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Journal articles on the topic 'Riesz Functional'

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Published: 4 June 2021
Last updated: 18 February 2022

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1

Buskes, G., B. de Pagter, and A. van Rooij. "Functional calculus on Riesz spaces." Indagationes Mathematicae 2, no. 4 (1991): 423–36. http://dx.doi.org/10.1016/0019-3577(91)90028-6.

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2

Batko, Bogdan. "On Approximate Solutions of Functional Equations in Vector Lattices." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/547673.

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We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra). The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equationF(x+y)+F(x)+F(y)≠0⇒F(x+y)=F(x)+F(y)in Riesz spaces, the Cauchy equation with squaresF(x+y)2=(F(x)+F(y))2inf-algebras, and the quadratic functional equationF(x+y)+F(x-y)=2F(x)+2F(y)in Riesz spaces.
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3

Talakua, Mozart W., and Stenly J. Nanuru. "TEOREMA REPRESENTASI RIESZ–FRECHET PADA RUANG HILBERT." BAREKENG: Jurnal Ilmu Matematika dan Terapan 5, no. 2 (December 1, 2011): 1–8. http://dx.doi.org/10.30598/barekengvol5iss2pp1-8.

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Hilbert space is a very important idea of the Davids Hilbert invention. In 1907, Riesz and Fréchet developed one of the theorem in Hilbert space called the Riesz-Fréchet representationtheorem. This research contains some supporting definitions Banach space, pre-Hilbert spaces, Hilbert spaces, the duality of Banach and Riesz-Fréchet representation theorem. On Riesz-Fréchet representation theorem will be shown that a continuous linear functional that exist in the Hilbert space is an inner product, in other words, there is no continuous linear functional on a Hilbert space except the inner product.
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4

Conway, John B., Domingo A. Herrero, and Bernard B. Morrel. "Completing the Riesz-Dunford functional calculus." Memoirs of the American Mathematical Society 82, no. 417 (1989): 0. http://dx.doi.org/10.1090/memo/0417.

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5

Yang, Zhuyuan, and Luoqing Li. "Approximation by Riesz means on the rotation group SO(3)." International Journal of Wavelets, Multiresolution and Information Processing 15, no. 04 (April 19, 2017): 1750035. http://dx.doi.org/10.1142/s0219691317500357.

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This paper deals with the approximation properties of Riesz means in [Formula: see text]. The convergence rate of the Riesz means is obtained. The equivalence of Riesz means and the Peetre [Formula: see text]-functional on the rotation group [Formula: see text] is established.
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6

Kalauch, Anke, Bas Lemmens, and Onno van Gaans. "Riesz completions, functional representations, and anti-lattices." Positivity 18, no. 1 (May 16, 2013): 201–18. http://dx.doi.org/10.1007/s11117-013-0240-x.

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7

XIAO, XIANG-CHUN, YU-CAN ZHU, and XIAO-MING ZENG. "GENERALIZED p-FRAME IN SEPARABLE COMPLEX BANACH SPACES." International Journal of Wavelets, Multiresolution and Information Processing 08, no. 01 (January 2010): 133–48. http://dx.doi.org/10.1142/s0219691310003419.

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The concept of g-frame and g-Riesz basis in a complex Hilbert space was introduced by Sun.18 In this paper, we generalize the g-frame and g-Riesz basis in a complex Hilbert space to a complex Banach space. Using operators theory and methods of functional analysis, we give some characterizations of a g-frame or a g-Riesz basis in a complex Banach space. We also give a result about the stability of g-frame in a complex Banach space.
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8

Xiao, Xue Mei. "Perturbation Theorems for Frames and Riesz Bases." Applied Mechanics and Materials 433-435 (October 2013): 44–47. http://dx.doi.org/10.4028/www.scientific.net/amm.433-435.44.

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This paper gives a perturbation theorem for frames in a Hilbert space which is a generalization of a result by Ping Zhao. It is proved that the condition a linear operator is invertible can be weakened to be surjective, and a similar result also be obtained for a Riesz basis. The perturbation theorems for frames and Riesz bases in a Hilbert space were studied by operator theory in functional analysis.
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9

Batko, Bogdan. "Stability of the Exponential Functional Equation in Riesz Algebras." Abstract and Applied Analysis 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/848540.

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We deal with the stability of the exponential Cauchy functional equationF(x+y)=F(x)F(y)in the class of functionsF:G→Lmapping a group (G, +) into a Riesz algebraL. The main aim of this paper is to prove that the exponential Cauchy functional equation is stable in the sense of Hyers-Ulam and is not superstable in the sense of Baker. To prove the stability we use the Yosida Spectral Representation Theorem.
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10

Miana, Pedro J. "Algebra Structure of Operator-Valued Riesz Means." Journal of Operators 2014 (May 18, 2014): 1–7. http://dx.doi.org/10.1155/2014/923616.

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We characterize operator-valued Riesz means via an algebraic law of composition and establish their functional calculus accordingly. With this aim, we give a new integral expression of the Leibniz derivation rule for smooth functions.
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11

NG, CHI-KEUNG. "On quaternionic functional analysis." Mathematical Proceedings of the Cambridge Philosophical Society 143, no. 2 (September 2007): 391–406. http://dx.doi.org/10.1017/s0305004107000187.

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AbstractIn this paper, we will show that the category of quaternion vector spaces, the category of (both one-sided and two sided) quaternion Hilbert spaces and the category of quaternion B*-algebras are equivalent to the category of real vector spaces, the category of real Hilbert spaces and the category of real C*-algebras respectively. We will also give a Riesz representation theorem for quaternion Hilbert spaces and will extend the main results in [12] (namely, we will give the full versions of the Gelfand–Naimark theorem and the Gelfand theorem for quaternion B*-algebras). On our way to these results, we compare, clarify and unify the term ‘quaternion Hilbert spaces’ in the literatures.
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12

Colombo, Fabrizio, and Jonathan Gantner. "Formulations of the -functional calculus and some consequences." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 146, no. 3 (May 5, 2016): 509–45. http://dx.doi.org/10.1017/s0308210515000645.

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In this paper we introduce the two possible formulations of the -functional calculus that are based on the Fueter–Sce mapping theorem in integral form and we introduce the pseudo--resolvent equation. In the case of dimension 3 we prove the -resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the -functional calculus.
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13

Zhang, Junli, and Pengcheng Niu. "Hölder Regularity of Quasiminimizers to Generalized Orlicz Functional on the Heisenberg Group." Journal of Function Spaces 2020 (November 25, 2020): 1–13. http://dx.doi.org/10.1155/2020/8838654.

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In this paper, we apply De Giorgi-Moser iteration to establish the Hölder regularity of quasiminimizers to generalized Orlicz functional on the Heisenberg group by using the Riesz potential, maximal function, Calderón-Zygmund decomposition, and covering Lemma on the context of the Heisenberg Group. The functional includes the p -Laplace functional on the Heisenberg group which has been studied and the variable exponential functional and the double phase growth functional on the Heisenberg group that have not been studied.
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14

Bagheri-Bardi, Ghorban Ali, Akaram Elyaspour, Somayeh Javani, and Minoo Khosheghbal-Ghorabayi. "An extension of Riesz dual pairing in non-commutative functional analysis." Colloquium Mathematicum 151, no. 1 (2018): 147–55. http://dx.doi.org/10.4064/cm6777-3-2017.

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15

Polat, Faruk. "Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras." Abstract and Applied Analysis 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/653508.

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16

Wróbel, Błażej. "Dimension freeLpestimates for single Riesz transforms via anH∞joint functional calculus." Journal of Functional Analysis 267, no. 9 (November 2014): 3332–50. http://dx.doi.org/10.1016/j.jfa.2014.07.005.

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17

Ross, David. "Yet another short proof of the Riesz representation theorem." Mathematical Proceedings of the Cambridge Philosophical Society 105, no. 1 (January 1989): 139–40. http://dx.doi.org/10.1017/s0305004100001456.

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F. Riesz's ‘Representation Theorem’ has been proved by methods classical [11, 12], category-theoretic [7], and functional-analytic [2, 9]. (Garling's remarkable proofs [5, 6] owe their brevity to the combined strength of these and other methods.) These proofs often reveal a connection between the Riesz theorem and some unexpected area of mathematics.
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18

Assaad, Joyce. "Riesz transforms, fractional power and functional calculus of Schrödinger operators on weightedLp-spaces." Journal of Mathematical Analysis and Applications 402, no. 1 (June 2013): 220–33. http://dx.doi.org/10.1016/j.jmaa.2013.01.024.

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19

Grobler, J. J. "ON THE FUNCTIONAL CALCULUS IN ARCHIMEDEAN RIESZ SPACES WITH APPLICATIONS TO APPROXIMATION THEOREMS." Quaestiones Mathematicae 11, no. 3 (January 1988): 307–21. http://dx.doi.org/10.1080/16073606.1988.9632147.

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20

Kisil, Vladimir V., and Enrique Ramírez de Arellano. "The Riesz-Clifford Functional Calculus for Non-Commuting Operators and Quantum Field Theory." Mathematical Methods in the Applied Sciences 19, no. 8 (May 25, 1996): 593–605. http://dx.doi.org/10.1002/(sici)1099-1476(19960525)19:8<593::aid-mma783>3.0.co;2-#.

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21

Veghova, Ivana, and Jozef Sumec. "Viscoelastic Effects in Solid Phase Continuous Media - Stress-Strain Relations." Applied Mechanics and Materials 837 (June 2016): 121–26. http://dx.doi.org/10.4028/www.scientific.net/amm.837.121.

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Mathematical modeling of boundary value problems in linear theory of viscoelasticity. Definitions and basic principles in the mathematical modeling theory. Constitutive functional and its transformation into a form of Stieltjes integral. Application of theory of algebraic sets and corresponding subsets. Riesz theory of representation and its application for derivation of constitutive equations. Integral and differential operator forms of stress-strain relationships for a solid-phase continuous media.
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22

Sumec, Jozef, and Lubos Hrustinec. "Time-Dependent Response of Mass Continuous Solid Phase Media by Integral Form of Constitutive Equations - Mathematical Modeling." Applied Mechanics and Materials 837 (June 2016): 163–66. http://dx.doi.org/10.4028/www.scientific.net/amm.837.163.

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Mathematical modeling of boundary value problems in linear theory of viscoelasticity. Definitions and basic principles in the mathematical modeling theory. Constitutive functional and its transformation into a form of Stieltjes integral. Application of theory of algebraic sets and corresponding subsets. Riesz theorem of representation and its application for derivation of constitutive equations. Integral operator forms of stress-strain relationships for a solid-phase continuous media.
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23

LEE, YUH-JIA, and CHEN-YUH SHIH. "THE RIESZ REPRESENTATION THEOREM ON INFINITE DIMENSIONAL SPACES AND ITS APPLICATIONS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 05, no. 01 (March 2002): 41–59. http://dx.doi.org/10.1142/s0219025702000705.

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Let H be a real separable Hilbert space and let E⊂H be a nuclear space with the chain {Em: m=1,2,…} of Hilbert spaces such that E = ∩m∈ℕEm. Let E* and E-m denote the dual spaces of E and Em, respectively. For γ > 0, let [Formula: see text] be the collection of complex-valued continuous functions f defined on E* such that [Formula: see text] is finite for every m. Then [Formula: see text] is a complete countably normed space equipping with the family {‖·‖m,γ : m = 1,2,…} of norms. Using a probabilistic approach, it is shown that every continuous linear functional T on [Formula: see text] can be represented uniquely by a complex Borel measure νT satisfying certain exponential integrability condition. The results generalize an infinite dimensional Riesz representation theorem given previously by the first author for the case γ = 2. As applications, we establish a Weierstrass approximation theorem on E* for γ≥1 and show that the space [Formula: see text] spanned by the class { exp [i(x,ξ)] : ξ ∈ E} is dense in [Formula: see text] for γ>0. In the course of the proof we give sufficient conditions for a function space on which every positive functional can be represented by a Borel measure on E*.
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24

Li, Luoqing. "The Peetre K-Functional and the Riesz Summability Operator for the Fourier–Legendre Expansions." Journal of Approximation Theory 99, no. 1 (July 1999): 112–21. http://dx.doi.org/10.1006/jath.1998.3309.

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25

Chen, Qing Jiang, and Lie Ya Yan. "The Decription of a Sort of Two-Directional Biorthogonal Trivariate Wavelet Packets with Finite Support." Key Engineering Materials 439-440 (June 2010): 902–7. http://dx.doi.org/10.4028/www.scientific.net/kem.439-440.902.

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The advantages of wavelets and their promising features in various application have attracted a lot of interest and effort in recent years. In this article, the notion of two-directional biorthogonal finitely supported trivariate wavelet packets with multiscale is developed. Their properties is investigated by virtue of algebra theory, time-frequency analysis method and functional analysis method. In the final, new Riesz bases of space are constructed from these wavelet packets. Three biorthogonality formulas regarding these wavelet packets are established
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26

Elst, A. F. M. Ter, and Humberto Prado. "Gaussian bounds for reduced heat kernels of subelliptic operators on nilpotent Lie groups." MATHEMATICA SCANDINAVICA 90, no. 2 (June 1, 2002): 251. http://dx.doi.org/10.7146/math.scand.a-14373.

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We obtain Gaussian estimates for the kernels of the semigroups generated by a class of subelliptic operators $H$ acting on $L_p(\boldsymbol R^k)$. The class includes anharmonic oscillators and Schrödinger operators with external magnetic fields. The estimates imply an $H_\infty$-functional calculus for the operator $H$ on $L_p$ with $p \in \langle 1,\infty\rangle$ and in many cases the spectral $p$-independence. Moreover, we show for a subclass of operators satisfying a homogeneity property that the Riesz transforms of all orders are bounded.
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27

Zhang, Jian, and Shui Wang Guo. "A Study of Matrix-Valued Trivariate Wavelet Packets Associated with an Orthogonal Matrix-Valued Trivariate Scaling Function." Key Engineering Materials 439-440 (June 2010): 1165–70. http://dx.doi.org/10.4028/www.scientific.net/kem.439-440.1165.

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Wavelet analysis has become a developing branch of mathematics for over twenty years. In this paper, the notion of matrix-valued multiresolution analysis of space is introduced. A method for constructing biorthogonal matrix–valued trivariate wavelet packets is developed and their properties are discussed by means of time-frequency analysis method, matrix theory and functional analysis method. Three biorthogonality formulas concerning these wavelet packets are provided. Finally, new Riesz bases of space is obtained by constructing a series of subspaces of biorthogonal matrix-valued wavelet packets.
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28

Kurdyumov, V. P., and A. P. Khromov. "Riesz bases formed by root functions of a functional-differential equation with a reflection operator." Differential Equations 44, no. 2 (February 2008): 203–12. http://dx.doi.org/10.1134/s0012266108020079.

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29

Alpay, Daniel, Fabrizio Colombo, Jonathan Gantner, and David P. Kimsey. "Functions of the infinitesimal generator of a strongly continuous quaternionic group." Analysis and Applications 15, no. 02 (January 25, 2017): 279–311. http://dx.doi.org/10.1142/s021953051650007x.

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The quaternionic analogue of the Riesz–Dunford functional calculus and the theory of semigroups and groups of linear quaternionic operators have recently been introduced and studied. In this paper, we suppose that [Formula: see text] is the quaternionic infinitesimal generator of a strongly continuous group of operators [Formula: see text] and we show how we can define bounded operators [Formula: see text], where [Formula: see text] belongs to a class of functions that is larger than the one to which the quaternionic functional calculus applies, using the quaternionic Laplace–Stieltjes transform. This class includes functions that are slice regular on the [Formula: see text]-spectrum of [Formula: see text] but not necessarily at infinity. Moreover, we establish the relation between [Formula: see text] and the quaternionic functional calculus and we study the problem of finding the inverse of [Formula: see text].
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30

Colombo, Fabrizio, and Irene Sabadini. "The -spectrum and the -functional calculus." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 3 (June 2012): 479–500. http://dx.doi.org/10.1017/s0308210510000338.

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In some recent papers (called $\mathcal{S}$-functional calculus) for n-tuples of both bounded and unbounded not-necessarily commuting operators. The $\mathcal{S}$-functional calculus is based on the notion of $\mathcal{S}$-spectrum, which naturally arises from the definition of the $\mathcal{S}$-resolvent operator for n-tuples of operators. The $\mathcal{S}$-resolvent operator plays the same role as the usual resolvent operator for the Riesz–Dunford functional calculus, which is associated to a complex operator acting on a Banach space. When one considers commuting operators (bounded or unbounded) there is the possibility of simplifying the computation of the $\mathcal{S}$-spectrum. In fact, in this case we can use the F-spectrum, which is easier to compute than the $\mathcal{S}$-spectrum. In the case of commuting operators, our functional calculus is based on the $\mathcal{F}$-spectrum and will be called $\mathcal{SC}$-functional calculus. We point out that for a correct definition of the $\mathcal{S}$-resolvent operator and of the $\mathcal{SC}$-resolvent operator in the unbounded case we have to face different extension problems. Another reason for a more detailed study of the $\mathcal{F}$-spectrum is that it is related to the $\mathcal{F}$-functional calculus which is based on the integral version of the Fueter mapping theorem. This functional calculus is associated to monogenic functions constructed by starting from slice monogenic functions.
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31

Yu, Yu Min, and Yu Qing Zhu. "Properties and Constructing of a Kind of Four-Dimensional Vector Wavelet Packets According to a Dilation Matrix." Key Engineering Materials 439-440 (June 2010): 920–25. http://dx.doi.org/10.4028/www.scientific.net/kem.439-440.920.

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In this paper, we introduce a sort of vector four-dimensional wavelet packets according to a dilation matrix, which are generalizations of univariate wavelet packets. The definition of biortho- gonal vector four-dimensional wavelet packets is provided and their biorthogonality quality is researched by means of time-frequency analysis method, vector subdivision scheme and functional analysis method. Three biorthogonality formulas regarding the wavelet packets are established. Finally, it is shown how to draw new Riesz bases of space from these wavelet packets. The sufficient condition for the existence of four-dimensional wavelet packets is established based on the multiresolution analysis method.
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32

Zhou, Jian Feng. "The Research of a Two-Directional Vector-Valued Quarternary Wavelet Packets and Applications in Material Engineering." Advanced Materials Research 712-715 (June 2013): 2487–92. http://dx.doi.org/10.4028/www.scientific.net/amr.712-715.2487.

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In this paper, we introduce a class of vector-valued four-dimensional wavelet packets according to a dilation matrix, which are generalizations of univariate wavelet packets. The defini -tion of biorthogonal vector four-dimensional wavelet packets is provided and their biorthogonality quality is researched by means of time-frequency analysis method, vector subdivision scheme and functional analysis method. Three biorthogonality formulas regarding the wavelet packets are established. Finally, it is shown how to draw new Riesz bases of space from these wavelet packets. The sufficient condition for the existence of four-dimensional wavelet packets is established based on the multiresolution analysis method.
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33

Burlutskaya, M. Sh. "On Riesz bases of root functions for a class of functional-differential operators on a graph." Differential Equations 45, no. 6 (June 2009): 779–88. http://dx.doi.org/10.1134/s0012266109060019.

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34

Savchuk, A. M., and I. V. Sadovnichaya. "Spectral Analysis of One-Dimensional Dirac System with Summable Potential and Sturm- Liouville Operator with Distribution Coefficients." Contemporary Mathematics. Fundamental Directions 66, no. 3 (December 15, 2020): 373–530. http://dx.doi.org/10.22363/2413-3639-2020-66-3-373-530.

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We consider one-dimensional Dirac operatorLP,U with Birkhoff regular boundary conditions and summable potential P(x) on[0, ]. We introduce strongly and weakly regular operators. In both cases, asymptotic formulas for eigenvalues are found. In these formulas, we obtain main asymptotic terms and estimates for the second term. We specify these estimates depending on the functional class of the potential: Lp[0,] with p [1,2] and the Besov space Bp,p'[0,] with p [1,2] and (0,1/p). Additionally, we prove that our estimates are uniform on balls Pp,R Then we get asymptotic formulas for normalized eigenfunctions in the strongly regular case with the same residue estimates in uniform metric on x [0,]. In the weakly regular case, the eigenvalues 2n and 2n+1 are asymptotically close and we obtain similar estimates for two-dimensional Riesz projectors. Next, we prove the Riesz basis property in the space (L2[0,])2 for a system of eigenfunctions and associated functions of an arbitrary strongly regular operatorLP,U. In case of weak regularity, the Riesz basicity of two-dimensional subspaces is proved. In parallel with theLP,U operator, we consider the SturmLiouville operator Lq,U generated by the differential -y'' + q(x)y expressionwith distribution potential q of first-order singularity (i.e., we assume that the primitive u = q(1) belongs to L2[0, ]) and Birkhoff-regular boundary conditions. We reduce to this case -(1y')'+i(y)'+iy'+0y, operators of more general form where '1,,0(-1)L2and 10. For operator Lq,U, we get the same results on the asymptotics of eigenvalues, eigenfunctions, and basicity as for operator LP,U . Then, for the Dirac operator LP,U, we prove that the Riesz basis constant is uniform over the ballsPp,R for p1 or 0. The problem of conditional basicity is naturally generalized to the problem of equiconvergence of spectral decompositions in various metrics. We prove the result on equiconvergence by varying three indices: fL[0,] (decomposable function), PL[0,] (potential), and Sm-Sm00,m in L[0,] (equiconvergence of spectral decompositions in the corresponding norm). In conclusion, we prove theorems on conditional and unconditional basicity of the system of eigenfunctions and associated functions of operator LP,U in the spaces L[0,],2, and in various Besov spaces Bp,q[0,].
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35

Kurdyumov, V. P., and A. P. Khromov. "The Riesz bases consisting of eigen and associated functions for a functional differential operator with variable structure." Russian Mathematics 54, no. 2 (January 14, 2010): 33–45. http://dx.doi.org/10.3103/s1066369x10020052.

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36

Kurdyumov, V. P., and A. P. Khromov. "On Riesz Basises of the Eigen and Associated Functions of the Functional-Differential Operator with a Variable Structure." Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics 7, no. 2 (2007): 20–25. http://dx.doi.org/10.18500/1816-9791-2007-7-2-20-25.

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37

Indumathi, A., Nagarajan Subramanian, and Ayhan Esi. "Geometric difference of six-dimensional Riesz almost lacunary rough statistical convergence in probabilistic space of 𝜒𝑓 3." Analysis 39, no. 1 (March 1, 2019): 7–17. http://dx.doi.org/10.1515/anly-2018-0051.

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Abstract In this paper we study the concept of generalized geometric difference of six-dimensional rough almost lacunary statistical Cesáro of {\chi^{3}} over probabilistic space P is defined by Musielak–Orlicz function. Since the study of convergence in Probabilistic space P is fundamental to probabilistic functional analysis, we feel that the concept of generalized geometric difference of six-dimensional rough almost lacunary statistical Cesáro of {\chi^{3}} over probabilistic space P defined by Musielak in a probabilistic space P would provide a more general framework for the subject.
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38

Marletta, Marco, Andrei Shkalikov, and Christiane Tretter. "Pencils of differential operators containing the eigenvalue parameter in the boundary conditions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 133, no. 4 (August 2003): 893–917. http://dx.doi.org/10.1017/s0308210500002730.

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The paper deals with linear pencils N − λP of ordinary differential operators on a finite interval with λ-dependent boundary conditions. Three different problems of this form arising in elasticity and hydrodynamics are considered. So-called linearization pairs (W, T) are constructed for the problems in question. More precisely, functional spaces W densely embedded in L2 and linear operators T acting in W are constructed such that the eigenvalues and the eigen- and associated functions of T coincide with those of the original problems. The spectral properties of the linearized operators T are studied. In particular, it is proved that the eigen- and associated functions of all linearizations (and hence of the corresponding original problems) form Riesz bases in the spaces W and in other spaces which are obtained by interpolation between D(T) and W.
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39

Gui, Changfeng, and Hui Guo. "On Nodal Solutions of the Nonlinear Choquard Equation." Advanced Nonlinear Studies 19, no. 4 (November 1, 2019): 677–91. http://dx.doi.org/10.1515/ans-2019-2061.

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AbstractThis paper deals with the general Choquard equation-\Delta u+V(|x|)u=(I_{\alpha}*|u|^{p})|u|^{p-2}u\quad\text{in }\mathbb{R}^{N},where {V\in C([0,\infty),\mathbb{R}^{+})} is bounded below by a positive constant, and {I_{\alpha}} denotes the Riesz potential of order {\alpha\in(0,N)}. In view of the convolution term, the nonlocal property makes the variational functional completely different from the one for local pure power-type nonlinearity. By combining the Brouwer degree and developing some new techniques, a family of radial solutions with a prescribed number of zeros is constructed for {p\in[2,\frac{N+\alpha}{N-2})}, while the degeneracy happens for {p\in(\frac{N+\alpha}{N},2)}. This result complements and improves the ones in the literature in the aspect of the range of p.
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40

Alpay, Daniel, Palle E. T. Jorgensen, and David P. Kimsey. "Moment problems in an infinite number of variables." Infinite Dimensional Analysis, Quantum Probability and Related Topics 18, no. 04 (December 2015): 1550024. http://dx.doi.org/10.1142/s0219025715500241.

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Let [Formula: see text]. Given a closed set [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the set of tuples of nonnegative integers [Formula: see text] with [Formula: see text] for finitely many [Formula: see text], the [Formula: see text]-moment problem on [Formula: see text] entails determining whether or not there exists a measure [Formula: see text] on [Formula: see text] so that [Formula: see text] and [Formula: see text] We prove that [Formula: see text] exists if and only if a natural analogue of the Riesz–Haviland functional [Formula: see text] is [Formula: see text]-positive, i.e. if [Formula: see text] is any polynomial which is nonnegative for all [Formula: see text], then [Formula: see text] We will also provide a sufficient condition for [Formula: see text] to be unique, an analogue of a celebrated theorem of K. Schmüdgen and an application to stochastic processes.
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41

BADRA, MEHDI, FABIEN CAUBET, and MARC DAMBRINE. "DETECTING AN OBSTACLE IMMERSED IN A FLUID BY SHAPE OPTIMIZATION METHODS." Mathematical Models and Methods in Applied Sciences 21, no. 10 (October 2011): 2069–101. http://dx.doi.org/10.1142/s0218202511005660.

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The paper presents a theoretical study of an identification problem by shape optimization methods. The question is to detect an object immersed in a fluid. Here, the problem is modeled by the Stokes equations and treated as a nonlinear least-squares problem. We consider both the Dirichlet and Neumann boundary conditions. Firstly, we prove an identifiability result. Secondly, we prove the existence of the first-order shape derivatives of the state, we characterize them and deduce the gradient of the least-squares functional. Moreover, we study the stability of this setting. We prove the existence of the second-order shape derivatives and we give the expression of the shape Hessian. Finally, the compactness of the Riesz operator corresponding to this shape Hessian is shown and the ill-posedness of the identification problem follows. This explains the need of regularization to numerically solve this problem.
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42

Radzievskii, G. V. "On the norms of riesz projections onto subspaces of root functions of a boundary value problem for a functional-differential expression." Differential Equations 42, no. 1 (January 2006): 54–67. http://dx.doi.org/10.1134/s0012266106010058.

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43

Alpay, Daniel, Fabrizio Colombo, and Irene Sabadini. "Perturbation of the generator of a quaternionic evolution operator." Analysis and Applications 13, no. 04 (April 28, 2015): 347–70. http://dx.doi.org/10.1142/s0219530514500249.

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The theory of slice hyperholomorphic functions, introduced in recent years, has important applications in operator theory. The quaternionic version of this function theory and its Cauchy formula yield to a definition of the quaternionic version of the Riesz–Dunford functional calculus which is based on the notion of S-spectrum. This quaternionic functional calculus allows to define the quaternionic evolution operator which appears in the quaternionic version of quantum mechanics proposed by J. von Neumann and later developed by S. L. Adler. Generation results such as the Hille–Phillips–Yosida theorem have been recently proved. In this paper, we study the perturbation of the generator. The motivation of this study is that, as it happens in the classical case of closed complex linear operators, to verify the generation conditions of the Hille–Phillips–Yosida theorem, in the concrete cases, is often difficult. Thus in this paper we study the generation problem from the perturbation point of view. Precisely, given a quaternionic closed operator T that generates the evolution operator [Formula: see text] we study under which condition a closed operator P is such that T + P generates the evolution operator [Formula: see text]. This paper is addressed to people working in different research areas such as hypercomplex analysis and operator theory.
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44

Macías-Díaz, Jorge E. "Numerically Efficient Methods for Variational Fractional Wave Equations: An Explicit Four-Step Scheme." Mathematics 7, no. 11 (November 13, 2019): 1095. http://dx.doi.org/10.3390/math7111095.

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In this work, we investigate numerically a one-dimensional wave equation in generalized form. The system considers the presence of constant damping and functional anomalous diffusion of the Riesz type. Reaction terms are also considered, in such way that the mathematical model can be presented in variational form when damping is not present. As opposed to previous efforts available in the literature, the reaction terms are not only functions of the solution. Instead, we consider the presence of smooth functions that depend on fractional derivatives of the solution function. Using a finite-difference approach, we propose a numerical scheme to approximate the solutions of the fractional wave equation. Along with this integrator, we propose discrete forms of the local and the total energy operators. In a first stage, we show rigorously that the energy properties of the continuous system are mimicked by our discrete methodology. In particular, we prove that the discrete system is dissipative (respectively, conservative) when damping is present (respectively, absent), in agreement with the continuous model. The theoretical numerical analysis of this system is more complicated in light of the presence of the functional form of the anomalous diffusion. To solve this problem, some novel technical lemmas are proved and used to establish the stability and the quadratic convergence of the scheme. Finally, we provide some computer simulations to show the capability of the scheme to conserve/dissipate the energy. Various fractional problems with functional forms of the anomalous diffusion of the solution are considered to that effect.
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45

Vandana, Dr, _. Deepmala, N. Subramanian, and Vishnu Narayan Mishra. "Riesz triple probabilisitic of almost lacunary ces$\acute{A}$ro $C_{111}$ statistical convergence of $\chi^{3}$ defined by a Musielak Orlicz function." Boletim da Sociedade Paranaense de Matemática 36, no. 4 (October 1, 2018): 23–32. http://dx.doi.org/10.5269/bspm.v36i4.32870.

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In this paper we study the concept of almost lacunary statistical Ces$\acute{a}$ro of $\chi^{3}$ over probabilistic $p-$ metric spaces defined by Musielak Orlicz function. Since the study of convergence in PP-spaces is fundamental to probabilistic functional analysis, we feel that the concept of almost lacunary statistical Ces$\acute{a}$ro of $\chi^{2}$ over probabilistic $p-$ metric spaces defined by Musielak in a PP-space would provide a more general framework for the subject.
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46

Yang, Dachun, Junqiang Zhang, and Ciqiang Zhuo. "Variable Hardy Spaces Associated with Operators Satisfying Davies–Gaffney Estimates." Proceedings of the Edinburgh Mathematical Society 61, no. 3 (May 21, 2018): 759–810. http://dx.doi.org/10.1017/s0013091517000414.

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AbstractLetLbe a one-to-one operator of type ω inL2(ℝn), with ω∈[0, π/2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. Letp(·): ℝn→(0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space$H_L^{p(\cdot )} ({\open R}^n)$associated withL. By means of variable tent spaces, the authors establish the molecular characterization of$H_L^{p(\cdot )} ({\open R}^n)$. Then the authors show that the dual space of$H_L^{p(\cdot )} ({\open R}^n)$is the bounded mean oscillation (BMO)-type space${\rm BM}{\rm O}_{p(\cdot ),{\kern 1pt} L^ * }({\open R}^n)$, whereL* denotes the adjoint operator ofL. In particular, whenLis the second-order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of$H_L^{p(\cdot )} ({\open R}^n)$and show that the fractional integralL−αfor α∈(0, (1/2)] is bounded from$H_L^{p(\cdot )} ({\open R}^n)$to$H_L^{q(\cdot )} ({\open R}^n)$with (1/p(·))−(1/q(·))=2α/n, and the Riesz transform ∇L−1/2is bounded from$H_L^{p(\cdot )} ({\open R}^n)$to the variable Hardy spaceHp(·)(ℝn).
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47

Zhuo, Ciqiang, and Dachun Yang. "Variable weak Hardy spaces WH L p(·)(ℝ n ) associated with operators satisfying Davies–Gaffney estimates." Forum Mathematicum 31, no. 3 (May 1, 2019): 579–605. http://dx.doi.org/10.1515/forum-2018-0125.

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Abstract Let {p(\,\cdot\,)\colon\mathbb{R}^{n}\to[0,1]} be a variable exponent function satisfying the globally log-Hölder continuous condition, and L a one-to-one operator of type ω in {L^{2}({\mathbb{R}}^{n})} , with {\omega\in[0,\pi/2)} , which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. In this article, we introduce the variable weak Hardy space {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} , associated with L via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space {\mathrm{WT}^{p(\,\cdot\,)}(\mathbb{R}_{+}^{n+1})} , which is also obtained in this article. In particular, when L is non-negative and self-adjoint, we obtain the atomic characterization of {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} . As an application of the molecular characterization, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, we prove that the associated Riesz transform {\nabla L^{-1/2}} is bounded from {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} to the variable weak Hardy space {\mathrm{WH}^{p(\,\cdot\,)}(\mathbb{R}^{n})} . Moreover, when L is non-negative and self-adjoint with the kernels of {\{e^{-tL}\}_{t>0}} satisfying the Gaussian upper bound estimates, the atomic characterization of {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} is further used to characterize this space via non-tangential maximal functions.
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48

Fel'shtyn, Alexander, and Evgenij Troitsky. "Geometry of Reidemeister classes and twisted Burnside theorem." Journal of K-theory 2, no. 3 (March 4, 2008): 463–506. http://dx.doi.org/10.1017/is008001006jkt028.

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AbstractThe purpose of the present mostly expository paper (based mainly on [17, 18, 40, 16, 11]) is to present the current state of the following conjecture of A. Fel'shtyn and R. Hill [13], which is a generalization of the classical Burnside theorem.Let G be a countable discrete group, φ one of its automorphisms, R(φ) the number of φ-conjugacy (or twisted conjugacy) classes, and S(φ) = #Fix the number of φ-invariant equivalence classes of irreducible unitary representations. If one of R(φ) and S(φ) is finite, then it is equal to the other.This conjecture plays a important role in the theory of twisted conjugacy classes (see [26], [10]) and has very important consequences in Dynamics, while its proof needs rather sophisticated results from Functional and Noncommutative Harmonic Analysis.First we prove this conjecture for finitely generated groups of type I and discuss its applications.After that we discuss an important example of an automorphism of a type II1 group which disproves the original formulation of the conjecture.Then we prove a version of the conjecture for a wide class of groups, including almost polycyclic groups (in particular, finitely generated groups of polynomial growth). In this formulation the role of an appropriate dual object plays the finite-dimensional part of the unitary dual. Some counter-examples are discussed.Then we begin a discussion of the general case (which also needs new definition of the dual object) and prove the weak twisted Burnside theorem for general countable discrete groups. For this purpose we prove a noncommutative version of Riesz-Markov-Kakutani representation theorem.Finally we explain why the Reidemeister numbers are always infinite for Baumslag-Solitar groups.
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49

Bui, The Anh, Jun Cao, Luong Dang Ky, Dachun Yang, and Sibei Yang. "Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates." Analysis and Geometry in Metric Spaces 1 (February 7, 2013): 69–129. http://dx.doi.org/10.2478/agms-2012-0006.

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Abstract Let X be a metric space with doubling measure and L a one-to-one operator of type ω having a bounded H∞ -functional calculus in L2(X) satisfying the reinforced (pL; qL) off-diagonal estimates on balls, where pL ∊ [1; 2) and qL ∊ (2;∞]. Let φ : X × [0;∞) → [0;∞) be a function such that φ (x;·) is an Orlicz function, φ(·;t) ∊ A∞(X) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index l(φ) ∊ (0;1] and φ(·; t) satisfies the uniformly reverse Hölder inequality of order (qL/l(φ))′, where (qL/l(φ))′ denotes the conjugate exponent of qL/l(φ). In this paper, the authors introduce a Musielak-Orlicz-Hardy space Hφ;L(X), via the Lusin-area function associated with L, and establish its molecular characterization. In particular, when L is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of Hφ,L(X) is also obtained. Furthermore, a sufficient condition for the equivalence between Hφ,L(ℝn) and the classical Musielak-Orlicz-Hardy space Hv(ℝn) is given. Moreover, for the Musielak-Orlicz-Hardy space Hφ,L(ℝn) associated with the second order elliptic operator in divergence form on ℝn or the Schrödinger operator L := −Δ + V with 0 ≤ V ∊ L1loc(ℝn), the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors show that the Riesz transform ∇L−1/2 is bounded from Hφ,L(ℝn) to the Musielak-Orlicz space Lφ(ℝn) when i(φ) ∊ (0; 1], from Hφ,L(ℝn) to Hφ(ℝn) when i(φ) ∊ (; 1], and from Hφ,L(ℝn) to the weak Musielak-Orlicz-Hardy space WHφ(ℝn) when i(φ)=is attainable and φ(·; t) ∊ A1(X), where i(φ) denotes the uniformly critical lower type index of φ
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50

Chen, Li, José María Martell, and Cruz Prisuelos-Arribas. "Conical square functions for degenerate elliptic operators." Advances in Calculus of Variations 13, no. 1 (January 1, 2020): 75–113. http://dx.doi.org/10.1515/acv-2016-0062.

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AbstractThe aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second-order divergence form degenerate elliptic operators. More precisely, let {L_{w}=-w^{-1}\mathop{\rm div}(wA\nabla)}, where {w\in A_{2}} and A is an {n\times n} bounded, complex-valued, uniformly elliptic matrix. Cruz-Uribe and Rios solved the {L^{2}(w)}-Kato square root problem obtaining that {\sqrt{L_{w}}} is equivalent to the gradient on {L^{2}(w)}. The same authors in collaboration with the second named author of this paper studied the {L^{p}(w)}-boundedness of operators that are naturally associated with {L_{w}}, such as the functional calculus, Riesz transforms, and vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in {L^{p}(v\,dw)} for {v\in A_{\infty}(w)}), and in particular a class of “degeneracy” weights w was found in such a way that the classical {L^{2}}-Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on {L^{p}(w)} and on {L^{p}(v\,dw)}, with {v\in A_{\infty}(w)}, of the conical square functions that one can construct using the heat or Poisson semigroup associated with {L_{w}}. As a consequence of our methods, we find a class of degeneracy weights w for which {L^{2}}-estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with {L_{w}}.
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