Mechanical Models for Hermite Interpolation on the Unit Circle

Berriochoa, E.; Cachafeiro, A.; García-Rábade, Héctor; García-Amor, J.

doi:10.3390/math9091043

Cited by 3 publications

(7 citation statements)

References 16 publications

(41 reference statements)

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“…Remark 2. In [17], it was proved property (11) when considering as nodal polynomials the paraorthogonal polynomials related to measures in the Szegő class with the Szegő function having analytic extension outside the unit disk (see [1,19]). Here, we have proved it by using only the separation properties satisfied by the nodal points.…”

Section: Nodal Systems: Properties and Auxiliary Resultsmentioning

confidence: 99%

“…Important extensions of the theory managing to circumvent the link between nodal systems and measures are the normal or strongly normal nodal arrays in the real case (see [9]) and the perturbed roots of the unity in the case of the unit circle (see [10]). In [11] and previous papers, we continued with these ideas, studying the classical Lagrange and Hermite problems and working with nodal distributions characterized by a good property of separation between the nodes, which can be obtained through a perturbation of the uniform distribution.…”

Section: Introductionmentioning

confidence: 99%

“…Our first example uses a measuring instrument based on a Cardan device. The full description of the nodal system and the proof that the system satisfies, (7), can be found in [11]. We choose…”

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confidence: 99%

“…Our second example uses a measuring device based on a countdown process. The full description of the nodal system and the proof that it satisfies, (7), can be found in [11]. As the test function, we take…”

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confidence: 99%

“…The third example is based on the analysis of certain positions of a natural satellite orbiting its planet. A detail description of the nodal system and the proof that ( 7) is satisfied can be found in [11]. We choose F(z) = e z + e 1 z as the test function.…”

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A Type of Interpolation between Those of Lagrange and Hermite That Uses Nodal Systems Satisfying Some Separation Properties

Berriochoa,

Cachafeiro,

García Rábade

et al. 2024

Mathematics

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In this paper, we study a method of polynomial interpolation that lies in-between Lagrange and Hermite methods. The novelty is that we use very general nodal systems on the unit circle as well as on the bounded interval only characterized by a separation property. The way in which we interpolate consists in considering all the nodes for the prescribed values and only half for the derivatives. Firstly, we develop the theory on the unit circle, obtaining the main properties of the nodal polynomials and studying the convergence of the interpolation polynomials corresponding to continuous functions with some kind of modulus of continuity and with general conditions on the prescribed values for half of the derivatives. We complete this first part of the paper with the study of the convergence for smooth functions obtaining the rate of convergence, which is slightly slower than that when equidistributed nodal points are considered. The second part of the paper is devoted to solving a similar problem on the bounded interval by using nodal systems having good properties of separation, generalizing the Chebyshev–Lobatto system, and well related to the nodal systems on the unit circle studied before. We obtain an expression of the interpolation polynomials as well as results about their convergence in the case of continuous functions with a convenient modulus of continuity and, particularly, for differentiable functions. Finally, we present some numerical experiments related to the application of the method with the nodal systems dealt with.

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Section: Nodal Systems: Properties and Auxiliary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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A Type of Interpolation between Those of Lagrange and Hermite That Uses Nodal Systems Satisfying Some Separation Properties

Berriochoa,

Cachafeiro,

García Rábade

et al. 2024

Mathematics

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Preface to “Mathematical Methods, Modelling and Applications”

Jódar

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2022

Mathematics

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Review of Methods, Applications and Publications on the Approximation of Piecewise Linear and Generalized Functions

2022

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Approximation of piecewise linear and generalized functions is an important and difficult problem. These functions are widely used in mathematical modeling of various processes and systems, such as: automatic control theory, electrical engineering, radio engineering, information theory and transmission of signals and images, equations of mathematical physics, oscillation theory, differential equations and many others. The widespread use of such functions is explained by their positive properties. For example, piecewise linear functions are characterized by a simple structure over segments. However, these features also have disadvantages. For example, in the case of using piecewise linear functions, solutions have to be built in segments. In this case, the problem of matching the obtained solutions at the boundaries of the segments arises, which leads to the complication of the research results. The use of generic functions has similar disadvantages. To eliminate shortcomings in practice, one resorts to the approximation of these functions. There are a large number of well-known methods for approximating piecewise linear and generalized functions. Recently, new methods for their approximation have been developed. In this study, an attempt was made to generalize and discuss the existing methods for approximating the considered functions. Particular emphasis is placed on the description of new approximation methods and their applications in various fields of science and technology. The publication-based review discusses the strengths and weaknesses of each method, compares them, and considers suitable application examples. The review will undoubtedly be interesting not only for mathematicians, but also for specialists and scientists working in various applied fields of research.

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Mechanical Models for Hermite Interpolation on the Unit Circle

Cited by 3 publications

References 16 publications

A Type of Interpolation between Those of Lagrange and Hermite That Uses Nodal Systems Satisfying Some Separation Properties

A Type of Interpolation between Those of Lagrange and Hermite That Uses Nodal Systems Satisfying Some Separation Properties

Preface to “Mathematical Methods, Modelling and Applications”

Review of Methods, Applications and Publications on the Approximation of Piecewise Linear and Generalized Functions

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