And it all makes for some very frustrating trial and error. Existence Theorem: If f is a continuous function in an open rectangle R = {(x, y) | a x b and c y d } that contains a point (xo, yo), then the initial value problem y’ = f(x, y), y(xo) = yo has atleast a solution in some open sub-interval of (a, b) which contains the point xo. Such is the case when the
H
(
k
/
N
)
{\displaystyle H(k/N)}
sequence is obtained by directly sampling the DTFT of the infinitely long §Discrete Hilbert transform impulse response.
A time-domain derivation proceeds as follows:
A frequency-domain derivation follows from §Periodic data, which indicates that the DTFTs can be written as:
(5a)The product with
H
(
f
)
{\displaystyle H(f)}
is thereby reduced to a discrete-frequency function:
We can also verify the inverse DTFT of (5b):
There is also a convolution theorem for the inverse Fourier transform:
The convolution theorem extends to tempered distributions.
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Hence, I. Basically, the algorithm I get from D. ) then we can uniquely determine the electric
field. Hence the existence and uniqueness theorem ensures that in some open interval centred at 1, the solution of the given ODE exists.
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4c)This form is often used to efficiently implement numerical convolution by computer. But when the non-zero portion of the
g
(
n
)
directory {\displaystyle g(n)}
or
h
(
n
)
{\displaystyle h(n)}
sequence is equal or longer than
N
,
{\displaystyle N,}
some distortion is inevitable. 1a)Applying the inverse Fourier transform
F
1
{\displaystyle {\mathcal {F}}^{-1}}
, produces the corollary:2eqs. .