Jökull - 01.12.1973, Page 45

assume that v(x, h) is band-limited and that its
Fourier transform satisfies the condition
V(k, h) = 0 for |k| > kc (39)
Mathematically, this is equivalent to multiply-
ing equations (37) and (38) with a cutoff func-
tion
C(k, k0) = ( 1 !k! ^ k° (40)
V °’ l 0 |k| > k0 v y
resulting in a band-limited restriction of U(k)
Ub(k) = U(k)C(k,k0) =
(41)
2exp(|h|k)V(k, h)C(k, k0)
The expression on the right of (41) can now
be Fourier inverted by expanding the expon-
ential factor in series in k and inverting term
by term. The result is (see Snow, 1923 and
Bodvarsson, 1971, 1973)
tion analogous to the matrix operation B in
equation (33). Moreover, ub(x) given by equa-
tion (42) is a generalized solution in the sense
of (23) of the overnegated and, therefore, under-
determined problem resulting from the band-
limited restriction of equation (34) expressed
in its transformed version by equation (41).
This becomes clear by observing that the
Fourier-components of a given function be-
longing to non-overlapping frequency bands are
orthogonal (Papoulis, 1966). Hence, the splitt-
ing of u(x) into two components
u(x) = ub(x) + uh(x) (43)
where ub(x) is the same band-limited restriction
as above, and uh(x) contains only components
with |k| > k0, leads to the following relation
for the U-norms
lul2 = !ubl2+luhl2 (44)
ub(x):
42<-
n = 0
i)"
h2n 2h r
—— D2n vb (x, h)---------j
(2n)! 7T J
yb(x'’ h)dx'
h2 + (x - x')2
(42)
where vb(x, h) is the band-limited restriction of
v(x, h) to the frequency band |k| íj kQ and D
ls the derivative with respect to x. Since the
derivatives of a bounded band-limited function
are bounded (Arsac, 1966) the infinite series on
the right of (42) will converge for all bounded
vb(x> h). The solution is well defined.
The infinite series in (42) is a clear indica-
tion of the improperly posed character of equa-
tton (34). From the numerical point of view,
series of this type are extremely unpleasant to
handle, in particular, when v(x, h) is of experi-
Hiental origin. In practice, as a matter of course,
°nly a finite number of terms can be handled.
The band-limiting approximation restricting
v(x> h) to vb(x, h) has the consequence that
mformation is lost and equation (34) for u(x)
becomes underdetermined in much the same
sense as equation (8) when m < n. Only the
restriction ub(x) of u(x) can be obtained by
(42). The band-limiting approximation obtain-
ed by multiplying equation (38) by C(k, k0),
arid resulting in equation (41), is an overnega-
The band-limiting multiplication operator
C(k, k0) has a null-space composed only of the
function uh(x). Hence, equadon (44) can be
interpreted in the same sense as the vector-
norm equation (14) above. This indicates the
analogy between ub(x) given by (42) and x0
given by (23).
It is of interst to note that according to the
sampling theorem (Arsac, 1966), a band-limited
function with a maximum frequency k0 is com-
pletely determined by its values at equidistant
points with the spacing 7r/k0. Hence, a periodic
band-limited function is completely determined
by a finite number of equidistant values.We have
above assumed that u(x) is of bounded support,
that is, u(x) = 0 for x outside a finite interval
(0, L). If h « L then for the present purpose
we may extend u(x) outside (0, L) to a periodic
function with a period L. The same applies
to v(x, h). Both functions u(x) and v(x, h) are
then totally determined by n = Lk0/7r equi-
distant values. As a consequence the sampling
theorem and the use of correct interpolation
functions (Arsac, 1966) allow us in principle to
JÖKULL 23. ÁR 43