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 Carpathian Mathematical PublicationsNumber of Followers: 0     Open Access journal ISSN (Print) 2075-9827 Published by Vasyl Stefanyk Precarpathian National University  [2 journals]
• The use of the isometry of function spaces with different numbers of
variables in the theory of approximation of functions

• Authors: D.M. Bushev; F.G. Abdullayev, I.V. Kal'chuk, M. Imashkyzy
Abstract: In the work, we found integral representations for function spaces that are isometric to spaces of entire functions of exponential type, which are necessary for giving examples of equality of approximation characteristics in function spaces isometric to spaces of non-periodic functions. Sufficient conditions are obtained for the nonnegativity of the delta-like kernels used to construct isometric function spaces with various numbers of variables.
PubDate: Thu, 30 Dec 2021 15:44:35 +000

• An efficient hybrid technique for the solution of fractional-order partial
differential equations

• Authors: H.K. Jassim; H. Ahmad, A. Shamaoon, C. Cesarano
Abstract: In this paper, a hybrid technique called the homotopy analysis Sumudu transform method has been implemented solve fractional-order partial differential equations. This technique is the amalgamation of Sumudu transform method and the homotopy analysis method. Three examples are considered to validate and demonstrate the efficacy and accuracy of the present technique. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the exact solution which shows that the proposed method is efficient, reliable and easy to implement for various related problems of science and engineering.
PubDate: Thu, 30 Dec 2021 15:11:02 +000

• Generalization of Szász operators: quantitative estimate and bounded
variation

• Authors: K. Bozkurt; M.L. Limmam, A. Aral
Abstract: Difference of exponential type Szász and Szász-Kantorovich operators is obtained. Similar estimates are given for higher order $\mu$-derivatives of the Szász operators and the Szász-Kantorovich type operators acting on the same order $\mu$-derivative of the function. These differences are given in quantitative form using first modulus of continuity. Convergence in variation of the operators in the space of functions with bounded variation with respect to the variation seminorm is obtained. The results propose a general framework covering the results provided by previous literature.
PubDate: Thu, 30 Dec 2021 14:51:19 +000

• Convergence properties of generalized Lupaş-Kantorovich operators

• Authors: M. Qasim; A. Khan, Z. Abbas, M. Mursaleen
Abstract: In the present paper, we consider the Kantorovich modification of generalized Lupaş operators, whose construction depends on a continuously differentiable, increasing and unbounded function $\rho$. For these new operators we give weighted approximation, Voronovskaya type theorem, quantitative estimates for the local approximation.
PubDate: Thu, 30 Dec 2021 00:00:00 +000

• Non-symmetric approximations of functional classes by splines on the real
line

• Authors: N.V. Parfinovych
Abstract: Let $S_{h,m}$, $h>0$, $m\in {\mathbb N}$, be the spaces of polynomial splines of order $m$ of deficiency 1 with nodes at the points $kh$, $k\in {\mathbb Z}$. We obtain exact values of the best $(\alpha, \beta)$-approximations by spaces $S_{h,m}\cap L_1({\mathbb R})$ in the space $L_1({\mathbb R})$ for the classes $W^r_{1,1}({\mathbb R})$, $r\in {\mathbb N}$, of functions, defined on the whole real line, integrable on ${\mathbb R}$ and such that their $r$th derivatives belong to the unit ball of $L_1({\mathbb R})$. These results generalize the well-known G.G. Magaril-Ilyaev's and V.M. Tikhomirov's results on the exact values of the best approximations of classes $W^r_{1,1}({\mathbb R})$ by splines from $S_{h,m}\cap L_1({\mathbb R})$ (case $\alpha=\beta=1$), as well as are non-periodic analogs of the V.F. Babenko's result on the best non-symmetric approximations of classes $W^r_1({\mathbb T})$ of $2\pi$-periodic functions with $r$th derivative belonging to the unit ball of $L_1({\mathbb T})$ by periodic polynomial splines of minimal deficiency. As a corollary of the main result, we obtain exact values of the best one-sided approximations of classes $W^r_1$ by polynomial splines from $S_{h,m}({\mathbb T})$. This result is a periodic analogue of the results of A.A. Ligun and V.G. Doronin on the best one-sided approximations of classes $W^r_1$ by spaces $S_{h,m}({\mathbb T})$.
PubDate: Thu, 30 Dec 2021 00:00:00 +000

• Approximation of classes of periodic functions of several variables with
given majorant of mixed moduli of continuity

• Authors: O.V. Fedunyk-Yaremchuk; S.B. Hembars'ka
Abstract: In this paper, we continue the study of approximation characteristics of the classes $B^{\Omega}_{p,\theta}$ of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers. We obtain the exact-order estimates of the orthoprojective widths of the classes $B^{\Omega}_{p,\theta}$ in the space $L_{q},$ $1\leq p<q<\infty,$ and also establish the exact-order estimates of approximation for these classes of functions in the space $L_{q}$ by using linear operators satisfying certain conditions.
PubDate: Thu, 30 Dec 2021 00:00:00 +000

• Approximation characteristics of the isotropic Nikol'skii-Besov
functional classes

• Authors: S.Ya. Yanchenko; O.Ya. Radchenko
Abstract: In the paper, we investigates the isotropic Nikol'skii-Besov classes $B^r_{p,\theta}(\mathbb{R}^d)$ of non-periodic functions of several variables, which for $d = 1$ are identical to the classes of functions with a dominant mixed smoothness $S^{r}_{p,\theta}B(\mathbb{R})$. We establish the exact-order estimates for the approximation of functions from these classes $B^r_{p,\theta}(\mathbb{R}^d)$ in the metric of the Lebesgue space $L_q(\mathbb{R}^d)$, by entire functions of exponential type with some restrictions for their spectrum in the case $1 \leqslant p \leqslant q \leqslant \infty$, $(p,q)\neq \{(1,1), (\infty, \infty)\}$, $d\geq 1$. In the case $2<p=q<\infty$, $d=1$, the established estimate is also new for the classes $S^{r}_{p,\theta}B(\mathbb{R})$.
PubDate: Thu, 30 Dec 2021 00:00:00 +000

• Approximation by trigonometric polynomials in the variable exponent
weighted Morrey spaces

• Authors: Z. Cakir; C. Aykol, V.S. Guliyev, A. Serbetci
Abstract: In this paper we investigate the best approximation by trigonometric polynomials in the variable exponent weighted Morrey spaces ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$, where $w$ is a weight function in the Muckenhoupt $A_{p(\cdot)}(I_{0})$ class. We get a characterization of $K$-functionals in terms of the modulus of smoothness in the spaces ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$. Finally, we prove the direct and inverse theorems of approximation by trigonometric polynomials in the spaces ${\mathcal{\widetilde{M}}}_{p(\cdot),\lambda(\cdot)}(I_{0},w),$ the closure of the set of all trigonometric polynomials in ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$.
PubDate: Wed, 29 Dec 2021 20:00:27 +000

• A solution of the fractional differential equations in the setting of
$b$-metric space

• Authors: H. Afshari; E. Karapinar
Abstract: In this paper, we study the existence of solutions for the following differential equations by using a fixed point theorems $\begin{cases} D^{\mu}_{c}w(\varsigma)\pm D^{\nu}_{c}w(\varsigma)=h(\varsigma,w(\varsigma)),& \varsigma\in J,\ \ 0<\nu<\mu<1,\\ w(0)=w_0,& \ \end{cases}$ where $D^{\mu}$, $D^{\nu}$ is the Caputo derivative of order $\mu$, $\nu$, respectively and $h:J\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous. The results are well demonstrated with the aid of exciting examples.
PubDate: Wed, 29 Dec 2021 19:47:06 +000

• Lupaş post quantum Bernstein operators over arbitrary compact
intervals

• Authors: A. Khan; M. Iliyas, M.S. Mansoori, M. Mursaleen
Abstract: This paper deals with Lupaş post quantum Bernstein operators over arbitrary closed and bounded interval constructed with the help of Lupaş post quantum Bernstein bases. Due to the property that these bases are scale invariant and translation invariant, the derived results on arbitrary intervals are important from computational point of view. Approximation properties of Lupaş post quantum Bernstein operators on arbitrary compact intervals based on Korovkin type theorem are studied. More general situation along all possible cases have been discussed favouring convergence of sequence of Lupaş post quantum Bernstein operators to any continuous function defined on compact interval. Rate of convergence by modulus of continuity and functions of Lipschitz class are computed. Graphical analysis has been presented with the help of MATLAB to demonstrate approximation of continuous functions by Lupaş post quantum Bernstein operators on different compact intervals.
PubDate: Tue, 28 Dec 2021 11:18:02 +000

• Lipschitz symmetric functions on Banach spaces with symmetric bases

• Authors: M.V. Martsinkiv; S.I. Vasylyshyn, T.V. Vasylyshyn, A.V. Zagorodnyuk
Abstract: We investigate Lipschitz symmetric functions on a Banach space $X$ with a symmetric basis. We consider power symmetric polynomials on $\ell_1$ and show that they are Lipschitz on the unbounded subset consisting of vectors $x\in \ell_1$ such that $x_n \le 1.$ Using functions $\max$ and $\min$ and tropical polynomials of several variables, we constructed a large family of Lipschitz symmetric functions on the Banach space $c_0$ which can be described as a semiring of compositions of tropical polynomials over $c_0$.
PubDate: Mon, 13 Dec 2021 17:27:19 +000

• Fixed points of hesitant fuzzy set-valued maps with applications

• Authors: M.S. Shagari; A. Azam
Abstract: In this paper, the notion of hesitant fuzzy fixed points is introduced. To this end, we define Suzuki-type $(\alpha,\beta)$-weak contractions in the framework of hesitant fuzzy set-valued maps, thereby establishing some corresponding fixed point theorems. The presented concept herein is an extension of fuzzy set-valued and multi-valued mappings in the corresponding literature. Examples are provided to support the assertions and generality of our obtained ideas. Moreover, one of our results is applied to investigate sufficient conditions for existence of a class of functional equation arising in dynamic programming.
PubDate: Sun, 12 Dec 2021 18:40:52 +000

• Application of symmetric analytic functions to spectra of linear operators

• Authors: I. Burtnyak; I. Chernega, V. Hladkyi, O. Labachuk, Z. Novosad
Abstract: The paper is devoted to extension of the theory of symmetric analytic functions on Banach sequence spaces to the spaces of nuclear and $p$-nuclear operators on the Hilbert space. We introduced algebras of symmetric polynomials and analytic functions on spaces of $p$-nuclear operators, described algebraic bases of such algebras and found some connection with the Fredholm determinant of a nuclear operator. In addition, we considered cases of compact and bounded normal operators on the Hilbert space and discussed structures of symmetric polynomials on corresponding spaces.
PubDate: Sat, 11 Dec 2021 18:26:17 +000

• Direct and inverse theorems on the approximation of almost periodic
functions in Besicovitch-Stepanets spaces

• Authors: A.S. Serdyuk; A.L. Shidlich
Abstract: Direct and inverse approximation theorems are proved in the Besicovitch-Stepanets spaces $B{\mathcal S}^{p}$ of almost periodic functions in terms of the best approximations of functions and their generalized moduli of smoothness.
PubDate: Sat, 11 Dec 2021 18:08:10 +000

• Recovery of continuous functions of two variables from their Fourier
coefficients known with error

• Authors: K.V. Pozharska; A.A. Pozharskyi
Abstract: In this paper, we continue to study the classical problem of optimal recovery for the classes of continuous functions. The investigated classes $W^{\psi}_{2,p}$, $1 \leq p < \infty$, consist of functions that are given in terms of generalized smoothness $\psi$. Namely, we consider the two-dimensional case which complements the recent results from [Res. Math. 2020, 28 (2), 24-34] for the classes $W^{\psi}_p$ of univariate functions. As to available information, we are given the noisy Fourier coefficients $y^{\delta}_{i,j} = y_{i,j} + \delta \xi_{i,j}$, $\delta \in (0,1)$, $i,j = 1,2, \dots$, of functions with respect to certain orthonormal system $\{ \varphi_{i,j} \}_{i,j=1}^{\infty}$, where the noise level is small in the sense of the norm of the space $l_p$, $1 \leq p < \infty$, of double sequences $\xi=( \xi_{i,j} )_{i,j=1}^{\infty}$ of real numbers. As a recovery method, we use the so-called $\Lambda$-method of summation given by certain two-dimensional triangular numerical matrix $\Lambda = \{ \lambda_{i,j}^n \}_{i,j=1}^n$, where $n$ is a natural number associated with the sequence $\psi$ that define smoothness of the investigated functions. The recovery error is estimated in the norm of the space $C ([0,1]^2)$ of continuous on $[0,1]^2$ functions. We showed, that for $1\leq p < \infty$, under the respective assumptions on the smoothness parameter $\psi$ and the elements of the matrix $\Lambda$, it holds $\Delta( W^{\psi}_{2,p}, \Lambda, l_p)= \sup\limits_{ y \in W^{\psi}_{2,p} } \sup\limits_{\ \xi \ _{l_p} \leq 1} \Big\ y - \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n} \lambda_{i,j}^n ( y_{i,j} + \delta \xi_{i,j}) \varphi_{i,j} \Big\ _{C ([0,1]^2)} \ll \frac{ n^{\beta + 1 - 1/{p}}}{\psi(n)}.$
PubDate: Fri, 10 Dec 2021 07:30:04 +000

• Approximation properties of multivariate exponential sampling series

• Authors: S. Kurşun; M. Turgay, O. Alagöz, T. Acar
Abstract: In this paper, we generalize the family of exponential sampling series for functions of $n$ variables and study their pointwise and uniform convergence as well as the rate of convergence for the functions belonging to space of $\log$-uniformly continuous functions. Furthermore, we state and prove the generalized Mellin-Taylor's expansion of multivariate functions. Using this expansion we establish pointwise asymptotic behaviour of the series by means of Voronovskaja type theorem.
PubDate: Tue, 07 Dec 2021 12:09:27 +000

• Approximation properties of modified Jain-Gamma operators

• Authors: S. Erdogan; A. Olgun
Abstract: In the present paper, we study some approximation properties of a modified Jain-Gamma operator. Using Korovkin type theorem, we first give approximation properties of such operator. Secondly, we compute the rate of convergence of this operator by means of the modulus of continuity and we present approximation properties of weighted spaces. Finally, we obtain the Voronovskaya type theorem of this operator.
PubDate: Tue, 07 Dec 2021 11:33:27 +000

• On convergence of branched continued fraction expansions of Horn's
hypergeometric function $H_3$ ratios

• Authors: T.M. Antonova
Abstract: The paper deals with the problem of convergence of the branched continued fractions with two branches of branching which are used to approximate the ratios of Horn's hypergeometric function $H_3(a,b;c;{\bf z})$. The case of real parameters $c\geq a\geq 0,$ $c\geq b\geq 0,$ $c\neq 0,$ and complex variable ${\bf z}=(z_1,z_2)$ is considered. First, it is proved the convergence of the branched continued fraction for ${\bf z}\in G_{\bf h}$, where $G_{\bf h}$ is two-dimensional disk. Using this result, sufficient conditions for the uniform convergence of the above mentioned branched continued fraction on every compact subset of the domain $\displaystyle H=\bigcup_{\varphi\in(-\pi/2,\pi/2)}G_\varphi,$ where $\begin{split} G_{\varphi}=\big\{{\bf z}\in\mathbb{C}^{2}:&\;{\rm Re}(z_1e^{-i\varphi})<\lambda_1 \cos\varphi,\; {\rm Re}(z_2e^{-i\varphi}) <\lambda_2 \cos\varphi, \\ &\; z_k +{\rm Re}(z_ke^{-2i\varphi})<\nu_k\cos^2\varphi,\;k=1,2;\; \\ &\; z_1z_2 -{\rm Re}(z_1z_2e^{-2\varphi})<\nu_3\cos^{2}\varphi\big\}, \end{split}$ are established.
PubDate: Sun, 05 Dec 2021 15:10:38 +000

• Asymptotic properties of Urysohn type generalized sampling operators

• Authors: H. Karsli
Abstract: The concern of this study is to continue the investigation of convergence properties of Urysohn type generalized sampling operators, which are defined by the author in [Dolomites Res. Notes Approx. 2021, 14 (2), 58-67]. In details, the paper centers around to investigation of the asymptotic properties together with some Voronovskaya-type theorems for the linear and nonlinear counterpart of Urysohn type generalized sampling operators.
PubDate: Sat, 04 Dec 2021 13:19:02 +000

• Parabolic convergence regions of branched continued fractions of the
special form

• Authors: D.I. Bodnar; I.B. Bilanyk
Abstract: Using the criterion of convergence of branched continued fractions of the special form with positive elements, effective sufficient criteria of convergence for these fractions are established. To study the parabolic regions of convergence, the element regions and value regions technique was used. In particular, half-planes are considered as value regions. A multidimensional analogue of Tron's twin convergence regions for branched continued fractions of the special form is established. The obtained results made it possible to establish the conditions for the convergence of the multidimensional $S$-fractions with independent variables.
PubDate: Sat, 20 Nov 2021 10:27:26 +000

• Asymmetric circular graph with Hosoya index and negative continued
fractions

• Authors: T. Komatsu
Abstract: It has been known that the Hosoya index of caterpillar graph can be calculated as the numerator of the simple continued fraction. Recently in [MATCH Commun. Math. Comput. Chem. 2020, 84 (2), 399-428], the author introduces a more general graph called caterpillar-bond graph and shows that its Hosoya index can be calculated as the numerator of the general continued fraction. In this paper, we show how the Hosoya index of the graph with non-uniform ring structure can be calculated from the negative continued fraction. We also give the relation between some radial graphs and multidimensional continued fractions in the sense of the Hosoya index.
PubDate: Sat, 20 Nov 2021 09:55:11 +000

• Approximation of functions of several variables by multidimensional
$S$-fractions with independent variables

• Authors: R.I. Dmytryshyn; S.V. Sharyn
Abstract: The paper deals with the problem of approximation of functions of several variables by branched continued fractions. We study the correspondence between formal multiple power series and the so-called "multidimensional $S$-fraction with independent variables". As a result, the necessary and sufficient conditions for the expansion of the formal multiple power series into the corresponding multidimensional $S$-fraction with independent variables have been established. Several numerical experiments show the efficiency, power and feasibility of using the branched continued fractions in order to numerically approximate certain functions of several variables from their formal multiple power series.
PubDate: Sat, 20 Nov 2021 09:16:48 +000

• Interpolational $(L,M)$-rational integral fraction on a continual set of
nodes

• Authors: Ya.O. Baranetskij; I.I. Demkiv, M.I. Kopach, A.V. Solomko
Abstract: In the paper, an integral rational interpolant on a continual set of nodes, which is the ratio of a functional polynomial of degree $L$ to a functional polynomial of degree $M$, is constructed and investigated. The resulting interpolant is one that preserves any rational functional of the resulting form.
PubDate: Fri, 19 Nov 2021 13:03:05 +000

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