmajorandminorsemi-axeswhoselengths sum
ρ
toρ > 1,thenfor x∈ [−1,1],
2M(N +1)2n n 2ρ n+1−j
kf(n)(x)−P(n)f(x)k ≤ , (3.29)
N ∞ (ρN −ρ−N)(2n−1)!! Xj=0(cid:18)(ρ−1)2(cid:19)
where(2n−1)!! = 1·3·5···(2n−1)and(−1)!! = 1.
Based on the above Lemmas 3.1-3.3, we derive error bounds in inverse powers of ω in the
followingTheorems3.1-3.2. For f ∈C2[0,b],theerrorboundoftheCCFmethods(2.5)isshown
asfollows.
Theorem3.1 Assume that f ∈ C2[0,b]. Then the absolute error of the CCF methods (2.5) for
ω ≥ 1,α > −1andeveryfixed N,satisfies
bα+1||f(x)−P f(x)|| ,
α+1 N ∞
 
|I1[f]−I1CCF[f]| ≤ min bCC1(ω1()ω[|)f|′|(fb′()x−)−P′Pf′N(bf()|x+)||∞b(,1+ 3|α−m−1|)||f′′(x)−P′′f(x)|| ]. (3.30)
Proof: Fromthedefinition oωfP f(x),itisNobviousthat 2 N ∞ 
N
f(0)−P f(0) = f(b)−P f(b) = 0. (3.31)
N N
Inthefollowingtheproofwillbesplitintothreeparts.
(1)Forthefirstinequality in(3.30),itfollowsatoncefrom(3.22)that
b
|I [f]−ICCF[f]| = xα(f(x)−P f(x))J (ωx)dx
1 1 (cid:12)(cid:12)Z0 N m (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) b (cid:12)
≤ (cid:12) |xα(f(x)−P f(x))J (ωx)|dx(cid:12)
N m
Z
0
b
≤ xαdx||f(x)−P f(x)||
N ∞
Z
0
bα+1
= ||f(x)−P f(x)|| .
α+1 N ∞
(2)ByusingLemma3.2andtheidentities (3.31),thesecondinequality in(3.30)followsthat
b
|I [f]−ICCF[f]| = xα(f(x)−P f(x))J (ωx)dx
1 1 (cid:12)(cid:12)Z0 N m (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) b (cid:12)
≤ C(cid:12) (ω)(|f(b)−P f(b)|+ |f′(x(cid:12))−P′ f(x)|dx)
1 N N
Z
0
≤ bC (ω)||f′(x)−P′ f(x)|| .
1 N ∞
10