| Summary: | 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 <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> by the class of exponential functions <inline-formula><math display="inline"><semantics><mrow><msub><mi>σ</mi><mrow><mi>a</mi><mi>d</mi><mi>a</mi><mi>p</mi><mi>t</mi></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> about the Hausdorff metric. We prove upper and lower estimates for “saturation”—<i>d</i> (in the case <inline-formula><math display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>). New activation and “semi-activation” functions based on <inline-formula><math display="inline"><semantics><mrow><msub><mi>σ</mi><mrow><mi>a</mi><mi>d</mi><mi>a</mi><mi>p</mi><mi>t</mi></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are defined. Some related problems are discussed. We also consider modified families of functions with “polynomial variable transfer”. Numerical examples, illustrating our results using <i>CAS MATHEMATICA</i> are given.
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