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The Fourier Theory
Published in Steven G. Krantz, Complex Variables, 2019
The Fourier transform f^ of an integrable function f enjoys the property that f^ is continuous and vanishes at infinity. However f^ itself need not be integrable. In fact f^ can die arbitrarily slowly at infinity. This fact of life necessitates extra care in formulating results about the Fourier transform and its inverse.
Weakly nonlinear magnetic equatorial Kelvin waves in rotating spherical coordinates
Published in Geophysical & Astrophysical Fluid Dynamics, 2018
The last four integrals on the right-hand side of (48) were chosen to be particular anti-derivatives that vanish as goes to infinity. We can apply L'Hopital's rule to the product of each of these integrals with to show that each one must vanish in the limit as goes to infinity. The first integral on the right-hand side of (48) contains the product . Applying L'Hopital to this product shows that it behaves like as goes to infinity. Therefore, the first integral on the right-hand side of (48) grows like as goes to infinity so that Φ must be set to zero in order for the product of with that integral to vanish at infinity. We cannot use the undetermined functions of s and τ that arise in the indefinite integrations in this problem to suppress that the solution at infinity as we have done previously. Setting in (48) gives the following linear partial differential equation for the amplitude function : which can be solved to give where and are arbitrary functions of their arguments.
A high dimensional functional time series approach to evolution outlier detection for grouped smart meters
Published in Quality Engineering, 2023
A. Elías, J. M. Morales, S. Pineda
Formally, let FD be a general integrated functional depth. This statistic evaluates the centrality of a given function y from a sample of functions with respect to the center of symmetry of its empirical distribution PT. This empirical distribution belongs to a functional random variable taking values in a space of continuous functions defined in a domain For we denote as the marginal distribution of PT at slice x and its cumulative marginal distribution as In practice, functional data is observed as a set of discrete points or evaluations of the curves. To simplify the exposition, we assume that the points in the grid at which the curves are observed/evaluated are common for all the functions and equal to However, as we said above, there are partially observed integrated functional depths applicable when the collection of discretized points varies from curve to curve without the need of preprocessing (Elías et al. 2022). Then, the empirical integrated functional depth is defined as being a weighting function that sums up to 1, and D a suitable depth. More specifically, according to Zuo and Serfling (2000), a depth function should be affine invariant, maximal at the center, monotone with respect to the deepest point, and should vanish at infinity. In the same vein, Nagy et al. (2016) and Gijbels and Nagy (2017) study, adapt and complete the counterparts of these properties for functional depths.
Limit behaviour of diffusion in high-contrast periodic media and related Markov semigroups*
Published in Applicable Analysis, 2019
A. Piatnitski, S. Pirogov, E. Zhizhina
Denote by the Banach space of continuous functions that vanish at infinity with the norm .