Constructive approximation of level continuous fuzzy functions1

AbstractLet X be a complete metric space. Viewing continuous real functions on X as closed subsets of X × R, equipped with Hausdorff distance, we show that C(X, R) is completely metrizable provided X is complete and sigma compact. Following the Bulgarian school of constructive approximation theory, a bounded discontinuous function may be identified with its completed graph, the set of points between the upper and lower envelopes of the function. We show that the space of completed graphs, too, is completely metrizable, provided X is locally connected as well as sigma compact and complete. In the process, when X is a Polish space, we provide a simple answer to the following foundational question: which subsets of X × R arise as completed graphs?

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Constructive approximation and numerical methods in geodetic research today – an attempt at a categorization based on an uncertainty principle

Journal of Geodesy ◽

10.1007/pl00004001 ◽

1999 ◽

Vol 73 (9) ◽

pp. 452-465 ◽

Cited By ~ 41

Author(s):

W. Freeden ◽

V. Michel

Keyword(s):

Numerical Methods ◽

Uncertainty Principle ◽

Constructive Approximation

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Recent Advances in Constructive Approximation Theory

10.1007/978-3-319-92165-5 ◽

2018 ◽

Cited By ~ 17

Author(s):

Vijay Gupta ◽

Themistocles M. Rassias ◽

P. N. Agrawal ◽

Ana Maria Acu

Keyword(s):

Approximation Theory ◽

Recent Advances ◽

Constructive Approximation

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Decorrelative Mollifier Gravimetry

10.5194/egusphere-egu21-409 ◽

2021 ◽

Author(s):

Willi Freeden

Keyword(s):

Inverse Problems ◽

Density Contrast ◽

Disturbing Potential ◽

Book Series ◽

Inverse Gravimetry ◽

Constructive Approximation ◽

Vening Meinesz ◽

Inverse Gravimetry Problem

<p>The lecture highlights arguments that, coming from multiscale mathematics, have fostered the advancement of gravimetry, as well as those that, generated by gravimetric problems, have contributed to the enhancement in constructive approximation and numerics. Inverse problems in gravimetry are delt with multiscale mollifier decorrelation strategies. Two examples are studied in more detail: (i) Vening Meinesz multiscale surface mollifier regularization to determine locally the Earth's disturbing potential from deflections of vertical, (ii) Newton multiscale volume mollifier regularization of the inverse gravimetry problem to derive locally the density contrast distribution from functionals of the Newton integral and to detect fine particulars of geological relevance. All in all, the Vening Meinesz medal&#160; lecture is meant as an&#160; \lq \lq appetizer'' served to enjoy the tasty meal "Mathematical Geoscience Today'' to be shared by geoscientists and mathematicians in the field of gravimetry. It provides innovative concepts and locally relevant applications presented in a monograph to be published by Birkh&#228;user in the book series &#8220;Geosystems Mathematics&#8221; (2021).</p>

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On the constructive approximation of non-linear operators in the modelling of dynamical systems

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000009188 ◽

1997 ◽

Vol 39 (1) ◽

pp. 1-27 ◽

Cited By ~ 12

Author(s):

A. P. Torokhti ◽

P. G. Howlett

Keyword(s):

Dynamical Systems ◽

Real Number ◽

Input Signal ◽

Linear Operators ◽

Non Linear ◽

Constructive Approximation ◽

Modelling Process ◽

Theoretical Procedure ◽

Small Disturbances

AbstractIn this paper we propose a systematic theoretical procedure for the constructive approximation of non-linear operators and show how this procedure can be applied to the modelling of dynamical systems. We extend previous work to show that the model is stable to small disturbances in the input signal and we pay special attention to the role of real number parameters in the modelling process. The implications of computability are also discussed. A number of specific examples are presented for the particular purpose of illustrating the theoretical procedure.

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A Single Hidden Layer Feedforward Network with Only One Neuron in the Hidden Layer Can Approximate Any Univariate Function

Neural Computation ◽

10.1162/neco_a_00849 ◽

2016 ◽

Vol 28 (7) ◽

pp. 1289-1304 ◽

Cited By ~ 25

Author(s):

Namig J. Guliyev ◽

Vugar E. Ismailov

Keyword(s):

Continuous Function ◽

Finite Interval ◽

Activation Function ◽

Feedforward Network ◽

Arbitrary Continuous Function ◽

The Real ◽

Univariate Function ◽

Constructive Approximation ◽

Hidden Layer ◽

Sigmoidal Activation Function

The possibility of approximating a continuous function on a compact subset of the real line by a feedforward single hidden layer neural network with a sigmoidal activation function has been studied in many papers. Such networks can approximate an arbitrary continuous function provided that an unlimited number of neurons in a hidden layer is permitted. In this note, we consider constructive approximation on any finite interval of [Formula: see text] by neural networks with only one neuron in the hidden layer. We construct algorithmically a smooth, sigmoidal, almost monotone activation function [Formula: see text] providing approximation to an arbitrary continuous function within any degree of accuracy. This algorithm is implemented in a computer program, which computes the value of [Formula: see text] at any reasonable point of the real axis.

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