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Journal articles on the topic 'Approximation of derivatives of functions'

Author: Grafiati

Published: 5 June 2025

Last updated: 16 July 2025

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1

Jetpisbayeva, A. E., and A. A. Jumabayeva. "Upper Estimates of the angle best approximations of generalized Liouville-Weyl derivatives." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 103, no. 3 (2021): 54–67. http://dx.doi.org/10.31489/2021m3/54-67.

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In this article we consider continuous functions f with period 2π and their approximation by trigonometric polynomials. This article is devoted to the study of estimates of the best angular approximations of generalized Liouville-Weyl derivatives by angular approximation of functions in the three-dimensional case. We consider generalized Liouville-Weyl derivatives instead of the classical mixed Weyl derivative. In choosing the issues to be considered, we followed the general approach that emerged after the work of the second author of this article. Our main goal is to prove analogs of the results of in the three-dimensional case. The concept of general monotonic sequences plays a key role in our study. Several well-known inequalities are indicated for the norms, best approximations of the r-th derivative with respect to the best approximations of the function f. The issues considered in this paper are related to the range of issues studied in the works of Bernstein. Later Stechkin and Konyushkov obtained an inequality for the best approximation f^(r). Also, in the works of Potapov, using the angle approximation, some classes of functions are considered. In subsection 1 we give the necessary notation and useful lemmas. Estimates for the norms and best approximations of the generalized Liouville-Weyl derivative in the three-dimensional case are obtained.

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2

Lamboni, Matieyendou. "Optimal Estimators of Cross-Partial Derivatives and Surrogates of Functions." Stats 7, no. 3 (2024): 697–718. http://dx.doi.org/10.3390/stats7030042.

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Computing cross-partial derivatives using fewer model runs is relevant in modeling, such as stochastic approximation, derivative-based ANOVA, exploring complex models, and active subspaces. This paper introduces surrogates of all the cross-partial derivatives of functions by evaluating such functions at N randomized points and using a set of L constraints. Randomized points rely on independent, central, and symmetric variables. The associated estimators, based on NL model runs, reach the optimal rates of convergence (i.e., O(N−1)), and the biases of our approximations do not suffer from the curse of dimensionality for a wide class of functions. Such results are used for (i) computing the main and upper bounds of sensitivity indices, and (ii) deriving emulators of simulators or surrogates of functions thanks to the derivative-based ANOVA. Simulations are presented to show the accuracy of our emulators and estimators of sensitivity indices. The plug-in estimates of indices using the U-statistics of one sample are numerically much stable.

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3

Mamman, John Ojima. "Computational Algorithm for Approximating Fractional Derivatives of Functions." Journal of Modeling and Simulation of Materials 5, no. 1 (2022): 31–38. http://dx.doi.org/10.21467/jmsm.5.1.31-38.

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This paper presents an algorithmic approach for numerically solving Caputo fractional diﬀerentiation. The trapezoidal rule was modiﬁed, the new modiﬁcation was used to derive an algorithm to approximate fractional derivatives of order α > 0, the fractional derivative used was based on Caputo deﬁnition for a given function by a weighted sum of function and its ordinary derivatives values at speciﬁed points. The trapezoidal rule was used in conjunction with the finite difference scheme which is the forward, backward and central difference to derive the computational algorithm for the numerical approximation of Caputo fractional derivative for evaluating functions of fractional order. The study was conducted through some illustrative examples and analysis of error.

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4

Arsent’eva, E. P., and Yu K. Dem’yanovich. "Approximation of functions with growing derivatives." Journal of Mathematical Sciences 178, no. 6 (2011): 565–75. http://dx.doi.org/10.1007/s10958-011-0570-9.

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5

Vinogradov, Oleg L. "On the constants in the inverse theorems for the first derivative." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 8, no. 4 (2021): 559–71. http://dx.doi.org/10.21638/spbu01.2021.401.

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The known proofs of the inverse theorems of the theory of approximation by trigonometric polynomials and by functions of exponential type are based on the idea of S. N. Bernstein to expand a function in a series containing its functions of best approximation. In this paper, a new method to establish the inverse theorems is introduced. We establish simple identities that immediately imply the inverse theorems mentioned and, moreover, with better constants. This method can be applied to derivatives of arbitrary order (not necessarily an integer one) and (with certain modifications) to estimates of some other functionals in terms of best approximations. In this paper, the case of the first derivative of a function itself and of its trigonometrically conjugate is considered.

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6

Singh, Satwinder Jit, and Anindya Chatterjee. "Beyond fractional derivatives: local approximation of other convolution integrals." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2114 (2009): 563–81. http://dx.doi.org/10.1098/rspa.2009.0378.

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Dynamic systems involving convolution integrals with decaying kernels, of which fractionally damped systems form a special case, are non-local in time and hence infinite dimensional. Straightforward numerical solution of such systems up to time t needs computations owing to the repeated evaluation of integrals over intervals that grow like t . Finite-dimensional and local approximations are thus desirable. We present here an approximation method which first rewrites the evolution equation as a coupled infinite-dimensional system with no convolution, and then uses Galerkin approximation with finite elements to obtain linear, finite-dimensional, constant coefficient approximations for the convolution. This paper is a broad generalization, based on a new insight, of our prior work with fractional order derivatives ( Singh & Chatterjee 2006 Nonlinear Dyn. 45 , 183–206). In particular, the decaying kernels we can address are now generalized to the Laplace transforms of known functions; of these, the power law kernel of fractional order differentiation is a special case. The approximation can be refined easily. The local nature of the approximation allows numerical solution up to time t with computations. Examples with several different kernels show excellent performance. A key feature of our approach is that the dynamic system in which the convolution integral appears is itself approximated using another system, as distinct from numerically approximating just the solution for the given initial values; this allows non-standard uses of the approximation, e.g. in stability analyses.

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7

Ditzian, Z., and D. Jiang. "Approximation of Functions by Polynomials in C[-L, 1]." Canadian Journal of Mathematics 44, no. 5 (1992): 924–40. http://dx.doi.org/10.4153/cjm-1992-057-2.

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AbstractA pointwise estimate for the rate of approximation by polynomials , For 0 ≤ ƛ ≤ 1, integer r, and δn(x) = n-1 + φ(x), is achieved here. This formula bridges the gap between the classical estimate mentioned in most texts on approximation and obtained by Timan and others (ƛ = 0) and the recently developed estimate by Totik and first author (ƛ = 1 ). Furthermore, a matching converse result and estimates on derivatives of the approximating polynomials and their rate of approximation are derived. These results also cover the range between the classical pointwise results and the modern norm estimates for C[— 1,1].

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8

Hornik, Kurt, Maxwell Stinchcombe, Halbert White, and Peter Auer. "Degree of Approximation Results for Feedforward Networks Approximating Unknown Mappings and Their Derivatives." Neural Computation 6, no. 6 (1994): 1262–75. http://dx.doi.org/10.1162/neco.1994.6.6.1262.

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Recently Barron (1993) has given rates for hidden layer feedforward networks with sigmoid activation functions approximating a class of functions satisfying a certain smoothness condition. These rates do not depend on the dimension of the input space. We extend Barron's results to feedforward networks with possibly nonsigmoid activation functions approximating mappings and their derivatives simultaneously. Our conditions are similar but not identical to Barron's, but we obtain the same rates of approximation, showing that the approximation error decreases at rates as fast as n−1/2, where n is the number of hidden units. The dimension of the input space appears only in the constants of our bounds.

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9

Jacobs, B. A. "A New Grünwald-Letnikov Derivative Derived from a Second-Order Scheme." Abstract and Applied Analysis 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/952057.

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A novel derivation of a second-order accurate Grünwald-Letnikov-type approximation to the fractional derivative of a function is presented. This scheme is shown to be second-order accurate under certain modifications to account for poor accuracy in approximating the asymptotic behavior near the lower limit of differentiation. Some example functions are chosen and numerical results are presented to illustrate the efficacy of this new method over some other popular choices for discretizing fractional derivatives.

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10

Chang, Xirong. "Approximation of locally integrable functions on the real line." International Journal of Wavelets, Multiresolution and Information Processing 12, no. 05 (2014): 1461002. http://dx.doi.org/10.1142/s0219691314610025.

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The aim of this paper is to extend (ψ, β)-derivatives to [Formula: see text]-derivatives for locally integrable functions on the real line and then investigate problems of approximation of the classes of functions determined by these derivatives with the use of entire functions of exponential type.

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11

Petersen, Philipp. "Shearlet approximation of functions with discontinuous derivatives." Journal of Approximation Theory 207 (July 2016): 127–38. http://dx.doi.org/10.1016/j.jat.2016.02.004.

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12

KAPOOR, G. P., and SRIJANANI ANURAG PRASAD. "CUBIC SPLINE SUPER FRACTAL INTERPOLATION FUNCTIONS." Fractals 22, no. 01n02 (2014): 1450005. http://dx.doi.org/10.1142/s0218348x14500054.

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In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [x0, xN], the interpolating Cubic Spline SFIF[Formula: see text] and their derivatives [Formula: see text] converge respectively to the data generating function y(x) ≡ y(0)(x) and its derivatives y(j)(x) at the rate of h2-j+ϵ(0 < ϵ < 1), j = 0, 1, 2, as the norm h of the partition of [x0, xN] approaches zero. The convergence results for Cubic Spline SFIF found here show that any desired accuracy can be achieved in the approximation of a regular data generating function and its derivatives by a Cubic Spline SFIF and its corresponding derivatives.

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13

Tang, Bowei. "On the Continuity of Functions and Their Application in Polynomial Approximation." Highlights in Science, Engineering and Technology 88 (March 29, 2024): 220–27. http://dx.doi.org/10.54097/j728na47.

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Researchers continue to explore the topological properties of continuity, including the use of open sets, neighborhoods, and limit points to define and characterize continuous functions. For other aspects, scholars have made achievements in both linear operators and functionals in the context of functional analysis, including applications to integral and differential equations. The development would also see achievements in studies with partial derivatives, directional derivatives, and differentiability. This study primarily discusses how to understand the continuity of one-variable and multivariable functions and introduces some methods for determining continuity. It then explores the application of function continuity in polynomial approximation, specifically focusing on approximating continuous functions in different scenarios using polynomials.

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14

Anastassiou, George A., and Dimitra Kouloumpou. "Neural Network Approximation for Time Splitting Random Functions." Mathematics 11, no. 9 (2023): 2183. http://dx.doi.org/10.3390/math11092183.

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In this article we present the multivariate approximation of time splitting random functions defined on a box or RN,N∈N, by neural network operators of quasi-interpolation type. We achieve these approximations by obtaining quantitative-type Jackson inequalities engaging the multivariate modulus of continuity of a related random function or its partial high-order derivatives. We use density functions to define our operators. These derive from the logistic and hyperbolic tangent sigmoid activation functions. Our convergences are both point-wise and uniform. The engaged feed-forward neural networks possess one hidden layer. We finish the article with a great variety of applications.

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15

Radziyevska, Elena, and Iryna Kovalska. "The Inverse Theorem for the Generalized Derivative in Banach Spaces." Mathematical and computer modelling. Series: Physical and mathematical sciences 24 (December 5, 2023): 101–8. http://dx.doi.org/10.32626/2308-5878.2023-24.101-108.

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Establishing the properties of the approximation characteristics of the studied functions is one of the main tasks of the theory of approximations. If, based on information about the behavior of the generalized derivative of some function f, it is possible to predict the behavior of the sequence of the best approximations of this function by polynomials, then we are talking about stating and proving direct theorems of the theory of approximations. If the properties of the function f Î X itself and its generalized derivatives are studied, relying on the behavior of the sequence best approximations, i.e., the differential-difference characteristics of the function f are established based on the study of the behavior of the sequence of its best approximations, then we speak of the proof of inverse theorems of approximation theory. The study of direct and inverse theorems begins with the works of Bernstein, Valle Poussin, Jackson and others in 1910-1912. They were continued by many scientists (N. I. Ahiezer, M. G. Crane, J. Favar, B. V. Stechkin, S. M. Nikolskyi, A. F. Timan, A. Zygmund, V. K. Dzyadyk, O. I. Stepanets). There are still many important and unsolved problems in the theory of approximations, in particular, such as extending direct and inverse theorems to new classes of functions and establishing the best values of constants in the corresponding inequalities. At the same time, it becomes possible to formulate new problems, in particular, problems of mathematical modeling already for whole classes of functions, that describe the studied processes. This article considers the inverse theorem – based on the properties of the sequence of best approximations, a conclusion is made about the properties of the element f of some Banach space X and its generalized derivatives. As well as the relations between Szego constants for different equivalent systems of elements of the Banach space are established

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16

Akobirshoev, M. O. "On the best simultaneous “angle” approximation in the mean of periodic functions of two variables from some classes." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 7 (August 21, 2024): 24–36. http://dx.doi.org/10.26907/0021-3446-2024-7-24-36.

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In the L2 metric, we obtain sharp inequalities between the best joint approximations of 2π-periodic functions f(x, y) differentiable in each of the variables and their successive derivatives f (μ,ν) (x, y) (μ = 0, 1, . . . , r; ν = 0, 1, . . . , s) by trigonometric “angles” with double integrals containing mixed moduli of continuity of higher orders of higher derivatives. The sharp values of the upper bound of the best joint approximation of some classes of functions given by the specified moduli of continuity are found.

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17

Shabozov, M. Sh, and Kh M. Khuromonov. "On the best polynomial approximation of analytical functions in the Bergman space B2." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 4 (April 25, 2025): 90–103. https://doi.org/10.26907/0021-3446-2025-4-91-103.

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In this paper a number of extreme problems related to the best polynomial approximation of analytical in a circle $U:=\{z\in\mathbb{C}:|z|<1\}$ functions belonging to the Bergman's space $B_2$ are being solved. The bilateral inequality is proved, which is a generalization of the result of periodic functions $f\in L_{2}$, by M.Sh.Shabozov--G.A.Yusupov obtained for the class $L_{2}^{(r)}[0,2\pi]$-in which $(r-1)$ the derivative of $f^{(r-1)}$ is absolutely continuous, and the derivative of $r $ is order of $f^{(r)}\ in L_{2}$ in the case of a polynomial approximation of $f\in \mathcal{A}(U)$ belonging to $B_{2}^{(r)}(U)$.A number of cases are given when the bilateral inequality turns into equality. For some classes of functions belonging to $B_2$, the exact values of the known $n$-diameters are found, and the problem of joint approximation of functions and their intermediate derivatives is solved.

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18

Sorych, Viktor, та Nina Sorych. "Extreme Values of the Best Approximations of Linear Combinations оf Harmonic Functions". Mathematical and computer modelling. Series: Physical and mathematical sciences 24 (5 грудня 2023): 108–18. http://dx.doi.org/10.32626/2308-5878.2023-24.108-118.

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Extreme problems and their practical applications have been under the scrutiny of mathematicians since ancient times. An important step in the development of extreme problems was made by P. L. Chebyshev, who in the 50s of the 19th century laid the foundations of a section of destructive function theory – the theory of approximation. A significant role of the formation of the theory of approximation of functions was played by Carl Weierstrass’s theorem on the convergence to zero of best approximations by polynomials of a continuous function. As is well known, Weierstrass’s theorem is not constructive – it does not contained estimates of the approach speed. Thanks to the work of D. Jackson, S. N. Bernstein, Vallee-Poussin and others, such estimates began to appear in works on approximation theory. At the same time, at the first stages of the development of the theory of approximation, approximations of individual functions were studied. That beginning of a new period, a dipper study of the deviation values of functions from their approximating polynomials, dates back to the 30s and 40s of the 20th century and is associated with the names of A. M. Kolmogorov, S. M. Nikolsky, J. Favard, N. I. Achieser, M. G. Crane and B. Nagy. Thanks to their works, the main emphasis in the theory of approximations is shifted to the study of the best approximations or other approximation characteristics of functions that have certain differential-difference or smoothness properties. In particular, in 1936, J. Favard calculated the exact values of the best uniform approximations by trigonometric polynomials of order no higher than n – 1 on classes of differentiable 2π-periodic functions, whose r-th (r – natural) derivatives are in a unit sphere of the space of essentially bounded functions. The problem of obtaining exact values of the best approximations in uniform and integral metrics for various functional compacts was in sight of many prominent mathematicians of the XX century. General issues related to the study of the best approximation functional: the existence of a polynomial of the best approximation, its characteristic properties, are destribed in detail in many monographs, in particular, for example, in the book by M. P. Korneichuk [1]. In the 80s and 90s of the XX century, O. I. Stepanets (see, [2, section III]) developed a new approach to the classification of periodic functions, which allowed for a fairly fine classification of extremely wide sets of periodic functions. At the same time, the results obtained for these classes are, on the one hand, general, and on the other hand, they give a number of new, hitherto unknown results that were impossible to obtain on previously known classes. Following the approaches to the requirements of function classification, we can consider a linear combination of function classes of a more complex nature. And then the problem of finding the exact values of the upper bounds of the best joint approximations will be reduced to the problem of the best approximation of this composite class corresponding to convolutions with the composite kernel.

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19

Min, G. "On approximation of functions and their derivatives by quasi-Hermite interpolation." International Journal of Mathematics and Mathematical Sciences 19, no. 2 (1996): 279–86. http://dx.doi.org/10.1155/s0161171296000385.

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In this paper, we consider the simultaneous approximation of the derivatives of the functions by the corresponding derivatives of quasi-Hermite interpolation based on the zeros of(1−x2)pn(x)(wherepn(x)is a Legendre polynomial). The corresponding approximation degrees are given. It is shown that this matrix of nodes is almost optimal.

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20

Fayyadh, Mohammed Hamad, та Alaa Adnan Auad. "Trigonometric Approximation by Modulus of Smoothness in Lp,α (X)". Al-Mustansiriyah Journal of Science 32, № 3 (2021): 20. http://dx.doi.org/10.23851/mjs.v32i3.953.

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In this paper, we study the approximate properties of functions by means of trigonometric polynomials in weighted spaces. Relationships between modulus of smoothness of function derivatives and those of the jobs themselves are introduced. In the weighted spaces we also proved of theorems about the relationship between the derivatives of the polynomials for the best approximation and the best approximation of the functions

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21

Khuromonov, Khuromon Mamadamonovich. "The best joint approximation of some classes of functions in the Bergman space 𝐵_2". Chebyshevskii Sbornik 25, № 5 (2025): 183–94. https://doi.org/10.22405/2226-8383-2024-25-5-183-194.

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In this paper, we study several extreme problems related to the best joint approximation of certain classes of analytical functions in the unit circle given by higher-order continuity modules in the Bergman space 𝐵_2. It should be noted that for the first time the problem of joint approximation of periodic differentiable functions and their consecutive derivatives by trigonometric polynomials and their corresponding derivatives in a uniform metric wasinvestigated by A.L.Garkavi [1]. The results obtained in [1] were generalized by A.F.Timan [2] for a class of integer functions of exponential type on the entire line. In the monograph [3].The problems of joint approximation are generalized to some classical theorems of the theory of approximation of functions. However, in the listed works, only asymptotically accurate results were obtained. In this paper, we prove a number of exact theorems for the joint approximation of analytic functions in the unit circle belonging to the Bergman space 𝐵_2, complementing the results of M.Sh.Shabozov [4].

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22

Hào, Dinh Nho, H. J. Reinhardt, and A. Schneider. "Stable approximation of fractional derivatives of rough functions." BIT Numerical Mathematics 35, no. 4 (1995): 488–503. http://dx.doi.org/10.1007/bf01739822.

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23

Berezovskii, A. I., and N. E. Nechiporenko. "Optimal accuracy approximation of functions and their derivatives." Journal of Soviet Mathematics 54, no. 2 (1991): 799–803. http://dx.doi.org/10.1007/bf01097590.

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24

George, A. Anastassiou. "q-Deformed and β-parametrized half hyperbolic tangent based Banach space valued ordinary and fractional neural network approximation". Annals of Communications in Mathematics 6, № 1 (2023): 1–16. https://doi.org/10.5281/zenodo.10059043.

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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. 

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25

Rao, Nadeem, Mohammad Farid, and Mohd Raiz. "On the Approximations and Symmetric Properties of Frobenius–Euler–Şimşek Polynomials Connecting Szász Operators." Symmetry 17, no. 5 (2025): 648. https://doi.org/10.3390/sym17050648.

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This study focuses on approximating continuous functions using Frobenius–Euler–Simsek polynomial analogues of Szász operators. Test functions and central moments are computed to study convergence uniformly, approximation order by these operators. Next, we investigate approximation order uniform convergence via Korovkin result and the modulus of smoothness for functions in continuous functional spaces. A Voronovskaja theorem is also explored approximating functions which belongs to the class of function having first and second order continuous derivative. Further, we discuss numerical error and graphical analysis. In the last, two dimensional operators are constructed to discuss approximation for the class of two variable continuous functions.

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26

Babenko, V. F., and D. A. Levchenko. "The best approximation of certain unbounded operators on classes of multivariable functions." Researches in Mathematics 20 (August 10, 2012): 34. http://dx.doi.org/10.15421/241205.

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We obtain the value of the best approximation of the linear combination, with non-negative coefficients, of the second partial derivatives and mixed derivatives of the second order on the class of multivariable functions with bounded third partial derivatives.

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27

Gupta, Vijay, Th M. Rassias, P. N. Agrawal, and Meenu Goyal. "Approximation with certain genuine hybrid operators." Filomat 32, no. 6 (2018): 2335–48. http://dx.doi.org/10.2298/fil1806335g.

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In the present article, we introduce a general sequence of summation-integral type operators. We establish some direct results which include Voronovskaja type asymptotic formula, point-wise convergence for derivatives, error estimations in terms of modulus of continuity and weighted approximation for these operators. Furthermore, the convergence of these operators and their first order derivatives to certain functions and their corresponding derivatives respectively is illustrated by graphics using Matlab algorithms for some particular values of the parameters c and ?.

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Anastassiou, George A. "Abstract Univariate Neural Network Approximation Using a q-Deformed and λ-Parametrized Hyperbolic Tangent Activation Function". Fractal and Fractional 7, № 3 (2023): 208. http://dx.doi.org/10.3390/fractalfract7030208.

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In this work, we perform univariate approximation with rates, basic and fractional, of continuous functions that take values into an arbitrary Banach space with domain on a closed interval or all reals, by quasi-interpolation neural network operators. These approximations are achieved by deriving Jackson-type inequalities via the first modulus of continuity of the on hand function or its abstract integer derivative or Caputo fractional derivatives. Our operators are expressed via a density function based on a q-deformed and λ-parameterized hyperbolic tangent activation sigmoid function. The convergences are pointwise and uniform. The associated feed-forward neural networks are with one hidden layer.

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29

Krivovichev, Gerasim V., and Elena S. Marnopolskaya. "The approach to optimization of finite-difference schemes for the advective stage of finite-difference-based lattice Boltzmann method." International Journal of Modeling, Simulation, and Scientific Computing 11, no. 01 (2020): 2050002. http://dx.doi.org/10.1142/s1793962320500026.

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The approach to optimization of finite-difference (FD) schemes for the linear advection equation (LAE) is proposed. The FD schemes dependent on the scalar dimensionless parameter are considered. The parameter is included in the expression, which approximates the term with spatial derivatives. The approach is based on the considering of the dispersive and dissipative characteristics of the schemes as the functions of the parameter. For the proper choice of the parameter, these functions are minimized. The approach is applied to the optimization of two-step schemes with an asymmetric approximation of time derivative and with various approximations of the spatial term. The cases of schemes from first to fourth approximation orders are considered. The optimal values of the parameter are obtained. Schemes with the optimal values are applied to the solution of test problems with smooth and discontinuous initial conditions. Also, schemes are used in the FD-based lattice Boltzmann method (LBM) for modeling of the compressible gas flow. The obtained numerical results demonstrate the convergence of the schemes and decaying of the numerical dispersion.

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Ghorbani, Hojjat, Yaghoub Mahmoudi, and Farhad Dastmalchi Saei. "Numerical Study of Fractional Mathieu Differential Equation Using Radial Basis Functions." Mathematical Modelling of Engineering Problems 7, no. 4 (2020): 568–76. http://dx.doi.org/10.18280/mmep.070409.

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In this paper, we introduce a method based on Radial Basis Functions (RBFs) for the numerical approximation of Mathieu differential equation with two fractional derivatives in the Caputo sense. For this, we suggest a numerical integration method for approximating the improper integrals with a singularity point at the right end of the integration domain, which appear in the fractional computations. We study numerically the affects of characteristic parameters and damping factor on the behavior of solution for fractional Mathieu differential equation. Some examples are presented to illustrate applicability and accuracy of the proposed method. The fractional derivatives order and the parameters of the Mathieu equation are changed to study the convergency of the numerical solutions.

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Novikov, Oleg, and Olga Rovenska. "Integral presentation of deviations of rectangular linear means of Fourier series on classes of periodic differentiable functions." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 32 (December 28, 2018): 92–103. http://dx.doi.org/10.37069/1683-4720-2018-32-10.

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The paper deals with the problems of approximation in a uniform metric of periodic functions of many variables by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. Questions of asymptotic behavior of the upper bounds of deviations of linear operators generated by the use of linear methods of summation of Fourier series on the classes of periodic differentiable functions are studied in many works. Methods of investigation of integral representations of deviations of polynomials on the classes of periodic differentiable functions of real variable originated and received its development through the works of S.M. Nikol'skii, S.B. Stechkin, N.P.Korneichuk, V.K. Dzadik, A.I. Stepanets, etc. Along with the study of approximation by linear methods of classes of functions of one variable, are studied similar problems of approximation by linear methods of classes of functions of many variables. In addition to the approximative properties of rectangular Fourier sums, are studied approximative properties of other approximation methods: the rectangular sums of Valle Poussin, Zigmund, Rogozinsky, Favar. In this paper we consider the classes of \(\overline{\psi}\)-differentiable periodic functions of many variables, allowing separately to take into account the properties of partial and mixed \(\overline{\psi}\)-derivatives, and given by analogy with the classes of \(\overline{\psi}\)-differentiable periodic functions of one variable. Integral representations of rectangular linear means of Fourier series on classes of \(\overline{\psi}\)-differentiable periodic functions of many variables are obtained. The obtained formulas can be useful for further investigation of the approximative properties of various linear rectangular methods on the classes \(\overline{\psi}\)-differentiable periodic functions of many variables in order to obtain a solution to the corresponding Kolmogorov-Nikolsky problems.

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32

Setukha, A. V. "Ob approksimatsii poverkhnostnykh proizvodnykh funktsiy s primeneniem integral'nykh operatorov." Дифференциальные уравнения 59, no. 6 (2023): 828–42. http://dx.doi.org/10.31857/s0374064123060122.

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Integral formulas are presented for approximating the surface gradient (of a scalar function given on a surface) and divergence (of a tangent vector field given on a surface) that are analogs of the well-known formulas for the derivatives of a function on a plane. Estimates of the error in the approximation of these functions are obtained. The question of subsequent approximation of the integrals that give expression for the surface gradient and divergence by quadrature sums over the values of the function under study at the nodes selected on the cells of the unstructured grid approximating the surface is also considered.

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33

Mostapha Frih, El, and Paul M. Gauthier. "Approximation of a Function and its Derivatives by Entire Functions of Several Variables." Canadian Mathematical Bulletin 31, no. 4 (1988): 495–99. http://dx.doi.org/10.4153/cmb-1988-071-1.

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34

Anastassiou, George A. "q-Deformed and delta-parametrized A-generalized logistic function induced Banach space valued multivariate multi layer neural network approximations." Studia Universitatis Babes-Bolyai Matematica 69, no. 3 (2024): 587–612. http://dx.doi.org/10.24193/subbmath.2024.3.08.

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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.

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35

Sintsova, Ksenia А., and Nikolay A. Shirokov. "Approximation by polynomials composed of Weierstrass doubly periodic functions." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 10, no. 1 (2023): 61–72. http://dx.doi.org/10.21638/spbu01.2023.106.

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The problem of describing classes of functions in terms of the rate of approximation of these functions by polynomials, rational functions, splines entered in the theory of approximation more than 100 years ago and still retains its relevance. Among a large number of problems related to approximation, we considered the problem of polynomial approximation in two variables of a function defined on the continuum of an elliptic curve in C2 and holomorphic in its interior. The formulation of such a question led to the need to study the approximation of a function that is continuous on the continuum of the complex plane and analytic in its interior, using polynomials in doubly periodic Weierstrass functions and their derivatives. This work is devoted to the development of this topic.

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36

Acar, Tuncer, Lakshmi Narayan Mishra, and Vishnu Narayan Mishra. "Simultaneous Approximation for Generalized Srivastava-Gupta Operators." Journal of Function Spaces 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/936308.

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We introduce a new Stancu type generalization of Srivastava-Gupta operators to approximate integrable functions on the interval0,∞and estimate the rate of convergence for functions having derivatives of bounded variation. Also we present simultenaous approximation by new operators in the end of the paper.

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37

Anastassiou, George. "q-DEFORMED AND λ-PARAMETRIZED A-GENERALIZED LOGISTIC FUNCTION BASED COMPLEX VALUED TRIGONOMETRIC AND HYPERBOLIC NEURAL NETWORK HIGH ORDER APPROXIMATIONS". Journal of Mathematical Analysis 14, № 3 (2023): 1–29. https://doi.org/10.54379/jma-2023-3-1.

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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.

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38

Gadjiev, Akif, and Nursel Çetіn. "Approximation of analytic functions by sequences of linear operators." Filomat 28, no. 1 (2014): 99–106. http://dx.doi.org/10.2298/fil1401099g.

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In the present paper, we investigate approximation of analytic functions and their derivatives in a bounded simply connected domain by the sequences of linear operators without the properties of k?positivity.

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39

NAVASCUÉS, M. ANTONIA, and M. VICTORIA SEBASTIÁN. "SOME RESULTS OF CONVERGENCE OF CUBIC SPLINE FRACTAL INTERPOLATION FUNCTIONS." Fractals 11, no. 01 (2003): 1–7. http://dx.doi.org/10.1142/s0218348x03001550.

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Fractal interpolation functions (FIFs) provide new methods of approximation of experimental data. In the present paper, a fractal technique generalizing cubic spline functions is proposed. A FIF f is defined as the fixed point of a map between spaces of functions. The properties of this correspondence allow to deduce some inequalities that express the sensitivity of these functions and their derivatives to those changes in the parameters defining them. Under some hypotheses on the original function, bounds of the interpolation error for f, f′ and f′′ are obtained. As a consequence, the uniform convergence to the original function and its derivative as the interpolation step tends to zero is proved. According to these results, it is possible to approximate, with arbitrary accuracy, a smooth function and its derivatives by using a cubic spline fractal interpolation function (SFIF).

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40

George, A. Anastassiou. "Parametrized error function based Banach space valued univariate neural network approximation." Annals of Communications in Mathematics 6, no. 1 (2023): 31–43. https://doi.org/10.5281/zenodo.10059027.

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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 fractional 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. 

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Laurinčikas, Antanas, and Darius Šiaučiūnas. "Joint Approximation by Dirichlet L-Functions." Mathematica Slovaca 72, no. 1 (2022): 51–66. http://dx.doi.org/10.1515/ms-2022-0004.

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Abstract In the paper, collections of analytic functions are simultaneously approximated by collections of shifts of Dirichlet L-functions (L(s + iγ1(τ), χ 1),…, L(s + iγ r (τ), χr )), with arbitrary Dirichlet characters χ 1,…, χr . The differentiable functions γ1(τ), …, γ r (τ) and their derivatives satisfy certain growth conditions. The obtained results extend those of [PAŃKOWSKI, Ł.: Joint universality for dependent L-functions, Ramanujan J. 45 (2018), 181–195].

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42

Dosiyev, Adiguzel, and Hediye Sarikaya. "14-point difference operator for the approximation of the first derivatives of a solution of Laplace’s equation in a rectangular parallelepiped." Filomat 32, no. 3 (2018): 791–800. http://dx.doi.org/10.2298/fil1803791d.

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A 14-point difference operator is used to construct finite difference problems for the approximation of the solution, and the first order derivatives of the Dirichlet problem for Laplace?s equations in a rectangular parallelepiped. The boundary functions ?j on the faces ?j, j = 1,2,...,6 of the parallelepiped are supposed to have pth order derivatives satisfying the H?lder condition, i.e., ?j ? Cp,?(?j), 0 < ? < 1, where p = {4,5}. On the edges, the boundary functions as a whole are continuous, and their second and fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error uh - u of the approximate solution uh at each grid point (x1,x2,x3), ?uh-u?? c?p-4(x1,x2,x3)h4 is obtained, where u is the exact solution, ? = ? (x1, x2,x3) is the distance from the current grid point to the boundary of the parallelepiped, h is the grid step, and c is a constant independent of ? and h. It is proved that when ?j ? Cp,?, 0 < ? < 1, the proposed difference scheme for the approximation of the first derivative converges uniformly with order O(hp-1), p ? {4,5}.

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43

Anastassiou, George A. "Multiple general sigmoids based Banach space valued neural network multivariate approximation." Cubo (Temuco) 25, no. 3 (2023): 411–39. http://dx.doi.org/10.56754/0719-0646.2503.411.

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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.

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44

López-Moreno, Antonio-Jesús, and Vijay Gupta. "Exponential moments and simultaneous approximation properties for Durrmeyer type operators with weights of Szasz basis functions." Filomat 35, no. 5 (2021): 1465–75. http://dx.doi.org/10.2298/fil2105465l.

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The present paper deals with the approximation properties for exponential functions of general Durrmeyer type operators having the weights of Sz?sz basis functions. Here we give explicit expressions for exponential type moments by means of which we establish, for the derivatives of the operators, the Voronovskaja formulas for functions of exponential growth and the corresponding weighted quantitative estimates for the remainder in simultaneous approximation.

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45

Pezza, Laura, and Simmaco Di Lillo. "Fractional Dynamical Systems Solved by a Collocation Method Based on Refinable Spaces." Axioms 12, no. 5 (2023): 451. http://dx.doi.org/10.3390/axioms12050451.

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A dynamical system is a particle or set of particles whose state changes over time. The dynamics of the system is described by a set of differential equations. If the derivatives involved are of non-integer order, we obtain a fractional dynamical system. In this paper, we considered a fractional dynamical system with the Caputo fractional derivative. We collocated the fractional differential problem in dyadic nodes and used refinable functions as approximation functions to achieve a good degree of freedom in the choice of the regularity. The collocation method stands out as a particularly useful and attractive tool for solving fractional differential problems of various forms. A numerical result is presented to show that the numerical solution fits the analytical one very well. We collocated the fractional differential problem in dyadic nodes using refinable functions as approximation functions to achieve a good degree of freedom in the choice of regularity.

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46

Lal, Shyam, and Neha Patel. "Legendre wavelet approximation of functions having derivatives of Lipschitz class." Filomat 35, no. 2 (2021): 381–97. http://dx.doi.org/10.2298/fil2102381l.

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In this paper, Legendre Wavelet approximation of functions f having first derivative f' and second derivative f'' of Lip? class, 0 < ? ? 1, have been determined. These wavelet estimators are sharper, better and best possible in Wavelet Analysis. It is observed that the LegendreWavelet estimator of f whose f'' ? Lip? is sharper than the estimator of f having f ' ?Lip? class.

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47

(Sharma), Prerna Maheshwari, and Vijay Gupta. "On Rate of Approximation by Modified Beta Operators." International Journal of Mathematics and Mathematical Sciences 2009 (2009): 1–8. http://dx.doi.org/10.1155/2009/205649.

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48

Kim, Ji-Eun. "Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System." Axioms 10, no. 3 (2021): 206. http://dx.doi.org/10.3390/axioms10030206.

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The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains.

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49

Ovchintsev, Mikhail. "Optimal recovery of derivatives of Hardy class functions." E3S Web of Conferences 110 (2019): 01043. http://dx.doi.org/10.1051/e3sconf/201911001043.

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The paper considers the best linear method for approximating the values of derivatives of Hardy class functions in the unit circle at zero according to the information about the values of functions at a finite number of points z1,...,zn that form a regular polygon, and also the error of the best method is obtained. The introduction provides the necessary concepts and results from the papers of K.Yu. Osipenko. Some results of the studies of S.Ya. Khavinson and other authors are also mentioned here. The main section consists of two parts. In the first part of the second section, the research method is disclosed, namely, the error of the best method for approximating the derivatives at zero according to the information about the values of functions at the points z1,...,zn is calculated; the corresponding extremal function is written out. It is established that for p>1, the corresponding extremal function is unique up to a constant factor that is equal to one in modulus. For p=1, the corresponding extremal function is not unique. All such corresponding extremal functions are determined here. In the second part of the second section, it is proved that for all p (1≤p<∞), the best linear approximation method is unique, and the coefficients of the best linear recovery method are calculated. The expressions used to calculate the coefficients are greatly simplified. At the end of the paper, the obtained results are described, and possible areas for further research are indicated.

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50

Cheng, Rongjun, Fengxin Sun, and Jufeng Wang. "High-order numerical modeling for two-dimensional two-sided space-fractional wave equation based on meshless method." International Journal of Computational Materials Science and Engineering 06, no. 01 (2017): 1750002. http://dx.doi.org/10.1142/s2047684117500026.

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The Grunwald formula is the traditional method to deal with Riemann–Liouville fractional derivative, while its convergence is only [Formula: see text]. In this paper, a high-order polynomial approximation is presented for the Riemann–Liouville fractional derivative. The quadratic polynomial functions and their fractional derivatives with explicit expressions are constructed to approximate the fractional derivative instead of Grunwald formula or shifted Grunwald formula. We proved that this technique has convergence of [Formula: see text]. Based on the MLS approximation and the high-order polynomial approximation and center difference method, a meshless analysis is proposed for the two-dimensional two-sided space-fractional wave equations (SFWE). The SFWE is found to be very adequate in describing anomalous transport and dispersion phenomena. In the meshless method, the trial function for the SFWE is constructed by the MLS approximation and the Riemann–Liouville fractional derivative is approximated by the high-order polynomial approximation, and the essential boundary conditions can be directly and easily imposed on as finite element method. This technique avoids singular integral, and has high accuracy and efficiency. Numerical results demonstrate that this method is highly accurate and computationally efficient for space-fractional wave equations.

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