is a formula for the approximate
integrals such as
F(o) = f dte^fit),
which retains uniform accuracy even when u is so large that many oscillations of the
integrand occur within a given element oí of the range of integration.
Filon formula  was derived on the assumption that/(f),
rather than the complete
integrand, may be approximated
stepwise by parabolas, so that it may be called a
rule. More sophisticated
'Filon' rules have appeared (e.g. , and
the references quoted in  ), but in fact with fast computers it is more useful to go
in the other direction, towards the least sophisticated integration formula of all.
The ordinary trapezoidal rule gives as an approximation
(Not = T), with weights
Wn = 1, n 5¿ ±JV,
The simplicity of the weights makes this the most desirable formula to use when the
number 2JV 4- 1 of given data values f(nôt) is large; for instance the trapezoidal rule
is invariably used in power spectral analyses . However, formula (2) cannot be
used unless wSt <5C
1, since the whole integrand is supposed to vary linearly over an
element St. But it is an exceedingly simple task to derive a modification to (2) by
assuming only that/(£)
itself varies linearly over the element 5t. The analysis is
similar to that used to derive the Filon-Simpson rule, and will not be given here. The
formula is of the same form as (2), but the weights are now
functions of wot, namely
10-N = ( 1 + iuôt — e"" ' ) /to b% ,
iv„ = ((sin|coS¿)/2^02,
wN = ( 1 — iu8t — e~lu )/oi 5t.
Note that the new weights tend to the trapezoidal-rule
values (3) as wSt —> 0.
case of interest is when the range of integration
2T is infinite.
Suppose we define
FTRAF = St E
as the ordinary
for this case. Then the Filon-
Received August 3, 1966.
E. O. TUCK
values for the Fourier integral F(u) = /"w dt exp (— | t \ + tat)
rule states that an approximation
with uniform validity with respect
to oí is
FFILOn = ((sin^wiO/MO^TRAp.
applied to the results of the crude trapezoidal rule.
For example, suppose fit) = eH" and T = ». Then the exact Fourier transform
= 2/(1 4- co2).
The series (5) resulting from the application of the ordinary trapezoidal rule maybe
summed to give
1 - 2e~Íl cos"5i + e_25')'
whence the Filon-modified result follows immediately
from (6). Fig. 1 shows the
results. In order to exaggerate
ences between the three curves, an absurdly large value U = 1.0 has been used for
the interval of integration,
but even with this coarse subdivision the low-frequency
(co < 1 say) error in the two trapezoidal-rule
is less than 8% . Above
co = 1 the trapezoidal
rule begins to give ever-decreasing
accuracy. Indeed, it is clear
from Eq. (5) that the trapezoidal
to any Fourier integral is periodic
in to, with period 2ír/áí (twice the "Nyquist
) so that the monotone
decreasing nature of the true integral (7) cannot be achieved. On the other hand,
except for a relatively narrow "dropout"
result retains approximately
the same 8% accuracy over the whole frequency range
in the approximation
for relatively high frequencies. The modification can hardly do
harm, and there seems no reason why it could not be employed every time a Fourier
integral is to be computed by the trapezoidal rule. If, as in the example given, the
function f(t) is mathematically
defined, the benefits are immediately
are available no matter how high the frequency. On the other hand, when/( t) is (say)
function of an experimental
record, there are other factors 
which limit consideration to relatively low frequencies; nevertheless it may well be
that if Eq. (6) were used, one could approach a little closer to the Nyquist frequency
than has been customary.
David Taylor Model Basin
Washington, D. C.
1. L. N. G. Filon,
"On a quadrature
formula for trigonometric
Proc Roy. Soc
Edinburgh, v. 49, 1929, pp. 38-47.
2. A. I. van de Vooren
& H. J. Van Linde, "Numerical
of integrals with
strongly oscillating integrand,"
Math. Comp., v. 20, 1966, pp. 232-245.
3. R. B. Blackman
& J. W. Tukey,
The Measurement of Power Spectra, Dover Publica-
tions, New York, 1959. MR 21 * 1684.
* The width of this region decreases markedly if more realistic small values of U are used.