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Journal articles on the topic 'Multivariate Operator Theory'

Author: Grafiati

Published: 6 September 2023

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1

Xu, Yuan. "Multivariate orthogonal polynomials and operator theory." Transactions of the American Mathematical Society 343, no. 1 (January 1, 1994): 193–202. http://dx.doi.org/10.1090/s0002-9947-1994-1169912-x.

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2

Anastassiou, George A. "Multivariate and abstract approximation theory for Banach space valued functions." Demonstratio Mathematica 50, no. 1 (August 28, 2017): 208–22. http://dx.doi.org/10.1515/dema-2017-0020.

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Abstract Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.

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3

Filipiak, Katarzyna, Daniel Klein, and Erika Vojtková. "The properties of partial trace and block trace operators of partitioned matrices." Electronic Journal of Linear Algebra 33 (May 16, 2018): 3–15. http://dx.doi.org/10.13001/1081-3810.3688.

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The aim of this paper is to give the properties of two linear operators defined on non-square partitioned matrix: the partial trace operator and the block trace operator. The conditions for symmetry, nonnegativity, and positive-definiteness are given, as well as the relations between partial trace and block trace operators with standard trace, vectorizing and the Kronecker product operators. Both partial trace as well as block trace operators can be widely used in statistics, for example in the estimation of unknown parameters under the multi-level multivariate models or in the theory of experiments for the determination of an optimal designs under the linear models.

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4

Fuchs, David, Steven Sherwood, and Daniel Hernandez. "An Exploration of Multivariate Fluctuation Dissipation Operators and Their Response to Sea Surface Temperature Perturbations." Journal of the Atmospheric Sciences 72, no. 1 (January 1, 2015): 472–86. http://dx.doi.org/10.1175/jas-d-14-0077.1.

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Abstract The fluctuation–dissipation theorem (FDT) has been proposed as a method of calculating the mean response of the atmosphere to small external perturbations. This paper explores the application of the theory under time and space constraints that approximate realistic conditions. To date, most applications of the theory in the climate context used univariate, low-dimensional-state representations of the climate system and an arbitrarily long sample size. The authors explore high-dimensional multivariate FDT operators and the lower bounds of sample size needed to construct skillful operators. It is shown that the skill of the operator depends on the selection of variables and features representing the climate system and that these features change once memory (slab ocean) is added to the system. In addition, it is found that the FDT operator has skill in estimating the response to realistic sea surface temperature (SST) patterns, such as El Niño–Southern Oscillation (ENSO), despite the fact that these patterns were not part of the data used to produce the operator. The response of clouds is also studied; for variables that represent cloud properties, the decrease in skill in relation to decrease in sample size still maintains the key features of the response.

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5

Hiai, Fumio, Yuki Seo, and Shuhei Wada. "Ando–Hiai type inequalities for multivariate operator means." Linear and Multilinear Algebra 67, no. 11 (June 27, 2018): 2253–81. http://dx.doi.org/10.1080/03081087.2018.1488938.

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6

Neretin, Yury A. "Multi-operator colligations and multivariate characteristic functions." Analysis and Mathematical Physics 1, no. 2-3 (August 2, 2011): 121–38. http://dx.doi.org/10.1007/s13324-011-0009-y.

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7

Reich, Nils. "Anisotropic Operator Symbols Arising From Multivariate Jump Processes." Integral Equations and Operator Theory 63, no. 1 (December 22, 2008): 127–50. http://dx.doi.org/10.1007/s00020-008-1645-y.

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8

Zizler, Peter. "The spectral radius of a multivariate sampling operator." Linear Algebra and its Applications 385 (July 2004): 463–74. http://dx.doi.org/10.1016/j.laa.2003.12.042.

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9

Chen, Zhong Yin. "The Traits of Two-Directional Orthogonal Multivariate Small-Wave Packages with Finite Support." Advanced Materials Research 430-432 (January 2012): 543–46. http://dx.doi.org/10.4028/www.scientific.net/amr.430-432.543.

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In this paper, we introduce the notion of vector-valued multiresolution analysis and two-directional vector-valued multivariate wavelet packages associated with an integer-valued dilation matrix. A novel method for constructing multi-dimensional vector-valued wavelet packet is presented. Their characteristics are researched by means of operator theory, time-frequency analysis method and matrix theory. Three orthogonality formulas concerning the wavelet packets are establ- lished. Orthogonality decomposition relation formulas of the space are derived by constru- cting a series of subspaces of wavelet packets.

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10

Xia, Yin Hong, and Hua Li. "The Characterization of a Kind of Vector-Valued Multivariate Wavelet Packets with Composite Dilation Matrix." Key Engineering Materials 439-440 (June 2010): 932–37. http://dx.doi.org/10.4028/www.scientific.net/kem.439-440.932.

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In this article, the notion of a kind of multivariate vector-valued wavelet packets with composite dilation matrix is introduced. A new method for designing a kind of biorthogonal vector- valued wavelet packets in higher dimensions is developed and their biorthogonality property is inv- -estigated by virtue of matrix theory, time-frequency analysis method, and operator theory. Two biorthogonality formulas concerning these wavelet packets are presented. Moreover, it is shown how to gain new Riesz bases of space by constructing a series of subspace of wavelet packets.

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11

Zhang, Tong Qi. "The Information Optimal Algorithm Based on Poly-Scaled Wavelet Wraps and Wavelet Frames with Finite Support." Advanced Materials Research 204-210 (February 2011): 1759–62. http://dx.doi.org/10.4028/www.scientific.net/amr.204-210.1759.

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In this paper, we propose the notion of vector-valued multiresolution analysis and the vector-valued mutivariate wavelet wraps with multi-scale factor of spaceL2(Rn, Cv), which are ge- neralizations of multivariate wavelet wraps. An approach for designing a sort of biorthogonal vec- tor-valued wavelet wraps in higher dimensions is presented and their biorthogonality trait is charac- -terized by virtue of integral transform, matrix theory, and operator theory. Two biorthogonality formulas regarding these wavelet wraps are established.

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12

Chen, Qing Jiang, and Zong Tian Wei. "Generation and Characteristics of a Class of Four-Dimensional Multiwavelet Packets with Six-Scale Dilation Factor." Key Engineering Materials 460-461 (January 2011): 317–22. http://dx.doi.org/10.4028/www.scientific.net/kem.460-461.317.

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In this paper, we introduce a class of vector-valued wavelet packets of space , which are generalizations of multivariate wavelet packets. A procedure for constructing a class of biorthogonal vector-valued wavelet packets in higher dimensions is presented and their biorthogonality properties are characterized by virtue of matrix theory, time-frequency analysis method, and operator theory. Three biorthogonality formulas regarding these wavelet packets are derived. Moreover, it is shown how to gain new Riesz bases of space from these wavelet packets.

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13

Tian, Xin Xian, and Ai Lian Huo. "Construction and Characteristics of a Kind of Four-Dimensional Vector Wavelet Packets with Four-Scale Dilation." Applied Mechanics and Materials 20-23 (January 2010): 1053–59. http://dx.doi.org/10.4028/www.scientific.net/amm.20-23.1053.

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In this paper, we introduce a class of vector-valued wavelet packets of space , which are generalizations of multivariate wavelet packets. A procedure for constructing a class of biorthogonal vector-valued wavelet packets in higher dimensions is presented and their biorthogonality properties are characterized by virtue of matrix theory, time-frequency analysis method, and operator theory. Three biorthogonality formulas regarding these wavelet packets are derived. Moreover, it is shown how to gain new Riesz bases of space from these wavelet packets.

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14

Li, Hong Wu, and Dong Liao. "The Properties of a Class of Higher-Dimensional Composite Wavelet Packet Bases." Key Engineering Materials 439-440 (June 2010): 1099–104. http://dx.doi.org/10.4028/www.scientific.net/kem.439-440.1099.

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In this paper, we introduce a class of vector-valuedwavelet packets of space , which are generalizations of multivariate wavelet packets. A procedure for constructing a class of biorthogonal vector-valued wavelet packets in higher dimensions is presented and their biorthogonality properties are characterized by virtue of matrix theory, time-frequency analysis method and operator theory. Three biorthogonality formulas regarding these wavelet packets are derived. Moreover, it is shown how to gain new Riesz bases of space from these wavelet packets. obtained.

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15

Desta, Henok Desalegn, Deepak B. Pachpatte, Jebessa B. Mijena, and Tadesse Abdi. "Univariate and Multivariate Ostrowski-Type Inequalities Using Atangana–Baleanu Caputo Fractional Derivative." Axioms 11, no. 9 (September 19, 2022): 482. http://dx.doi.org/10.3390/axioms11090482.

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In this paper, we obtain some univariate and multivariate Ostrowski-type inequalities using the Atangana–Baleanu fractional derivative in the sense of Liouville–Caputo (ABC). The results obtained for both left and right ABC fractional derivatives can be applied to study further fractional inequalities and estimate various non-local function problems since the operator consists of a non-singular kernel. The obtained results are more generalized in nature.

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16

Pandey, Kshitij Kumar, and Puthan Veedu Viswanathan. "Multivariate Fractal Functions in Some Complete Function Spaces and Fractional Integral of Continuous Fractal Functions." Fractal and Fractional 5, no. 4 (October 25, 2021): 185. http://dx.doi.org/10.3390/fractalfract5040185.

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There has been a considerable evolution of the theory of fractal interpolation function (FIF) over the last three decades. Recently, we introduced a multivariate analogue of a special class of FIFs, which is referred to as α-fractal functions, from the viewpoint of approximation theory. In the current note, we continue our study on multivariate α-fractal functions, but in the context of a few complete function spaces. For a class of fractal functions defined on a hyperrectangle Ω in the Euclidean space Rn, we derive conditions on the defining parameters so that the fractal functions are elements of some standard function spaces such as the Lebesgue spaces Lp(Ω), Sobolev spaces Wm,p(Ω), and Hölder spaces Cm,σ(Ω), which are Banach spaces. As a simple consequence, for some special choices of the parameters, we provide bounds for the Hausdorff dimension of the graph of the corresponding multivariate α-fractal function. We shall also hint at an associated notion of fractal operator that maps each multivariate function in one of these function spaces to its fractal counterpart. The latter part of this note establishes that the Riemann–Liouville fractional integral of a continuous multivariate α-fractal function is a fractal function of similar kind.

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17

Power, S. C. "Spectral characterization of the Wold–Zasuhin decomposition and prediction-error operator." Mathematical Proceedings of the Cambridge Philosophical Society 110, no. 3 (November 1991): 559–67. http://dx.doi.org/10.1017/s0305004100070626.

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Over thirty years ago Wiener and Masani pointed out in the introduction of their celebrated paper [31] that for a general multivariate stationary stochastic process no relation had been given for the prediction-error matrix in terms of the spectrum of the process. In particular it was unknown how to characterize the rank of the process in spectral terms (cf. Masani[12], p. 369, question 1). Despite explicit progress in this connection with certain regular processes, such as the series representations in [11, 19, 22, 32], or the iterative approach of [28, 29], and despite progress in the structure theory of degenerate processes ([8, 10, 14, 15, 26]), a general relation or series expression has remained elusive.

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18

Chen, Qing Jiang, and Yu Ying Wang. "The Properties of Multidimensional Semi-Orthogonal Frame Wavelet Packets." Key Engineering Materials 439-440 (June 2010): 896–901. http://dx.doi.org/10.4028/www.scientific.net/kem.439-440.896.

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Wavelet analysis has become a popular subject in scientific research during the past twenty years. In this work, we introduce the notion of vector-valued multiresolution analysis and vector-valued multivariate wavelet packets associated with an integer-valued dilation matrix. A novel method for constructing multi-dimen- -sional vector-valued wavelet packet is presented. Their characteristics are researched by means of operator theory, time-frequency analysis method and matrix theory. Three orthogonality formulas concerning the wavelet packets are established. Orthogonality decomposition relation formulas of the space are derived by constructing a series of subspaces of wavelet packets. Finally, one new orthonormal wavelet packet bases of are constructed from these wavelet packets.

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19

Ustimenko, Vasyl, Urszula Romańczuk-Polubiec, Aneta Wróblewska, Monika Katarzyna Polak, and Eustrat Zhupa. "On the Constructions of New Symmetric Ciphers Based on Nonbijective Multivariate Maps of Prescribed Degree." Security and Communication Networks 2019 (April 1, 2019): 1–15. http://dx.doi.org/10.1155/2019/2137561.

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The main purpose of this paper is to introduce stream ciphers with the nonbijective encryption function of multivariate nature constructed in terms of algebraic graph theory. More precisely, we describe the two main symmetric algorithms for creation of multivariate encryption transformations based on three families of bipartite graphs with partition sets isomorphic to Kn, where K is selected as the finite commutative ring. The plainspace of the algorithm is Ω={x∣∑xi∈K⁎, x∈Kn}⊂Kn,Ω≅K⁎×Kn-1. The second algorithm is a generalization of the first one with using the jump operator, where generalized encryption map has an essentially higher degree in comparison with the previous version. Moreover, the degree of this generalized map is not bounded by some constant. This property guarantees resistance of the cipher to linearization attacks.

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20

Steinheuer, Julian, and Petra Friederichs. "Vertical profiles of wind gust statistics from a regional reanalysis using multivariate extreme value theory." Nonlinear Processes in Geophysics 27, no. 2 (April 23, 2020): 239–52. http://dx.doi.org/10.5194/npg-27-239-2020.

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Abstract. Many applications require wind gust estimates at very different atmospheric height levels. For example, the renewable energy sector is interested in wind and gust predictions at the hub height of a wind power plant. However, numerical weather prediction models typically only derive estimates for wind gusts at the standard measurement height of 10 m above the land surface. Here, we present a statistical post-processing method to derive a conditional distribution for hourly peak wind speed as a function of height. The conditioning variables are taken from the COSMO-REA6 regional reanalysis. The post-processing method was trained using peak wind speed observations at five vertical levels between 10 and 250 m from the Hamburg Weather Mast. The statistical post-processing method is based on a censored generalized extreme value (cGEV) distribution with non-homogeneous parameters. We use a least absolute shrinkage and selection operator to select the most informative variables. Vertical variations of the cGEV parameters are approximated using Legendre polynomials, such that predictions may be derived at any desired vertical height. Further, the Pickands dependence function is used to assess dependencies between gusts at different heights. The most important predictors are the 10 m gust diagnostic, the barotropic and the baroclinic mode of absolute horizontal wind speed, the mean absolute horizontal wind at 700 hPa, the surface pressure tendency, and the lifted index. Proper scores show improvements of up to 60 % with respect to climatology, especially at higher vertical levels. The post-processing model with a Legendre approximation is able to provide reliable predictions of gusts' statistics at non-observed intermediate levels. The strength of dependency between gusts at different levels is non-homogeneous and strongly modulated by the vertical stability of the atmosphere.

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21

Pitrik, József, and Dániel Virosztek. "A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means." Linear Algebra and its Applications 609 (January 2021): 203–17. http://dx.doi.org/10.1016/j.laa.2020.09.007.

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22

Shi, Kaihe, Huiru Liu, Li Zhang, and Qing Li. "Analysis of Air Quality Evolution Trends in the Chinese Air Pollution Transmission Channel Cities under Socioeconomic Development Scenarios." Sustainability 15, no. 3 (January 22, 2023): 2118. http://dx.doi.org/10.3390/su15032118.

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In this paper, based on the traditional grey multivariate convolutional model, the concept of a buffer operator is introduced to construct a single-indicator buffered grey multivariate convolutional model applicable to air quality prediction research. The construction steps of the model are described in detail in this paper, and the stability of the model is analyzed based on perturbation theory. Furthermore, the model was applied to predict the air quality composite index of the “2 + 26” Chinese air pollution transmission corridor cities based on different socioeconomic development scenarios in a multidimensional manner. The results show that the single-indicator buffered grey multivariate convolutional model constructed in this paper has better stability in predicting with a small amount of sample data. From 2020 to 2025, the air quality of the target cities selected in this paper follows an improving trend. The population density, secondary industry, and urbanization will not have a significant negative impact on the improvement of air quality if they are kept stable. In the case of steady development of secondary industry, air quality maintained a stable improvement in 96.4% of the “2 + 26” cities. The growth rate of population density will have an inverted U-shaped relationship with the decline in the city air quality composite index. In addition, with the steady development of urbanization, air quality would keep improving steadily in 71.4% of the “2 + 26” cities.

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23

Popescu, Gelu. "Noncommutative Multivariable Operator Theory." Integral Equations and Operator Theory 75, no. 1 (December 8, 2012): 87–133. http://dx.doi.org/10.1007/s00020-012-2012-6.

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24

Rosenkranz, Markus, Xing Gao, and Li Guo. "An algebraic study of multivariable integration and linear substitution." Journal of Algebra and Its Applications 18, no. 11 (August 19, 2019): 1950207. http://dx.doi.org/10.1142/s0219498819502074.

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We set up an algebraic theory of multivariable integration, based on a hierarchy of Rota–Baxter operators and an action of the matrix monoid as linear substitutions. Given a suitable coefficient domain with a bialgebra structure, this allows us to build an operator ring that acts naturally on the given Rota–Baxter hierarchy. We conjecture that the operator relations are a noncommutative Gröbner–Shirshov basis for the ideal they generate.

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25

Kaptanoğlu, H. Turgay. "Aspects of multivariable operator theory on weighted symmetric Fock spaces." Communications in Contemporary Mathematics 16, no. 05 (August 29, 2014): 1350034. http://dx.doi.org/10.1142/s021919971350034x.

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We obtain all Dirichlet spaces ℱq, q ∈ ℝ, of holomorphic functions on the unit ball of ℂN as weighted symmetric Fock spaces over ℂN. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the ℱq. We pick out the analytic Hilbert modules from among the ℱq. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each ℱq. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C*-algebras generated by the shift operators on the ℱq fit in exact sequences that are in the same Ext class. We identify the groups K0 and K1 of the Toeplitz algebras on the ℱq arising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces ℱb that depend on a weight sequence b.

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26

Shakil, M., and M. Ahsanullah. "Some inferences on the distribution of the Demmel condition number of complex Wishart matrices." Special Matrices 5, no. 1 (January 26, 2017): 127–38. http://dx.doi.org/10.1515/spma-2017-0011.

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Abstract In recent years, many researchers have studied the distributions of the Demmel (or the scaled) condition numbers (DCN) of complex Wishart matrices. In this paper, several new distributional properties of the distribution of the Demmel condition number of complex Wishart matrices are presented. The limiting distributions of the standardized extreme order statistics are given. Since the truncated distributions arise in practical statistics where the ability of record observations is limited to a given threshold or within a specified range, there has been a great interest, in recent years, in the characterizations of probability distributions by truncated moments. Before a particular probability distribution model is applied to fit the realworld data, it is necessary to confirm whether the given continuous probability distribution satisfies the underlying requirements by its characterization. Therefore, in this paper, some characterizations of the distribution of DCN are also shown. We hope that the findings of this paper will be quite useful in many fields of pure and applied sciences, such as, probability, statistics, multivariate statistics, linear algebra, operator algebra theory, actuarial science, physics, wireless communications, and polarimetric synthetic aperture radar (PolSAR), among others.

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Shakil, M., and M. Ahsanullah. "A note on the characterizations of the distributions of the condition numbers of real Gaussian matrices." Special Matrices 6, no. 1 (July 1, 2018): 282–96. http://dx.doi.org/10.1515/spma-2018-0022.

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Abstract Many researchers and authors have studied the distributions of the condition numbers of real Gaussian matrices, which appear in many fields of pure and applied sciences, such as, probability, statistics, multivariate statistics, linear algebra, operator algebra theory, actuarial science, physics, wireless communications, and polarimetric synthetic aperture radar (PolSAR). Motivated by this, in this paper, we first present several new distributional properties of the distributions of the condition numbers of real Gaussian matrices. Since it is important to know the percentage points of a given distribution for any statistical application, we have also computed percentiles of the said distributions of the condition numbers. Before a particular probability distribution model is applied to fit the real world data, it is necessary to confirm whether the given continuous probability distribution satisfies the underlying requirements by its characterizations. Also, the truncated distributions arise in practical statisticswhere the ability of record observations is limited to a given threshold or within a specified range. In view of these facts, some characterizations by truncated first moment are also presented. We hope that the findings of this paper will be quite useful to the researchers in various fields of pure and applied sciences as stated above.

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28

Altwaijry, Najla, Silvestru Sever Dragomir, and Kais Feki. "New Results on Boas–Bellman-Type Inequalities in Semi-Hilbert Spaces with Applications." Axioms 12, no. 7 (June 27, 2023): 638. http://dx.doi.org/10.3390/axioms12070638.

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In this article, we investigate new findings on Boas–Bellman-type inequalities in semi-Hilbert spaces. These spaces are generated by semi-inner products induced by positive and positive semidefinite operators. Our objective is to reveal significant properties of such spaces and apply these results to the field of multivariable operator theory. Specifically, we derive new inequalities that relate to the joint A-numerical radius, the joint operator A-seminorm, and the Euclidean A-seminorm of tuples of semi-Hilbert space operators. We assume that A is a nonzero positive operator. Our discoveries provide insights into the structure of semi-Hilbert spaces and have implications for a broad range of mathematical applications and beyond.

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29

Greene, Devin C. V. "Free resolutions in multivariable operator theory." Journal of Functional Analysis 200, no. 2 (June 2003): 429–50. http://dx.doi.org/10.1016/s0022-1236(02)00043-5.

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30

Popescu, Gelu. "Unitary invariants in multivariable operator theory." Memoirs of the American Mathematical Society 200, no. 941 (2009): 0. http://dx.doi.org/10.1090/memo/0941.

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Altwaijry, Najla, Silvestru Sever Dragomir, and Kais Feki. "Inequalities and Reverse Inequalities for the Joint A-Numerical Radius of Operators." Axioms 12, no. 3 (March 22, 2023): 316. http://dx.doi.org/10.3390/axioms12030316.

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In this paper, we aim to establish several estimates concerning the generalized Euclidean operator radius of d-tuples of A-bounded linear operators acting on a complex Hilbert space H, which leads to the special case of the well-known A-numerical radius for d=1. Here, A is a positive operator on H. Some inequalities related to the Euclidean operator A-seminorm of d-tuples of A-bounded operators are proved. In addition, under appropriate conditions, several reverse bounds for the A-numerical radius in single and multivariable settings are also stated.

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32

Previato, Emma. "Multivariable Burchnall–Chaundy theory." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, no. 1867 (June 22, 2007): 1155–77. http://dx.doi.org/10.1098/rsta.2007.2064.

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Burchnall & Chaundy (Burchnall & Chaundy 1928 Proc. R. Soc. A 118 , 557–583) classified the (rank 1) commutative subalgebras of the algebra of ordinary differential operators. To date, there is no such result for several variables. This paper presents the problem and the current state of the knowledge, together with an interpretation in differential Galois theory. It is known that the spectral variety of a multivariable commutative ring will not be associated to a KP-type hierarchy of deformations, but examples of related integrable equations were produced and are reviewed. Moreover, such an algebro-geometric interpretation is made to fit into A.N. Parshin's newer theory of commuting rings of partial pseudodifferential operators and KP-type hierarchies which uses higher local fields.

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33

Arveson, Whilliam. "Subalgebras of C*-algebras III: Multivariable operator theory." Acta Mathematica 181, no. 2 (1998): 159–228. http://dx.doi.org/10.1007/bf02392585.

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34

Popescu, Gelu. "Noncommutative Berezin transforms and multivariable operator model theory." Journal of Functional Analysis 254, no. 4 (February 2008): 1003–57. http://dx.doi.org/10.1016/j.jfa.2007.06.004.

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35

Chaurasia, V. B. L., and Vinod Gill. "New fractional calculus results involving Srivastava’s general class of multivariable polynomials and The H̅ - function." Journal of Applied Mathematics, Statistics and Informatics 11, no. 1 (May 1, 2015): 19–32. http://dx.doi.org/10.1515/jamsi-2015-0002.

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Abstract A significantly large number of earlier works on the subjects of fractional calculus give interesting account of the theory and applications of fractional calculus operators in many different areas of mathematical analysis. In the present paper, we study and develop an important result involving a fractional differential operator for the product of general multivariable polynomials, general polynomial set and two -functions. The result discussed here can be used to investigate a wide class of new and known results, hitherto scattered in the literature. For the sake of illustration, six interesting special case have also been recorded here of our main findings.

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36

Alomari, Mohammad W., Gabriel Bercu, Christophe Chesneau, and Hala Alaqad. "On Some Inequalities for the Generalized Euclidean Operator Radius." Axioms 12, no. 6 (May 31, 2023): 542. http://dx.doi.org/10.3390/axioms12060542.

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In the literature, there are many criteria to generalize the concept of a numerical radius; one of the most recent and interesting generalizations is the so-called generalized Euclidean operator radius, which reads: ωpT1,⋯,Tn:=supx=1∑i=1nTix,xp1/p,p≥1, for all Hilbert space operators T1,⋯,Tn. Simply put, it is the numerical radius of multivariable operators. This study establishes a number of new inequalities, extensions, and generalizations for this type of numerical radius. More precisely, by utilizing the mixed Schwarz inequality and the extension of Furuta’s inequality, some new refinement inequalities are obtained for the numerical radius of multivariable Hilbert space operators. In the case of n=1, the resulting inequalities could be considered extensions and generalizations of the classical numerical radius.

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37

Anastassiou, George A. "General uniform approximation theory by multivariate singular integral operators." Annales Polonici Mathematici 103, no. 1 (2012): 15–25. http://dx.doi.org/10.4064/ap103-1-2.

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38

Richards, Donald St P. "Applications of Invariant Differential Operators to Multivariate Distribution Theory." SIAM Journal on Applied Mathematics 45, no. 2 (April 1985): 280–88. http://dx.doi.org/10.1137/0145015.

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39

Kribs, David W. "On bilateral weighted shifts in noncommutative multivariable operator theory." Indiana University Mathematics Journal 52, no. 6 (2003): 1595–614. http://dx.doi.org/10.1512/iumj.2003.52.2375.

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40

Lee, Sang Hoon, Woo Young Lee, and Jasang Yoon. "Subnormality of Powers of Multivariable Weighted Shifts." Journal of Function Spaces 2020 (November 27, 2020): 1–11. http://dx.doi.org/10.1155/2020/5678795.

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Given a pair T ≡ T 1 , T 2 of commuting subnormal Hilbert space operators, the Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for the existence of a commuting pair N ≡ N 1 , N 2 of normal extensions of T 1 and T 2 ; in other words, T is a subnormal pair. The LPCS is a longstanding open problem in the operator theory. In this paper, we consider the LPCS of a class of powers of 2 -variable weighted shifts. Our main theorem states that if a “corner” of a 2-variable weighted shift T = W α , β ≔ T 1 , T 2 is subnormal, then T is subnormal if and only if a power T m , n ≔ T 1 m , T 2 n is subnormal for some m , n ≥ 1 . As a corollary, we have that if T is a 2-variable weighted shift having a tensor core or a diagonal core, then T is subnormal if and only if a power of T is subnormal.

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41

Dunkl, Charles F. "Integral Kernels with Reflection Group Invariance." Canadian Journal of Mathematics 43, no. 6 (December 1, 1991): 1213–27. http://dx.doi.org/10.4153/cjm-1991-069-8.

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Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group-invariant measure. The theory has been developed in several papers of the author [4,5,6,7]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.

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Beasley, LeRoy B., and Sang-Gu Lee. "Linear operators preserving multivariate majorization." Linear Algebra and its Applications 304, no. 1-3 (January 2000): 141–59. http://dx.doi.org/10.1016/s0024-3795(99)00227-x.

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43

Annaby, M. H. "MULTIVARIATE SAMPLING THEOREMS ASSOCIATED WITH MULTIPARAMETER DIFFERENTIAL OPERATORS." Proceedings of the Edinburgh Mathematical Society 48, no. 2 (May 23, 2005): 257–77. http://dx.doi.org/10.1017/s0013091504000100.

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AbstractWe investigate the multivariate sampling theory associated with multiparameter eigenvalue problems. A several-variable counterpart of the classical sampling theorem of Whittaker, Kotel’nikov and Shannon is given. It arose when the multiparameter system has order one. Two-dimensional sampling theorems associated with two-parameter systems of second-order differential operators will be established. The sampling formulae are of multivariate non-uniform Lagrange interpolation type. Unlike many of the known formulae, the interpolating functions are not necessarily products of single variable functions.

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Lei, Tao, Yi Wang, and Weiwei Luo. "Multivariate Self-Dual Morphological Operators Based on Extremum Constraint." Mathematical Problems in Engineering 2015 (2015): 1–16. http://dx.doi.org/10.1155/2015/596348.

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Self-dual morphological operators (SDMO) do not rely on whether one starts the sequence with erosion or dilation; they treat the image foreground and background identically. However, it is difficult to extend SDMO to multichannel images. Based on the self-duality property of traditional morphological operators and the theory of extremum constraint, this paper gives a complete characterization for the construction of multivariate SDMO. We introduce a pair of symmetric vector orderings (SVO) to construct multivariate dual morphological operators. Furthermore, utilizing extremum constraint to optimize multivariate morphological operators, we construct multivariate SDMO. Finally, we illustrate the importance and effectiveness of the multivariate SDMO by applications of noise removal and segmentation performance. The experimental results show that the proposed multivariate SDMO achieves better results, and they suppress noises more efficiently without losing image details compared with other filtering methods. Moreover, the proposed multivariate SDMO is also shown to have the best segmentation performance after the filtered images via watershed transformation.

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45

Dynin, Alexander. "Multivariable wiener-hopf operators I. Representations." Integral Equations and Operator Theory 9, no. 4 (July 1986): 537–56. http://dx.doi.org/10.1007/bf01204627.

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Dynin, A. "Multivariable Wiener-Hopf operators I. Representations." Integral Equations and Operator Theory 10, no. 6 (November 1987): 904. http://dx.doi.org/10.1007/bf01196126.

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Upmeier, Harald. "An index theorem for multivariable Toeplitz operators." Integral Equations and Operator Theory 9, no. 3 (May 1986): 355–86. http://dx.doi.org/10.1007/bf01199351.

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Costarelli, Danilo, and Gianluca Vinti. "Multivariate sampling Kantorovich operators: from the theory to the Digital Image Processing algorithm." PAMM 15, no. 1 (October 2015): 655–56. http://dx.doi.org/10.1002/pamm.201510317.

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Anastassiou, George A. "Approximation by multivariate sublinear and max-product operators." Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 113, no. 2 (January 18, 2018): 507–40. http://dx.doi.org/10.1007/s13398-018-0491-2.

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Erkuş-Dumanß, Esra, and Oktay Duman. "Integral-type generalizations of operators obtained from certain multivariate polynomials." Calcolo 45, no. 1 (March 2008): 53–67. http://dx.doi.org/10.1007/s10092-008-0143-6.

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