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Electronic Transactions on Numerical Analysis.
Volume 14, pp. 36-44, 2002.
Copyright  2002, Kent State University.
ISSN 1068-9613.
Kent State University
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UNIFORM APPROXIMATION BY MINIMUM NORM INTERPOLATION∗
FRANZ-JÜRGEN DELVOS
†
Abstract. Harmonic Hilbert spaces were introduced as an extension of periodic Hilbert spaces introduced by
Babus̆ka [1] to the non-periodic case [6]. In this paper we will investigate approximation by minimum norm interpolation in harmonic Hilbert spaces.
Key words. minimum norm interpolation, harmonic Hilbert spaces, remainders.
AMS subject classifications. 41A05, 41A25, 41A65, 65D05, 65T99.
1. Introduction. Periodic Hilbert spaces were introduced by Babus̆ka [1] and minimum
norm interpolation was investigated in these spaces [9, 3, 4]. We extended the construction of
periodic Hilbert spaces to the non-periodic case by introducing harmonic Hilbert spaces and
investigated minimum norm interpolation in these spaces [5]. In [6, 7] we studied approximation properties of classical cardinal interpolation in harmonic Hilbert spaces. We will investigate uniform approximation by minimum norm interpolation in the associated harmonic
Hilbert space.
2. Minimum norm interpolation. The Wiener algebra A(R) consists of those bounded
continuous functions f which can represented as Fourier integrals of functions F ∈ L 1 (R) :
Z
f (x) =
F (t)eixt dt.
R
Let b > 0. Then the series
X
F (t + 2bk)
k∈Z
converges absolutely almost everywhere and defines a measurable function
F2b ∈ L1 ([−b, b])
which is called the periodization of F ∈ L1 (R) (see [2]). The periodization F2b ∈ L1 ([−b, b])
and the function f are related by
Z
π
π
(2.1)
f (r ) =
F2b (t)eir b t dt, r ∈ Z.
b
[−b,b]
(See [2], p.33.)
The function of exponential-type
(2.2)
Tb (f )(x) =
Z
χ[−b,b] (t)F2b (t)eixt dt
R
is called the exponential-type interpolant of f in view of
(2.3)
∗ Received
† Lehrstuhl
π
π
Tb (f )(r ) = f (r ), r ∈ Z.
b
b
October 3, 2000. Accepted for publication May 10, 2001. Communicated by Sven Ehrich.
für Mathematik I, Universität Siegen, Walter-Flex-Straße 3, Germany.
36
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MINIMUM NORM INTERPOLATION
37
For F2b ∈ L2 ([−b, b]) it is just the cardinal function C(f, πb ) (see [11]). It possesses the
cardinal series representation
X
π
f (kh)S(k, h)(x) = C(f, h)(x), h = ,
Tb (f )(x) =
b
k∈Z
where
π
S(k, h)(x) = sinc( (x − kh))
h
is the shifted Sinc function with step size h = πb .
The harmonic Hilbert space HD (R) is the subspace of functions f ∈ A(R) such that
Z
R
|F (t)|2
dt < ∞,
D(t)
where D ∈ L1 (R) is assumed to be non negative and bounded. D is called the defining function (see also [10] where HD (R) is called a weighted L2 −space). Note that if D(t) vanishes
at a set M of posivite measure the Fourier transform F has to vanish almost everywhere on
the same set.
It turns out to be useful to add the additional assumptions:
(2.4)
D(−t) = D(t), D(t) ≥ D(t + h) ≥ 0,
and
(2.5)
0 < D(b) ≤ D2b (t) ≤
X
D(kb) < ∞,
k≥0
for almost all t, h ≥ 0.
It was shown in [6] that for any f ∈ HD (R) we have
X
2
|f (kh)| < ∞, h =
k∈Z
π
,
b
which implies F2b ∈ L2 ([−b, b]) and
(2.6)
Tb (f ) = C(f, h), f ∈ HD (R) .
The Fourier integral of D
d(x) =
Z
D(t)eixt dt
R
is called the generating function of the harmonic Hilbert space HD (R). With respect to the
inner product
Z
(f, g)D =
F (t)G(t)/D(t)dt
R
we have
f (x) = (f, d(· − x))D
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F.-J. DELVOS
which shows that HD (R) is a reproducing kernel Hilbert space (see also [10]).
Minimum norm interpolation in HD (R) with respect to the uniform mesh kπ
b , k ∈ Z, is
solved theoretically by the projection theorem. Consider the closed linear subspace
kπ
D
Nb = f ∈ HD (R) : f ( ) = 0, k ∈ Z ,
b
and its orthogonal complement (NbD )⊥ which is the closure of the linear subspace generated
by shifted generating functions d(· − k kπ
b ), k ∈ Z :
(NbD )⊥
kπ
= lin d(· − k ) : k ∈ Z
b
−
.
There is a unique orthogonal projector TbD with range (NbD )⊥ and kernel NbD . The projection
theorem states that TbD (f ) is the unique function of minimum norm in the linear manifold
f + NbD .
P ROPOSITION 2.1 (Minimum norm interpolation).
Given f ∈ HD (R), its projection TbD (f ) on (NbD )⊥ is the unique minimum norm interpolant
of f ∈ HD (R) , i. e., it minimizes the norm kgkD among all functions g ∈ HD (R) satisfying
kπ
g( kπ
b ) = f ( b ), k ∈ Z.
The minimum norm interpolant of the Sinc function S(0, h) is denoted by
S D (0, h) = TbD (S(0, h)) =: gbD .
The generalized Sinc function S D (0, h) is given by the Fourier integral
Z
D(t) ixt
π
1
e dt = gbD (x) , h = .
S D (0, h)(x) =
2b R D2b (t)
b
It was shown in [6] that for any f ∈ HD (R) the associated minimum norm interpolant
TbD (f ) possesses a generalized cardinal series C D (f, h) :
X
π
TbD (f )(x) =
f (kh)gbD (x − kh) = C D (f, h)(x), h = .
b
k∈Z
Note that for D(t) = χ[−b,b] (t) we obtain C D (f, h) = C(f, h). For deriving error estimates
an integral representation of the minimum norm interpolant TbD (f ) is of importance.
P ROPOSITION 2.2. The generalized cardinal series C D (f, h) possesses the integral
representation
Z
π
F2b (t)
D(t)eixt dt = C D (f, h)(x),
h= .
(2.7)
TbD (f )(x) =
D
(t)
b
2b
R
Proof. Put
CnD (f, h)(x) =
n
X
k=−n
f (kh)gbD (x − kh),
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MINIMUM NORM INTERPOLATION
n
F2b
(t) =
n
1 X
π
f (kh)e−ikht , h = .
2b
b
k=−n
Since
Z
CnD (f, h)(x) =
R
n
F2b
(t)
D(t)eixt dt,
D2b (t)
we can conclude
D
Tb (f )(x) − CnD (f, h)(x) ≤
≤
Z
n
|F2b (t) − F2b
(t)|
R
Z
=
≤
|F2b (t) −
D(t)
dt
D2b (t)
2
n
F2b
(t)|
R
Z
|F2b (t) −
[−b,b]
1
D(b)
Z
D(t)
dt
D2b (t)2
2
n
F2b
(t)|
12 Z
D(t)dt
R
! 12 Z
21
1
D(t)dt
dt
D2b (t)
R
! 21 Z
1
2
2
[−b,b]
21
n
|F2b (t) − F2b
(t)| dt
D(t)dt
.
R
This shows
C D (f, h)(x) = lim CnD (f, h)(x) = TbD (f )(x).
n→∞
3. Approximation by minimum norm interpolation. We start with approximation by
functions of exponential type which are defined as Fourier integrals
Z
Sb (f )(x) =
χ[−b,b] (t)F (t)eixt dt.
R
P ROPOSITION 3.1. For any f ∈ HD (R) the error estimate

|f (x) − Sb (f )(x)| ≤ 2b
holds.
Proof . Since
|f (x) − Sb (f )(x)| ≤
Z
=
ZR
≤
Z
X
k≥1
 12
Z
D(kb)
1 − χ[−b,b] (t)
R
1 − χ[−b,b] (t) |F (t)| dt
1 − χ[−b,b] (t)
R
|F (t)|
1
D(t) 2
D(t)
1
12
Z
R
|F (t)|2
1 − χ[−b,b] (t)
dt
D(t)
and
Z
R
1 − χ[−b,b] (t) D(t)dt = 2
Z
dt
! 21
D(t) 2 dt
1 − χ[−b,b] (t) D(t)dt
R
|F (t)|2
b
∞
D(t)dt ≤ 2b
∞
X
k=1
D(bk),
! 12
,
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the proof is complete.
P ROPOSITION 3.2. Let f ∈ HD (R) and define h(x) = Sb (f )(x). Then the error
estimate
 21

X
h(x) − TbD (h)(x) ≤ 2 2b
D(kb)
k≥1
Z
|F (t)|2
χ[−b,b] (t)
dt
D(t)
R
! 12
holds.
Proof. Note that
h(x) =
Z
H(t)eixt dt, H(t) = χ[−b,b] (t)F (t).
R
Moreover, we have
H2b (t)χ[−b,b] (t) = χ[−b,b] (t)F (t)
for almost all t. Then we can conclude
Z χ[−b,b] (t)F (t) − H2b (t) D(t) dt
D
(t)
2b
R
Z
H2b (t)
≤
χ[−b,b] (t) F (t) −
D(t) dt
D
(t)
2b
R
Z
H2b (t)
D(t) dt
+
1 − χ[−b,b] (t) D
(t)
2b
R
Z
D2b (t) − D(t)
|F (t)| dt
=
χ[−b,b] (t)
D2b (t)
R
Z
Z H2b (t) H2b (t) +
D(t)dt − χ[−b,b] (t) D2b (t) D(t)dt
R
R D2b (t)
Z
D2b (t) − D(t)
=
χ[−b,b] (t)
|F (t)| dt
D2b (t)
R
Z
Z
D(t)
+
χ[−b,b] (t) |H2b (t)| dt − χ[−b,b] (t)
|H2b (t)| dt
D
2b (t)
R
R
Z
D2b (t) − D(t)
=
χ[−b,b] (t)
|F (t)| dt
D2b (t)
R
Z
Z
D(t)
|F (t)| dt
χ[−b,b] (t) |F (t)| dt − χ[−b,b] (t)
+
D2b (t)
R
R
Z
|F (t)|
= 2 χ[−b,b] (t) (D2b (t) − D(t))
dt.
D2b (t)
R
h(x) − T D (h)(x) ≤
b
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MINIMUM NORM INTERPOLATION
This shows
Z
h(x) − TbD (h)(x) ≤ 2
≤2
Z
≤2
Z

χ[−b,b] (s) |D2b (t) − D(t)|
R
χ[−b,b] (s) |D2b (t) − D(t)| dt
R
≤ 2 2b
X
k≥1
 21
D(kb)
Z
21
12
|F (t)|2
χ[−b,b] (s)
dt
D2b (t)
R
! 21
Z
|F (t)|2
χ[−b,b] (s)
dt
D(t)
R
1
χ[−b,b] (s) |D2b (t) − D(t)|
dt
D
(t)
2b
R
2
|F (t)|
dt
D2b (t)
2
|F (t)|
χ[−b,b] (t)
dt
D(t)
R
Z
! 12
! 21
which completes the proof.
Our main result is the error estimate for minimum norm interpolation in the associated
harmonic Hilbert space.
T HEOREM 3.3. Let f ∈ HD (R). Then the estimate
 12

X
f (x) − TbD (f )(x) ≤ 4 b
D(kb)
k≥1
Z
2
R
|F (t)|
dt
D(t)
! 21
holds.
Proof . We have

|f (x) − Sb (f )(x)| ≤ 2b
X
k≥1
 21
D(kb)
Z
R
|F (t)|2
1 − χ[−b,b] (t)
dt
D(t)

 21
X
Sb (f )(x) − T D (Sb (f ))(x) ≤ 2 2b
D(kb)
b
k≥1
Z
2
! 12
|F (t)|
χ[−b,b] (t)
dt
D(t)
R
,
! 12
Moreover,
D
Tb (g)(x) ≤
Z
R
|G2b (t)|
D(t)dt =
D2b (t)
Z
χ[−b,b] (t) |G2b (t)| dt ≤
R
Z
|G(t)| dt.
R
.
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Then we obtain
f (x) − TbD (f )
≤ |f (x) − Sb (f )(x)| + Sb (f )(x) − TbD (Sb (f ))(x) + TbD (Sb (f ))(x) − TbD (f )(x)
Z
1 − χ[−b,b] (t) |F (t)| dt + Sb (f )(x) − TbD (Sb (f ))(x)
≤2
R

≤ 2 2b

≤ 4 b
X
k≥1
X
k≥1
 12 
Z
D(kb) 
 21
Z
D(kb)
R
|F (t)|2
1 − χ[−b,b] (t)
dt
D(t)
R
|F (t)|2
dt
D(t)
! 21
! 12
Z
+
2
|F (t)|
χ[−b,b] (t)
dt
D(t)
R
! 12 
.
We consider two examples.
C OROLLARY 3.4. Let f ∈ HD (R) and D(t) = 1 + t2r
f (x) − T D (f )(x) ≤ 4b−r+1/2
b
holds.
2r
2r − 1
21 Z
−1
, r ≥ 1. Then the estimate
|F (t)|
2
1+t
R
2r
dt
12
Proof: We have
X
D(kb) ≤ D(b) +
k≥1
Z
∞
1
1 + (bt)
2r
−1
dt ≤ b
−2r
+
Z
∞
(bt)
−2r
dt = b−2r
1
2r
.
2r − 1
C OROLLARY 3.5. Let f ∈ HD (R) and D(t) = e−α|t| , α > 0. Then the estimate
f (x) − TbD (f )(x) ≤ 4e−αb/2
αb + 1
α
12 Z
2 α|t|
|F (t)| e
R
dt
21
holds.
Proof: We have
X
k≥1
D(kb) ≤ e−αb +
Z
∞
1
1
αb + 1 −αb
e−αbt dt = e−αb 1 +
e
.
=
αb
αb
4. Improved error estimates. We recall that the error estimate is based on investigating
Z
|f (x) − Sb (f )(x)| ≤
1 − χ[−b,b] (t) |F (t)| dt,
R
D
Tb (f )(x) − TbD (Sb (f ))(x) ≤
Sb (f )(x) − TbD (Sb (f ))(x) ≤ 2
Z
Z
R
1 − χ[−b,b] (t) |F (t)| dt,
χ[−b,b] (t) (D2b (t) − D(t))
R
|F (t)|
dt.
D2b (t)

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Following the idea of Golomb [8, 3, 4] for the periodic case, we assume that the function
Z
f (x) =
F (t)eixt dt
R
satisfies the smoothness condition
Z
|F (t)| /D(t)dt < ∞.
R
Then we can conclude
|f (x) − Sb (f )(x)| ≤ D(b)
Z
R
1 − χ[−b,b] (t) |F (t)| /D(t)dt ,
D
Tb (f )(x) − TbD (Sb (f ))(x) ≤ D(b)
Z
R
1 − χ[−b,b] (t) |F (t)| /D(t)dt ,
∞
X
Sb (f )(x) − TbD (Sb (f ))(x) ≤ 2
D(kb)
k=1
Z
χ[−b,b] (t) |F (t)| /D(t)dt.
R
This implies
P ROPOSITION 4.1. Assume that f satisfies the smoothness condition
Z
|F (t)| /D(t)dt < ∞.
R
Then the error estimate
∞
X
f (x) − TbD (f )(x) ≤ 2
D(kb)
k=1
holds.
Z
|F (t)| /D(t)dt
R
We consider two examples.
−1
C OROLLARY 4.2. Let D(t) = 1 + t2r
, r ≥ 1 and assume that f satisfies
Z
|F (t)| 1 + t2r dt < ∞.
R
Then the estimate
f (x) − T D (f )(x) ≤ b−2r
b
holds.
4r
2r − 1
Z
R
|F (t)| 1 + t2r dt
Proof . We have
X
k≥1
D(kb) ≤ b−2r
2r
.
2r − 1
43
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C OROLLARY 4.3. Let D(t) = e−α|t| , α > 0, and assume that f satisfies
Z
|F (t)| eα|t| dt < ∞.
R
Then the estimate
holds.
Z
f (x) − TbD (f )(x) ≤ e−αb 2 + 2
|F (t)| eα|t| dt < ∞
αb
R
Proof . We have
X
k≥1
D(kb) ≤
αb + 1 −αb
e
.
αb
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