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: * One might argue that this follows from l'Hopital's rule, but in point of fact this is a fake
application: one already needs to evaluate the limit when proving the formula for the derivative
of the sine function!
1/12/04 508 © 2003 Peter J...
In particular, exponentials
e-ikx Q? "negative" frequency can all be converted into positive versions, namely e1 (n~fe)a;j
by the same sampling argument A3.3)...
Since we cannot distinguish sampled exponentials of frequency higher than n, we only need
consider a finite sum
n-l
f(x) ~ p(x) = co + Cleix + c2e2ix+ ¦¦¦ + сгг_1е{п-1У1Х = J^ ckeikx A3.5)
using the first n exponentials A3.4)...
Proof: The crux of the matter relies on properties of the remarkable complex numbers
С = e2^'n = cos — + i sin — , A3.10)
where n = 1,2,3,......
In Figure 13.3 we compare the function, with the interpolation points indicated, and
discrete Fourier representations for both n = 4 and n = 16 points...
Orthog-
Orthogonality is the first and most important feature of any linear algebra algorithm; were it not
for the dramatic simplification afforded by the orthogonal basis formulae, Fourier analysis
would not have evolved into today's essential tool...
If n = 2r is a power of 2, then this game can be played all the way back to the start,
beginning with the trivial order 1 discrete Fourier series which just samples the function
at a single point...
The breakthrough came in
1988, when, in her thesis, the Dutch mathematician Ingrid Daubechies produced the first
1/12/04 530 © 2003 Peter J...
First, localization of the wavelets requires that the scaling function has
bounded support, and so <p(x) = 0 when x lies outside some bounded interval [a, b]...
Solving the Dilation Equation
Let us now discuss how to solve the dilation equation A3.45)...
The proof of formula for
the norms of the mother and daughter wavelets is left as an exercise for the reader...
With the values of the Daubechies scaling function on a sufficiently dense set of dyadic
points in hand, the consequential values of the mother wavelet are given by A3.49), which,
1/12/04 537 © 2003 Peter J...
Moreover, the
frequencies represented in the Fourier transform are no longer dictated by the length of
the interval, and so we are effectively decomposing a quite general, non-periodic function,
into a "continuous linear combination" or, rather, integral, of trigonometric functions of
arbitrary frequency...
Thus, by linearity,
•^ ^ ^ 1 1 2o
/.<*) = Л<*) + /,<*) = ^—k + ^^T* = TFeT*' A3'85>
which is real and even because fe is an even function; see Exercise I...
However, the latter statement must be taken with
a grain of salt, since the integral does not converge in the normal (Riemann or Lebesgue)
sense...
For example, the bound electrons in an atom
are localized by the electrostatic attraction of the nucleus...
As a function of x, it satisfies the homogeneous differential
equation — и" + со2 и = 0, except at the point x = у when its derivative has a jump
discontinuity of unit magnitude...
In particular, the solution formula A3.112) is equal to the convulution of an even
exponential pulse g(x) = — е~ш\х\ with the forcing function:
u(x) = g(x) * h(x) = g(x- y) h(y) dy = g * h(x)
¦J —CO
On the other hand, its Fourier transform A3.109) is the ordinary multiplicative product
u(k) =g(k)h(k)
of the Fourier transforms of g and h...
It turns out the proper setting for the rigorous theory is the Hilbert space of complex-
valued square-integrable functions, the same infinite-dimensional vector space that lies at
the foundation of modern quantum mechanics...