1
THE STRANG-FIX INTERPOLATION ERROR BOUND IN AN OPERATOR SETTING Luc Knockaert Abstract A generalization of the important Strang-Fix bound in the domain of L2 interpolation and quasi-interpolation is presented. The results presented are obtained by straightforward and well-known operator and algebra norm techniques.
I. INTRODUCTION
I
NTERPOLATION is one of the basic operations in signal processing. A well-known example is the cardinal series representation of band-limited functions with the sinc kernel [1]. The concept of quasi-interpolation was introduced
by Strang and Fix [2] as a theoretical tool for deriving error bounds for the approximation of functions using a general convolution-based interpolation method. Specifically, these authors derived an important error bound for the interpolation of smooth L2 −functions. The Strang-Fix error bound was further studied and refined in [3]-[5]. Although explicit L2 bounds were derived in [4], the mathematical techniques utilized to obtain them were rather lengthy and not prone to generalization. In this letter we obtain a generalization of the Strang-Fix bound by rather simple and well-known operator and algebra norm techniques. The key part is played by Vidav’s theorem [6], which relates the spectral and numerical radii of Hermitian operators. II. ALGEBRA NORMS AND GREEN’S OPERATORS Consider the normed linear space B of complex-valued functions over the real line R with vector norm k · k. An algebra norm [7] (we use the same notation k · k, since in general there can be no confusion) of a linear operator L from B to B satisfies the usual norm requirements, together with the multiplicative condition kL 1 L2 k ≤ kL1 kkL2 k. If kIk = 1, where I is the identity operator, the algebra norm is called unital. For instance the algebra norm induced by the vector norm kLk = sup x6=0
kLxk kxk
(1)
EDICS SPL.SP.2.5. Luc Knockaert is with IMEC-INTEC, St. Pietersnieuwstraat 41, B-9000 Gent, Belgium. Tel: +32 9 264 33 43, Fax: +32 9 264 35 93. E-mail:
[email protected]
2
is unital. The spectral radius of L is defined as ρ(L) = inf kLn k1/n = lim kLn k1/n = n→∞
n≥1
sup λ∈ Sp(L)
|λ|
(2)
In definition (2) Sp(L) is the spectrum of L and k · k is an arbitrary unital algebra norm. The null-space of L, denoted N (L) is defined as N (L) = {x|Lx = 0}
(3)
We suppose that L admits a right-inverse G, also called the Green’s operator [8], obtained via the defining equation LG = I. We have the following Proposition 1: Given an operator L, its Green’s operator G, and any operator P such that the induced algebra norm satisfies kP Gk < ∞. Then N (L) ⊂ N (P ) and kP xk ≤ kP GkkLxk Proof: We have from definition (1) that kP Gyk ≤ kP Gkkyk
(4)
The solution to equation Lx = y is given by x = Gy + x0 where x0 ∈ N (L). Hence kP x − P x0 k ≤ kP GkkLxk
(5)
kP zk ≤ kP GkkLzk
(6)
Taking x − x0 = z we have
Hence Lz = 0 must imply P z = 0 and the proof is complete. When working in a Hilbert space with scalar product h·|·i, the numerical radius of an operator [9] is defined as ¯ ¯ ¯ hLx|xi ¯ ¯ ¯ v(L) = sup ¯ ¯ x6=0 hx|xi
(7)
Hence, the induced algebra norm in Hilbert space is
kLkH = sup x6=0
s
where LA represents the adjoint operator. This leads to
hLx|Lxi = hx|xi
q
v(LA L)
(8)
Proposition 2: kLk2H ≤ kLA kkLk, where k · k is an arbitrary unital algebra norm. Proof: We have kLk2H = v(LA L). Since LA L is self-adjoint or Hermitian, we have by Vidav’s theorem [6], [9] kLk2H = v(LA L) = ρ(LA L) ≤ kLA Lk ≤ kLA kkLk
(9)
It should be noted that a matrix analog of Proposition 2 can be found in [10]. III. STRANG-FIX CONDITION In this section we work in the usual Hilbert space `2 with scalar product hx|yi = p hx|xi. Consider the convolution operator
Lx =
Z
R∞
−∞
x(t)y ∗ (t)dt and norm kxk2 =
∞ −∞
`(t − t0 )x(t0 )dt0
(10)
3
and the interpolation error operator P x = x(t) −
X
x(hk)φ
k∈Z
µ
t −k h
¶
where φ(t) is an interpolational kernel and h is the interpolation step. By means of the Fourier transform Z ∞ e−iωt x(t)dt x ˆ = Fx =
(11)
(12)
−∞
ˆ and Pˆ defined as the operators L and P admit the Fourier domain representations L ˆ ˆ x, FLx = LFx = Lˆ
FP x = Pˆ x ˆ
(13)
ˆ The convolution operator L transforms to the multiplication operator L ˆ x(ω) ˆ x = `(ω)ˆ Lˆ
(14)
and likewise the Green’s operator becomes ˆ x = ϑ(ω)ˆ Gˆ x(ω) =
1 x ˆ(ω) ˆ `(ω)
(15)
By the Poisson summation formula the interpolation error operator transforms into ¶ X µ 2π ˆ ˆ Px ˆ=x ˆ(ω) − φ(hω) x ˆ ω−k h
(16)
k∈Z
We have
Theorem: The interpolation error satisfies the bound ˆ H kLxk2 ≤ kP xk2 ≤ kPˆ Gk where the coefficients Ck are defined as C0
=
Ck
=
Ã
X
Ck
k∈Z
!
kLxk2
ˆ sup |1 − φ(hω)||ϑ(ω)|
(17)
(18)
ω
ˆ sup |φ(hω)||ϑ(ω − 2πk/h)| ω
k ∈ Z0 = Z\{0}
(19)
√ ˆ H. Proof: The scaled Fourier transform F/ 2π being an isometry in `2 , it is easy to show that kP GkH = kPˆ Gk Proposition 1 amounts to say that the bound kP xk2 ≤ kP GkH kLxk2 holds, provided kP GkH < ∞. To find an upper bound for kP GkH we utilize Proposition 2 applied to the unital algebra norm [11] Z ∞ |a(ω, ω 0 )|dω 0 kAk1 = sup ω
(20)
−∞
ˆ is where a(ω, ω 0 ) is the integral kernel of A. The integral kernel of B = Pˆ G ( µ ¶) X 2π 0 0 0 ˆ ˆ b(ω, ω ) = [1 − φ(hω)]δ(ω − ω ) − φ(hω) δ ω−ω −k ϑ(ω 0 ) h
(21)
k∈Z0
The theorem then follows as a consequence of Proposition 2 with the norm k · k 1 in the r.h.s.
ˆ As a simple application of the bound (17), consider the differential operator L defined by `(ω) = (iω)N . It is clear that Ck = hN ck , where c0
=
ck
=
−N ˆ sup |1 − φ(ω)||ω|
(22)
ω
ˆ sup |φ(ω)||ω − 2πk|−N ω
k ∈ Z0
(23)
4
yielding the original Strang-Fix bound kP xk2 ≤ hN c kLxk2 , where c = c was obtained by other means in [5].
P
ck . A similar, but different, bound coefficient
Note that a necessary condition for all ck < ∞ is that φˆ(l) (2πk) = δ0,k δ0,l Of course we also need c =
P
k∈Z
0≤l
