Abstract
In this paper, we study some properties of an exponentially optimal filter proposed by Tadmor and Tanner. More precisely, we consider the problem for approximating the function of rectangular type F(t) by the class of exponential functions σadapt(t) about the Hausdorff metric. We prove upper and lower estimates for “saturation”—d (in the case q=2). New activation and “semi-activation” functions based on σadapt(t) are defined. Some related problems are discussed. We also consider modified families of functions with “polynomial variable transfer”. Numerical examples, illustrating our results using CAS MATHEMATICA are given.
Highlights
In [1,2], the authors construct a new class of accurate filters for processing piecewise smooth spectral data
Questions related to the synthesis and analysis of transfer functions, radiation diagrams with algebraic and trigonometric polynomials about Hausdorff distance are elaborated in detail in the monograph in [8]
In this connection, the specialists working in the indicated field have the floor
Summary
Various modifications of this “powerful” class of functions have been proposed and studied by a number of researchers. This study represents a certain interesting problem for approximating the function F (t) with the specified class of exponential functions σadapt (t) in the Hausdorff sense. The basic approaches for approximation of functions and point sets of the plane by algebraic and trigonometric polynomials in respect to Hausdorff distance (H–distance) are connected to the work and achievements of Bl. Sendov who established a Bulgarian school in Approximation theory, for developing the theory of Hausdorff approximations. A number of modified adaptive functions have been proposed that can find application in the field of antenna-feeder analysis. The specialists working in this direction will assess which of the new models have the right to exist
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