Some Intrinsic Properties of Tadmor–Tanner Functions: Related Problems and Possible Applications

: 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 σ adapt ( 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 σ adapt ( 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.


Introduction
In antenna-feeder techniques, the most often occurred signals F(t) are of rectangular type, as shown in Figure 1. In [1,2], the authors construct a new class of accurate filters for processing piecewise smooth spectral data. Definition 1. The exponentially optimal filter is defined as [1,2]: 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 σ adapt (t) in the Hausdorff sense. Definition 2 ( [3,4]). The Hausdorff distance (the H-distance) [3] ρ( f , g) between two interval functions f , g on Ω ⊆ R, is the distance between their completed graphs F( f ) and F(g) considered as closed subsets of Ω × R. More precisely, wherein ||.|| is any norm in R 2 -e. g. the maximum norm ||(t, x)|| = max{|t|, |x|}-hence, the distance between the points A = (t A , 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, particularly for developing the theory of Hausdorff approximations.
In this paper we consider some intrinsic properties of function (1). A number of modified adaptive functions have also been proposed that can find application in the field of antenna-feeder analysis. Of course, the specialists working in this direction will assess which of the new models have the right to exist.

Some Intrinsic Properties
We will consider only the case q-even (see, Figure 2). Let t 0 be the value for which σ adapt (t 0 ) = 1 2 -i.e., t 0 is the solution of the nonlinear equation: The Hausdorff distance d between F(t) and σ adapt (t) satisfies the relation: For large values of q, the coefficient c q grows very rapidly, which severely limits the computational calculations and visualization of the results in any Computer Algebraic Platform. The same remark remains valid when solving the nonlinear Equation (4) for the magnitude of the best Hausdorff approximation-d. This requires obtaining precise two-sided estimates for d. We will sketch the idea, for example, for q = 2. Let (see (4)) The following theorem gives upper and lower bounds for d.

Theorem 1.
Let q = 2. The "saturation"-d satisfies the following inequalities where Proof. From G (d) > 0, we conclude that the function G is strictly monotonically increasing. Consider the function From Taylor expansion, we obtain Figure 4). In addition, G 1 (d) > 0. Further, we have This completes the proof of the theorem.
From (5), we see that for The reader may formulate the corresponding approximation problem for arbitrary q following the ideas given in this note, and this will be omitted. Some techniques for obtaining accurate two-sided estimates can be found in [7].

Some Remarks
The basic problems considered in [8][9][10]) are an approximation of functions and point sets by algebraic and trigonometric polynomials in the Hausdorff metric as well as their applications in the field of antenna-feeder techniques, analysis and the synthesis of antenna patterns and filters, including noise minimization by the suitable approximation of impulse functions.
The application of Fourier analysis for the approximation of impulse and transmitting functions is compared to this, which can be obtained using the elements of the best Hausdorff approximation.
Let t = b 0 (π cos(θ) + a), where θ is azimuthal angle and a is the phase difference.
Numerical examples are presented using CAS MATHEMATICA.

• The New Activation Function
We define the following activation function based on σ adapt (t) (1): In antenna-feeder technique most often occurred signals are of types shown in Figures 9 and 10. For q even, the corresponding approximation using model (9) is shown in Figure 9. For q odd, the corresponding approximation using new activation function F 3 (t) is shown in Figure 10.  The reader may formulate the corresponding approximation problem following the ideas given in this article, and this will be omitted. The results have important role in the fields of Population Dynamics, Biostatistics, Signal Theory, Reliability Analysis and Life testing experiments. For other results, see [12][13][14][15]. Another application of the Hausdorff metric can be found in [16].

Another activation function
Formally, we define the following activation function where a > 0.
The shifted Heaviside step function is defined by Obviously, the new function F 5 (t) can be used successfully to approximate the Heaviside step function (see Figure 12). Approximation of shifted Heaviside step function h t 0 (t) by sigmoidal function F 5 (t) for: (a) q = 6: a = 1000, t 0 = 0.282245; Hausdorff distance d = 0.0580217; (b) q = 3 a = 700, t 0 = 0.0919762; Hausdorff distance d = 0.0460022 is visualized on Figure 12.
The function F 5 (t) can be used in the field of population dynamics and biostatistics [7]. •

The new function
Formally, we define the following function: where H(t) and G(t) are polynomials.
In some cases, the function F 6 (t) can be used for analysis of "rectangular signals".
A "typical radiation pattern" F 6 (θ) for b = 0.132 and a = 4.263 is depicted on Figures 14 and 15.
Unfortunately, these diagrams cannot always be realized in practice.  • Application to the "cut-off function" The analysis presented in this article can also be applied to the "cut-off function", see (1): where C > 0 and p is even. •

Approximation with restrictions
Often some restrictions must be imposed on the main lobe of the radiation pattern-for example, see Figure 16 The polynomial f (t) = 133.123786t 10 − 251.561221t 8 + 107.004415t 6 + 36.476971t 4 − 30.804320t 2 + 6.189054 computed by Remez's algorithm from [18] and shown in Figure 17, provides a reliable approximation of the window (Figure 16).
We note that the "adaptive function" with c 1 = 6.9 based on function of Tadmor and Tanner (1) is of the form shown in Figure 17.
Unfortunately, these diagrams cannot always be realized in practice.
Specialists working in these scientific fields have the final word.  We will look at another instructive example.
A typical diagram function R(t) with imposed constraints at the median level is shown in Figure 18. After serious analysis, it turned out that the function by Tadmor and Tanner could be modified in such a way as to obtain the following "differential analogue with step h": where σ adapt (t) := e cq t q t 2 −1 and c q = 2 q . 3 8 .
From the attached visualization (see Figures 19 and 20) it can be seen that the proposed new modification can be used successfully for the approximation of functions of type R(t).
Obviously, higher order differential analogues can be obtained, which we leave to the readers' attention.

Concluding Remarks
The aim of the present research is to define families of "adaptive functions" that could be used in various branches of scientific knowledge, in particular in the approximation of a number of classical impulse signals, point sets in the plane and supplemented graphs of discontinuous functions.
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 article we consider only some aspects related to the disclosure of intrinsic properties of some of the proposed families of "adaptive functions" in a purely methodological aspect.
Many of the proposed new models could not be used as "feasible models-diagrams" in practice.
In this connection, the specialists working in the indicated field have the floor. For some results, see .
Funding: This research received no external funding.