Jackson Type Research Articles

Shape Preserving Approximation of Periodic Functions: Conclusion

AbstractWe give here the final results about the validity of Jackson-type estimates in comonotone approximation of $$2\pi $$ 2 π -periodic functions by trigonometric polynomials ($$q=1$$ q = 1 ). For coconvex ($$q=2$$ q = 2 ) and the so called co-q-monotone, $$q>2$$ q > 2 , approximations, everything is known by now. Thus, this paper concludes the research on Jackson type estimates of Shape Preserving Approximation of periodic functions by trigonometric polynomials. It is interesting to point out that the results for comonotone approximation of a periodic function are substantially different than the analogous results for comonotone approximation of a continuous function on a finite interval, by algebraic polynomials.

Q-Deformed and delta-parametrized A-generalized logistic function induced Banach space valued multivariate multi layer neural network approximations

Abstract. Here we research the multivariate quantitative approximation of Banach space valued continuous multivariate functions on a box or RN, N ∈ N, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We investigate also the case of approximation by iterated multilayer neural network operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or 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 network are with one or multi hidden layers. Mathematics Subject Classification (2010): 41A17, 41A25, 41A30, 41A36. Keywords: Multi-layer approximation, q-deformed and λ-parametrized A- generalized logistic function, multivariate neural network approximation, quasi-interpolation operator, Kantorovich type operator, quadrature type operator, multivariate modulus of continuity, iterated approximation.

Approximation of time separating stochastic processes by neural networks revisited

Here we study the univariate quantitative approximation of time separating stochastic process over the whole real line by the normalized bell and squashing type neural network operators. Activation functions here are of compact support. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged stochastic function or its high order derivative. The approximations are pointwise and with respect to the Lp norm. The feed-forward neural networks are with one hidden layer. We finish with a great variety of special applications.

Mediterranean fruit fly population phenological patterns are strongly affected by elevation and host presence

The Mediterranean fruit fly (medfly) (Ceratitis capitata, Diptera: Tephritidae), is an extremely polyphagous pest that threatens the fruit production and trading industry worldwide. Monitoring C. capitata populations and analysing its dynamics and phenology is considered of outmost importance for designing and implementing sound management approaches. The aim of this study was to investigate the factors regulating the population dynamics of the C. capitata in a coastal and semi-mountainous area. We focused on effects of topography (e.g. elevation), host presence and seasonal patterns of ripening on the phenological patterns considering data collected in 2008. The experimental area is characterized by mixed fruit orchards, and Mediterranean climate with mild winters. Two trap types were used for population monitoring. The female targeted McPhail type and the male targeted Jackson type. Traps were placed in farms located at different elevations and landscape morphology (coastal and semi-mountainous areas). The main crops included citrus, apples, peaches, plums, pears, figs, quinces and apricots. Adult captures were first recorded in May, peaked in mid-summer and mid-autumn and almost ceased at the end of the season (January 2008). Captures in the coastal areas preceded that of highlands by 15 days. Most of the adults detected during the fruit ripening of late stone fruit cultivars (first peak) and citrus (second peak). The probability of capturing the first adults preceded almost three weeks the peak of adult captures either considering the elevation or host focus analyses. The results provide valuable information on the seasonal population trend of C. capitata in mixed fruit Mediterranean orchards and can support the set-up of IPM systems in areas with various landscapes and different hosts throughout the fruit growing season.

Trigonometric generated Lp degree of approximation

In this article we continue the study of smooth Picard singular integral operators that started in [3], see there chapters 10-14. This time the foundation of our research is a trigonometric Taylor’s formula. We establish the Lp convergence of our operators to the unit operator with rates via Jackson type inequalities engaging the first Lp modulus of continuity. Of interest here is a residual appearing term. Note that our operators are not positive.

Q-Deformed and L-parametrized hyperbolic tangent function relied complex valued multivariate trigonometric and hyperbolic neural network approximations

Here we study the multivariate quantitative approximation of complex valued continuous functions on a box of RN , 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 multidimen-sional Jackson type inequalities involving the multivariate moduli of continuity of the en- gaged function and its partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a q-deformed and λ-parametrized hyper-bolic tangent function, which is a sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network are with one or multi hidden layers. The basis of our theory are the introduced multivariate Taylor formulae of trigonometric and hyperbolic type.

Multiple general sigmoids based Banach space valued neural network multivariate approximation

Here we present multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or \(\mathbb{R}^{N},\) \(N\in \mathbb{N}\), by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by several different among themselves general sigmoid functions. This is done on the purpose to activate as many as possible neurons. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer. We finish with related \(L_{p}\) approximations.

Q-Deformed and λ-parametrized A-generalized logistic function based Banach space valued ordinary and fractional neural network approximation

Here we research the univariate quantitative approximation, ordinary and fractional, of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative of fractional derivatives. 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 Banach space valued feed-forward neural networks are with one hidden layer.

MULTIVARIATE FUZZY APPROXIMATION BY NEURAL NETWORK OPERATORS ACTIVATED BY A GENERAL SIGMOID FUNCTION

Here is studied in detail the multivariate fuzzy approximation to the multivariate unit by multivariate fuzzy neural network operators activated by a general sigmoid function. These operators are multivariate fuzzy analogs of earlier studied multivariate Banach space valued ones. The derived results generalize earlier Banach space valued ones into the fuzzy level. Here the high order multivariate fuzzy pointwise and uniform convergences with rates to the multivariate fuzzy unit operator are given through multivariate fuzzy Jackson type inequalities involving the multivariate fuzzy moduli of continuity of the mth order (m ≥ 0) H-fuzzy partial derivatives, of the involved multivariate fuzzy number valued function. The treated operators are of averaged, quasi-interpolation, Kantorovich and quadrature types at the multivariate fuzzy setting.

Hyperbolic Tangent Like Relied Banach Space Valued Neural Network Multivariate Approximations

Abstract Here we examine the multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝ N , N ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a hyperbolic tangent like sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Q-Deformed and β-parametrized half hyperbolic tangent based Banach space valued ordinary and fractional neural network approximation

Here we research the univariate quantitative approximation, ordinary and fractional, of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative of fractional derivatives. Our operators are defined by using a density function generated by a q-deformed and β-parametrized half hyperbolic tangent function, which is a sigmoid function. The approximations are pointwise and of the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer.

Fuzzy fractional more sigmoid function activated neural network approximations revisited

<p style='text-indent:20px;'>Here we study the univariate fuzzy fractional quantitative approximation of fuzzy real valued functions on a compact interval by quasi-interpolation arctangent-algebraic-Gudermannian-generalized symmetrical activation function relied fuzzy neural network operators. These approximations are derived by establishing fuzzy Jackson type inequalities involving the fuzzy moduli of continuity of the right and left Caputo fuzzy fractional derivatives of the involved function. The approximations are fuzzy pointwise and fuzzy uniform. The related feed-forward fuzzy neural networks are with one hidden layer. We study also the fuzzy integer derivative and just fuzzy continuous cases. Our fuzzy fractional approximation result using higher order fuzzy differentiation converges better than in the fuzzy just continuous case.</p>

Parametrized error function based Banach space valued univariate neural network approximation

Here we research the univariate quantitative approximation of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. We perform also the related Banach space valued ractional approximation. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative or fractional derivaties. Our operators are defined by using a density function induced by a parametrized error function. The approximations are pointwise and with respect to the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer. We finish with a convergence analysis.

Jackson type inequalities for differentiable functions in weighted Orlicz spaces

In the present work some Jackson Stechkin type direct theorems of trigonometric approximation are proved in Orlicz spaces with weights satisfying some Muckenhoupt A p A_p condition. To obtain a refined version of the Jackson type inequality, an extrapolation theorem, Marcinkiewicz multiplier theorem, and Littlewood–Paley type results are proved. As a consequence, refined inverse Marchaud type inequalities are obtained. By means of a realization result, an equivalence is found between the fractional order weighted modulus of smoothness and Peetre’s classical weighted K K -functional.

Richards’s curve induced Banach space valued multivariate neural network approximation

Here, we present multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or {mathbb {R}}^{N},Nin {mathbb {N}}, by the multivariate normalized, quasi-interpolation, Kantorovich-type and quadrature-type neural network operators. We examine also the case of approximation by iterated operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high-order Fréchet derivatives. Our multivariate operators are defined using a multidimensional density function induced by the Richards’s curve, which is a generalized logistic function. The approximations are pointwise, uniform and L_{p}. The related feed-forward neural network is with one hidden layer.

Richards's curve induced Banach space valued ordinary and fractional neural network approximation.

Here we perform the univariate quantitative approximation, ordinary and fractional, of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative or fractional derivatives. Our operators are defined by using a density function generated by the Richards curve, which is generalized logistic function. The approximations are pointwise and of the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer.

General multivariate arctangent function activated neural network approximations

Here we expose multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or \(\mathbb{R}^{N}\), \(N\in \mathbb{N}\), by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Frechet derivatives.
 Our multivariate operators are defined by using a multidimensional density function induced by the arctangent function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Bernstein-Jackson-type inequalities with exact constants in Orlicz spaces

We establish the Bernstein and Jackson type inequalities with exact constants for estimations of best approximations by exponential type functions in Orlicz spaces $L_M(\mathbb{R}^n)$. For this purpose, we use a special scale of approximation spaces $\mathcal{B}_\tau^s(M)$ that are interpolation spaces between the subspace $\mathscr{E}_M$ of exponential type functions and the space $L_M(\mathbb{R}^n)$. These approximation spaces are defined using a functional $E\left(t,f\right)$ that plays a similar role as the module of smoothness. The constants in obtained inequalities are expressed using a normalization factor $N_{\vartheta,q}$ that is determined by the parameters $\tau$ and $s$ of the approximation space $\mathcal{B}_\tau^s(M)$.

No Jackson-type estimates for piecewise q -monotone, q ≥ 3 , trigonometric approximation

UDC 517.5 We say that a function f ∈ C [ a , b ] is q -monotone, q ≥ 3 , if f ∈ C q - 2 ( a , b ) and f ( q - 2 ) is convex in ( a , b ) . Let f be continuous and 2 π -periodic, and change its q -monotonicity finitely many times in [ - π , π ] . We are interested in estimating the degree of approximation of f by trigonometric polynomials which are co- q -monotone with it, namely, trigonometric polynomials that change their q -monotonicity exactly at the points where f does. Such Jackson type estimates are valid for piecewise monotone ( q = 1 ) and piecewise convex ( q = 2 ) approximations. However, we prove, that no such estimates are valid, in general, for co- q -monotone approximation, when q ≥ 3 .

Algebraic function based Banach space valued ordinary and fractional neural network approximations

Here we research the univariate quantitative approximation, ordinary and fractional, of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative of fractional derivatives. Our operators are defined by using a density function generated by an algebraic sigmoid function. The approximations are pointwise and of the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer.