Best one-sided <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> approximation to the Heaviside and sign functions

Bustamante, Jorge; Quesada, José M.; Martínez-Cruz, Reinaldo

doi:10.1016/j.jat.2012.02.006

Cited by 15 publications

(2 citation statements)

References 2 publications

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“…Problems of one-sided weighted integral approximation to the characteristic function of an interval and to similar functions by algebraic or trigonometric polynomials arise in various areas of mathematics and have a rich history (see [1,2,5,6,11,14,16,17] and the references therein). Let us outline only several exact results on problem (2.4) closely related to the present paper; for a more complete presentation of the topic see [1,11].…”

Section: One-sided Approximation To the Characteristic Function Of Anmentioning

confidence: 99%

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One-Sided L-Approximation on a Sphere of the Characteristic Function of a Layer

Deikalova

Torgashova

2018

UMJ

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In the space L(S m−1) of functions integrable on the unit sphere S m−1 of the Euclidean space R m of dimension m ≥ 3, we discuss the problem of one-sided approximation to the characteristic function of a spherical layer G(J) = {x = (x 1 , x 2 ,. .. , xm) ∈ S m−1 : xm ∈ J}, where J is one of the intervals (a, 1], (a, b), and [−1, b), −1 < a < b < 1, by the set of algebraic polynomials of given degree n in m variables. This problem reduces to the one-dimensional problem of one-sided approximation in the space L φ (−1, 1) with the ultraspherical weight φ(t) = (1−t 2) α , α = (m−3)/2, to the characteristic function of the interval J. This result gives a solution of the problem of one-sided approximation to the characteristic function of a spherical layer in all cases when a solution of the corresponding one-dimensional problem known. In the present paper, we use results by A.G. Babenko, M.V. Deikalova, and Sz.G. Revesz (2015) and M.V. Deikalova and A.Yu. Torgashova (2018) on the one-sided approximation to the characteristic functions of intervals.

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Section: One-sided Approximation To the Characteristic Function Of Anmentioning

confidence: 99%

“…6) and other M − |u| nodes are chosen so that the formula have the highest degree of precision; here, |u| is the cardinality, i. e., the number of points of the set u. It is known (see, for example,[15, Ch.…”

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confidence: 99%

One-Sided L-Approximation on a Sphere of the Characteristic Function of a Layer

Deikalova

Torgashova

2018

UMJ

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Fuzzy modeling and control of a class of non‐differentiable multi‐input multi‐output nonlinear systems

Zare

Shasadeghi

Izadian

et al. 2021

Asian Journal of Control

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This paper presents a new approach in modeling and control of multi‐input multi‐output (MIMO) systems that have non‐differentiable operating points. A circle criterion is introduced at the non‐differentiable operating points to divide the entire operating region into two parts. Takagi‐Sugeno fuzzy models are developed in each part, and a switching framework is introduced to model the operating region. Accordingly, a sliding mode controller (SMC) is developed. The proposed modeling and controller are implemented on the benchmark quadruple tank process (QTP). It is demonstrated that the proposed modeling and controller design provides simplicity and universality in a set of nonlinear systems; it is robust with respect to various internal and external disturbances and model uncertainties which allows for accurate regulation and tracking; and it converges in finite time, is capable of controlling nonlinear systems with coupling dynamic, and enables fuzzy modeling of continuous‐time systems without imposing differentiability.

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Extremal Functions with Vanishing Condition

Littmann

Spanier

2015

Constr Approx

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For a Hermite-Biehler function E of mean type τ we determine the optimal (with respect to the de Branges measure of E) majorant M + E and minorant M − E of exponential type τ for the truncation of x → (x 2 + a 2 ) −1 . We prove thatwhere K is the reproducing kernel for the de Branges space H(E). As an application we determine the optimal majorant and minorant for the Heaviside function that vanish at a fixed point α = ia on the imaginary axis. We show that the difference of majorant and minorant has integral value (πa − tanh(πa)) −1 πa.

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Best one-sided L1 approximation to the Heaviside and sign functions

Cited by 15 publications

References 2 publications

One-Sided L-Approximation on a Sphere of the Characteristic Function of a Layer

One-Sided L-Approximation on a Sphere of the Characteristic Function of a Layer

Fuzzy modeling and control of a class of non‐differentiable multi‐input multi‐output nonlinear systems

Extremal Functions with Vanishing Condition

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