2.3 Trigonometric Interpolation
2.3.4 The Calculation of Fourier Coefficients. Attenuation Factors
2.3.4 The Calculation of Fourier Coefficients. Attenuation
βj= 1 N
f0
2 +f1e−jix1+· · ·+fN−1e−jixN−1+fN 2 e−jixN
approximating the integral (2.3.4.2), so that one might think of using the sums
(2.3.4.3) βj(f) =βj := 1 N
N−1 k=0
fke−jixk
for all integers j = 0, ±1, ±2, . . .as approximate values to the desired Fourier coefficients cj(f). This approach appears attractive, since fast Fourier transforms can be utilized to calculate the quantities βj(f) effi- ciently. However, for large indicesj the valueβj(f) is a very poor approx- imation tocj(f). Indeed, βj+kN =βj holds for all integersk, j, while on the other hand lim|j|→∞cj = 0. [This follows immediately from the con- vergence of the Fourier series (2.3.4.1) for the argumentx = 0.] A closer look also reveals that the asymptotic behavior of the Fourier coefficients cj(f) depends on the degree of differentiability of f:
(2.3.4.4) Theorem. If the2π-periodic functionf has an absolutely con- tinuousr thderivative f(r), then
|cj|=O 1
|j|r+1
.
Proof.Successive integration by parts yields cj = 1
2π
# 2π
0
f(x)e−jixdx
= 1
2πji
# 2π
0
f(x)e−jixdx
=. . .
= 1
2π(ji)r
# 2π
0
f(r)(x)e−jixdx
= 1
2π(ji)r+1
# 2π
0
e−jixdf(r)(x)
in view of the periodicity off. This proves the proposition.
To approximate the Fourier coefficients cj(f) by values which display the right asymptotic behavior, the following approach suggests itself: De- termine for given valuesfk,k= 0,±1,±2,. . ., as simple a functiong∈ K as possible which approximatesf in some sense (e.g., interpolatesf forxk) and share withf some degree of differentiability. The Fourier coefficients
cj(g) ofg are then chosen to approximate the Fourier coefficients cj(f) of the given functionf. In pursuing this idea, it comes as a pleasant surprise that even for quite general methods of approximating the function f by a suitable function g, the Fourier coefficients cj(g) of g can be calculated in a straightforward manner from the coefficients βj(f) in (2.3.4.3). More precisely, there are so-calledattenuation factorsτj,jinteger, which depend only on the choice of the approximation method and not on the particular function valuesfk,k= 0, ±1,. . ., and for which
cj(ϕ) =τjβj(f), j= 0,±1, . . . .
To clarify what we mean by an “approximation method”, we consider
— besides the setK of all absolutely continuous 2π-periodic functions f : IR→IR — the set
IF ={(fk)k∈ZZ|fk∈IR, fk+N =fkfork∈ZZ}, ZZ:={k|kinteger}, of allN-periodic sequences of real numbers
f = (. . . , f−1, f0, f1, . . .).
For convenience, we denote by f both the functionf ∈ K and its corre- sponding sequence (fk)k∈ZZ withfk =f(xk). The meaning off will follow from the context.
Any method of approximation assigns to each sequencef ∈IF a func- tiong=P(f) inK; it can therefore be described by a map
P : IF→ K.
K and IF are real vector spaces with the addition of elements and the multiplication by scalars defined in the usual straightforward fashion. It therefore makes sense to distinguishlinear approximation methodsP. The vector space IF is of finite dimension N, a basis being formed by the se- quences
(2.3.4.5) e(k)= e(k)j
j∈ZZ, k= 0,1, . . . , N−1, where
e(k)j := 1 ifk≡jmodN, 0 otherwise.
In both IF and K we now introduce translation operators E: IF→ IF andE:K → K, respectively, by
(Ef)k:=fk−1 for allk∈ZZ, iff ∈IF,
(Eg)(x) :=g(x−h) for allx∈IR, ifg∈ K,h:= 2π/N =x1. (For convenience, we use the same symbol for both kinds of translation op- erators.) We call an approximation methodP: IF→ Ktranslation invariant if
P(E(f)) =E(P(f))
for allf ∈IF, that is, a “shifted” sequence is approximated by a “shifted”
function.P(E(f)) =E(P(f)) yields P(Ek(f)) = Ek(P(f)), where E2 = E◦E,E3 =E◦E◦E, etc. We can now prove the following theorem by Gautschi and Reinsch [for further details see Gautschi (1972)]:
(2.3.4.6) Theorem. For each approximation method P: IF → K there exist attenuation factors τj,j∈ZZ, for which
(2.3.4.7) cj(P f) =τjβj(f) for all j∈ZZ and arbitraryf ∈IF if and only if the approximation methodP is linear and translation invari- ant.
Proof.Suppose that P is linear and translation invariant. Everyf ∈ IF can be expressed in terms of the basis (2.3.4.5):
f =
N−1 k=0
fke(k)=
N−1 k=0
fkEke(0). Therefore
g:=P f =
N−1
k=0
fkEkP e(0),
by the linearity and the translation invariance ofP. Equivalently, g(x) =
N−1 k=0
fkη0(x−xk),
where η0 :=P e(0) is the function which approximates the sequence e(0). The periodicity ofgyields
cj(P f) =cj(g) =
N−1 k=0
fk 2π
# 2π
0
η0(x−xk)e−jixdx
=
N−1 k=0
fk 2πe−jixk
# 2π
0
η0(x)e−jixdx
=τjβj(f), where
(2.3.4.8) τj:=N cj(η0).
We have thus found expressions for the attenuation factorsτjwhich depend only on the approximation methodP and the numberN of given function
valuesfk for argumentsxk, 0≤xk<2π. This proves the “if” direction of the theorem.
Suppose now that (2.3.4.7) holds for arbitraryf ∈IF. Since all functions inK can be represented by their Fourier series, and in particularP f ∈ K, (2.3.4.7) implies
(2.3.4.9) (P f)(x) = ∞ j=−∞
cj(P f)ejix= ∞ j=−∞
τjβj(f)ejix.
By the definition (2.3.4.3) of βj(f), βj is a linear operator on IF and, in addition,
βj(Ef) = 1 N
N−1 k=0
fk−1e−jixk = 1 Ne−jih
N−1
k=0
fke−jixk=e−jih·βj(f).
Thus (2.3.4.9) yields the linearity and the translation invariance ofP:
(P(E(f)))(x) = ∞ j=−∞
τjβj(f)eji(x−h)= (P f)(x−h) = (E(P(f)))(x).
As a by-product of the above proof, we obtained an explicit formula (2.3.4.8) for the attenuation factors. An alternative way of determining the attenuation factors τj for a given approximation methodP is to evaluate the formula
(2.3.4.10) τj= cj(P f)
βj(f) for a suitablef ∈IF.
Example 1.For a given sequencef∈IF, letg:=P f be the piecewise linear in- terpolation off, that is,gis continuous and linear on each subinterval [xk, xk+1], and satisfies g(xk) = fk for k = 0, ±1,. . . . This function g = P f is clearly absolutely continuous and has period 2π. It is also clear that the approximation methodP is linear and translation invariant. Hence Theorem (2.3.4.6) ensures the existence of attenuation factors. In order to calculate them, we note that for the special sequencef=e(0) of (2.3.4.5)
βj(f) = 1 N, P f(x) =
1−1h|x−xkN| if|x−xkN| ≤h,k= 0,±1,. . .,
0 otherwise,
cj(P f) = 1 2π
# 2π 0
P f(x)e−jixdx= 1 2π
# h
−h
1−|x|
h
e−jixdx.
Utilizing the symmetry properties of the above integrand, we find
cj(P f) = 1 π
# h 0
1−x
h
cosjx dx= 2 j2πhsin2
jh 2
. Withh= 2π/N the formula (2.3.4.10) gives
τj= sinz
z 2
with z:=πj
N, j= 0,±1, . . . .
Example 2.Letg:=P f be the periodic cubic spline function [see Section 2.4]
with g(xk) = fk, k = 0, ±1, . . . . Again, P is linear and translation invari- ant. Using the same technique as in the previous example, we find the following attenuation factors:
τj= sinz
z
4 3
1 + 2 cos2z, wherez:=πj N.