By A. Campillo
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Extra resources for Algebroid Curves in Positive Characteristic
If the substitution nπ/L = tn is employed, and δt = tn+1 − tn , then f (x) = 1 2L L −L f (s)ds + 1 π ∞ L n=1 −L f (s) cos[tn (s − x)]ds δt. 60) If f is integrable on R, then in the limit as L → ∞ the first term on the right-hand side of Eq. 60) vanishes, and the sum in Eq. 8 Basic results from complex variable theory 23 which can be rewritten as follows: ∞ 1 2π f (x) = −∞ ∞ dt −∞ f (s)eit(s−x) ds. 62) Both of the preceding two results are referred to as the Fourier integral formula. 61) is occasionally useful for the situation where f (x) has particular symmetry properties.
5) is frequently denoted by the function g( ). Clearly, the function g depends on the entire shape of f . In other words, g at some point x, g(x), is not determined simply by the value of the function f evaluated at the same point. That is, g has a non-local dependence on f . The situation where g(x) is determined directly by the value f (x) arises when there is a simple functional connection between f and g; for example, suppose g(x) = sin[ f (x)], then the value of g at the point x depends only on the value of f evaluated at x.
This notion has important consequences. A function f could be zero over a large region of the real axis and finite for a small region, but its Hilbert transform could be everywhere non-zero. Applications will be encountered later that reflect this type of behavior. 6) where a is a real positive constant. This functional form appears in several diverse applications, and is sometimes referred to as a Cauchy pulse, and in other applications is closely related to the Lorentzian profile. The Hilbert transform of this function is given by g(x) = Hf (x) = a2 x .
Algebroid Curves in Positive Characteristic by A. Campillo