Interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations
We consider interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations on the sphere. The asymptotic behavior of the mean-square interpolation error is investigated. The degree of convergence to zero of the meansquare interpolation error i...
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Інститут математики НАН України
2007
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| Цитувати: | Interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations / N. Semenovs’ka // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 234-242. — Бібліогр.: 3 назв.— англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860254232790171648 |
|---|---|
| author | Semenovs’ka, N. |
| author_facet | Semenovs’ka, N. |
| citation_txt | Interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations / N. Semenovs’ka // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 234-242. — Бібліогр.: 3 назв.— англ. |
| collection | DSpace DC |
| description | We consider interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations on the sphere. The asymptotic behavior of the mean-square interpolation error is investigated. The degree of convergence to zero of the meansquare interpolation error is obtained. Efficient volume of the set of observations is given.
|
| first_indexed | 2025-12-07T18:47:14Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.234-242
NATALIYA SEMENOVS’KA
INTERPOLATION OF HOMOGENEOUS AND
ISOTROPIC RANDOM FIELD IN THE CENTER
OF THE SPHERE BY UNIFORM DISTRIBUTED
OBSERVATIONS
We consider interpolation of homogeneous and isotropic random field
in the center of the sphere by uniform distributed observations on the
sphere. The asymptotic behavior of the mean-square interpolation
error is investigated. The degree of convergence to zero of the mean-
square interpolation error is obtained. Efficient volume of the set of
observations is given.
1. Introduction
In the paper [1] the problem of interpolation of homogeneous and isotropic
random field in in the center of the circle by observations on the circle
was considered. In the paper we consider a particle case of the problem
of interpolation of homogeneous and isotropic random field in an arbitrary
point inside of the sphere by observations on the sphere which had consid-
ered in paper [2]. That is the problem of interpolation of homogeneous and
isotropic random field in the center of the sphere by uniform distributed ob-
servations on the sphere. Namely, we investigate the order of convergence
of the interpolation error to zero. We obtain also efficient volume of the set
of observations.
So, we will use the results and notions of the aforementioned papers to
investigate our problem.
Give some initial notions.
2000 Mathematics Subject Classifications. Primary 60G60, 62M20; Secondary 60G25,
93E10.
Key words and phrases. Homogeneous and isotropic random field, interpolation in
the center of the sphere, mean-square interpolation error, asymptotic behavior.
234
INTERPOLATION OF RANDOM FIELD 235
2. The interpolation problem
At first, give the following notations.
Let ξ(x), x ∈ Rn, be a mean-square continuous, homogeneous and wide-
isotropic centered random field at the Euclidian space Rn, that is, Eξ(x) =
0, E|ξ(x)|2 < ∞ and the correlation function ϕ(|x − y|) = Eξ(x)ξ(y) de-
pends only on the distance |x − y| between the points x and y.
Denote by (r, ϕ) = (ρ, ϕ1, . . . , ϕn−1) the spherical coordinates of the
point x and let Sl
m(ϕ) be orthonormalized spherical harmonics of order m,
h(m, n) = (2m + n − 2) (m+n−3)!
(n−2)!m!
is their quantity.
The following spectral representation takes place [3].
ξ(x) = cn
∑
m≥0
h(m,n)∑
l=1
Sl
m(ϕ)ζ l
m(r),
ζ l
m(r) =
∞∫
0
J
m+ n−2
2
(λr)
(λr)
n−2
2
Z l
m(dλ),
(1)
where c2
n = 2n−1Γ
(
n
2
)
π
n
2 , and Z l
m(·) is a sequence of uncorrelated random
measures on (0, +∞) with spectral measure Φ:
EZ l
m(S) = 0
EZ l
m(S1)Z
q
p(S2) = δp
mδq
l Φ(S1∩S2)
(2)
for all S, S1, S2 ∈ B(0, +∞) and for all m, l, p, q ≥ 0.
bm(r, ρ) =
∞∫
0
Jm+ n−2
2
(λr)
(λr)
n−2
2
·
Jm+ n−2
2
(λρ)
(λρ)
n−2
2
dΦ(λ), (3)
where Jm is the Bessel function of order m, and bm(r) = bm(r, r).
From (2)it follows that ζ l
m(r) are uncorrelated random variables for dif-
ferent m and l:
Eζ l
m(r)ζq
p(r) = δp
mδq
l bm(r). (4)
Consider the problem of interpolation of field ξ(x) in an arbitrary point
y inside of the sphere Sn of radius r by observations on the sphere. This
problem reduces to that of finding of the projection ξ̂(y) of element ξ(y) at
the linear space Hξ(r) formed by the mean-square closure of the linear span
of random variables of the form ξ(x) : |x| = r :
ξ̂(y) ∈ Hξ(r) = Cl
{∑
αkξ(xk), |xk| = r
}
.
Such interpolation formula by Theorem 3 ([3], IV, § 1) has the form
ξ̂(y) = c(y)
∫
Sn
ξ(x)dμn, (5)
236 NATALIYA SEMENOVS’KA
where
c(y) =
1
ωn
· b0(r, ρ)
b0(r)
, (6)
μn is Lebesgue measure on the unit sphere in Euclidian space Rn and ωn is
that sphere surface square.
The corresponding interpolation error σ2 = E(ξ(y)− ξ̂(y))2 equals
σ2 = ϕ(0) − c2
n
ωn
· b2
0(r, ρ)
b0(r)
. (7)
When using the interpolation formula (5) in practice, the integral
∫
Sn
ξ(x)dμn
is replaced by the corresponding integral sum. Hence, we consider, instead
of the problem of interpolation based on the space
Hξ(r) = Cl
{∑
αkξ(xk), |xk| = r
}
,
the problem of interpolation of the value ξ(y) from observations on the space
HX(r) =
{∑
αkξ(xk), xk ∈ X
}
, (8)
where X = {x1, . . . , xN} is some finite subset of the sphere Sn formed by
observation points.
So, the integral (5) replaced by finite linear combination (8) of values of
the field ξ(x) in points of a certain set X. Therefore, we obtain the problem
of accuracy of that replacing, which equivalent to that one of accuracy of
the interpolation formula (5) approximation.
Fix a set X = {x1, . . . , xN}. Since HX ⊂ Hξ, the properties of mean-
square projection imply that for any estimate ξ̂X(y) ∈ HX the identity
σ2
X ≡ E(ξ(y)− ξ̂X(y))2 = σ2+ σ̂2
X ≡ E(ξ(y)− ξ̂(y))2+E(ξ̂(y)− ξ̂X(y))2 (9)
holds.
Definition 1. According to (9), a set X ⊂ Sn will be called efficient if
the error of approximation of the integral (5) by the integral sum does not
exceed the error of the corresponding interpolation, that is, σ̂2
X ≤ σ2. In
this case the total variance of the approximate solution of the interpolation
problem does not exceed the value σ2
X ≤ 2σ2, where σ2 is evaluated in (7).
The best mean-square interpolation of the value ξ(y) from values from
X is of the form
ξ̂(y) =
N∑
k=1
αkξ(xk), (10)
where αk, k = 1, . . . , N are weight coefficients.
INTERPOLATION OF RANDOM FIELD 237
Denote by
FN (ϕ) =
∑
k:ϕk≤ϕ
αk, ϕ1 ∈ [0, 2π], ϕi ∈ [0, π], i = 2, n − 1,
the cumulative ”distribution function” of weights {α1, . . . , αN} of points
{x1, . . . , xN} on the sphere. Here the inequality ϕk ≤ ϕ means the system
of step-by-step inequalities ϕ
(i)
k ≤ ϕi, i = 1, n − 1.
In the paper [2] we get the following result.
Corollary. For any sequence of series of sets XN = {x1, . . . , xN}, where
xk = (r, ϕk), 0 < ϕ
(i)
1 < ϕ
(i)
2 < · · · < ϕ
(i)
N = 2π, 0 < ϕ
(i)
1 < ϕ
(i)
2 < · · · < ϕ
(i)
N =
π, i = 2, n − 1, under condition
∑
m≥0
h(m,n)∑
l=1
∣∣∣clm(y)
∣∣∣2 < +∞, (11)
the mean-square approximating error σ̂2
X tends to zero as N → ∞, provided
that weights
αk = C(y, xk)
n−1∏
i=1
(ϕ
(i)
k − ϕ
(i)
k−1),
and max |
n−1∏
i=1
(ϕ
(i)
k − ϕ
(i)
k−1)| → 0 as N → ∞.
Here
C(y, x) =
1
ωn(n − 2)
∑
m≥0
bm(r, ρ)
bm(r)
(2m + n − 2)C
n−2
2
m (cos θ), (12)
θ is angle between vectors x and y and Cν
m(x), m ≥ 0, are Gegenbouer
polynomials which can be defined as coefficients in representation of function
(1 − 2tx + t2)−ν =
∞∑
m=0
Cν
m(x)tm, (13)
that is generatrices for those polynomials.
3. Asymptotic behavior of the error for the uniform
distribution
Let us consider the case where the points of the set XN = {x1, . . . , xN}
are uniformly distributed on the sphere Sn, that is, ϕ
(1)
k = 2π(k1−1)
M
, ϕ
(i)
k =
π(ki−1)
M
, i = 2, n, ki = 1, M . So, we have N = Mn−1 points. From
Corollary 1 follows, that choice of interpolation weights of the form αk =
238 NATALIYA SEMENOVS’KA
ϕ(r)l(ϕk)
c2nb0(r)
n−1∏
i=1
(ϕ
(i)
k −ϕ
(i)
k−1), where l(ϕ) = (sin ϕ2)
n−2(sin ϕ3)
n−3 . . . sin ϕn−1 and
ϕ(r) = 2
n−2
2 Γ(n/2)
∞∫
0
Jn−2
2
(λr)(λr)
2−n
2 dΦ(λ) is correlation function, mini-
mizes (moreover, reduces to zero) limit error σ̂2∞. Let’s investigate the
degree of convergence of σ̂2
X to zero in this case.
Since the random variables ζ l
m are uncorrelated in accordance with (4)
and interpolation formula for ξ̂(y) has the form (10), taking into account
the spectral representation (1), we have
σ̂2
X = E(ξ̂(y) − ξ̂X(y))2 =
= c2
n
∑
m≥0
h(m,n)∑
l=1
∣∣∣∣ N∑
k=1
αkS
l
m(ϕk) − bm(r,ρ)
bm(r)
Sl
m(ψ)
∣∣∣∣2 E|ζ l
m(r)|2 (14)
Evaluate equality in (14), taking into account the fact that the field is
extrapolating in the center of the sphere (it means that ρ = 0 and bm(r, ρ) =
δ0
m
ωn
c2n
ϕ(r)).
σ̂2
X = E(ξ̂(y)− ξ̂X(y))2 = c2
n
∑
m≥0
h(m,n)∑
l=1
∣∣∣∣ N∑
k=1
αkS
l
m(ϕk) − bm(r,ρ)
bm(r)
Sl
m(ψ)
∣∣∣∣2 bm(r)
= c2n
ωn
(
N∑
k=1
αk − ωnϕ(r)
c2nb0(r)
)2b0(r)+
+ c2n
ωn
2
(n−2)
N∑
k,j=1
αkαj
∞∫
0
∑
m≥1
(m + n−2
2
)
J2
m+ n−2
2
(λ r)
(λ r)n−2 C
n−2
2
m (cos θkj)dΦ(λ) =
= c2n
ωn
(S(α) − ωnϕ(r)
c2nb0(r)
)2b0(r) − c2n
ωn
b0(r)S
2(α)+
+ cn√
ωn
N∑
k,j=1
αkαj
∞∫
0
J n−2
2
(λRkj)
(λRkj)
n−2
2
dΦ(λ),
(15)
where S(α) =
N∑
k=1
αk, Rkj = 2r sin
θkj
2
, and θkj are angles between vectors
xk and xj .
Substitute values of interpolation weights αk in last equality.
σ̂2
X = c2n
ωn
(ωnϕ(r)
c2nb0(r)
− ωnϕ(r)
c2nb0(r)
)2b0(r) − c2n
ωn
b0(r)
(
ωnϕ(r)
c2nb0(r)
)2
+
+ cn√
ωn
N∑
k,j=1
(
2πn−1ϕ(r)
c2nNb0(r)
)2
l(ϕk)l(ϕj)
∞∫
0
J n−2
2
(2λr sin
θkj
2
)
(2λr sin
θkj
2
)
n−2
2
dΦ(λ) =
= −ωnϕ2(r)
c2nb0(r)
+ cn√
ωn
N∑
k,j=1
(
ϕ(r)2πn−1
b0(r)c2nN
)2
l(ϕk)l(ϕj)
∞∫
0
J n−2
2
(2λr sin
θkj
2
)
(2λr sin
θkj
2
)
n−2
2
dΦ(λ).
Denote by F (ϕ) = ϕ(r)
b0(r)c2n
∫ ϕ
0
l(u)du distribution function of weights {α1, . . . ,
INTERPOLATION OF RANDOM FIELD 239
αN}. If we pass in (15) to the limit as N → ∞ we obtain
ωnϕ2(r)
c2nb0(r)
= cn√
ωn
∞∫
0
(∫
Πn
∫
Πn
Jn−2
2
(2λr
√
2(1−cos θ))
(2λr
√
2(1−cos θ))
n−2
2
dF(u)dF(v)
)
dΦ(λ) =
= cn√
ωn
(
ϕ(r)
b0(r)c2n
)2 ∞∫
0
(∫
Πn
∫
Πn
l(u)l(v)
Jn−2
2
(2λr
√
2(1−cos θ))
(2λr
√
2(1−cos θ))
n−2
2
dudv
)
dΦ(λ),
where θ is the angle between u and v.
Recall that l(u) = (sin u2)
n−2(sin u3)
n−3 . . . sin un−1, and
cos (̂u, v) = cos u2 cos v2 + sin u2 sin v2 cos u3 cos v3+
+ sin u2 sin v2 sin u3 sin v3 cos u4 cos v4 + . . .+
+(cosu1 cos v1 + sin u1 sin v1)
n−1∏
i=2
sin ui sin vi =
= cos u2 cos v2 + sin u2 sin v2 cos u3 cos v3+
+ sin u2 sin v2 sin u3 sin v3 cos u4 cos v4 + . . .+
+ cos(u1 − v1)
n−1∏
i=2
sin ui sin vi
depends on cos(u1 − v1) and variables ǔ = (u2, . . . , un−1) and v̌ =
(v2, . . . , vn−1).
Denote H(cos(u1 − v1), ǔ, v̌) = cn√
ωn
∞∫
0
l(u)l(v)
J n−2
2
(2λr
√
2(1−cos θ))
(2λr
√
2(1−cos θ))
n−2
2
dΦ(λ) =
l(u)l(v)ϕ(2r
√
2(1 − cos θ)), ǩ = (k2, . . . , kn−1), Πn−1 = [0, π]n−2, and write
σ̂2
X =
(
ϕ(r)
b0(r)c2n
)2
(
( πn−2
Mn−2 )
2
M∑
ǩ,ǰ=1
(2π)2
M2
M∑
k1,j1=1
H(cos(ϕ
(1)
k − ϕ
(1)
j ), ϕ̌k, ϕ̌j)−
− ∫
Πn−1
∫
Πn−1
2π∫
0
2π∫
0
H(cos(u1 − v1), ǔ, v̌)du1dv1dǔdv̌
)
.
(16)
Taking into account evenness and summeriness of the function H(cos(u1 −
v1), ǔ, v̌) write
(2π)2
M2
M∑
k1,j1=1
H(cos(ϕ(1)
k − ϕ
(1)
j ), ϕ̌k, ϕ̌j) = (2π)2
M2
M∑
k1,j1=1
H(cos(k − j)2π
M , ϕ̌k, ϕ̌j) =
= (2π)2
M
∑
|s|<M
(1 − |s|
M )H(cos s2π
M , ϕ̌k, ϕ̌j) = 2 (2π)2
M
M−1∑
s=0
(1 − s
M )H(cos s2π
M , ϕ̌k, ϕ̌j),
and also
2π∫
0
2π∫
0
H(cos(u1 − v1), ǔ, v̌)du1dv1 = 2(2π)2
1∫
0
(1 − x)H(cos 2πx, ǔ, v̌)dx.
240 NATALIYA SEMENOVS’KA
Denote G(x, ǔ, v̌) = (1 − x)H(cos 2πx, ǔ, v̌), Πǩ =
n−1∏
i=2
[ (ki−1)π
M
, kiπ
M
], Π1/M =
[ π
M
, π
M
]n−2.
Evaluate the equality in (16).
σ̂2
X = 2
(
2πϕ(r)
b0(r)c2n
)2
(
( πn−2
Mn−2 )
2
M∑
ǩ,ǰ=1
1
M
M−1∑
s=0
G( s
M
, ϕ̌k, ϕ̌j)−
− ∫
Πn−1
∫
Πn−1
1∫
0
G(x, ǔ, v̌)dxdǔdv̌
)
=
= −2
(
2πϕ(r)
b0(r)c2n
)2 M∑
ǩ,ǰ=1
M−1∑
s=0
∫
Πǩ
∫
Πǰ
s+1
M∫
s
M
(
G(x, ǔ, v̌) − G( s
M
, ϕ̌k, ϕ̌j)
)
dxdǔdv̌ =
= −2
(
2πϕ(r)
b0(r)c2n
)2 M−1∑
ǩ,ǰ=0
M−1∑
s=0
∫
Π1/M
∫
Π1/M
1
M∫
0
(
G(x + s
M
, ǔ + ǩφ, v̌ + ǰφ)−
−G( s
M
, ǩφ, ǰφ)
)
dxdǔdv̌,
where φ = (π/M, . . . , π/M︸ ︷︷ ︸
n−2
).
σ̂2
X = −2
(
2πϕ(r)
b0(r)c2n
)2 M−1∑
ǩ,ǰ=0
M−1∑
s=0
∫
Π 1
M
∫
Π 1
M
1
M∫
0
(
G′( s
M , ǩφ, ǰφ)(x, ǔ, v̌) + o1( 1
M)
)
dxdǔdv̌.
Since∫
Π1/M
∫
Π1/M
1
M∫
0
G′
ui
(( s
M
, ǩφ, ǰφ)uidxdǔdv̌ = G′
ui
( s
M
, ǩφ, ǰφ) π
2M2
(
πn−2
Mn−2
)2
, i ≥ 2,
(similarly for
∫
Π1/M
∫
Π1/M
G′
vi
(kφ; jφ)vidudv),
∫
Π1/M
∫
Π1/M
1
M∫
0
G′
x((
s
M
, ǩφ, ǰφ)uidxdǔdv̌ = G′
x(
s
M
, ǩφ, ǰφ) 1
2M2
(
πn−2
Mn−2
)2
,
we have
σ̂2
X = − 1
M
(
2πϕ(r)
b0(r)c2n
)2 M−1∑
ǩ,ǰ=0
M−1∑
s=0
(
π
M
(
πn−2
Mn−2
)2 n−1∑
i=2
(G′
ui
( s
M
, ǩφ, ǰφ)+
+G′
vi
( s
M
, ǩφ, ǰφ)) + 1
M
(
πn−2
Mn−2
)2
G′
x(
s
M
, ǩφ, ǰφ)
)
+ o2(
1
M
) =
= − 1
M
(
2πϕ(r)
b0(r)c2n
)2 ∫
Πn−1
∫
Πn−1
1∫
0
{
π
n−1∑
i=2
(G′
ui
(x, ǔ, v̌) + G′
vi
(x, ǔ, v̌))+
+G′
x(x, ǔ, v̌)} dxdǔdv̌ + o2(
1
M
).
Note that for all i ≥ 2 integral
π∫
0
G′
ui
(x, ǔ, v̌)dui = G(x, ǔ, v̌) |π0= 0, since
sin ui is a component of the function G(x, ǔ, v̌).
INTERPOLATION OF RANDOM FIELD 241
(Similarly,
π∫
0
G′
vi
(x, ǔ, v̌)dvi = 0.)
1∫
0
G′
x(x, ǔ, v̌)dui = G(1, ǔ, v̌) − G(0, ǔ, v̌) = −H(1, ǔ, v̌)
So,
σ̂2
X =
1
M
(
2πϕ(r)
b0(r)c2
n
)2 ∫
Πn−1
∫
Πn−1
H(1, ǔ, v̌)dǔdv̌ + o(
1
M
). (17)
Taking into account obtained result we formulate the following theorem.
Theorem 1. Let the sphere Sn be uniformly divided by the set XN =
{x1, . . . , xN}, N = Mn−1, of points with spherical angles ϕ
(1)
k = 2π(k1 −
1)/M, ϕ
(i)
k = π(ki−1)/M, i = 2, n − 1. Let the coefficients αk = ϕ(r)l(ϕk)
b0(r)c2n
2πn−1
N
,
where l(ϕ) = (sin ϕ2)
n−2(sin ϕ3)
n−3 . . . sin ϕn−1. Then the asymptotic mean-
square error of the estimate
ξ̂X(0) =
ϕ(r)
b0(r)c2
n
· 2πn−1
N
N∑
k=0
l(ϕk)ξ(xk)
is
σ̂2
X =
1
n−1
√
N
(
ϕ(r)
b0(r)c2
n
)2
V ar
∫
Sn
ξ(0, ǔ)dμn(u) + o(
1
n−1
√
N
). (18)
Proof. Taking into account the value H(cos(u1−v1), ǔ, v̌), from (17) obtain
σ̂2
X = 1
M
(
2πϕ(r)
b0(r)c2n
)2 ∫
Πn−1
∫
Πn−1
l(u)l(v)ϕ(2r
√
2(1 − cos(̂̌u, v̌)))dǔdv̌ + o( 1
M
) =
= 1
M
(
ϕ(r)
b0(r)c2n
)2 ∫
Πn
∫
Πn
l(u)l(v)ϕ(|u − v|) |u1=v1=0 dudv + o( 1
M
) =
= 1
M
(
ϕ(r)
b0(r)c2n
)2 ∫
Sn
∫
Sn
ϕ(|u − v|) |u1=v1=0 dμn(u)dμn(v) + o( 1
M
) =
= 1
M
(
ϕ(r)
b0(r)c2n
)2
V ar
∫
Sn
ξ(0, ǔ)dμn(u) + o( 1
M
).
Corollary 1. The uniformly distributed set XN = {x1, . . . , xN}, N =
Mn−1, of observations on the sphere Sn with interpolation weights αk =
ϕ(r)l(ϕk)
b0(r)c2n
2πn−1
N
is efficient starting from the volume
N =
⎧⎪⎨⎪⎩
(
ϕ(r)
b0(r)c2
n
)2
·
V ar
∫
Sn
ξ(0, ǔ)dμn(u)
ϕ(0) − ϕ(r)
b0(r)
⎫⎪⎬⎪⎭
n−1
.
242 NATALIYA SEMENOVS’KA
Proof. The proof follows directly from the Definition 1, by which σ̂2
X � σ2,
and from formulas (18) and (7).
Bibliography
1. Kartashov, M.V., Finite-dimensional interpolation of a random field on the
plane, Probability Theory and Math. Statist., 51, (1994), 53–61.
2. Semenovs’ka, N., Interpolation problem for homogeneous and isotropic ran-
dom field, Probability Theory and Math. Statist., 74, (2006), 150–158
(Ukrainian).
3. Yadrenko, M.I. Spectral theory of random fields, Vischa Shkola, Kiev, (1980),
208 (Russian).
Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: semenovsky@voliacable.com
|
| id | nasplib_isofts_kiev_ua-123456789-4492 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T18:47:14Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Semenovs’ka, N. 2009-11-19T10:23:54Z 2009-11-19T10:23:54Z 2007 Interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations / N. Semenovs’ka // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 234-242. — Бібліогр.: 3 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4492 We consider interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations on the sphere. The asymptotic behavior of the mean-square interpolation error is investigated. The degree of convergence to zero of the meansquare interpolation error is obtained. Efficient volume of the set of observations is given. en Інститут математики НАН України Interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations Article published earlier |
| spellingShingle | Interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations Semenovs’ka, N. |
| title | Interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations |
| title_full | Interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations |
| title_fullStr | Interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations |
| title_full_unstemmed | Interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations |
| title_short | Interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations |
| title_sort | interpolation of homogeneous and isotropic random field in the center of the sphere by uniform distributed observations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4492 |
| work_keys_str_mv | AT semenovskan interpolationofhomogeneousandisotropicrandomfieldinthecenterofthespherebyuniformdistributedobservations |