One-Step Recurrences for Stationary Random Fields on the Sphere

0.47 MB·

by  R. K. Beatson

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 043, 19 pages
One-Step Recurrences for Stationary Random Fields
on the Sphere(cid:63)
R.K. BEATSON † and W. ZU CASTELL ‡§
† School of Mathematics and Statistics, University of Canterbury,
Private Bag 4800, Christchurch, New Zealand
E-mail: [email protected]
URL: http://www.math.canterbury.ac.nz/~r.beatson
6 ‡ Scientific Computing Research Unit, Helmholtz Zentrum Mu¨nchen,
1
Ingolst¨adter Landstraße 1, 85764 Neuherberg, Germany
0
2 E-mail: [email protected]

r URL: http://www.helmholtz-muenchen.de/asc
p
A § Department of Mathematics, Technische Universit¨at Mu¨nchen, Germany

8 Received January 28, 2016, in final form April 15, 2016; Published online April 28, 2016
2
http://dx.doi.org/10.3842/SIGMA.2016.043

]
A
Abstract. Recurrences for positive definite functions in terms of the space dimension have
C
been used in several fields of applications. Such recurrences typically relate to properties of
h. the system of special functions characterizing the geometry of the underlying space. In the
t caseofthesphereSd−1 ⊂Rdthe(strict)positivedefinitenessofthezonalfunctionf(cosθ)is
a
m determinedbythesignsofthecoefficientsintheexpansionoff intermsoftheGegenbauer
polynomials {Cλ}, with λ = (d−2)/2. Recent results show that classical differentiation
[ n
andintegrationappliedtof havepositivedefinitenesspreservingpropertiesinthiscontext.

3 However, in these results the space dimension changes in steps of two. This paper develops
v operatorsforzonalfunctionsonthespherewhichpreserve(strict)positivedefinitenesswhile
3
movingupanddownintheladderofdimensionsbystepsofone. Thesefractionaloperators
4
are constructed to act appropriately on the Gegenbauer polynomials {Cλ}.
7 n
7
Key words: positive definite zonal functions; ultraspherical expansions; fractional integra-
0
. tion; Gegenbauer polynomials
1
0 2010 Mathematics Subject Classification: 42A82; 33C45; 42C10; 62M30
6
1
:
v
i 1 Introduction
X
r
a This paper develops operators for zonal functions on the sphere which preserve (strict) posi-
tive definitenesss while moving up and down in the ladder of dimensions by steps of one. The
operators provide tools for forming families of (strictly) positive definite zonal functions. Such
(strictly) positive definite zonal functions can be used as covariance models for estimating re-
gionalized variables and also for interpolation on spheres.
Withinadeterministiccontext,zonalpositivedefinitefunctionsonthespherehavebeenused
for interpolation or approximation of scattered data (see [10, 11] and the references therein).
The standard ansatz in this setting is a linear combination of spherical translates of a fixed
(zonal) basis function. While the present paper could well have been stated within the context
of approximation on the sphere, we rather chose to provide a probabilistic framework, which is
to some extent is equivalent, i.e., the theory of regionalized variables.
(cid:63)ThispaperisacontributiontotheSpecialIssueonOrthogonalPolynomials,SpecialFunctionsandApplica-
tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html
2 R.K. Beatson and W. zu Castell
Regionalized variables on spherical domains can nicely be modeled using random fields on
spheres [6, 15]. Such a random field is given through a set of random variables, Z(x) say, where
x ∈ Sd−1. Assuming the field to be Gaussian, i.e., for every n ∈ N, (Z(x ),...,Z(x ))T has
1 n
a multivariate Gaussian distribution for any choice of x ,...,x ∈ Sd−1, the distribution can be
1 n
characterized by its first two moments.
Assuming second order (weak) stationarity, the covariance for an isotropic model is deter-
mined by a function
Cov(cid:0)Z(x),Z(y)(cid:1) = f(cosθ), x,y ∈ Sd−1,
where θ = θ(x,y) = arccos(cid:0)xTy(cid:1) is the geodesic distance between the points x and y on the
sphere Sd−1.
As a consequence of Kolmogorov’s extension theorem (see [5, Theorem 36.3]), the set of
isotropicGaussianrandomfieldscanbeidentifiedwiththesetofzonalpositivedefinitefunctions
on the sphere [12, 15]. We note in passing that L´evy named such processes Brownian motion.
Definition 1.1. A continuous function g: [0,π] → R is (zonal) positive definite on the sphe-
re Sd−1 if for all n ∈ N and all distinct point sets {x ,...,x } on the sphere, the inequality
1 n
n
(cid:88)
c c g(θ(x ,x )) ≥ 0
i j i j
i,j=1
holds true for all c ,...,c ∈ R. The function is (zonal) strictly positive definite on Sd−1 if the
1 n
inequality holds in the strict sense for all c ,...,c ∈ R not vanishing simultaneously.
1 n
Although the natural distance on the unit sphere is an angle in [0,π], it is convenient for the
purpose of this paper to consider functions in x = cosθ ∈ [−1,1], instead. Thus, by Λ we
d−1
will denote the cone of all functions f ∈ C[−1,1] such that f(cos ·) is positive definite on Sd−1.
Λ+ will denote the subcone of all strictly positive definite functions in Λ .
d−1 d−1
Gaussian random fields have been widely applied to statistically analyze spatial phenomena
[8, 16, 23]. In particular, kriging allows prediction of spatial variables from given samples at
arbitrary locations. The key ingredient for such an approach lies in determining a suitable
model for the covariance function of the spherical random field. Commonly, such a model can
be inferred from given data through fitting a parametric family of models (i.e., estimation of
the covariance).
Models for covariance functions have further been used for simulation of stationary random
fields. Matheron [17] suggested a method based on proper averaging of stationary random fields
onalowerdimensionalspace. IntheEuclideansettingthisso-calledturning bands method works
as follows:
Given a stationary random field Z on the real line with covariance function C and a ran-
1 1
domly chosen direction ξ ∈ Sd−1, Z = Z (xTξ) defines a stationary random field on Rd.
ξ 1
Averaging over all directions ξ leads to a stationary field on Rd the covariance function of
which, C say, relates to C via the so-called turning bands operator
d 1
(cid:90) ∞
C (t) = const (cid:0)1−τ2(cid:1)d−23C (tτ)dτ, t ∈ R .
d + 1 +
0
The turning bands operator represents one example out of a suite of operators, mapping radial
positive definite functions on Rd onto such functions on a higher or lower dimensional space.
Wendland [25], Wu [26] and Gneiting [13] used such operators to derive compactly supported
functions of a given smoothness. Recurrences for radial positive definite functions in general
have been studied by several authors [22, 27]. Due to Schoenberg’s characterization of radial
One-Step Recurrences for Stationary Random Fields on the Sphere 3
positive definite functions and the fact that scale mixtures of such functions preserve positive
definiteness, recurrence operators can be derived from corresponding relations between special
functions. InthecaseofradialfunctionsonRd, theappropriatefundamentalrelationisSonine’s
first integral for Bessel functions of the first kind (see [27]).
In a recent paper [4] the authors applied similar operators to derive parametrized families of
suitable locally supported covariance models for stationary random fields on the sphere Sd−1.
These operators are based on properties of Gegenbauer polynomials, appearing in Schoenberg’s
characterization [24] of zonal positive definite functions on the sphere.
Theorem 1.2. Let λ = (d−2)/2, and consider a continuous function f on [−1,1]. The function
f(cos ·) is positive definite on Sd−1, i.e., f ∈ Λ , if and only if f has an ultraspherical
d−1
expansion
∞
(cid:88)
f(x) ∼ a Cλ(x), x ∈ [−1,1], (1.1)
n n
n=0
in which all the coefficients a are nonnegative, and the series converges at the point x = 1. If
n
this is the case, the series converges absolutely and uniformly on the whole interval.
Chen, Menegatto and Sun [7] showed that a necessary and sufficient condition for f ◦cos to
be strictly positive definite on Sd−1, d ≥ 3, is that, in addition to the conditions of Theorem 1.2,
infinitely many of the Gegenbauer coefficients a with odd index, and infinitely many of those
n
with even index, are positive. In the case d = 2 the criteria is necessary but not sufficient
for f ◦ cos to be strictly positive definite. A characterization in this case has been given by
Menegatto, Oliveira & Peron [18], although the criterion is a little more involved (see also [3]
for further details on these issues).
In the same spirit as for the turning bands method, a zonal function defined on a lower
dimensional sphere Sd−κ can be lifted up to Sd−1 through averaging over the set of all copies
of Sd−κ contained in Sd−1. In [4] it is shown that the analogues for the sphere of Matheron’s
mont´ee and descente operators (see [16]) for Rd are the operators
(cid:90) x
(If)(x) = f(u)du, x ∈ [−1,1],
−1
and
(Df)(x) = f(cid:48)(x), x ∈ [−1,1].
Paralleling the behaviour of Matheron’s operators in the Euclidean case the operators move
in the ladder of dimensions by steps of two. Specifically, the I and D operators map zonal
positive definite functions f(cos·) on Sd onto ones on Sd−2 and Sd+2, respectively (see [4] for
details). Therefore, the natural question arises, whether it would also be possible to proceed
through steps by one within the ladder of dimensions. While in the Euclidean case this could
be achieved using fractional differentiation and integration (see [27]), the situation is more
intriguing in the spherical setting. The reason lies in the fact that the characterizing special
functions for the sphere are polynomials, which are not preserved through fractional integration.
Thus, one has to work with combinations of fractional operators in order to guarantee that the
operators are mapping into the space of polynomials.
Inthepresentpaper,weprovideasuiteoffouroperatorswhichcanbeusedtodefineaclavier
(cf.[16])forthesphere. ThemainresultsaregiveninTheorems2.3,2.4and2.8,below. Westart
with introducing the appropriate fractional operators in the following section and studying their
action on ultraspherical expansions. The action of the operators on Gegenbauer polynomials
shown in the last section is derived using properties of hypergeometric F -functions.
2 1
4 R.K. Beatson and W. zu Castell
2 Definition of the half-step operators
Intheexpansion(1.1)thedimensiondappearsintheparameterλ = (d−2)/2oftheGegenbauer
polynomials. This relation between λ and d will be fixed throughout the paper.
From DCλ = 2λCλ+1 (cf. [9, 10.9(23)]) we see that classical differentiation and its inverse,
n n−1
integration, alter the parameter λ by an integer. This is why the operators I and D traverse
the ladder of dimensions in steps of two (see [4]). At the same time, I and D change the degree
of polynomials by one. Therefore, in order to obtain a one-step operator in the dimension, we
have to consider fractional integration and differentiation, a fact which perfectly parallels the
Euclidean setting (see [20, 27]).
Wearenowreadytodefinethehalf-stepoperatorsanddiscusstheiractiononpositivedefinite
functions on Sd−1.
Definition 2.1. For f ∈ L1[−1,1] and λ ≥ 0, define
(cid:90) x
I+λf(x) = I+λ,12f(x) = (1+x)−λ+12 (x−τ)−12(1+τ)λf(τ)dτ, (2.1)
−1
(cid:90) 1
I−λf(x) = I−λ,21f(x) = (1−x)−λ+12 (τ −x)−12(1−τ)λf(τ)dτ. (2.2)
x
Using these, we further define
Iλ = Iλ +Iλ and Iλ = Iλ −Iλ. (2.3)
+ + − − + −
Apart from the additional factor (1±x)−λ+21 in front of the integral and the weight (1±τ)λ,
the operators Iλ are classical Riemann–Liouville fractional integrals of order 1 (cf. [21, De-
± 2
finition 2.1]) on the interval [−1,1]. To define inverse operators, we use the corresponding
Riemann–Liouville fractional derivates (cf. [21, Definition 2.2]).
Definition 2.2. Let f be absolutely continuous on [−1,1] and λ ≥ 0. Then
D+λf(x) = D+λ,21f(x) = (1+x)ddx (cid:26)(1+x)−λ(cid:90) x(x−τ)−21(1+τ)λ−12f(τ)dτ(cid:27),
−1
D−λf(x) = D−λ,21f(x) = (1−x)ddx (cid:26)(1−x)−λ(cid:90) 1(τ −x)−21(1−τ)λ−12f(τ)dτ(cid:27).
x
Using these, we further define
Dλ = Dλ +Dλ and Dλ = Dλ −Dλ.
+ + − − + −
Themainresultsofthispaperarethefollowingtwotheoremsgivingprecisestatementsofthe
dimension hopping and positive definiteness preserving properties of the operators Iλ and Dλ.
± ±
These are one-step analogues of Theorems 2.2 and 2.3 in [4]. Since in the light of Theorem 1.2
the statements can be considered as statements concerning ultraspherical expansions without
referring back to a sphere, we are considering m = d−1 to be a positive integer.
Theorem 2.3. Let m be a positive integer and λ = (m−1)/2.
(a) (i) Let f ∈ Λ , m ≥ 1. Then Iλf ∈ Λ .
m+1 ± m
(ii) Let f ∈ Λ+ , m ≥ 2. Then Iλf ∈ Λ+.
m+1 ± m
(b) Let m ≥ 1, f ∈ Λ+ be nonnegative, and f have Gegenbauer expansion,
m+1
∞
(cid:88) λ+1
f ∼ a C 2,
n n
n=0
One-Step Recurrences for Stationary Random Fields on the Sphere 5
with all coefficients, {a }∞ , positive. Then Iλf is also nonnegative, Iλf ∈ Λ+, and all
n n=0 + + m
the coefficients b in the expansion
n
∞
(cid:88)
Iλf ∼ b Cλ,
+ n n
n=0
are positive.
Proof. Theproofsforthestatementsarealmostidenticalwiththoseofthecorrespondingparts
of Proposition 2.2 in [4], provided that proper analogues for certain statements on Gegenbauer
polynomials are given. We therefore restrict ourselves to pointing out where adaptations of the
proof given in [4] are needed.
One of these details concerns the boundedness of the operators Iλ as operators from C[−1,1]
±
to C[−1,1]. This follows from the definitions of Iλ and Iλ in equations (2.1), (2.2) and (2.3),
± ±
combined with the beta integrals
(cid:90) x Γ(1)Γ(ν +1)
(x−τ)−12(1+τ)νdτ = (1+x)ν+12B(cid:0)1,ν +1(cid:1) = (1+x)ν+12 2 , (2.4)
2 Γ(ν + 3)
−1 2
and
(cid:90) 1 Γ(1)Γ(ν +1)
(τ −x)−12(1−τ)νdτ = (1−x)ν+12B(cid:0)1,ν +1(cid:1) = (1−x)ν+21 2 .
2 Γ(ν + 3)
x 2
Similarly, positivity of the operator Iλ follows from the definitions (2.1), (2.2) and (2.3). The
+
main ingredient thus remaining to be shown is the action of Iλ on the Gegenbauer polyno-
±
λ+1
mial C 2. This part is given in Theorem 3.3, below. Note that in contrast to the operators
n
studied in [4], there is no need to deal with an extra constant in statements (i) and (ii). This
follows from Theorem 3.3, showing that the operators Iλ do not introduce an additional con-
±
stant. (cid:4)
Theorem 2.4. Suppose that f ∈ Λ , m ≥ 1, and let λ = (m−1)/2. Then, if both functions
m
Dλf ∈ C[−1,1], then Dλf ∈ Λ . If, in addition, f ∈ Λ+, then Dλ ∈ Λ+ .
± ± m+1 m ± m+1
Remark 2.5. Since the operators defined above can be seen as standard operators of fractional
integration/differentiation,classicalresultsfromfractionalcalculuscanbeapplied. Forexample,
if (1+τ)−1/2f(τ) ∈ Lipα for some α > 1, in particular, if f ∈ Lipα and suppf ⊂ (−1,1], then
2
byTheorem19in[14]D0f existsandiscontinuous. Ananalogousstatementholdsforgeneralλ.
±
The proof of Theorem 2.4 depends heavily on a multiplier relationships between the Gegen-
bauer coefficients of f and those of Dλf. The details of these relationship, and the proof of
±
Theorem 2.4, will be deferred to the next subsection.
Let us finish the section with considering an example.
Example 2.6. Consider the operator Iλ. In view of its definition (2.1) this operator maps
+
functions locally supported near one to functions locally supported near one. Also, since Iλ =
+
(Iλ +Iλ)/2 this operator preserves (strict) positive definiteness by Theorem 2.3.
+ −
Note that by a change of variables
(cid:90) x (cid:90) 1
(x−τ)−21(1+τ)λf(τ)dτ = (x+1)λ+12 (1−s)−12sλf(cid:0)(x+1)s−1(cid:1)ds.
−1 0
Therefore, if f were such that f(cid:0)(x+1)s−1(cid:1) = (1−ys)−a, the integral becomes
(cid:90) 1(1−s)−12sλ(1−ys)−ads = Γ(Γλ(cid:0)+λ1+)Γ3(cid:0)(cid:1)12(cid:1) 2F1(cid:20)aλ,λ++31(cid:12)(cid:12)(cid:12)(cid:12)y(cid:21), (2.5)
0 2 2
being a special case of Euler’s integral for hypergeometric functions (cf. [19, (15.6.1)]).
6 R.K. Beatson and W. zu Castell
Now consider the Cauchy family
ϕα,β(r) = (1+rα)−αβ, 0 < α ≤ 2, β > 0,
which is strictly positive definite on Rd for all d ≥ 1 (cf. [13]). Choosing α = 2 and restricting
the function ϕ to the sphere, we obtain (setting τ = cosθ)
2,β
√
ϕβ(τ) = ϕ2,β(cid:0) 2−2cosθ(cid:1) = (3−2τ)−β2, β > 0.
Thus,
ϕβ(cid:0)(x+1)s−1(cid:1) = 5−β2 (cid:0)1− 25(x+1)s(cid:1)−β2 .
Therefore, from (2.5) we have that
√
I+λϕβ(x) = 5β2πΓΓ(cid:0)(λλ++13)(cid:1)(x+1)2F1(cid:20)β2λ,λ++321(cid:12)(cid:12)(cid:12)(cid:12) 25(x+1)(cid:21).
2
Since (see [19, (15.4.6)])
(cid:20) (cid:12) (cid:21)
(1−z)−a = 2F1 ab,b(cid:12)(cid:12)(cid:12)z , (2.6)
we can choose β = 2λ+3, yielding
(cid:114)
π Γ(λ+1)
Iλϕ (x) = (x+1)(3−2x)−(λ+1). (2.7)
+ 2λ+3 5Γ(cid:0)λ+ 3(cid:1)
2
Therefore, the function given in (2.7) is strictly positive definite on S2λ+1 by the remark at the
start of the example.
We can follow the same line of argument applying Iµ to Iλϕ . Again, the result is
+ + 2λ+3
a hypergeometric function. Interestingly, for µ = λ−3 the series is of the form (2.6) resulting in
2
I+λ−32(cid:0)I+λϕ2λ+3(cid:1)(x) = π5ΓΓ(cid:0)(cid:0)λλ++ 123(cid:1)(cid:1) (x+1)2(3−2x)−(λ+12).
2
In general, for a function
g (x) = (x+1)m(3−2x)−γ, m ∈ N , γ > 0,
m,γ 0
we obtain that
I+γ−m−23gm,γ(x) = (cid:114)π5Γ(cid:0)Γγ(−γ)12(cid:1)(x+1)m+1(3−2x)−(γ−12).
2.1 Ultraspherical expansions of f and Dλf
±
The main results of this section will be Theorems 2.7 and 2.8 giving multiplier relationships
between the Gegenbauer coefficients of the (formal) series of f and those of the (formal) series
of Dλf and Dλf. These relationships will later be used to show that the operators Dλ and Dλ
+ − + −
have the positive definiteness preserving properties given in Theorem 2.4.
ThefirststatementshowsthattheoperatorsD0,D0 canbeappliedtermbytermtoaCheby-
+ −
shev series to obtain the formal Legendre series of D0f.
±
One-Step Recurrences for Stationary Random Fields on the Sphere 7
Theorem 2.7. Let f ∈ C[−1,1] with (formal) Chebyshev series
∞
(cid:88)
f ∼ a T .
n n
n=0
If both functions D0f ∈ C[−1,1], then the (formal) Legendre series
±
∞
(cid:88)
D0f ∼ b P
+ n n
n=0
has coefficients
b = (n+1)πa , n ∈ N ,
n n+1 0
and the (formal) Legendre series
∞
(cid:88)
D0f ∼ c P
− n n
n=0
has coefficients
c = nπa , n ∈ N .
n n 0
Similar relations between coefficients in ultraspherical expansions hold for higher order
Gegenbauer polynomials.
Theorem 2.8. Let λ > 0, and let f ∈ C[−1,1] have a (formal) Gegenbauer series
∞
(cid:88)
f ∼ a Cλ.
n n
n=0
If both functions Dλf ∈ C[−1,1], then the (formal) Gegenbauer series
±
∞
Dλf ∼ (cid:88)b Cλ+12
+ n n
n=0
has coefficients
√
Γ(λ+ 1) π 2(n+2λ+1)
b = 2 a , n ∈ N , (2.8)
n n+1 0
Γ(λ) n+λ+1
and the (formal) Gegenbauer series
∞
Dλf ∼ (cid:88)c Cλ+12,
− n n
n=0
has coefficients
√
Γ(λ+ 1) π 2n
c = 2 a , n ∈ N . (2.9)
n n 0
Γ(λ) n+λ
8 R.K. Beatson and W. zu Castell
Remark 2.9. Theorem 2.7 is the limiting case of Theorem 2.8 under the limit
1
C0(x) = lim Cλ(x), n > 0, and C0(x) = T (x) = 1. (2.10)
n λ→0+ λ n 0 0
Furthermore, we have the special cases
2
C0(x) = T (x), n > 0,
n n n
while
1
C2(x) = P (x) and C1(x) = U (x), n ≥ 0.
n n n n
Before proving the theorems we state the following technical lemma.
Lemma 2.10. For λ > 0, n ∈ N and x ∈ [−1,1],
0
n−1
d (cid:8)(1+x)Cλ(x)(cid:9) = (n+1)Cλ(x)+2(cid:88)(k+λ)Cλ(x), (2.11)
dx n n k
k=0
n−1
d (cid:8)(1−x)Cλ(x)(cid:9) = −(n+1)Cλ(x)+2(cid:88)(−1)k+n+1(k+λ)Cλ(x). (2.12)
dx n n k
k=0
Proof. Formula (2.12) can be obtained from equation (2.11) by using the reflection formula for
Gegenbauer polynomials
Cλ(−x) = (−1)nCλ(x). (2.13)
n n
and the change of variables y = −x.
Fortheproofofformula(2.11)wewillusetworecurrencesinvolvingderivativesofGegenbauer
polynomials which can be found, for example in [9, 10.9(35)]. For notational convenience we use
the (non-standard) notation Dλ(x) = d Cλ(x) within the proof of the lemma. Then,
n dx n
nCλ(x) = xDλ(x)−Dλ (x), and (2.14)
n n n−1
(cid:0)1−x2(cid:1)Dλ(x) = (cid:0)1−x2(cid:1)2λCλ+1(x) = (n+2λ−1)Cλ (x)−nxCλ(x). (2.15)
n n−1 n−1 n
Turn now to an induction proof of formula (2.11). The statement is clearly true for Cλ(x) = 1,
0
adopting the convention that the sum then is empty.
Now assume that n ∈ N and that the first statement is true for n−1. Consider
d (cid:8)(1+x)Cλ(x)(cid:9) = Cλ(x)+(1+x)Dλ(x). (2.16)
dx n n n
Using (2.15) and then (2.14) we obtain that
(1+x)Dλ(x) = (cid:0)1−x2(cid:1)Dλ(x)+(1+x)xDλ(x)
n n n
= (n+2λ−1)Cλ (x)−nxCλ(x)+(1+x)(cid:0)nCλ(x)+Dλ (x)(cid:1)
n−1 n n n−1
= (n+2λ−1)Cλ (x)+nCλ(x)+(1+x)Dλ (x).
n−1 n n−1
Therefore, applying (2.16),
d (cid:8)(1+x)Cλ(x)(cid:9) = (n+2λ−1)Cλ (x)+(n+1)Cλ(x)+(1+x)Dλ (x)
dx n n−1 n n−1
= (n+2λ−1)Cλ (x)+(n+1)Cλ(x)
n−1 n
One-Step Recurrences for Stationary Random Fields on the Sphere 9
+ d (cid:8)(1+x)Cλ (x)(cid:9)−Cλ (x)
dx n−1 n−1
= (n+2λ−2)Cλ (x)+(n+1)Cλ(x)+ d (cid:8)(1+x)Cλ (x)(cid:9).
n−1 n dx n−1
Using the induction hypothesis gives
d (cid:8)(1+x)Cλ(x)(cid:9) = (n+1)Cλ(x)+(n+2λ−2)Cλ (x)
dx n n n−1
n−2
(cid:88)
+nCλ (x)+2 (k+λ)Cλ(x)
n−1 k
k=0
n−2
= (n+1)Cλ(x)+2(cid:2)(n−1)+λ(cid:3)Cλ (x)+2(cid:88)(k+λ)Cλ(x),
n n−1 k
k=0
which completes the proof. (cid:4)
Proof of Theorem 2.7. Note that the continuity of the two functions D0f implies that of the
±
functions D0f. Then, proceeding by integration by parts, the coefficient b of P in the formal
± n n
Legendre expansion of D0f is (2n+1)/2 times
+
(cid:90) 1
H = P (x)(cid:0)D0f(cid:1)(x)dx
+ n +
−1
(cid:90) 1 d (cid:18)(cid:90) x (cid:19)
= P (x)(1+x) (x−t)−1/2(1+t)−1/2f(t)dt dx
n
dx
−1 −1
(cid:20) (cid:90) x (cid:21)1
= P (x)(1+x) (x−t)−1/2(1+t)−1/2f(t)dt
n
−1 −1
−(cid:90) 1 (cid:90) x(x−t)−1/2(1+t)−1/2f(t)dt d (cid:8)P (x)(1+x)(cid:9)dx.
n
dx
−1 −1
In view of the formula [19, (5.12.1)],
(cid:90) x
(x−t)−1/2(1+t)−1/2dt = π,
−1
forall−1 < x ≤ 1,thelimitasxtendsto−1ofthequantitywithinthesquarebracketsvanishes.
Hence,
(cid:90) 1
H = 2P (1) (cid:0)1−t2(cid:1)−1/2f(t)dt
+ n
−1
(cid:90) 1 (cid:90) 1 d
− (x−t)−1/2 {P (x)(1+x)}dx(1+t)−1/2f(t)dt
n
dx
−1 t
(cid:90) 1 (cid:90) 1 (cid:34) n−1 (cid:35)
(cid:88)
= 2πa − (x−t)−1/2 (n+1)P (x)+ (2k+1)P (x) dx(1+t)−1/2f(t)dt,
0 n k
−1 t
k=0
where the last step follows from an application of formula (2.11).
Noting the relationship (cf. [19, (18.17.46)])
(cid:90) 1(x−t)−1/2P (x)dx = 1 √ 1 (cid:2)T (t)−T (t)(cid:3), (2.17)
k (k+ 1) 1−t k k+1
t 2
10 R.K. Beatson and W. zu Castell
after some straightforward calculation, the double integral above turns into the form
H = 2n (cid:90) 1 (cid:0)1−t2(cid:1)−12T (t)f(t)dt+ 2(n+1) (cid:90) 1 (cid:0)1−t2(cid:1)−12T (t)f(t)dt.
+ n n+1
2n+1 2n+1
−1 −1
Therefore,
n n+1
H = πa + πa .
+ n n+1
2n+1 2n+1
Analogously, we define
(cid:90) 1
H = P (x)(cid:0)D0f(cid:1)(x)dx.
− n −
−1
A similar integration by parts argument, but now using the formula (2.12), and the relationship
(cf. [19, (18.17.45)])
(cid:90) t (t−x)−1/2P (x)dx = 1 √ 1 (cid:2)T (t)+T (t)(cid:3), (2.18)
k (k+ 1) 1+t k k+1
−1 2
shows that
n n+1
H = − πa + πa .
− n n+1
2n+1 2n+1
Since b = 2n+1(H +H ) we finally obtain
n 2 + −
b = (n+1)πa , n ∈ N .
n n+1 0
This completes the proof of the part of the theorem concerning D0f. The proof of the part of
+
the theorem concerning D0f is similar and will be omitted. (cid:4)
−
The proof of Theorem 2.8 relies on a kind of fractional integration by parts. Before going
into details, we will state some technical lemmas.
Lemma 2.11. For λ ≥ 1/2, n ∈ N and x ∈ [−1,1],
0
d (cid:8)(1+x)λ+1(1−x)λCλ+12(x)(cid:9) = (1+x)(cid:0)1−x2(cid:1)λ−1Q (x), (2.19)
n n+1
dx
where
λ+1 2λ+n λ−1
Q (x) = (1−x)C 2(x)−(n+1) C 2(x).
n+1 n 2λ−1 n+1
Proof. Note the formula (see [1, (22.13.2)] or [19, (18.17.1)] for the general Jacobi case)
n(cid:16)1+ n (cid:17)(cid:90) x(cid:0)1−t2(cid:1)λ−12Cλ(t)dt = Cλ+1(0)−(cid:0)1−x2(cid:1)λ+12Cλ+1(x),
2λ n n−1 n−1
0
which implies
(cid:32) (cid:33)
d (cid:8)(cid:0)1−x2(cid:1)λCλ+21(x)(cid:9) = −(n+1) 1+ n+1 (cid:0)1−x2(cid:1)λ−1Cλ−21(x), λ > 1/2.
dx n 2(λ− 1) n+1
2