Convergence of polyharmonic splines on semi-regular grids Z x aZ^n  for a to 0

Let
p,n
∈
N
with 2
p
≥
n
+ 2
,
and let
I
a
be a polyharmonic spline of
order
p
on the grid
Z
×
a
Z
n
which satisfies the interpolating conditions
I
a
(
j,am
) =
d
j
(
am
) for
j
∈
Z
,m
∈
Z
n
where the functions
d
j
:
R
n
→
R
and the parameter
a>
0 are given. Let
B
s
(
R
n
) be the set of all integrable
functions
f
:
R
n
→
C
such that the integral
k
f
k
s
:=
Z
R
n
b
f
(
ξ
)
(1 +
|
ξ
|
s
)
dξ
is finite. The main result states that for given
σ
≥
0 there exists a
constant
c>
0 such that whenever
d
j
∈
B
2
p
(
R
n
)
∩
C
(
R
n
)
,j
∈
Z
,
satisfy
k
d
j
k
2
p
≤
D
·
(1 +
|
j
|
σ
) for all
j
∈
Z
there exists a polyspline
S
:
R
n
+1
→
C
of order
p
on strips such that
|
S
(
t,y
)
−
I
a
(
t,y
)
|≤
a
2
p
−
1
c
·
D
·
(1 +
|
t
|
σ
)
for all
y
∈
R
n
,t
∈
R
and all 0
<a
≤
1.