Olver P.J., Shakiban C. Прикладная математика (проект, 2004), Olver P.J., Shakiban C. Applied mathematics (draft, 2004)(T)(1165s).djvu:

Olver P.J., Shakiban C. Прикладная математика (проект, 2004)

Olver P.J., Shakiban C. Applied mathematics (draft, 2004)(T)(1165s).djvu

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Date Dec 31, 2004

Cites: Thus, to prove A2.106), it suffices to show that
A2.107)
The proofs of the two formulae are identical, and so we concentrate on the first...
Here, though, it will be more convenient to take
the basic interval of definition to be [0, 2тг] instead of [ —тг,тг]...
* In computer graphics, the term "aliasing" is used in a much broader sense that covers a
variety of artifacts introduced by discretization — particularly, the jagged appearance of lines
and smooth curves on a digital monitor...
On the
other hand, if the periodic extension of f(x) is discontinuous, then one observes a discrete
version of the Gibbs phenomenon at the points of discontinuity...
The Fast Fourier Transform
While we may appreciate a theoretical algorithm for its intrinsic elegance, in the real
world, the bottom line is always efficiency of implementation: the less total computation,
the faster the processing, and hence the more extensive the range of applications...
Note particularly that the bit reversal map depends
upon the original choice of r = log2 n...
However, this does not significantly alter the final operation count.) There
are r = log2 n different stages, and so we require a total of rn = nlog2n complex addi-
additions/subtractions and the same number of multiplications...
Therefore, by A3.41),
= / wj,k(x)wl,rn(x)dx = ± / wl,rn(x) dx = °-
JO JO
1/12/04
529
© 2003 Peter J...
Up to constant multiple, the only "solution"
of the functional equation A3.54) with bounded support is the delta function 8(x); see
Exercise I for a justification...
Fourier series and their generalizations were originally designed to solve boundary
value problems on bounded intervals...
For an interval of length ?, the frequencies A3.70) required to represent a function are
equally spaced, with
Ak = ku+1 — ku = — ...
Olver
Identifying с = а/2тг leads to the formula for A3.91), and we conclude that the Fourier
transform of A3.90) is
2|fe|...
As in the previous
case, the delta function singularity at к = 0 reflects the lack of decay of the constant
function...
On the other hand, since the odd exponential pulse
has a jump discontinuity of magnitude 2 at x = 0, its derivative contains a delta function,
and is equal to
/» = -ae~a\x\ +2S(x) = - afe(x) + 2 5(x)...
Practical applications rely on tables (or, even better, computer algebra
systems) that recognize a wide variety of transforms for basic functions of importance in ap-
applications...
The remaining integral can be evaluated using a partial fraction decomposition
11
1/12/04 550 © 2003 Peter J...
In fact, according to the last property,
the delta function plays the role of the "convolution unit":
••oo
f(x) * 8(x) = / f(x - y) 8(y) dy = f(x
J — oo
by the fundamental property A1.37) of the delta function...
We adapt Definition 12.32 to the entire
real line, and so define the Hilbert space L2 = L2( —oo, oo) to be the infinite-dimensional
vector space consisting of all complex-valued functions f(x) which are defined for all x G Ш
and have finite L2 norm
POO
|2= / \f(x)\2dx<oo...
In quan-
quantum theory, each of the paired quantities, e.g., position and momentum, are interrelated
by the Fourier transform...
s — l w s2 + ш2
Taking real and imaginary parts of this identity, we discover the formulae for the Laplace
transforms of the basic trigonometric functions:
?[coscut] = 9 S 9 , ?[smcut] = _ Ш ...
The following result, which follows from the Fourier transform Theorem 13.27,
says that the Laplace transform also uniquely determines the function...