Trigonometric and Hyperbolic Generated Approximation Theory

The following sections are included:

• Dedication
• Preface
• About the Author
• Contents

Based on Ali and Pales (2022), we produce a collection of interesting trigonometric and hyperbolic Taylor formulae with integral remainders. Using these we derive Opial and Ostrowski-type corresponding inequalities of various kinds and norms. It follows Anastassiou (2023).

Here, we research the univariate quantitative approximation of complex-valued continuous functions on a compact interval by complex-valued neural network operators. These approximations are derived by establishing Jackson-type inequalities involving the modulus of continuity of the engaged function’s high order derivatives. The nature of our approximations are trigonometric and hyperbolic. Our operators are defined by using a density function generated by a q-deformed and λ-parametrized A-generalized logistic function, which is a sigmoid function. The approximations are pointwise and of the uniform norm. The related complex-valued feed-forward neural networks are with one hidden layer. It follows Anastassiou (2023d).

Here we research the multivariate quantitative approximation of complex-valued continuous functions on a box of ℝ^N, N ∈ ℕ, by the multivariate normalized-type neural network operators. We investigate also the case of approximation by iterated multilayer neural network operators. These approximations are achieved by establishing multidimensional Jackson-type inequalities involving the multivariate moduli of continuity of the engaged function and its partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a q-deformed and λ-parametrized A-generalized logistic function, which is a sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural networks are with one or multiple hidden layers. The basis of our theory is the introduced multivariate Taylor formulae of trigonometric and hyperbolic type. It follows Anastassiou (2023d).

Here we study the univariate quantitative approximation of complex-valued continuous functions on a compact interval by complex-valued neural network operators. These approximations are derived by establishing Jackson-type inequalities involving the modulus of continuity of the used function’s high order derivatives. The kinds of our approximations are trigonometric and hyperbolic. Our operators are defined by using a density function generated by a q-deformed and λ-parametrized hyperbolic tangent function, which is a sigmoid function. The approximations are pointwise and of the uniform norm. The related complex-valued feed-forward neural networks are with one hidden layer. It follows Anastassiou (2023d).

Here we study the multivariate quantitative approximation of complex-valued continuous functions on a box of ℝ^N, N ∈ ℕ, by the multivariate normalized-type neural network operators. We investigate also the case of approximation by iterated multilayer neural network operators. These approximations are achieved by establishing multidimensional Jackson-type inequalities involving the multivariate moduli of continuity of the engaged function and its partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a q-deformed and λ-parametrized hyperbolic tangent function, which is a sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural networks are with one or multiple hidden layers. The basis of our theory is the introduced multivariate Taylor formulae of trigonometric and hyperbolic type. It follows Anastassiou (2023d).

Here we research the univariate quantitative approximation of complex-valued continuous functions on a compact interval by complex-valued neural network operators. These approximations are derived by establishing Jackson-type inequalities involving the modulus of continuity of the used function’s high order derivatives. The kinds of our approximations are trigonometric and hyperbolic. Our operators are defined by using a density function generated by a general sigmoid function. The approximations are pointwise and of the uniform norm. The related complex-valued feed-forward neural networks are with one hidden layer. It follows Anastassiou (2023d).

Here we research the multivariate quantitative approximation of complex-valued continuous functions on a box of ℝ^N, N ∈ ℕ, by the multivariate normalized-type neural network operators. We investigate also the case of approximation by iterated multilayer neural network operators. These approximations are achieved by establishing multidimensional Jackson-type inequalities involving the multivariate moduli of continuity of the engaged function and its partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by general multiple sigmoid functions. The approximations are pointwise and uniform. The related feed-forward neural networks are with one or multiple hidden layers. The basis of our theory is the introduced multivariate Taylor formulae of trigonometric and hyperbolic type. It follows Anastassiou (2023).

Here we study the multivariate approximation of time separating random functions over a box or all of ℝ^N, N ∈ ℕ, by quasi-interpolation neural network operators. These approximations are derived by establishing Jackson-type inequalities involving the multivariate modulus of continuity of the engaged random function or its high order partial derivatives. Our operators are defined by using density functions induced by the logistic and hyperbolic tangent activation sigmoid functions. The approximations are pointwise and with respect to the uniform norm. The feed-forward neural networks are with one hidden layer. We finish with a lot of interesting applications. It follows Anastassiou and Kouloumpou (2023).

The first author recently derived several approximation results by neural network operators, see the new monograph Anastassiou (2023d). There, the approximation methods are derived from the parametrized and deformed neural networks induced by the q-deformed and λ-parametrized logistic and hyperbolic tangent activation functions. The results we apply here are univariate on a compact interval, regular and fractional. The outcome is the quantitative approximation of Brownian motion over the three-dimensional sphere. We derive several Jackson-type inequalities estimating the degree of convergence of our neural network operators to a general expectation function of Brownian motion. We give a detailed list of approximation applications regarding the expectation of well-known functions of Brownian motion. Smoothness of our functions is taken into account producing higher speeds of approximation. It follows Anastassiou and Kouloumpou (2024).

The first author recently derived several approximation results by neural network operators, see his new monograph Anastassiou (2023d). There, the approximation methods were derived from the parametrized and deformed neural networks induced by the parametrized error and q-deformed and β-parametrized half-hyperbolic tangent activation functions. The results we apply here are univariate on a compact interval, regular and fractional. The outcome is the quantitative approximation of Brownian motion on simple graphs, in particular over a system S of semiaxes emanating from a common origin radially arranged and a particle moving randomly on S. We derive several Jackson-type inequalities estimating the degree of convergence of our neural network operators to a general expectation function of Brownian motion. We give a detailed list of approximation applications regarding the expectation of well-known functions of this Brownian motion. Smoothness of our functions is taken into account producing higher speeds of approximation. It follows Anastassiou and Kouloumpou (2023).

In this chapter, we estimate the degree of approximation of multivariate pointwise and uniform convergences in the p-mean to the fuzzy-random unit operator of multivariate fuzzy-random quasi-interpolation perturbed activation functions-based neural network operators. These multivariate fuzzy-random operators arise in a natural way among multivariate fuzzy-random neural networks. The rates are given through multivariate probabilistic Jackson-type inequalities involving the multivariate fuzzy-random modulus of continuity of the engaged multivariate fuzzy-random function. It follows Anastassiou and Kouloumpou (2023e).

Based on trigonometric and hyperbolic Taylor-type formulae, we establish related Shisha–Mond form inequalities leading to interesting Korovkin theorems. We deal with the high order of approximation of positive linear operators to the unit operator. The results are quantitative via the modulus of continuity. We finish with applications to Bernstein operators. It follows Anastassiou (2023).

Here, we first describe the new conformable derivatives and integrals (Anderson and Ulness 2015). Then, we establish Opial, Ostrowski, Poincaré, Sobolev, Polya and Hilbert–Pachpatte-type integral inequalities. It follows Anastassiou (2023).

Left and right Caputo fractional calculus is developed further with new interesting representation formulae. In that spirit, we introduce and develop in great generality the corresponding left and right proportional Caputo style fractional calculus. We establish the analogous proportional fractional representation formulae. Based on all the above, we derive left and right fractional integral inequalities of Opial, Poincaré, Sobolev and Hilbert– Pachpatte types. It follows Anastassiou (2023).

Here we present general multivariate radial mixed Ostrowski type inequalities over balls. The proofs are derived by implementation of some essential estimates out of some new trigonometric and hyperbolic Taylor’s formulae (Anastassiou 2023a) and reducing the multivariate problem to a univariate one via general polar coordinates. It follows Anastassiou (2023b).

Here we present general multivariate mixed Ostrowski-type inequalities over spherical shells and balls. We cover the radial and not necessarily radial cases. The proofs are derived by the use of some estimates coming out of some new trigonometric and hyperbolic Taylor’s formulae (Anastassiou 2023a) and reducing the multivariate problem to a univariate one via general polar coordinates. It follows Anastassiou (2023c).

Here based on trigonometric and hyperbolic-type Taylor formulae, we derive Polya-type inequalities in a number of cases. It follows Anastassiou (2024).

Here we present general multivariate Opial and Polya-type inequalities over spherical shells. The proofs are derived by the use of some estimates coming out of some new trigonometric and hyperbolic Taylor’s formulae (Ali and Pales 2022, Anastassiou 2023) and reducing the multivariate problem to a univariate one via general polar coordinates. It follows Anastassiou (2024).

In this chapter, based on trigonometric and hyperbolic-type Taylor formulae, we establish Poincaré, Sobolev and Hilbert–Pachpatte-type inequalities of different kinds, specific and general. It follows Anastassiou (2023).

In this chapter, we continue the study of smooth Picard singular integral operators that started in Anastassiou (2011), see there Chapters 10–14. This time, the foundation of our research is a trigonometric Taylor’s formula. We establish the convergence of our operators to the unit operator with rates via Jackson-type inequalities engaging the first modulus of continuity. Of interest, here is a residual appearing term. Note that our operators are not positive. Our results are pointwise and uniform. It follows Anastassiou (2023b).

In this chapter, we continue the study of smooth Picard singular integral operators that started in Anastassiou (2011), see there Chapters 10–14. This time, the foundation of our research is a trigonometric Taylor’s formula. We establish the L_p convergence of our operators to the unit operator with rates via Jackson-type inequalities engaging the first L_p modulus of continuity. Of interest, here is a residual appearing term. Note that our operators are not positive. It follows Anastassiou (2023).

In this chapter, we continue the research on the smooth Picard singular integral operators that started in Anastassiou (2011), see there Chapters 10–14. This time, the foundation of our research is a parametrized trigonometric Taylor’s formula. We establish the convergence of our operators to the unit operator with rates via Jackson-type inequalities engaging the first modulus of continuity. Of interest, here is a parametrized residual appearing term. Note that our operators are not positive. Our results are pointwise and uniform. It follows Anastassiou (2023b).

In this chapter, we continue the research on the smooth Picard singular integral operators that started in Anastassiou (2011), see there Chapters 10–14. This time, the foundation of our research is a parametrized trigonometric Taylor’s formula. We establish the L_p, p ≥ 1, convergence of our operators to the unit operator with rates via Jackson-type inequalities engaging the first L_p modulus of smoothness. Of interest, here is a parametrized residual appearing term. Note that our operators are not positive. It follows Anastassiou (2023b).

In this chapter, we reexamine the uniform approximation properties of smooth Picard multivariate singular integral operators over ℝ^N, N ≥ 1. We establish their convergence to the unit operator with rates. The estimates are pointwise and uniform. The established inequalities involve the multivariate first modulus of continuity. Our approach is based on a new multivariate trigonometric Taylor formula. At first, we present in detail the general theory of uniform approximation by general smooth multivariate singular integral operators, which then is applied to the Picard operator case. It follows Anastassiou (2023b).

In this chapter, we reexamine the L_p, 1 ≤ p < ∞, approximation properties of general smooth multivariate singular integral operators over ℝ^N, N ≥ 1. It is a trigonometric-based approach with detailed applications to the corresponding smooth multivariate Picard singular integral operators. The results are quantitative via Jackson-type inequalities involving their first L_p modulus of continuity. It follows Anastassiou (2023a).

In this chapter, we continue the study of approximation of various smooth singular integral operators. This time, the foundation of our research is a trigonometric Taylor’s formula. We establish the convergence of our operators to the unit operator with rates via Jackson-type inequalities engaging the first modulus of continuity. Of interest, here is a residual appearing term. Note that our operators are not positive. Our results are pointwise and uniform. The studied operators here are of the following types: Gauss–Weierstrass, Poisson–Cauchy and trigonometric. It follows Anastassiou (2024).

In this chapter, we continue with the study of smooth Gauss–Weierstrass, Poisson–Cauchy and trigonometric singular integral operators that started in Anastassiou (2011), see there Chapters 10–14. This time, the foundation of our research is a trigonometric Taylor’s formula. We establish the univariate L_p convergence of our operators to the unit operator with rates via Jackson-type inequalities involving the first L_p modulus of continuity. Of interest, here is a residual appearing term. Note that our operators are not positive. It follows Anastassiou (2024).

In this chapter, we continue with the study of smooth Gauss–Weierstrass, Poisson–Cauchy and trigonometric singular integral operators that started in Anastassiou (2011), see there Chapters 10–14. This time, the foundation of our research is a trigonometric Taylor’s formula. We prove the parametrized univariate uniform convergence of our operators to the unit operator with rates via Jackson-type parametrized inequalities involving the first modulus of continuity. Of interest, here is a residual appearing term. Note that our operators are not in general positive. It follows Anastassiou (2024).

In this chapter, we continue with the study of smooth Gauss–Weierstrass, Poisson–Cauchy and trigonometric singular integral operators that started in Anastassiou (2011), see there Chapters 10–14. This time, the foundation of our research is a trigonometric Taylor’s formula. We prove the parametrized univariate L_p convergence of our operators to the unit operator with rates via Jackson-type parametrized inequalities involving the first L_p modulus of continuity. Of interest, here is a residual appearing term. Note that our operators are not in general positive. It follows Anastassiou (2024).

In this chapter, we employ the uniform and L_p, 1 ≤ p < ∞, approximation properties of general smooth multivariate singular integral operators over ℝ^N, N ≥ 1. It is a trigonometric-relied approach with detailed applications to the corresponding smooth multivariate Gauss–Weierstrass singular integral operators. The results are quantitative via Jackson-type inequalities involving the first uniform and L_p moduli of continuity. It follows Anastassiou (2024).

In this chapter, we apply the uniform and L_p, 1 ≤ p < ∞, approximation properties of general smooth multivariate singular integral operators over ℝ^N, N ≥ 1. It is a trigonometric-based approach with detailed applications to the corresponding smooth multivariate Poisson–Cauchy singular integral operators. The results are quantitative via Jackson-type inequalities involving the first uniform and L_p moduli of continuity. It follows Anastassiou (2023).

In this chapter, we apply the uniform and L_p, 1 ≤ p < ∞ approximation properties of general smooth multivariate singular integral operators over ℝ^N, N ≥ 1. It is a trigonometric-based approach with detailed applications to the corresponding smooth multivariate trigonometric singular integral operators. The results are quantitative via Jackson-type inequalities involving the first uniform and L_p moduli of continuity. It follows Anastassiou (2024).

The following sections are included:

• Conclusion
• Index