attenuation factors in practical fourier analysis

Numer. Math. t8, 373-400 (t972) �9 by Springer-Verlag 1972

Attenuation Factors in Practical Fourier Analysis*

WALTER GAUTSCHI

Depar tmen t of Computer Sciences, Purdue Universi ty , Lafayette , I n d i a n a 47907

Received May 26, 1971

S u m m a r y . Given a 2 ~t-periodic funct ion [, i t is desired to approximate its n - th Four ier coefficient c n (/) in terms of funct ion values [~ a t N equidis tant abscisses

x g = p 2 ~ / N , p = O , 1 . . . . . N - - t .

A t ime-honored procedure consists in in terpola t ing / a t these points by some 2 7t- periodic funct ion q0 and approx imat ing c n (]) by c n (9 ) . I n a n u m b e r of cases, where ~0 is piecewise polynomial , i t has been known tha t %(~0)-----7:n~ n (l), where ~n (]) is the t rapezoidal rule approximat ion of c n (f) and v n is independen t of [. Our interes t is in the factors r n, called a t t enua t ion factors. We first clarify the condit ions on the approximat ion process P : / - + ~ under which such a t t enua t ion factors arise. I t tu rns ou t tha t a necessary and sufficient condi t ion is l inear i ty and t rans la t ion invar iance of P . The la t te r means t ha t shif t ing the periodic da ta ]----{] } one place to the r ight /g . . has the effect of shift ing ~0 = P [ by the same amount . An exphcl t formula for T n is obta ined for a n y process P which is l inear and t rans la t ion invar ian t . For interpola- t ion processes i t suffices to ob ta in a factorizat ion cn(q~ ) = o~(n)v?f(n), where to does no t depend on ] and ~! (n) has period N. This also implies existence of a t t enua t ion factors T n, which are expressible in terms of ~o. The results can be extended in two directions: First , the process P m a y also approximate successive derivat ive values ]h~_ (K) x = 0, t , h - - 1, of the funct ion ~, in which case formulas of the type c n (~0) ---- . . . ,

~=~0xn,~ ~n(](~}) emerge. Secondly,r_1 P m a y be t rans la t ion inva r i an t over r subintervals ,

r > i , in which case cn(q~ ) = ~. vn, e c n + e g l , ( ] ) " All results are i l lustrated b y a n u m b e r

of examples, in which ~0 are polynomial and nonpolynomia l spline interpolants , including deficient splines, as well as other piecewise polynomial interpolants . These include approx iman t s of low and med ium con t inu i ty classes permi t t ing arbi t rar i ly high degree of approximat ion.

1. Introduction

The p r o b l e m of p rac t ica l F o u r i e r ana lys i s o f t en p resen t s i tself in the fol lowing form. I t is des i red to ca lcu la te the Fo u r i e r coeff icients

2 n

0

of a 2 ~t-periodic r ea l -va lued f u n c t i o n ] which is k n o w n , or can be ca lcula ted , on a set of discrete po in t s

2~t (t .2) x~ = p --if-, p = 0, 1 . . . . . N - - t .

* This work was carried ou t while the au thor was Visi t ing Professor a t the Mathemat ica l Ins t i tu te of the Technical Univers i ty of Munich, Germany. The work was supported in pa r t by a Fu lb r igh t research grant.

2 6 N u m e r . M a t h . , B d . t 8

374 W. Gautschi :

Derivative values, in addition to function values, may also be available at these points.

If no information is known about / , other than the values

(1.3) 1~ =/(x~,), /z = 0 , t . . . . . N - - 1 ,

a reasonable approximation to c. (/) is given by

N - - 1 de/ I - - " 0.4) 22 l,,e ' " " .

/.L=O

This may be justified by noting that for each integer s ~ 0, with 2s + 1 --_ N, the trigonometric polynomial of order s,

0.5) = y. n = - - $

formed with the coefficients (1.4), approximates / best among all polynomials of

(7 ')' the same type in the sense of the discrete L2-norm H ~ = Ig(,,./I �9 if 0

2s +1 =N, then (t.5) in fact represents the unique interpolation polynomial belonging to the data (t.3). A more practical consideration in support of c~ is the fact that sums such as those in (1.4) can be calculated on binary digital computers very efficiently by the algorithm of Cooley and Tukey [4], now com- monly known as the "fast Fourier transform" [5, 9]. I t may also be noted that the approximations ~,, share with the exact Fourier coefficients c~ the symmetry property

(t.6) 3 . = ~ , all n.

In the presence of additional information about the function /, however, the choice (1.4) may fall short in reflecting essential properties of Fourier coefficients. If it is known, e.g., that / has an (r--l)-st derivative which is absolutely continuous on the real line, then c, = o (n- ' ) as n-->oo. The approximations 3~, on the other hand, have period N,

(1.7) c.+N=c'., all n,

and are thus unable to simulate the asymptotic behavior of c~.

For this reason one often attempts to approximate / by some ~0 which shares with / some of its smoothness properties, and then takes c. (9) to approximate e. ([). In many cases where q~ interpolates / at the points x~, and is made up of polynomial pieces over the subintervals (xv, xv+z), it has been found that

(t.8) c. (q0) = it. ~,, (/), all n,

where z. are certain universal factors not depending o n / . These are referred to as attenuation/actors. By choosing q0 judiciously one can arrange these factors to go to zero at a rate comparable to that of the c. (/). The fact that a relation of the type (t .8) still permits use of the fast Fourier transform is another attractive feature of (t .8).

Attenuation Factors in Practical Fourier Analysis 375

An instance of (~.8), as pointed out by Yugkov [19~, was already discovered in 1898 by Onmoff [t2]. He considers the broken line interpolant T and states (without proof) that (t.8) is valid with

(t.9) ~n = [ sin (r*n/N) ]

The same result was rediscovered by D~tllenbach [6] in 192t, based on calculations of H. Weyl, and again much later by Chao [3]. D~llenbacb also considers a cubic interpolation scheme and determines the associated attenuation factors. Further instances of (t .8) are discussed by Eagle [7].

In the terminology of spline functions the broken line interpolant is a polynomial spline interpolation function of degree t. I t is interesting to note that Runge [t4, p. 193ff.] already in t904 constructed periodic spline interpolants of degree 2 (without, of course, naming them as such) in connection with Fourier analysis, although he did not obtain the respective attenuation factors. Spline interpolants of higher degrees were used systematically by Eagle [7] in 1928, and ten years later, independently, by Quade and Collatz [13]. Eagle gives a remark- ably elegant derivation of the attenuation factors in the case of splines (which he calls " la th functions"). A more lengthy derivation is given by Quade and Collatz, whose major concern, however, are the approximation-theoretic aspects of the problem. These are also discussed more recently by Ehlich [8] and Golomb [10]. Further short derivations of attenuation factors can be found in Bauer and Stetter [2], who also consider the generalization to Fourier transforms.

In choosing 9 it does not suffice, of course, to simulate the smoothness properties of 1. One also wants local accuracy, i.e., 9 should approximate / as closely as possible on each subinterval (x~,, x,+l). In this respect, the splines have the (perhaps undesirable) feature of correlating accuracy and smoothness: increasing the degree of the spline also increases its degree of smoothness. I t seems worthwhile to make available a repertoire of other approximants 9 which are capable of fitting / to high accuracy and yet have low, or only moderate, degrees of smoothness. We will partially meet this need in Section 5 where examples are provided of piecewise polynomial approximants 9 whose polynomial degrees are 2r--t, and whose degrees of smoothness are t (Example 5.2) and r (Example 5.3). We shall also give an example of a nonpolynomial approximant 9, viz., a generalized spline function belonging to a linear differential operator with constant coefficients (Example 5.4). In all cases, the respective attenuation factors will be specified explicitly.

In view of the multitude of possible attenuation factors the question naturally arises as to the precise conditions on the approximation process P : /--~9 in order to yield a formula of the type (1.8). We shall give a complete answer to this question in Section 3, Theorems 3.t and 3.2. We show, in essence, that for (t.8) to hold, it is necessary and sufficient that P be linear and translation invariant over an interval of length h = 2 ~[N. The latter means that by shifting the periodic data /={1 ,} one place to the right, the (periodic) function ~0 =P/ i s shifted by the same amount. An explicit formula for the attenuation factor ~, is also obtained for any linear process P which is translation invariant in this sense.

26*

376 W. Gautschi:

These results extend readily to Hermite-type approximation processes (in- volving any number of successive derivative values), where formulas of the type

(1. to) = Y s

emerge. This is again illustrated in Section 5 (Example 5.6), where q~ is taken to be a periodic spline interpolant of degree 2 r--1 and deficiency k (in a terminology of Ahlberg et al. [1]). As special cases this includes ordinary splines (k----t) and Hermite interpolation polynomials (k----r).

For interpolation processes it will also be shown (Theorem 3.3) that for (t.8) to hold, it suffices to find a factorization

(t .tt) c.@)

where ~o! (n) is an arbitrary N-periodic function, and co (n) is independent o f / . The attenuation factor ~, can then be expressed in terms of co. This result is often more convenient for deriving attenuation factors than the explicit formula provided in Theorem 3.1.

Besides (t.8), formulas of a somewhat different character exist in the literature. For example, if N is even, and / is interpolated by quadratic polynomials over panels of two consecutive subintervals, then Yu~kov [t9] has shown that

(t.12) all n.

There are now two attenuation factors associated with this interpolation process. Using cubic interpolation over three consecutive intervals gives rise to three attenuation factors (Yu~kov [20]). We prove in section 4 the considerably more general result that a formula of the type

r - - 1

0.13) = Z Q ~ 0

is valid precisely when the process P : /-->~ is linear and translation invariant over r subintervals. This is once more illustrated in Section 5 (Example 5.5), where the formulas of Yu~kov (and more general formulas) are rederived in a particularly transparent manner.

Section 2 contains some auxiliary results concerning the functions g~(z)=

~, [z/O, +z)] k+l, which will find use in the subsequent sections.

2. Mathematical Preliminaries

In the following sections some properties of the functions

(2.t) o~(z) :~_o0~-~+---[) ; k =0 , t , 2 . . . . ; z~=0 (mod t)

will be needed. (If k----0, the summation is to be understood in the sense of a principal value.) The functions (2.1) have been introduced and studied by Ehlich [8], who also gives explicit expressions for t ~ k ~ t l and numerical tables. For k --0, as is well known,

(2.2) go (z) = Z~z cot Z~z.


Proposition 2.1.

Z t ~+~ (z) = ~k (z) - ~ 4 - ( ~ (z),

Prool. F r o m

one gets

k = 0 , t , 2, . . . .

Z--r Crk(Z) = ~, (~+Z)--Ik+ll

d [ z - l k + ~ ) a k C z ) ] = _ ( k + t ) ~ (~+z)-Ik+~> dz

= - (k + t)z-lk+s~ ~k+l (z).

Solving for ak+l, and carrying out the differentiation, gives the desired result.

Proposition 2.2. / ~ , ~k+a , ,

ak(z) = ~ ) qk-l~cos ~zl , k = t, 2 . . . . .

where qk-1 (t) is a polynomial o/degree k - - t . In /ac t ,

- - t s , . qo(t) : -1 , qk(t) = t q k _ l ( t ) + ~+-~qk_l(t} , k = l , 2, 3 . . . . .

so that qh (t) is even [odd] i ] k is even [odd].

Proo]. By Proposi t ion 2.t , and (2.2),

( "~ ~=( "" 1' al (z) -----nz cot n z - - z ~ cot ~z sin2 nz] \ sin n z / "

The assertion is thus t rue for k = t, wi th qo(t)= t . Proceeding by induction, assume tha t the proposit ion holds for some k. Then, by Proposi t ion 2A, af ter some e lementary computa t ion ,

ak+l (z) = ak (z) k + 1 ak(z)

_ ( ,~/ ,+,~cos~z q,_l(COS~,)+ t-cos. . , . } -- \sln ~ Z l ( - k - ~ qk_l(COS ~z) ,

f rom which the assert ion follows.

Proposition 2.3. For z real and not an integer the matrix

[aS(") O's+l (Z) ... O'S+p__l (*) ]

/~ .3 / ~,.,(z~=]?.+~.(~/.. 7:+7(71. ..:.. ~.,+~.~1. .

L(Ts+p-I(2) as+p(g) ... O's+sp_z(g )

is positive definite i / s is an odd integer >= t and p is any integer > t. In particular,

(2.4) det H,,p(z) > 0 , z 4 : 0 ( m o d i ) , s(odd) > t , p-->__t.

378 W. Gautschi :

Proo] 1. Let x r ---- [~1, $~ . . . . , ~p]. Then

p

i ,j=l P ~. [ Z \s--l+i+j - -

i ,j=l

( z ~ S + l ~ , ~ i ~ Z t i - l p . I Z \~--1

--~=__ookY-~g] i : l $ i k V ~ - g ] [ $

The last expression is nonnegative, if s is odd, and can only vanish if the poly- p

nomial ~0(t) = ~. ~ t ~-1 vanishes a t all points z/(v +z) , v = 0 , 4-1, 4-2 . . . . . i.e., i f ~O (t) ~ 0- i = 1

Although not needed in the sequel, the following result is offered because of possible independent interest.

Proposition 2.4. The sequence {ak+ x (z)}k~176 0 has the generating funct ion

~, ~z sin ~ z y (2.4) ak+l (z)Yk - - sin (r~z(l -- y)) sin ~z" ~=0 Y

Proof . By (2A),

= , oo ~ k ~v+z] =,=_oo V-#-~

,=-oo ~ + z - - z y

I z y

~--bz

-y v + z + v + z - - z y "

Therefore, using (2.2),

k =O O'k+l (z) y = ~ - [cot (:~z (1 - - y ) ) - - cot fez/,

which is the same as (2.4).

3. Single Attenuation Factors

We denote by N the set of integers, N = { 0 , • 4-2 . . . . }, and by R the set of reals. The set of periodic da ta will be denoted by

F = [{/,,}m~2V: /mER, /m+N = / m , all m E N 1 .

Each m e m b e r of F m a y be thought of as the sequence of function values/m = / (Xm), X m = m 2 ~ / N , m -----0, i l , 4-2 . . . . , for some 2 ~-pefiodic func t ion / (x ) . Evident ly ,

t The author is indebted to Professor W. B. Gragg for suggesting the idea for the proof.


F is an N-dimensional linear vector space under the usual definitions of addit ion and scalar mult ipl icat ion of sequences. As basis we take eo, e 1 . . . . . eN_l, where the m- th component of % is given by

140 if m=p(modN), (3.t) (%)m = otherwise.

E a c h / E F then has the representat ion

N--1

(3.2) 1 = Y, l.e , l={l=)eF. /J=0

Besides F, we consider the linear space of 2 re-periodic continuous real-valued functions, which we denote b y ~.. Our approximat ion process is then thought of as an opera tor P : F - ->#- from F into ~ . I f 9 ~ P f satisfies 9 ( x v ) = f u , ju ---- 0, 1 . . . . . N - - 4, we call P an interpolation operator.

~re define the shift opera tor E in F as usual by (El), , = [~+a , all mEN, and in o*- b y (Eg)(x) = ~ 0 ( x + h ) , all xER, where h=2~/N.

Definition. An opera tor P : F--->#- is called translation invariant if

(3.3) P(E/)=E(PJ), all fEF.

Clearly, (3.3) implies the more general ident i ty

(3.4) P(E~/)=E~(P]), all 2 ~ N .

Also note from (3.1) t ha t

(3.5) el, =E-Ueo, ~u = 0 , 4, 2, . . . , N - 1.

Theorem 3.1. Let P: F-->o~ be a linear operator from F into ~ and 9 =P[. I] P is translation invariant, then

(3.6) c . ( 9 ) = r . ~ n ( [ ) , all hEN, all /cF,

where

(3.2) z~ ---- Nc~ (~0) , ~o =Peo.

Proo/. L e t / - - { / ~ } be an a rb i t r a ry element of F. F rom (3.2) we get b y the l ineari ty of P, and using (3.4), (3.5),

N--I N--I

~o = P ] = Z l~,Pe~,= ~, [vP(E-"eo) ~ = 0 /z=O

N--1

= Y 1,, (Peo), # = 0

t ha t is, the following representa t ion of ~0 as a discrete convolution,

N--1 = X

380 W. Gautschi:

Therefore,

fL~

1 f 9(x)e_in~d x c. (9) = - ~ 0

2n N--X t ~ i v y ~o(X_xu)e_i,Xd x

2 ~ 0 0

lUt

N--I f t N -- n ,~=ol~ e-~'~" - ~ ~o(X--x,)e-~"c"-~, I dx

0

= ~. (/). Ne~(~o),

since 70, as an alement of ~,, is 2z~-periodic. This proves (3.6), (3.7).

Theorem 3.t admits the following converse.

Theorem 3.2. Suppose each element o/ ~o< ~" has an unilormly convergent Fourier series. Let P: Ig--~ ~o be an operator such that

(3.8) c.(Pt)=~,~.(t), all n~X, all IcF. Then the operator P is necessarily linear and translation invariant.

Proof. By assumption,

(x) = (P/) (x) = ~ c.(9) e'"" = ~ 3. ~.(/)e'"', n ~ - - o o n ~ - - o o

from which the linearity of P is evident. Moreover,

P(E/)(x)= F, ~,~.(E/le'"" n ~ - - O 0

= ~ ~.e'"~e.(/)e'"'= ~, ~.~.(l)e'"~'+" n ~ - - 0 0 n ~ - - O 0

= [E (Pl) l (x),

i.e., P is translation invariant.

Both theorems can readily be extended to include approximation processes which involve not only function values, but also any number of successive derivative values.

The data space F then consists of elements / which are k-tuples of N-periodic sequences,

r / : l (39, / = / ( ' /

F has thus dimension kN. Defining #- to be the linear space of 2=-periodic ( k - 1)-times continuously differentiable functions, an interpolation process P is an operator from Iv into #" such that 9 = P ] satisfies 9(")(x~)=/(~1, n- -0 , t , . . . , k - - L g = 0 , t . . . . . N - - t .

Attenuation Factors in Practical Fourier Analysis 38t

The definition of translation invariance of an operator P : F - + # - remains as before, if the shift operator E in F is understood to act simultaneously on all sequences {/~} in (3-9).

Theorem 3.t now extends as follows. I / the operator P: F - + ~ is linear and translation invariant, and cp = P/ , then

k--1

(3.1o) c.(~0) = Y, ~.,. c'~ (1~'~), S = 0

where

all hEN,

z.,s = N c. (*7o,s) ,

(3.tl) p 0 ~o,o----P , ~ o , x = P . . . . . ~o,k-1 = �9

The proof is virtually the same as before. Also the analogue of Theorem 3.2 is immediate.

Eq. (%7), in principle, can be used to calculate the attenuation factors for any linear process P which is translation invariant. In simple cases this approach has already been taken by Eagle [71- In more complicated situations the theorem which follows may be more convenient.

For the purpose of this theorem we define #" to be the linear space of 2 n- periodic continuous functions ~0 whose Fourier series converge at each x~,

(3.t2) ~o(xA= c.(~o)e '~'~, ~ = o , t . . . . . N - - I .

(The summation in (3.12) and in similar formulas in the sequel is always to be understood in the sense of a principal value.)

If 9 E ~ , then by (t.4), (3.t2), for every hEN,

= 1 N - I

/ Since for any p EN,

1 N-1 1t if p = 0 ( m o d N ) (3.13) ~ - ~ e ~p*~= ~

~ o tu if p4=o(modN) ,

it follows, as is well known, that eo

(3.14) ~.(~o)= ~ C.N+.(Cp), all hEN, all ~ E ~ .

Theorem 3.3. Let P: I ~ - - ~ be an arbitrary interpolation operator, and cp = P / . Then/or

(3.15) c . ( 9 ) = ~ . 3 . ( / ) , all n r all / r

to hold with some ~ not depending on / it is necessary and su/ficient that two/unctions co (n), ~pt(n) exist having the/ollowing properties:

382 W. Gautschi:

(i) o~ (n) is defined for all n E N \ N o and is independent o / / , where N o is either the empty set, or a subset o] N with the property that n E N o implies n + v N CNo, all

(ii) ~l(n + N ) =~0l(n ), all n~N, al l /EF;

(iio) ~t(n) --0, all h E N o, all ]EF;

(iii) c~(9) =~o(n)~t(n), all h E N \ N o , all /EF.

I] 09 (n), ~! (n) with the stated properties exist, then in/act

~o (n) all n CN o (mod N) ~, (3.t6)

where the series in the denominator converges and has sum 4: O. Moreover,

T , = I , all n E No , (3.t6o)

r ~ + , ~ = 0 , all h E N o, vEN\{O}.

Remarks. t. The function ~o (n) in (iii) is not uniquely determined, but only up to an arbitrary N-periodic (nonvanishing) factor. Such a factor, of course, does not affect the value of Tn in (3.t6).

2. If (3-t 5) holds for an interpolation process P which carries / ~ t into ~0 ----- 1, then (3A6o) necessarily holds with N o ----{0}. To see this, simply apply (3.t 5) with 9 ~ 1, ] ~ t , and observe that for this choice

c n = if n 4 : 0 , c n = if n 4 = 0 ( m o d N ) .

3- If (3.15) holds for an interpolation process P which preserves symmetry about n, i.e., for which

(3.17) / , = / N - , implies ~ ( x ) : ~ ( 2 n - - x ) ,

then z~ is necessarily real-valued. This follows from the fact that both 3, (/) and c, (9) are real for a n y / , 9 : P / satisfying (3.17). All examples considered later will meet the condition in (3.17). I t is not difficult, however, to construct interpolation processes which violate (3.17). For example, if the restriction of ~0 to [x,, x~+l], /z=O, 1 . . . . . N - - I , is taken to be the quadratic polynomial interpolating / at x~, x~+~, x~+ 2, then symmetry is clearly destroyed. Accordingly, one finds in this case that

N--1

'(r ') z = d l ~ # _ 1 e p

and o9 (n) is indeed complex-valued.

Proo/ o] Theorem 3.8. (a) Sufficiency. Assume that oJ(n), ~0t(n ) with the properties (i)-(iii) exist. According to Off) we have that

=,o(n),eAn), 9; = P / ,

2 If N o is empty, the quantifier is to be interpreted as "all n e N", otherwise "all n 4= no (mod N)" where n o is an arbitrary integer in N o.


holds for all ] EF and all n E N \ N o. Consequently, by (3. t 4), noting that 3. (9) = 3. (/) and using (ii), we get

~,,(/)---- ~, c,,v+,,(9)= ~, coCvN+n)yJt(vN+n) • = - - o o v ~ - - o o

(3.t8) o o

= ~. [oJ(vN+n)yJl(n)], all nCNo(modN), all /EF, v = - - o o

where the last series converges (possibly trivially, if y~l(n)=0).

We show that in fact

(3.t9) ~ o)(vN +n)

converges and has a nonzero sum. For this it suffices to exhibit an ] Eti' for which c~ (/) ~ O. Because then, ~0t(n ) ~ 0 for this particular ], by virtue of (3.18), and the assertion follows, again from (3.18). The choice ]EF, however, such that

]o=1, / . = 0 for / ~ = t , 2 . . . . . N - - t ,

will do, since then 3~ (/) ---- 1/N ~= O.

Convergence of (3A9) being assured, we now conclude from (3A8) that

~ . ( / ) = +n) ~vl(n ), all nCNo(modN), all /EF.

Multiplying through by o~ (n), we get

~(/)co(n)-= I .~oow(vN +n)]o~(n)~,(n)

= [~=~_oo~~ +n)lc~(9),

proving (3.t 5) for n r N o (mod N), with ~ as defined in (3.16). Assuming now nENo, we have by (iio) that ~vt(n ) =0 , all /EF, and by (ii)

that ~vi(n +vN ) =0 , all ]EF, yEN. Since by assumption n +vNCN o for v4=0, it follows from (iii) that c~+.2v(9)----0, all /OF, n~Xo, v~X\(0} . This proves (3.15) for integers of the form n+vN, nEN o, v4=O, with Tn+.N =0 . From this, and (3A4), noting again that 3~(9)=c~(/), it follows further that 3~(])=c~(9), nENo, proving (3.t5) for nEN o, with T . = t .

(b) The necessity of (i)-(iii) is trivial, since we may take o~ (n) = z., ~v! (n) ---- 3. (]), and N o the empty set. Theorem 3-3 is proved.

4. Several Attenuation Factors

The concept of translation invariance introduced in the previous section will be generalized as follows.

Definition. An operator P: F - ~ is called r-translation invariant (r an integer), if

(4.t) P(E'I)=E'(P/), all tEF.

384 W. Gautschi:

Translation invariance in the previous sense thus coincides with t-translation invariance. I t implies r-translation invariance for each integer r. On the other hand, P may be r-translation invariant, for some fixed r > t, without being t-translation invariant. Note that (4.t) implies

(4.2) P(Ea ' / )=EX ' (P] ) , all 2EN.

Theorem 4.1. Let r ~ t be an integer and N be divisible by r,

(4.3) N =rq .

Let P: F - - ~ be a linear operator/tom F into ~,, and ~ = P]. I] P is r-translation invariant, then

r--1

(4.4) c.(~v) = Z r.,Qc.+Qq(t), all hEN, all IEF, 0 = 0

where

(4.5) r--1

N ~. e'l.+Qq~x, c.(~~ rio = Peo. OmO

Remarks. t. If P is t-translation invariant, and thus also r-translation invariant for any r ~ t, then (4.4) reduces to (3.6). In fact, observing that

r i o = P e o = E - " P e o, rlo(x ) =rio (x --xo), and thus

t e.(ri~) = -~ f rio(x--xo)e-'"~dx =e-'"~"c.(rio),

o we find from (4.5),

N ,-1 { : ' ( r io ) if Q=O ~",~ ----7 c"(ri~ z~ eieq'~= ~ if O < Q = < r - - l .

2. The result (4.4) may be given a slightly different form by breaking up the r - 1 [r/n] r - 1

sum on the fight of (4.4) into two pieces, according to ~ = Y. + Y. , and o~0 o = o o = [#2]+1

introducing the new variable of summation e ' = p - r in the second piece. Using (t.7), one gets

It~hi

~,,(~) = ,=t~,~+ i,,,o~,,+o,(l), (4.4')

where

(4.5') A =/r . ,Q if ~___0

Tn'e t~,Q+, if o < O .

Pro# o/ Theorem 4.1. Using the representation (3.2) we have for each /EF,

N--1 r--1 q--1 / = Y / . e , = Z Z/ ,+, .e ,+~.

~ 0 Q~O ~ 0

r--I q--1

= Z Z t#+~ . , .E -~I ' e# �9


By linearity of P, and (4.2), r - - I q--I

~ : P / = Z Z / e + a , P ( E - a ' % ) Q ~ 0 J , = 0

r--1 q--1

= Z Z Ie+ .E- ' (P~e) , e=O J.=O

i.e.,

Therefore,

Letting

r - i q - - I

(~1 = Z Z re+ ~.~o (~ - ~,)-

c. I r:x - a ]

r--1 - - I = y

r--1 q--I

= r co e ' . , ,

q--1

d e = d e , . = E l~+a,e -~""+~', e = 0 , 1, . . . , r - - l ,

we thus have r--1

(4.6) c, (q~) = Z c,, ( ~ Q ) e ' ' d e.

Now observe that 1 r--1

~.(/) = ~ Y.ae.

More generally, r--1 q--1

t (4.7) ~.+o,(t) = ~ - Y, Z / , + - e-'("+~

e~O ~.=0

Since

a = 0 , t . . . . , r - - t .

(n + a q ) %+ a, =nxe+a, +aq(o + A t ) 2n rq

=nxe+a, + a q x o + a 2 . 2~ ,

Eq. (4.7) simplifies to r - - i q-- I

N Y.,.a g Z .t e-inxp+xt l o + a v

i.e., to

(4.8) r--1

N 2.+~r (/) = Z e-'~e2"/'de, e z 0

a ~ 0 , 1, . . . , r - - t .

We have obtained a system of r linear equations in r "unknowns" de. The coefficient matrix is the Vandermonde matrix

V ( I , e -~ ' i l r , e -2"2~'qr . . . . , e -(r-t)~=~l~)

386 W. Gautschi :

in the r-th roots of unity. Since the inverse of V is readily found to be

_ I eae2ail, V - l = [ v ~ , v ~ Q - ,

one obtains from (4.8),

and thus from (4.6),

N ~*-1 - - ~ eaO2'tilr ~ H~ d~ -- y ~ .+QqvJ,

0 = 0

r - - 1

o = 0

r - - I r - - I N - �9

- ; , c.(,~o)e . . . . * e"~='/'c.+~(l) a ~ 0 0 = 0

r - l / N r-1 einX*'+~e2ni/r )

This proves (4.4) and (4.5).

The following converse of Theorem 4A is proved in the same manner as Theorem 3.2.

Theorem 4.2. Suppose each element of ~ o ( ~ has an unilormly convergent Fourier series. Let P: F-+ ~o be an operator such that

r o l

(4.9) c~(P/)= X *..o~.+,~(1). atl nEX. all I~F. #=0

Then the operator P is necessarily linear and r-translation invariant.

It is clear that in analogy to the discussion in Section 3 we could also treat Hermite approximation processes which are r-translation invariant. However, we shall not pursue this here any further.

5. Examples

We begin by recalling a well-known representation for Fourier coefficients. Suppose g is a 2 ~r-periodic function which, together with all derivatives of orders up to and including the (k +1)-st , is piecewise continuous. Let ~ denote the points of discontinuity of gCS) in the half-open interval [0, 2z t), and 8g~) the respective jumps

(5.t) ~oc ' l~ l im Egcsl (~,1 + x) _gr (~1 _ x)] . t ' ~ a X ~ 0

Then, for n 4: 0,

(5.2) ~

, f ~gt, k)e- ~ + (in)k+1 + (in)k+1 Z i~r162 gCk+tl(x)e-i,Xdx. ff

0

The proof follows directly from a repeated application of integration by parts.


Eq. (5.2), applied to g = 9, will be useful in deriving the factorization in (iii) of Theorem 3-3.

We note that the validity of (5.2) is not restricted to integer values of n. The result holds for arbitrary real n, provided the " jumps" at ~0----0 are defined by

(5.3) ($g(o0 d~f lim e-~ ~i~ g(~) $ o [g(~) (x) - - ( - - x ) ] ,

regardless of whether g(') is continuous at 0 or not.

5.1. Single Attenuation Factors

For completeness we include as first example the well-known case where 9 = P / i s a polynomial spline function (Eagle [7], Quade and Collatz [t3], Bauer and Stetter [2], Ehlich [8], Golomb [10]).

Example 5.1. Given/EF, let 9 = P~ denote the periodic spline interpolant of degree 2r --1, i.e.,

9~C~'-~(-- oo, oo),

9(x+2~r)----9(x), all x6R, (S .4)

9(x,,) = l . , ~ = o , t . . . . . N - - t , 9 (2') (x) = o , x 4= x . .

I t is known that 9 exists uniquely.

We first illustrate the use of Theorem 3.3. Applying (5.2) to 9, with k = 2r - -2 , we get

2~t 1

2~rc'~(9)-- (in)"-' f 9"'-" 0

N - - 1 xla+t 1 = ,,.,..-. Y 9 , . . - . , (x .+ �89 e-,-,dx, . . o ,

p = 0 x .

9 {~'-1) being piecewise constant. Thus,

with c. (9) = o (n)~t (n), n ~ 0,

t o ( n ) - n " ' ~I(n) = - -

, u = 0

Clearly, ~! (0) = 0, all /EF, and the relation

x~+x

in . f e-inXd x ~ e -inx~" --e -inx~'§

shows that ~/(n) has period N. All conditions (i)-(iii) No ={0}, being verified, we conclude that

N - - 1 x~+t

(-t/'2,~ in ~, 9 ( " - ' ) ( x . + ~ - XN

of Theorem 3.3, with

c.(9)=%,cn(/), all nEN, all /EF,

388 W. Gautschi:

with 3 o = t , r,N----O for v~=0, and

t T~ = for n 4:=0 (mod N).

n*" ~ (v N + n)-:"

In terms of the funefions a k (z) defined in (2.t), we can write more briefly

(5-5) "rn-- aet_x(z) ' z - - N ' n=[=O (modN).

The s~me result can also be derived from Theorem 3.13. Let, in fact,

~(u)=[ sin(u~2) 1", ~, u[2 ~ (U) = ~v (u + 2 ~ ~).

Schoenberg [t 6] has shown that o o

1 f ~(u) e~.,du (5.6) L , ( t ) = ~ - d ~(u) - - C O

is a cardinal spline interpolant of degree 2 r - - t , i.e., of continuity class C*'-z(--o% co), a polynomial of degree 2 r - l on each open interval (m, m + 1), m fiN, and satisfying the unit interpolation conditions

{t0 if m = 0 L,(m) = if m4=0, mEN.

I t follows from this that

so that (3.7) gives g ~ - - O 0

N f e_inZd x T. = - ~ ~o ( x) o

0

- - h .-~-o~ L, +~N e-in"dx

oo (x+XI. 'Y oo

-~- . ~ L'(t)e-in'tdt = f L,(t) e-in''dt"

Inverting the Fourier transform in (5.6), on the other hand, shows that

~b(u) -- f L'(t)e-i 'tdt' ~ 0 0

so that 1F (n h) ~ (2 ~zZ) 1 t

- - ~ - - co t z \ ~ r - - a 2 T _ i ( z ) '

in agreement with (5.5). 3 The derivation which follows is due to Dr. Christian H. Reinsch.


Example 5.2. Let ]EF, and p(x) =pu(x ) be the polynomial of degree 2 r - - t interpolating ] at the 2 r points xu+ a, 2 = -- r + t, - - r + 2 . . . . , r - - 1, r. Define qJ(x) = p v ( x ) on [xu, X v + l ] , / z = 0 , -4-1, ~ 2 . . . . .

Clearly, 9EC(--oo, co). All derivatives 9(s) s = 1 , 2 . . . . . 2 r - - t , are continuous on each open interval (x,, xv+l), but will have jumps at the points xu. Since q~I~") (x) ---- 0, a.e., we have from (5.2), applied to %

2r--1 N--I (5.7) 2~cn(q~) Z t (in)*+z Z O (s) -i.x~, = ~u e , n # : O .

s = l p = 0

We proceed to calculate the jumps 8q~ s). Let 2~

P ( l ) = p ~ ( x ~ + l h ) , h - - N "

By Newton 's interpolation formula we have

,1(,+:, I P(O = Z k=O

Therefore,

(5.80) = Z

s = t , 2, ..., 2 r - - 1 . 2r--1 /

h=, k ],=~ t"- '+: '

We obtain h s ~ 9~ ) by subtract ing from (5.8o) the relation (5.81) with p replaced b y / t - - l ,

, I , l _ A % . _ . + I - - A k l , - . �9 hag.--~=st\ k ]t=o \ k / t=l

The sum on the right, however, "collapses", since

/t=o k ]t=l Ak/~-"

= ( , + , - , y , , _ I ( , + , - , y , , - t ,,, .

/t=o k - - 1 ]t=o zl~/"-''

As a result,

(5.9) ' (s)=(t+r-- l l ( ' ) h dgu \ 2 r - - I ]t=o A2"/"-'' s = t , 2 . . . . . 2 r - - t .

Since

2 r - - 1 } - - (2r--t)l [t2--(r--t)2] "'" [ t~-- t ] t

is an odd function, all derivatives of even order at t = 0 vanish in (5.9). Let t ing

(t + r - - l l ( " - l ' Y " = ( - - t ) ' + ' \ 2 r - - 1 ]t=o ' s = t , 2 , . . . , r ,

~t7 Numer. Math,, Bd. t8

390 W. Gautschi :

we thus obtain f rom (5.7) r N - - 1 2~Cn(~)=(__,l_)rhs,~=l Yrs X (AS'/ ~e-'"" (nh)2s , , . _ . , n 4=0. -- I t = 0

This is the desired factorization c~ (9) -----w (n)~pt(n), with

s=l ( hn)2s ' n + 0 ,

( _ _ l ) r h N - - 1

~/(n)- 2~ ~ (A"'/, - ,):~"""

I t is indeed evident tha t ~Pt (n) has period N. Moreover, since with ] also all differences AS/have period N, one has with g~, =AS'-x[~,_. t ha t

N--1 N--I

Z AS'/.-. = Z Ag.=gu--go=O, /~=O /~=O

i.e., ~vl(O ) =O. Theorem 3.3 is thus applicable with N O ={O}, giving

~(~)=r~(/), all ~ X , with

and

(5.~o)

T O : I , "gv N = 0 for

all /EF,

v # 0 ,

Y X Y,s (2 •z) , , - ss

s = l

Tn= ~ rrs(2~zl ir-zsa~'s-X(Z) , Z - - N ' n=~O (modN) . S = I

A short table of the coefficients Y,s follows.

Table 5.t. The coeJficients Yrs in (5.10)

, 4

2 t i6 1 3 ~/3o ~/4 t 4 t/t40 7/120 1/3 I

(5.1ol)

(5.1os)

(5.1o8)

(5.to4)

Using Proposi t ion 2.2 on obtains from (5.10), af ter some computa t ion ,

( s i n ~ z / ~ [ , q - - ( ~ z ) ' + 8 (~Z)'] (r-~--3),

"r,~=\ nz / t+T(=z)Z+T5(~z)4+~(~zz )6 ( r = 4 ) .

Attenuation Factors in Practical Fourier Analysis 39t

The first of these, (5.t01), of course, agrees with the attenuation factor for spline interpolants of degree one; the second, for r = 2, is due to D~illenbach [6].

Comparing (5.101)-(5.t04) with the corresponding attenuation factors for splines,

( s in ~ z / 2 , Tn= \ ~ 1 qar_z(COSnZ)

[cf. (5-5) and Proposition 2.2], one notes that the polynomial factors in the brackets of (5.t01)-(5.104) are precisely the beginning terms of the power series expansion of t/q2,-2 (cos :~ z). Thus, for small z =n]N, the interpolation polynomials of Example 5.2 give only slightly different Fourier coefficients compared to the spline interpolants of Example 5.1. For large z, however, the attenuation factors show different behavior, as they should, the two approximants belonging to different continuity classes.

Example 5.3. Let ~ denote the central difference operator, <Syu =yu+�89 - -Yu-t , : , ~ k - 1 + ~2k-1 �9 and ~2k-lyv �89 y~_~ yv+�89 the mean odd differences. There are

unique finite difference expressions

r--1 r--1 a ~ ' (5.11) (L2sY),,,= ~, sk Y~,, (L2,-xY).= ~, b,,k~'~k-aY~,

k = s k = s

such that

(5.12) h~ = (LoY),, e = 0 , t . . . . . 2 r - - 2

is valid for any polynomial y of degree --<2r--2. The coefficients a,k and b~k in (5.11) indeed [1t, p. t36] are the coefficients in the power series expansions

(5.t3) where

[ 1 g ] is--1 r z ] 2 s oo [2 s i n h - ~-1 [2 sinhq-~ -] = Za.kzZh. - - / , b.k.z ~k-1

k = , VI + z* /4 k = , "

Given ]EF, let now p (x ) :p , ( x ) be the unique polynomial of degree 2 r - - t satisfying

1,.pC.) (x.) = ( L . / ) . . h sP C') ( x . + d = (L. 1 ) . + 1 . s = 0. t . . . . . r - - t .

where h =2n/N, and let 9(x) =pv(x) on [x~, x~+x], p = 0 , + t , + 2 . . . . . Clearly, 9 E C ' -z (-- o% oo), so that 9 has now a degree of smoothness which is about midway between those in Examples 5A and 5.2.

Letting P (t) =p, (x~, + th), the well-known formula for Hermite interpolation gives

r--1 r--1 P(t) = E h,( t)(L,/) ,+ E ( - - t )*h,( / - - t ) (Lsl) t ,+l ,

s ~ O s ~ O

t p(1_t),'-s-1/r+a--t)_ ~--o ~ t~ s : O , l . . . . . r - - l . h, (t) = ~ a

Similarly as in the previous example, we may now calculate the jumps ~ ~0~ ~ for r ~< ~ =< 2 r - 1, and then use (5.2) to find a factorization for c~(9). We omit the somewhat lengthy calculations and content ourselves in stating the final result.

27*

392 W. Gautschi:

One finds that

where

(5.t4)

and

(5.15)

Here,

(5.16)

(5.t7)

where

(5.t8)

c.(,p) =o(n)~An) , n 4=0,

N--1 v,~(n) = Z t .~- '"~"

p=O

,-1 Adz) ._L " ~ Bq(z) n O9(")= Ze=[ L~] (2~z)'e+x " e ~ ] ( 2 z ~ z ) " + ( ' ' Z-- N"

[(r--1)/2] Aq(z) =2( - -1 ) e sin 2z~z ~, h(~2so)(1)~(sin ~z)

s =0 IriS] }

+ cos ~ Z Eh(,~(l) cos 2 ~ - h ~ ' s ~ ( o ) ? tL(sin ~ ) ,

[ [(r-- 1)/2] Bq(z)=Z(--t)e+I I s~=o [h~e+l)(o) --,.,s~('q+l)(l) cos 2ztz]~(sin ~z)

[rill } + cos ~tz sin 2 zz ~ h~Q_ +1) (t) fls (sin zz) ,

v--1 r--1 ~s( z) --- Z ( - - t ) k a , k(2z) "k, fls(z) = Z (--l)kbsk(Zz) ~-1 .

k ~ S k m s

It is evident that ~vt(n ) =N~. ( / ) in (5A4) has period N. Also, Ae(z ) and Bq(z ) both have a factor sin~z~z. This follows for A e from the identity

he(t) = hi(t) - ~ ( t - t ) + 1 - t , , which implies

h~,e) (t) =h(?e)(t) - h ~ ~) (o), all q > o, and for B e from

which implies ho(O =1 --hoO -t),~

h~+l)(0)----h~e+s)(t), all 0=>0.

The common factor sinZ(z~n/N) (which is N-periodic and vanishes at n =0) can be transferred from o9 (n) to ~pt(n), with the result that Theorem 3.3 becomes applicable with N o ={0}. The first few attenuation factors, which follow from (3.t6) and (5AS)-(5A8) are listed below:

__ (sin ~tz ]' (r = t ) , (5.t91) ~n--~ ~tz /

(sin z~z ]' (5.t9~) 7. = ~ / [3 - - 2z t z cot ztz]

(5.t98)

(r = 2),

"r,----(sinztz]e~tz } [25--7(~z) ' - -24~zcotz~z] ( r = 3 ) ,

/ s in ~ z \ s r _ 734 (~z),__/_ 116 a) ] (5.19,) "r, = [ ~ ) [ 6 2 3 - - - ~ - ~ 0 2 2 ~ z - - ~ - - ( : g z ) c o t ztz (r =4) .

4 This is most quickly seen by checking that the function on the right-hand side has the same interpolatory properties as ho(t), viz., h~ s) (0)=~so, h(oSlO)=0, s = O , l , . . . , r - - t .


The first of these is again the attenuation factor for broken line approximants; the second is due to Eagle [71.

Interestingly enough, the expressions in brackets in (5.191_,) have power series expansions whose initial terms are precisely those given in the brackets of (5.101_,).

Example 5.4. Given I~F, we now take for ~0 a generalized periodic spline interpotant corresponding to the linear differential operator

(5.20) L = D " + a l D ' - I + ... +a , , D = d / d x ,

where % are real constants. This means that

~ O ' - ~ ( - oo, oo),

~0(x+2~t)=~0(x), all xER , (5.21)

~ ( x . ) = l . , ~ = o , t . . . . . N - - ~ ,

(Z*Lg)(x) = 0 , x # x , ,

where L* denotes the formal adjoint of L,

(5.20*) L* = ( - - I ) 'D" + ( - -1) ' -1a l D ' - I + ..- + a , .

The existence of such a spline interpolant is assured if L has the property that

L y = O , y (xu)=O ~ = 0 , t . . . . . N) implies y = 0

(Ahlberg el a/. [t], p. 199). This is the case, e.g., if N =>r and if we assume that the characteristic polynomial of L,

0t(t) = t ' + a ~ t ' -1 + .-. + a , ,

has distinct zeros tQ, ~ = 1, 2 . . . . , r, such that

(5.22) t o - - to4=O(modiN ) for Q # a .

The fundamental solution set y o = e x p (tex), 0 = 1 , 2 . . . . . r, of L y = O is then indeed unisolvent on any interval. We also assume that none of the nonvanishing zeros t o is an integer multiple of i. This implies that

(5.23) ~(t) = ( - - t ) ' ~ (t)~(--t) = t ~" + l l t 2"-~ + ... + l , ,

which (apart from the sign) is the characteristic polynomial of L ' L , does not vanish at an integer multiple of i, except possibly at zero.

Applying now (5.2) to 99 (*Q), Q = 0 , t . . . . . r - - t , we get for n:~O,

N--1 $~ t ( 2 r - l ) -i~x. - - t f 2~Cn(~(2~ ( in )" - 'o ~' ~ " e " • d 9(2")(x)e-inXdx"

.u=O 0

Q = 0 , t , . . . , r - - t .

Using the last relation in (5.21) and the fact that ~ (t) in (5.23) is the characteristic polynomial of L ' L , we can write

t (5.24) c" (qal=~ - - ( i n ) " - ' Q ('Pl (n) - - r") ' q = 0 , t . . . . . r - - l ,

394 W. Gautschi:

where

(5.25) N--1

1 ~t( n )= ~ z~ ~ (,,-11 -.,,~,. O ~% e , l a = o

9"n r--1 1

(5.26) y . = E,. ( , ) + . . . = y , 0 Q = 0

Multiplying (5.24) by I,_Q, ~ = 0 , t . . . . . r - - t , and adding up the results, we get

I>: , , . ] Y~= (in), ,- ,o (~~176 Lq=O

i.e., since by assumption 2 (in) q: O,

r~-= [1 - ( i ~ ) , , 1 a( in) ] Wfl n)"

Now (5.24), with ~ =0 , gives the desired relation

(5.27) c,(~) =ro(n)yt(n ) , n:4:0,

with t

(5 .28) o (n) - a ( i ,0 "

Clearly, ~o/(n) in (5.25) has period N. If 2(0) 4=0, Theorem 3.3 applies with the empty set for N o. (Eq. (5.27) then also holds for n =0 , as can be concluded from (5.27') below by letting n-~O.) Whether or not this case has any practical merits is questionable, since the interpolation process P in this case does not reproduce the f u n c t i o n / ~ t, since L * L 9 =O has no nontrivial constants among its solutions.

If, on the other hand, 2(O)=0, then y l (O)=0, all /~F, as we now proceed to show. We have noted earlier that the result (5.2) holds for arbitrary real n, if one observes the definition (5.3). The preceding derivation, therefore, can be carried through under this more general assumption on n, giving in place of (5.27)

t l - - e -2,"n 2 ( in )Fo_ l,_~ +y)l(n) n(real) :~0, (5.27') c . ( 9 ) = Z(in) 2zcin = '

where

0 t 9~(2e+1) 1 9j(~r_2) ~ = ~ ( ~ ) ( ) + T g ( 0 ) + . . . + (in)*'- '~-~ (0), e = 0 , 1 . . . . . r - - l ,

and the definition (5.25) of ~vl(n ) is to be adapted in accordance with (5.3). Since for ~-----0, t, . . . , r - - l ,

1,_~F~=l,_~(in)~Fo+O(n), as n-->0, we see that

r--1

~(i , , )Fo- Y, l ,_o~ Q=O

- - t ( in)Fo- [t (/n) - (r ~o + o (n) =(in)z 'Fo+O(n)=O(n) , as n -+0 ,

proving indeed that ~v1(n ) -+0 as n-+0, all tEF. Thus, Theorem 3.3 applies with Xo ={o}.


(5.29)

with

(5.30)

In summary, then,

c.(9)----~.~.(/), all hEN, all /EF ,

1 T n l oo ,~(in)

Z ~(i(vN+.)) v = - - c o

for all hEN, if 2(0) 4=0, and for all n 4:0 (rood N), if .~(0) = 0 . In the latter case. 3o = t , and z. u ---- O, v 4= O.

For the "splines in tension", considered by Schweikert [t7], we have L=D(D--a) , and thus L*L=D4--~2D ~, i.e., 9(x)=exe*~+c~e-~+e3+c~x on each subinterval (x,, xu+a). In this case, 2( in)=n4+~n ~.

5.2. Several Attenuation Factors The following example generalizes constructions and results due to Yugkov [20].

Example 6.6. Let r --> 2 be an integer, and assume N divisible by r, say, N :rq . Letting p (x) = p , (x) be the unique polynomial of degree --< r satisfying

(5.31) p(x~,+,)=l~,+s, s = o , t . . . . . r, we define 9(x) = p , ( x ) on [x, , , x(~+l), ] ,/~ = 0 , l . . . . . q - - t .

The interpolation process 9 = P/of Example 5.5 is clearly linear and r-translation invariant. Hence, by Theorem 4.1,

r--1

(5.32) e . ( 9 ) = Z z. ,~.+~q(]) , all hEN, all /EF . e--O

In order to calculate the attenuation factors ~.,~, we denote by

o = o , , . . . . . r z,.~ = ~ 8 t-Q 04=0

the fundamental Lagrange interpolation polynomials belonging to the set of abscissas {0, 1 . . . . . r}. Then, with h----2 ~/N,

[1,,o(t), x=th , for O<=t<=r, I

,~o(x) =~l,,o(--t), x~-2n+th , for --r<_t<_O, lo for x. ~ x ~ x(q_l) ,,

so that 2 ~ , :

0

1/ : } _ h l, o(t)e-inhtdt + l, o(--t)e-~'q~'~+th)dt - - r

- - 2~h f l,,o(t) [e_inht +einhtldt, 0

396 W. Gautschi:

i.e., f

hf c.(~o ) = ~ l,,o(t ) cos nht dt.

h / l , , . ( t ) e_ i .h td t ' 0 < a = < r - - t . c, 07~ = ~-~ 0

Therefore, applying (4.5),

{/ / } 1 E giOa2~]r --" -- (5.33) T.,~ = - r 2 l,,o(t ) co snh t+ l,,~(t)e ,.hit *)dr �9 a = l 0

Observing t ha t

(5.34) l,.o(t) = z . . . . . ( r - t ) , ~ = o , t . . . . . r, we can t ransform the sum in (5.33) as follows,

r - - 1 r

Y. e ~0~"/' f z, ,o(t)e -~"hc'-~ at 0 = 1 0

= Z e~Ql'-~2"/" f l, . . . . (t)e-~"hC'-'+~ a = l 0

�9 --I r

= Y e -~~ f l .... a(r--#e-i"~C-'+aldT a = l 0

r - - 1

= ~' e-i~~ f l,,o(t) elnh(t-*)dt. ~ 1 0

This shows tha t the sum in (5.33) is real, so tha t (5.33) simplifies to {/ 1; } t 2 l,,o(t ) cos nht + ~, 1.,o(t) cos [nh(l--a) --Oa2~/r]dt z.,o= ~-

~ 0

Again using (5.34), we can fur ther write f

T,,o= ~, lf,o(t) COS [nh(t--a) --ea2z~/r]dt,

or, since symmet r i c te rms (with indices a and r - a) are equal,

(5.35) v., o - ; /,,~(t) cos [nh(t--a)--Oa2~/r]dt if r is odd, 0

~ . . ~ = 7 (_~)o l,.,l~(t )cos [nh(~--r/2)ldt 0

(5.359 {,/2)-i " } + 2 ~ ] l , ~ cos [nh( t - -a) - -ea2~/r]d t if r i s even .

O

Similarly, for 0 < a ~ r - - t, = ~ / , , , ( t ) , x = t h , for O<~t~r,

~o(X) /o for x,<_x<=2~, giving


For r = 2 , e.g., one obtains from (5.35') by an elementary computation,

---_ (sin 0zzla[cos ( :~z+o~]2) + :zz sin (z~z+oz~/2)] ( r = 2 ; ~ = 0 , 1 ) ,

where, as before, z ~-nJN. Similarly, (5.35) for r = 3 gives

, [ ~,,,e = -- 36(~z)~ 2 cos 6:~z--9 cos (4~z --Q2n]3)

+ 18 cos ( 2 z ~ z + o 2 z ~ / 3 ) - t t [

t [sin 6 ~ z - - 4 sin (4~zz sin (2~z +02z~/3)] (5.3%) + ~ - e2=/3) + 5

1 [cos 6 z~z -- 3 cos (4 ~zz -- 0 2 ~z/3) + 24(=zp k

+ 3 c o s ( 2 z ~ z + 9 2 n / 3 ) - - t ] ( r = 3 ; 0 = 0 , 1 , 2 ) .

Both results (5.352) and (5-353), in a somewhat different form, were already obtained by Yugkov [20].

The case r = N ~ 8 is considered by Salzer [15 ] who has numerical tables for 0 ~<n ~24 .

5.3. Attenuation Factors Associated with Derivatives

Example 5.6. Let k and r be integers with I <_k<_r. Define ~ as follows:

~0~C~'- l -h( - ~ , o~),

( x + 2 ~ ) = ~ 0 ( x ) , all xER,

(5.36) 91~)(x~)-----/~ ~), ~-----0,1 . . . . . k - - l ; t t : 0 , 1 , . . . , N - - t ,

91~,1 (x) = 0, x 4= x~.

In the terminology of spline functions, ~0 is a periodic spline interpolant of degree 2 r - - t and deficiency k. I t exists and is uniquely determined (Ahlberg et al. [t, pp. i67-168]). The special case k ~ t gives ordinary splines, considered in Example 5.1. If k ~ r , we are dealing with Hermite interpolation of order r - t on each subinterval Ix,, xu+t].

Applying (5.2) to 9 (*), s =0 , t . . . . . k - - l , we get for n~=0,

k--1 N--1

(5.37) c , ( ~ c , ~ ) - ~ ~, 1 (in)ir+l+~c--s--/~ E ~ {2r+x--~)e--inx~ - - ~ v _ , s = 0 , ] , . . . , k - - l .

For n =~ 0 (mod N) one computes

. . . . [i(vN+n)]k+a t -- (in)l,+1 ek(z), z = ~ , k -~0, t , 2 . . . . .

Moreover, by construction,

~(~r s = o , t . . . . . k - - t .

398 W. Gautschi :

1 d~-- 2~t

we can write

Applying (3.t4) to 9('), we thus obtain

~.(1%= ~, c,N+,,(~ ~sl) v~--oo

(5.38) k--1 I ~ -~ a2r+~-s-~(z) N-I -- ~ % e , 2~'~__o (2r+x--k) --inx. (in)tit+l+ . . . . /~ Z p~0

Defining N--1 t

(in)2v+l+"--k

s = o , 1 . . . . . k - - t .

~%, o , z = _ ,

k--1 (5.39) c,,(9) = Z d,,,

~ = 0

by virtue of (5.37) with s = 0 . On the other hand, (5.38) represents a system of k linear equations in the k unknowns d,,

k--1 (5.40) ~, (in)'a=,+ . . . . k(z)d,, =3,(1(')), s = 0 , 1 . . . . . k - - t .

Inverting this system, and substituting the result in (5-39) gives c, (~0) as a linear combination of the cn (/(s)),

k--1 Tn, S

(5.4t) c,,(~) :s~=o~)~c,,(f(s)), n 4=0 (mod N).

The coefficient matrix of the system (5.40) is given by

A = [i L "-k+' ' "-'1 'r. 0(77-1 . ? ' ? .: :::/ (in')k-lJ La2r-2k+l az,-2~+2 ... a2,-~J

Except for the ordering of the rows, the second factor of A is identical with the matrix Hs,p of Proposition 2.3, if we define s = 2 r - - 2 k + t and p = k . Since s ~ l is odd, and p >_ t, it follows from (2.4) that det A 4= 0 for n 4= 0 (mod N). One checks easily that the attenuation factors V~,s in (5.41), s-----0, t, . . . , k - - t , are just the column sums of H -1 ~,-~k+l,~(z), taken in the order from right to left.

A slightly more explicit expression for v,,s may be obtained from Proposi- tion 2.2. In fact, if we let

q2,-k-1 q2,-~ ..- q2,-2 ]

Q(g)=lq2r?k?2 ~2:--k.:. ":'. q,2r_3. [ Q-I(z ) k--1 - , = [~ ] . . . . o ,

Lq2,-~k q2,-2k+l ... q2,-~-x-I

where qs =qs (cos zz) , then an elementary calculation yields

(5.42) = k ~ l (sin zrz/2,+~_k_s+l "~,,,s ~ _ ~ 0 ~ / %s (cos zz), s = 0 , t . . . . . k - - t .


For r = k = 2 , e.g., one finds

3s in~z ( sin nz) T,, o -- (nz) 3 cos ~tz ~z '

(5.43) 3 1:2 , sin z~z \ (r = k = 2 ) . ~",~ -- (~z)* ~ cos~ ~z + 3 m cos zrz),

This corresponds to cubic Hermite interpolation on each subinterval. The procedure has already been discussed by Serebrennikov Et8], who expresses the result in the form of an additive correction term to D~illenbach's result, bu t does not note the connection with a t tenuat ion factors.

The following limiting relations are worth noting,

~ , ,0-+1, (in)-XT,,1-~0 as n - + 0 , (5.44)

�9 1 3 "c,,,o--+O, (,n)-~:,,x--~(n~-~-(ivN)-~ as n-+vN, v#:O.

I t can be verified tha t with these limiting values Eq. (5.41) (for r = k = 2 ) also holds true when n = 0 (mod N).

For r = 3, k = 2, Eq. (5.42) gives

45 sin ~z I Zn,o= - - (z~z)S (1 + sin* nz) [qs (cos ~z)

(5.45) 4s [ %~,1 = (~z)4(t _t_sin2~z) q4( e~ ~z)

sin nz z~z q~(cos z~z)],

sin nz nz qa (cos z~z)], ( r = 3 , k = 2 ) ,

where again limit relations similar to those in (5.44) are valid.

For r = k = 3, finally,

,5 [ sinrrZ_cos~z_3(sin~z]~l z ~ , ~ (~z) ~ s i n2 :~z+3 ~z \ - ~ ! / '

(5.46) Z~,x-- (~z)3314sinzrzcos~tz +--~(l +lncos*~z)-- (~z)wsinzrzcosztz] 3 [ z~,~= (nz) ~ t + s i n S ~z + 4 sinz~znZ /sin nz\~l

One checks tha t

�9 2 1

which is in agreement with the relation

c0(~) =c0(l) + ~--6 h~~

obtained by 5-th degree Hermite interpolation.

Acknowledgment. The author is greatly indebted to Dr. Christian H. Reinsch for many valuable discussions which helped clarify and simplify the exposition at several places. In particular, the present version of Theorem 3.1, as well as Theorem 3.:2, are due to Dr. Reinsch. He clearly recognized the role of translation invariance for the existence of attenuation factors, which the author had expressed only implicitly in a preliminary version of Theorem 3.1.

400 W. Gautschi: Attenuation Factors in Practical Fourier Analysis

References

1. Ahlberg, J .H . , Nilson, E .N. , ~Valsh, J. L. : The theory of splines and their application. NewYork-London: Academic Press 1967.

2. Bauer, F .L . , Stetter, H. J.: Zur numerischen Fourier-Transformation. Numer. Math. 1, 208-220 (t959).

3. Chao, F. H. : A new method of practical harmonic analysis [Chinese]. Acta Math. Sinica 6, 433-451 (t956).

4. Cooley, J. w. , Tukey, J. w . : An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19, 297-3ol (t965).

5. -- Lewis, P. A. W., Welch, P. D. : The fast Fourier transform and its applications. I E E E Trans. Education E-12, 27-34 (t969).

6. DRllenbach, W. : Versch~rftes rechnerisches Verfahren der harmonischen Analyse. Arch. Elektrotechnik 10, 277-282 (1921).

7. Eagle, A. : On the relations between the Fourier constants of a periodic function and the coefficients determined by harmonic analysis. Philos. Mag. 5 (7), t 13-132 (1928).

8. Ehlich, H.: Untersuchungen zur numerischen Fourieranalyse. Math. Z. 91, 380-420 (1966).

9. Gentleman, W. M., Sande, G.: Fast Fourier transforms--for fun and profit, t966 Fall Joint Computer Conference, AFIPS Proc., vol. 29. Washington D. C. : Spartan 1966.

10. Golomb, M. : Approximation by periodic spline interpolants on uniform meshes. J. Approximation Theory 1, 26-65 (t968).

11. Hildebrand, F. B. : Introduction to numerical analysis. NewYork: McGraw-Hill 1956.

12. Oumoff, N. : Sur l 'application de la m~thode de Mr. Ludimar Hermann k l 'analyse des courbes p6riodiques. Le Physiologiste Russe 1, 52-64 (1898/99).

13. Quade, W., Collatz, L. : Zur Interpolationstheorie tier reellen periodischen Funk- tionen. Sitzungsber. Preuss. Akad. Wiss. 30, 383-429 (1938).

t4. Runge, C. : Theorie und Praxis tier Reihen. Leipzig: G. J. G6schen'sche Verlags- handlung t 904.

15. Salzer, H. E. : Formulas for calculating Fourier coefficients. J. Math. Phys. 36, 96-98 0957).

t6. Schoenberg, I. J. : On spline interpolation at all integer points of the real axis. Colloquium on the Theory of Approximation of Functions (Cluj, 1967). Mathe- matica (Cluj) 10 (33), t 51-t 70 (1968).

17. Schweikert, D. G. : An interpolation curve using a spline in tension. J. Math. and Phys. 45, 3 t2-3t7 (t966).

18. Serebrennikov, M. G. : A more exact method of harmonic analysis of empirical periodic curves [Russian]. Akad. Nauk SSSR. Prikl. Mat. Meh. 12, 227-232 (1948).

19. Yugkov, P. P. : The practical harmonic analysis of empirical functions when the given curve is replaced by another approximating the given one by tracing [Russian]. Akad. Nauk SSSR. In~. Sbornik 6, 197-210 (1950).

20. -- On the correction of the coefficients obtained in the usual practical harmonic analysis [Russian]. Akad. Nauk SS SR. In~. Sbornik 10, 2t 3-222 (195 ~).

Prof. Dr. Walter Gautschi Dept. of Computer Sciences Purdue University Lafayette, Indiana 47907/USA

attenuation factors in practical fourier analysis

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