where we made use of the definition (28) and K ( 0 ) is a suitable integration constant.
For this purpose, we will make use of the definition of Frechet derivatives and we recall Lemma 3.2 of [5] here.
Our answer makes use of the definition of oscillation given at the end of Section 5.1: The solutions of equation (61), the Chebyshev polynomials, are defined in the closed interval ([-1, 1]); therefore, the lowest-order solution that can be called oscillatory according to our definition is the (n=3) polynomial.
Here we have made use of the definitions of,,, in (1.9), (1.11) and.
And satisfies being a constant; here we have made use of the definitions of in (1.5), (1.7), and.
On making use of the series definition (1.5), the above expression becomes.
The latter states that for a given pair of random variables, namely X and Y, variable X has a causal influence on Y if a manipulation of the values of X leads to a change in the probability distribution of Y. Making use of the aforementioned definition of causation, it can be easily observed that each defined variable C L j i has a causal influence on every following variable C L j i + 1, ∀ j.
Making use of the correct definition of Δ z D, this equation can be re-expressed as where k c is the instantaneous central wavenumber, z φn is the phase pathlength corresponding to the nth sample reflector, z φ-r0 is the phase pathlength of the reference reflector at the central wavelength, and M is the slope of the wavelength-dependent reference phase pathlength (in mm/nm).
Here we make use of the following definition, which is a modification of the one given in [5], and is equivalent to the classical one, but it is more suitable for some applications to difference equations, see [6].
