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Description and some formal properties of beta filters; Compact support quasi-Gaussian convolution operators with applications to the construction of spatial covariances
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    The beta distributions are a standard class of smooth two-parameter probability densities on a finite line segment, so named because their normalizing constants on a unit interval are the Euler beta functions of the two parameters. Of particular interest and utility are the symmetric distributions obtained when the two parameters are equal and nonnegative integers since, in these cases, the distributions each take the form of a polynomial over the interval. When the shared integer parameter is positive, the function is unimodal, approximating a Gaussian with greater fidelity as the parameter increases. Owing to its computationally convenient form, and the properties we have mentioned, the radially-symmetric beta filter, whose kernel’s radial profile is such an integer parameter beta distribution, provides a very attractive choice for synthezing spatial covariances, especially in a multigrid framework where the quasi-Gaussian components of different scales and amplitudes can be superposed to form more general covariance profiles in a controlled way. This note describes the formal properties of the beta filters, including their first few moments, their Fourier or Bessel transforms, and the analytic formulas of the kernels of the homogeneous and self-adjoint covariances that can be constructed by self-convolution of the radially-symmetric filters.
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