Table Of ContentA fully discretised filtered polynomial approximations on
spherical shells
Yoshihito Kazashi
27th January 2017
7
1
0
2 Abstract
n
A fully implementable filtered polynomial approximation on spherical shells is con-
a
J sidered. The method proposed is a quadrature-based version of a filtered polynomial
6 approximation. The radial direction and the angular direction of the shells are treated
2 separately with constructive filtered polynomial approximation. The approximation error
withrespecttothesupremumnormisshowntodecayalgebraicallyforfunctionsinsuitable
] differentiability classes. Numerical experiments support the results.
A
N
Keywords: Filtered polynomial approximation, Filtered hyperinterpolation, Spherical
h. shells
t
a
m
1 Introduction
[
1 This paper is concerned with constructive global polynomial approximation on spherical shells.
v
Problems on such domains naturally arise in a wide range of geosciences, and numerous com-
0
putational methods are proposed [7, 8, 13, 20, 24]. Nonetheless, theoretical analysis does not
2
6 seemtohaveattractedmuchattention. Onerecentresultconsideredin[12]isafullydiscretised
7
polynomial approximation on spherical shells. The method considered there can be seen as an
0
. approximation of the L2-orthogonal projection. However, L2-projection is not the best choice
1
when one wants a small point-wise error—recall how the Fourier series of f on the torus may
0
7 fail to converge on any measure zero set if f is merely continuous [11]. This paper considers
1
a method with good uniform convergence using filtering. Our ultimate goal is to construct a
:
v fully discretised filtered polynomial approximation method and analyse the errors.
i
X A classical remedy for the failure of the Fourier series on the torus mentioned above is to
r use a smoothing (or filtering) process such as Ces´aro sums, Lanczos smoothing, or the raised
a
cosine smoothing [2]. An underlying idea is to smoothly truncate the series by multiplying the
Fouriercoefficientsofhigherorderbyasuitablysmallfactor. Filteredapproximationshavealso
been considered for other regions, including the sphere. (See [18, 21] and references therein.)
In most realistic applications the Fourier coefficients are not known, as integrals are not
computable exactly. As an alternative, quadrature-based approximations of these filtered
methods have been considered for various settings, particularly on the sphere [15, 22]. The
present paper considers a quadrature-based filtered polynomial approximation for spherical
shells S = {x ∈ R3 | r ≤ (cid:107)x(cid:107) ≤ r = r +ε}, where r ≤ 1 ≤ r and r −r = ε > 0
ε in 2 out in in out out in
1
as a domain. That is, given a function f, we smoothly truncate its Fourier series by a suit-
able ‘filter’ h, and approximate Fourier coefficients by quadrature rules. The motivation is to
propose an implementable technique with a good point-wise convergence. Our results give,
to the best of our knowledge, the first theoretical results on constructive filtered polynomial
approximation on spherical shells. Our method requires only point values of the function we
approximate, and thus can be implemented exactly in a real number model of computation.
We regard S as a product of the interval [r ,r ] in the radial direction and the unit
ε in out
sphere S2 in the angular direction. The product setting is natural, since in practice functions
on a spherical shell vary on different scales in the radial and angular directions. For example,
the mantle can be seen as a set of spherical layers with different characteristics [1, 10, 5]. Some
properties of the atmosphere, such as the ionization rate [9, p. 151], electric field [9, p. 155],
depend strongly on the altitude, and hence vary rapidly in the radial direction.
For a continuous function f ∈ C(S ) we consider the approximation taking the following
ε
(α,β)
form. Let J (α,β > −1) be the Jacobi polynomial of degree k mapped affinely to [r ,r ]
k in out
from [−1,1], and Y be the spherical harmonics of degree (cid:96) and order m. More detailed
(cid:96)m
definitions are given later. Let h: [0,∞) × [0,∞) → [0,∞) be a function with a compact
support that is non-increasing in each variable. Then, the method we propose takes the form
∞ (cid:96) (cid:18) (cid:19)
(cid:88) (cid:88) k (cid:96) (α,β)
V f := h , c J Y . (1.1)
KL K N k(cid:96)m k (cid:96)m
k,(cid:96)=0m=−(cid:96)
Note that this is actually a finite sum. Here, the coefficients {c } ⊂ R are quadrature
k(cid:96)m
approximations of Fourier coefficients,
(cid:90)
1
(α,β)
c ≈ fJ Y .
k(cid:96)m γ2 k (cid:96)m
k Sε
The quadrature approximation, the measure used in the integral, and the normalising constant
γ are defined later.
k
Following [22], we shall call V f filtered hyperinterpolation of f on S , if the quadrature
KL ε
is of suitably high polynomial precision.
Our main result Corollary 5.4 gives error convergence orders of the method we propose in
termsofthesupremumnorm. Theerrorisanalysedbytreatingtheradialandangulardirection
separately. We introduce the filtered hyperinterpolation operator R in the radial direction
K
and A in the angular direction. The error f −V f in terms of the supremum norm turns
L KL
out to be bounded by the sum of the error bounds for each direction.
The outline of this paper is as follows. Section 2 introduces notations we need. In Section 3
and 4, we introduce the filtered hyperinterpolation approximations in the radial direction and
the angular direction. Section 5 develops the filtered hyperinterpolation on spherical shells and
analyses the error. We give numerical results in Section 6, and Section 7 concludes the paper.
2 Preliminaries
We set up some notations and introduce the problem we consider.
With r ∈ (0,1] and r ∈ [1,∞) (r (cid:54)= r ), we consider a spherical shell S = {x ∈
in out in out ε
R3 | r ≤ (cid:107)x(cid:107) ≤ r }. We assume r −r = ε > 0. We use the spherical coordinate system
in 2 out out in
x=rσ= (rsinθcosϕ,rsinθsinϕ,rcosθ) (r ∈ [0,∞), θ ∈ [0,π], ϕ ∈ [0,2π)),
2
where r = (cid:107)x(cid:107) , σ = x, and for θ ∈ {0,π} we let ϕ = 0.
2 r
For σ ∈ S2, we often write a function f(θ,ϕ) on the unit sphere S2 as f(σ).
In the following, we introduce orthogonal polynomials on the interval and the sphere. Fur-
ther, we introduce the approximation method we consider.
2.1 Orthogonal polynomials
Let J∗ = J∗(α,β) be the Jacobi polynomial of degree k with the parameters α,β > −1 on
k k
[−1,1]. Define J (k = 0,...,K) by
k
(cid:18) (cid:19)
2r−(r +r )
J (r) = J∗ in out , r ∈ [r ,r ]
k k r −r in out
out in
Let w∗(x) := (1−x)α(1+x)β (x ∈ (−1,1)) be the weight function associated with {J∗} =
k
(cid:16) (cid:17)1
{J∗(α,β)}, that is, with γ = (cid:82)1 (J∗(x))2w∗(x)dx 2 we have
k k −1 k
(cid:90) 1
J∗(x)J∗(x)w∗(x)dx = δ γ2, (2.1)
j k jk k
−1
whereδ = 1ifj = k andδ = 0otherwise. Forexample,theweightassociatedwithLegendre
jk jk
polynomials (α = β = 0) is w∗ (x) = 1 (x ∈ (−1,1)), and for Chebyshev polynomials
Legendre
(α = β = −1) we have w∗ (x) = √ 1 (x ∈ (−1,1)). We always consider a fixed pair
2 Chebyshev 1−x2
of parameters (α,β), and thus we omit them in the notation J∗, and J .
k k
Define the measure µrad on [r ,r ] by
in out
(cid:90) (cid:18) (cid:19)
2r−(r +r ) 2
µrad(A) = w∗ out in dr,
r −r r −r
A out in out in
for any Lebesgue measurable set A in [r ,r ]. Then, we have
in out
(cid:90) r (cid:90) 1
out
J (r)J (r)dµrad(r) = J∗(x)J∗(x)w∗(x)dx = δ γ2. (2.2)
j k j k jk k
r −1
in
Let Y (θ,ϕ) be the real spherical harmonics on the unit sphere S2 defined by
(cid:96),m
1
Y(cid:96),0(θ,ϕ) = √2πP(cid:96),0(cosθ)
1
Y (θ,ϕ) = √ P (cosθ)cosmϕ (m = 1,...,(cid:96))
(cid:96),m π (cid:96),m
1
Y(cid:96),−m(θ,ϕ) = √πP(cid:96),m(cosθ)sinmϕ (m = 1,...,(cid:96)),
where P are defined as follows. Consider the associated Legendre polynomials
(cid:96),m
P0(x) = P (x) (m = 0)
(cid:96) (cid:96)
dmP
Pm(x) = (1−x2)m/2 (cid:96)(x) (m = 1,...,(cid:96))
(cid:96) dxm
((cid:96)−m)!
P−m(x) = (−1)m Pm(x) (m = 1,...,(cid:96)),
(cid:96) ((cid:96)+m)! (cid:96)
3
whereP (x)(x ∈ [−1,1])istheLegendrepolynomialofdegree(cid:96) ∈ {0}∪N. Then,{P (x)}
(cid:96) (cid:96),m (cid:96)=0,1,2,...
m=−(cid:96),...,(cid:96)
are defined by
(cid:40) (cid:115) (cid:41)
2(cid:96)+1((cid:96)−m)!
P (x) ≡ Pm(x) .
(cid:96),m 2 ((cid:96)+m)! (cid:96)
|m|≤(cid:96),(cid:96)=0,1,2,···
(cid:82)2π(cid:82)π (cid:82) (cid:82)
We often write the integral f(θ,ϕ)sinθdθdϕ as f(θ,ϕ)dS or f(σ)dS. The
0 0 S2 S2
above normalisation gives us
(cid:90)
Y (θ,ϕ)Y (θ,ϕ)dS = δ δ . (2.3)
(cid:96),m µ,ν lµ mν
S2
Finally, we let P ([r ,r ]), and P (S2) be the space of polynomials of degree ≤ k on [r ,r ],
k in out (cid:96) in out
andrespectivelythespaceofsphericalpolynomialofdegree≤ (cid:96)onS2. Fordetailsoforthogonal
polynomials, see, for example, [23, 25]. Consider functions hang,hrad: [0,∞) → [0,∞) with
supp(hang) ⊂ [0,a], and supp(hrad) ⊂ [0,b] (a,b ∈ (1,2]). Further, we assume hang(s) =
hrad(t) = 1 for s,t ∈ [0,1]. Let us define the filter function h: [0,∞)×[0,∞) → [0,∞) by
h: (s,t) (cid:55)→ hrad(s)hang(t). (2.4)
We consider an approximation of a real-valued function f on the shell [r ,r ]×S2 of the form
in out
∞ (cid:96) (cid:18) (cid:19)
(cid:88) (cid:88) k (cid:96)
V f := h , c J (r)Y (θ,ϕ), (2.5)
KL K L k(cid:96)m k (cid:96),m
k,(cid:96)=0m=−(cid:96)
where coefficients {c } ⊂ R shall be defined in Section 5, (5.1). They are approximations of
k(cid:96)m
Fourier coefficients, that is,
1 (cid:90) 2π(cid:90) π(cid:90) rout
c ≈ f(r,θ,ϕ)J (r)Y (θ,ϕ)dµrad(r)sinθdθdϕ. (2.6)
k(cid:96)m γ2 k (cid:96),m
k 0 0 rin
3 Filtered hyperinterpolation on the radial interval
In this section we define filtered hyperinterpolation in the radial direction, and we will see that
it is bounded as an operator from C(S ) to C(S ).
ε ε
In order to develop properties of the filtered hyperinterpolation, as an intermediate step we
define the continuous filtered approximation in the radial direction.
Let (cid:104)f,g(cid:105) := (cid:82)routfgdµrad, and K(a) = max{(cid:100)aK(cid:101)−1,K} with a ∈ (1,2]. For
L2µ([rin,rout]) rin
f ∈ C([r ,r ]), we define the filtered approximation R f (K ≥ 1) by
in out K
∞ (cid:18) (cid:19)(cid:28) (cid:29)
(cid:88) k J J (r)
R f(r) = hrad f, k k (3.1)
K K γ γ
k=0 k L2µ([rin,rout]) k
K(a) (cid:18) (cid:19)(cid:28) (cid:29)
(cid:88) k J J (r)
= hrad f, k k (3.2)
K γ γ
k=0 k L2µ([rin,rout]) k
= (cid:104)f,G (·,r)(cid:105) , (3.3)
K L2([r ,r ])
µ in out
4
where
K(a) (cid:18) (cid:19)
(cid:88) k J (s)J (r)
G (s,r) := hrad k k for (s,r) ∈ [r ,r ]×[r ,r ]. (3.4)
K K γ γ in out in out
k k
k=0
Note that this is a finite sum.
The following proposition is standard.
Proposition 3.1. For R defined by (3.1) we have
K
(cid:90) r
out
(cid:107)R (cid:107) = sup |G (s,r)|dµrad(s). (3.5)
K C([r ,r ])→C([r ,r ]) K
in out in out
r∈[r ,r ] r
in out in
Proof. Clearly, we have
(cid:90) r
out
(cid:107)R (cid:107) ≤ sup |G (s,r)|dµrad(s),
K C([r ,r ])→C([r ,r ]) K
in out in out
r∈[r ,r ] r
in out in
(cid:16) (cid:17)
since |R f(r)| ≤ sup |f(t)| (cid:82)rout|G (s,r)|dµrad(s) for each r ∈ [r ,r ]. Con-
K t∈[rin,rout] rin K in out
versely, let
(cid:40)
G (s,r)
K G (s,r) (cid:54)= 0,
sgn (s) := |GK(s,r)| K
r
0 otherwise.
Since G (·,r) is measurable, so is sgn (·). Clearly, sgn (·) ∈ L1 := L1 ([r ,r ]). Thus,
K r r µ µrad in out
there exist a sequence {g } ⊂ C([r ,r ]) such that g → sgn (n → ∞) in L1. Therefore,
n n in out n r µ
there exist a subsequence {g } ⊂ {g } such that
n,k k n
g (s) → sgn (s) (k → ∞) for µrad-almost every s.
n,k r
Replacing,ifnecessary,eachfunctiong withmin{max{−1,g },1},wecanassume|g | ≤ 1
n,k n,k n,k
independently of k, so that
|g (·)G (·,r)| ≤ max |G (s,t)|.
n,k K K
s,t∈[r ,r ]
in out
Thus, from the dominated convergence theorem we have
(cid:90) r (cid:90) r
out out
|G (s,r)|dµrad(s) = sgn (s)G (s,r)dµrad(s)
K r K
r r
in in
(cid:90) r
out
= lim g (s)G (s,r)dµrad(s)
n,k K
k→∞ r
in
= lim R g (r) ≤ (cid:107)R (cid:107) .
K n,k K C([r ,r ])→C([r ,r ])
k→∞ in out in out
The following result essentially due to Mhaskar [14] establishes the uniform boundedness
of {R } .
K K
5
Theorem 3.2 ([14]). Let the parameters (α,β) for the Jacobi polynomial satisfy α,β ≥ −1,
2
and let ι ≥ α + β + 1 be an integer. Suppose that the filter function hrad: [0,∞) → [0,∞)
satisfies hrad(s) = 1 for s ∈ [0,1], supp(hrad) ⊂ [0,a] for some a ∈ (1,2]. Further, suppose
that hrad and its derivatives of all orders up to ι−1 are absolutely continuous, and the ι-th
derivative is of bounded variation. Then, R : C([r ,r ]) → C([r ,r ]) defined by (3.1)
K in out in out
satisfies
sup(cid:107)R (cid:107) < ∞. (3.6)
K C([r ,r ])→C([r ,r ])
in out in out
K≥1
Proof. Following [14, Proof of Theorem 3.1], we see that the current assumption on the filter
hrad implies [14, Theorem 3.1], and thus from [14, (2.10)] we have
(cid:90) r
out
sup |G (s,r)|dµrad(s) < ∞.
K
r∈[r ,r ] r
in out in
Then, the statement follows from Proposition 3.1.
The filtered hyperinterpolation defined in the next section is obtained by approximating
(cid:104)·,·(cid:105) by the Gauss-type quadrature.
L2([r ,r ])
µ in out
3.1 Filtered hyperinterpolation
(cid:108) (cid:109)
K(a)+K−1
We first recall the Gauss-Jacobi quadrature rule. Let κ := , and {r } ⊂
Q 2 j j=0,...,κQ
(α,β)
[r ,r ]bethezerosofaJacobipolynomialJ = J . Then,thereexiststhecorrespond-
in out κQ+1 κQ+1
ing positive weight {wrad} that defines a quadrature rule Q (2κ +1) with the preci-
j j=0,...,κQ rad Q
sion 2κ +1(≥ K(a)+K). That is, for f ∈ C([r ,r ]) let Q (2κ +1)f := (cid:80)κQ f(r )wrad.
Q in out rad Q j=0 j j
Then, we have
(cid:90) rout (cid:18)2r−(r +r )(cid:19) 2
Q (2κ +1)q = q(r)w∗ out in dr
rad Q r −r r −r
r out in out in
in
(cid:90) r
out
= q(r)dµrad(r), (3.7)
r
in
for any q ∈ P ([r ,r ]), in particular, for any q ∈ P ([r ,r ]).
2κQ+1 in out K(a)+K in out
In the following, for a function f: S → R on S we often write f(rσ) (rσ ∈ S ) as f(r,σ).
ε ε ε
For the sake of simplicity, we introduce the notation
κ
Q
(cid:88)
(cid:104)f(·,σ),g(·,σ)(cid:105)rad := wradf(r ,σ)g(r ,σ), f,g ∈ C(S ), (3.8)
Q (2κ +1) j j j ε
rad Q
j=0
where a quadrature rule Q (2κ +1) is used in the right hand side.
rad Q
We defined the filtered approximation R as (3.5) with the calligraphic character. Here,
K
we define the discretised filtered approximation operator R : C(S ) → C(S ) for the radial
K ε ε
direction, a quadrature-based approximation of R . For a function f in C(S ), we define the
K ε
operator R (K ≥ 1) as
K
6
K(cid:88)(a) (cid:18) k (cid:19)(cid:28) J (cid:29)rad J (r)
R f(r,σ) = hrad f(·,σ), k k . (3.9)
K K γ γ
k=0 k Qrad(2κQ+1) k
Following [22], we call R f filtered hyperinterpolation of f in the radial direction. We now
K
develop a bound of R on the interval [r ,r ] in terms of the supremum norm over [r ,r ].
K in out in out
Later, we use the result to analyse the error of filtered hyperinterpolation on spherical shells.
3.2 Supremum norm bound of R f
K
In this section, we obtain a bound of R f in terms of the supremum norm. We have the
K
following bound.
Proposition 3.3. Let the parameters (α,β) for the Jacobi polynomial satisfy α,β ≥ −1,
2
and let ι ≥ α + β + 1 be an integer. Suppose that the filter function hrad: [0,∞) → [0,∞)
satisfies hrad(s) = 1 for s ∈ [0,1], supp(hrad) ⊂ [0,a] for some a ∈ (1,2]. Further, suppose
that hrad and its derivatives of all orders up to ι−1 are absolutely continuous, and the ι-th
derivative is of bounded variation. For f ∈ C(S ), let R f (K ≥ 1) be defined by (3.9) with
ε K
K(a) = max{(cid:100)aK(cid:101)−1,K} (a ∈ (1,2]). Then, for each σ ∈ S2 we have
sup |R f(r,σ)| ≤ C sup |f(r,σ)|, (3.10)
K 1
r∈[r ,r ] r∈[r ,r ]
in out in out
where the constant C is independent of K and f.
1
Proof. Fix σ ∈ S2. Clearly, we have
κ
Q
(cid:88)
|RKf(r,σ)| ≤ ( sup |f(r,σ)|) wjrad|GK(rj,r)|. (3.11)
r∈[r ,r ]
in out j=0
Note that G (·,r) is a polynomial of degree ≤ K(a). From a non-trivial result by Nevai which
K
gives a bound on Gauss-Jacobi quadrature formulae in terms of the integral the quadrature
approximates, (see, for example [16, p. 35, theorem 4.7.4]) we have
(cid:88)κQ (cid:18) K(a) (cid:19)(cid:90) rout
wrad|G (r ,r)| ≤ C +1 |G (s,r)|dµrad(s), (3.12)
j K j κ +1 K
Q r
j=0 in
where C depends only on the measure µrad.
K(a)+K−1
Since ≤ κ , we have
2 Q
K(a) 2K(a)
≤ ≤ 2, (3.13)
(κ +1) K +K(a)
Q
and thus
κQ (cid:90) r
(cid:88) out
wrad|G (r ,r)| ≤ 3C |G (s,r)|dµrad(s). (3.14)
j K j K
r
j=0 in
7
This, together with (3.5) and (3.11), we have
sup |R f(r,σ)|
K
r∈[r ,r ]
in out
(cid:18) (cid:19) (cid:90) r
out
≤ 3C sup |f(r,σ)| sup |G (s,r)|dµrad(s) (3.15)
K
r∈[r ,r ] r∈[r ,r ] r
in out in out in
(cid:18) (cid:19)
= 3C sup |f(r,σ)| (cid:107)R (cid:107) . (3.16)
K C([r ,r ])→C([r ,r ])
in out in out
r∈[r ,r ]
in out
In view of Theorem 3.2, we can conclude that
sup |R f(r,σ)| ≤ C sup |f(r,σ)|, (3.17)
K 1
r∈[r ,r ] r∈[r ,r ]
in out in out
where the constant C is independent of K and f.
1
4 Filtered hyperinterpolation on sphere
We now introduce filtered hyperinterpolation in the angular direction. Let L(b) = max{(cid:100)bL(cid:101)−
1,L} with b ∈ (1,2]. Let Q (L(b)+L) be a positive-weight (ν +1)-point spherical numerical
ang Q
integration rule
ν
Q
Q (L(b)+L)f(r) := (cid:88)wangf(r,σ ), f ∈ C(S ), (4.1)
ang j j ε
j=0
with points σ ,...,σ ∈ S2 and corresponding weights wang,...,wang, which integrates all
0 νQ 0 νQ
spherical polynomials of degree ≤ L(b)+L exactly. That is, we have
(cid:90)
pdS = Q (L(b)+L)p, p ∈ P (S2). (4.2)
ang L(b)+L
S2
Usingthisquadraturerule,letusdefineabilinearmap(cid:104)·,·(cid:105)ang : C(S )×C(S ) → C(S )
Q (L(b)+L) ε ε ε
ang
by
ν
Q
(cid:104)f,g(cid:105)ang (r) := (cid:88)wangf(r,σ )g(r,σ ), f,g ∈ C(S ). (4.3)
Q (L(b)+L) j j j ε
ang
j=0
Similarly to the radial direction, we consider a filter function hang: [0,∞) → [0,∞) with
hang(s) = 1 for s ∈ [0,1] and supp(hang) ⊂ [0,b] with a suitable smoothness. We define
the filtered hyperinterpolation operator A : C(S ) → C(S ) in the angular direction by
L ε ε
∞ (cid:96) (cid:18) (cid:19)
(A f)(r,σ) := (cid:88) (cid:88) hang (cid:96) (cid:10)f(r,·),Y (·)(cid:11)ang Y (σ) (4.4)
L L (cid:96),m Qang(L(b)+L) (cid:96),m
(cid:96)=0m=−(cid:96)
L(b) (cid:96) (cid:18) (cid:19)
= (cid:88) (cid:88) hang (cid:96) (cid:10)f(r,·),Y (·)(cid:11)ang Y (σ). (4.5)
L (cid:96),m Qang(L(b)+L) (cid:96),m
(cid:96)=0m=−(cid:96)
8
4.1 Supremum norm estimate on the sphere
To obtain our error estimate on spherical shells, we need the following supremum norm error
estimate for the hyperinterpolation in the angular direction.
Theorem 4.1 (supremumnormestimateonS2, [26]). Let r ∈ [r ,r ]. Suppose that the filter
in out
function hang: [0,∞) → [0,∞) satisfies hang(s) = 1 for s ∈ [0,1], supp(hang) ⊂ [0,b] for some
b ∈ (1,2]. Further, suppose that hang is absolutely continuous and its derivative is of bounded
variation. Then, for f(r,·) ∈ C(S2) and L ≥ 1, we have
sup |f(r,σ)−A f(r,σ)| ≤ C inf sup |f(r,σ)−p(σ)|. (4.6)
L 2
σ∈S2 p∈PL(S2)σ∈S2
where the constant C is independent of f and L.
2
Proof. From [26, Proof of Theorem 1.1], for any g ∈ C(S2) we have
sup |A g(σ)| ≤ C sup |g(σ)|,
L
σ∈S2 σ∈S2
with a constant C > 0 independent of g and L. Now, observe A p(σ) = p(σ) for any p ∈
L
P (S2). Then, the statement follows using the standard technique
L
|f(r,σ)−A f(r,σ)| ≤ |f(r,σ)−p(σ)|+|A (p(σ)−f(r,σ))|,
L L
for arbitrary p ∈ P (S2).
L
5 Filtered hyperinterpolation on spherical shells
We finally define the filtered hyperinterpolation operator on S , and give an error estimate in
ε
terms of the supremum norm over S .
ε
First, note that for f ∈ C(S ), we have A R f = R A f. We define the operator V on
ε L K K L KL
C(S ) as
ε
(V f)(r,σ)
KL
(cid:88)∞ (cid:88)(cid:96) (cid:18) k (cid:96)(cid:19)(cid:32)(cid:88)κQ (cid:88)νQ Jk(rj) (cid:33)
= h , W f(r ,σ ) Y (σ ) J (r)Y (σ), (5.1)
K L jn j n γ2 (cid:96),m n k (cid:96),m
k,(cid:96)=0m=−(cid:96) j=0n=0 k
(cid:108) (cid:109)
where h(cid:0)m, (cid:96)(cid:1) = hrad(cid:0)k(cid:1)hang(cid:0)(cid:96)(cid:1), W = wradwang, κ = K(a)+K−1 so that the radial
M L K L jn j n Q 2
quadrature has the precision K(a)+K, and ν is taken so that the angular quadrature has
Q
the precision L+L(b).
We estimate the error by following decomposition. For an arbitrary norm (cid:107)·(cid:107), from V =
KL
R A we have
K L
(cid:107)f −V f(cid:107) = (cid:107)f −R A f +R f −R f(cid:107)
KL K L K K
≤ (cid:107)f −R f(cid:107)+(cid:107)R (f −A f)(cid:107). (5.2)
K K L
We derive estimates for both term (cid:107)f −R f(cid:107) and (cid:107)R (f −A f)(cid:107), with (cid:107)·(cid:107) being the
K K L
supremum norm.
9
5.1 Best approximation by polynomials
Werecordclassicalresultsofestimatesonbestapproximationbypolynomials. Later,wereduce
the error of the filtered hyperinterpolation approximation to the best approximation error.
Ontheinterval, wehavethefollowingwell-knownresults(seeforexample, [19,pp.196–197,
p. 26]).
Theorem 5.1. Let
Erad (f) = inf sup |f(r)−p(r)|.
K,∞
p∈PK([rin,rout])r∈[rin,rout]
Then, for f ∈ Cη([r ,r ]) with η ∈ {1,...,K}, we have
in out
(cid:18) (cid:19)
(K −η)!
Erad (f) ≤ Cη sup |f(η)(r)| , (5.3)
K,∞ 3 K!
r∈[r ,r ]
in out
where the constant C is independent of f and K.
3
On the sphere, we have the following classical result by Pawelke [17]. Note that we could
also use the resent result by Dai and Xu [3, corollary 3.7], see also [4, corollary 4.5.6.].
Theorem 5.2 (Pawelke [17]). Let
Eang(f) = inf sup |f(σ)−p(σ)|.
L,∞ p∈PL(S2)σ∈S2
Then, for each f ∈ C2t(S2), (t = 1,2,...), there exists a constant C > 0 independent of f
4
and L, such that
(cid:18) (cid:19)
Eang(f) ≤ CtL−2t sup |∆t f(σ)| (5.4)
L,∞ 4 S
σ∈S2
holds, where ∆ is the Laplace–Beltrami operator on S2.
S
5.2 Error estimate
We have the following estimate.
Theorem5.3. Forf ∈ C(S ), letV f bedefinedby (5.1). Then, underthesameassumptions
ε KL
as Proposition 3.3 and Theorem 4.1, we have
sup |f(r,σ)−V f(r,σ)|
KL
(r,σ)∈S
ε
≤ (1+C ) sup Erad (f(·,σ))+C (1+C ) sup Eang(f(r,·)), (5.5)
1 K,∞ 1 5 L,∞
σ∈S2 r∈[r ,r ]
in out
for f ∈ C(S ), where the constants C , C > 0 are independent of K, L, and f.
ε 1 5
10
