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Journal articles on the topic 'Inner product spaces and their generalizations'

Author: Grafiati

Published: 4 June 2021

Last updated: 5 February 2022

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1

Jayaraman, Sachindranath. "Nonnegative generalized inverses in indefinite inner product spaces." Filomat 27, no. 4 (2013): 659–70. http://dx.doi.org/10.2298/fil1304659j.

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The aim of this article is to investigate nonnegativity of the inverse, the Moore-Penrose inverse and other generalized inverses, in the setting of indefinite inner product spaces with respect to the indefinite matrix product. We also propose and investigate generalizations of the corresponding notions of matrix monotonicity, namely, o-(rectangular) monotonicity, o-semimonotonicity and ?-weak monotonicity and its interplay with nonnegativity of various generalized inverses in the same setting.

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2

Ebanks, B. R., PL Kannappan, and P. K. Sahoo. "A Common Generalization of Functional Equations Characterizing Normed and Quasi-Inner-Product Spaces." Canadian Mathematical Bulletin 35, no. 3 (September 1, 1992): 321–27. http://dx.doi.org/10.4153/cmb-1992-044-6.

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AbstractWe determine the general solutions of the functional equation for ƒi: G → F (i = 1,2,3,4), where G is a 2-divisible group and F is a commutative field of characteristic different from 2. The motivation for studying this equation came from a result due to Dry gas [4] where he proved a Jordan and von Neumann type characterization theorem for quasi-inner products. Also, this equation is a generalization of the quadratic functional equation investigated by several authors in connection with inner product spaces and their generalizations. Special cases of this equation include the Cauchy equation, the Jensen equation, the Pexider equation and many more. Here, we determine the general solution of this equation without any regularity assumptions on ƒi.

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3

Ceno, Stela. "Some properties of the superior and inferior semi inner product function associated to the 2-norm." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 5 (June 30, 2016): 6254–60. http://dx.doi.org/10.24297/jam.v12i5.322.

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Special properties that the scalar product enjoys and its close link with the norm function have raised the interest of researchers from a very long period of time. S.S. Dragomir presents concrete generalizations of the scalar product functions in a normed space and deals with the interesting properties of them. Based on S.S. Dragomirs idea in this paper we treat generalizations of superior and inferior scalar product functions in the case of semi-normed spaces and 2-normed spaces.

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4

Kechriniotis, AristidesI, and KonstantinosK Delibasis. "On Generalizations of Grüss Inequality in Inner Product Spaces and Applications." Journal of Inequalities and Applications 2010, no. 1 (2010): 167091. http://dx.doi.org/10.1155/2010/167091.

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5

Alomari, Mohammad W. "On Pompeiu Cebysev Type Inequalities for Positive Linear Maps of Selfadjoint Operators in Inner Product Spaces." JOURNAL OF ADVANCES IN MATHEMATICS 15 (December 1, 2018): 8081–92. http://dx.doi.org/10.24297/jam.v15i0.7927.

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In this work, generalizations of some inequalities for continuous synchronous (h-asynchronous) functions of linear bounded selfadjoint operators under positive linear maps in Hilbert spaces are proved.

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6

Pecaric, J. "On Hua's inequality in real inner product spaces." Tamkang Journal of Mathematics 33, no. 3 (September 30, 2002): 265–68. http://dx.doi.org/10.5556/j.tkjm.33.2002.265-268.

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7

DRAGOMIR, S. S., and G. S. YANG. "ON HUA'S INEQUALITY IN REAL INNER PRODUCT SPACES." Tamkang Journal of Mathematics 27, no. 3 (September 1, 1997): 227–32. http://dx.doi.org/10.5556/j.tkjm.27.1996.4338.

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8

RADAS, SONJA, and TOMISLAV SIKIC. "A NOTE ON THE GENERALIZATION OF HUA'S INEQUALITY." Tamkang Journal of Mathematics 28, no. 4 (December 1, 1997): 321–23. http://dx.doi.org/10.5556/j.tkjm.28.1997.4310.

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In this paper we establish Lo-Keng Hua's inequality for linear operators in real inner product spaces. Our result generalizes Hua's inequality in real inner product spaces, obtained recently by S. S. Dragomir and G.-S. Yang.

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9

Malejki, Renata. "Stability of a generalization of the Fréchet functional equation." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 14, no. 1 (December 1, 2015): 69–79. http://dx.doi.org/10.1515/aupcsm-2015-0006.

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AbstractWe prove some stability and hyperstability results for a generalization of the well known Fréchet functional equation, stemming from one of the characterizations of the inner product spaces. As the main tool we use a fixed point theorem for some function spaces. We end the paper with some new inequalities characterizing the inner product spaces.

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10

Dragomir, S. S. "Generalizations of Buzano inequality for $n$-tuples of vectors in inner product spaces with applications." Tbilisi Mathematical Journal 10, no. 2 (February 2017): 29–42. http://dx.doi.org/10.1515/tmj-2017-0023.

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11

Zaheer, Neyamat. "A generalization of Lucas' theorem to vector spaces." International Journal of Mathematics and Mathematical Sciences 16, no. 2 (1993): 267–76. http://dx.doi.org/10.1155/s0161171293000316.

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The classical Lucas' theorem on critical points of complex-valued polynomials has been generalized (cf. [1]) to vector-valued polynomials defined onK-inner product spaces. In the present paper, we obtain a generalization of Lucas' theorem to vector-valued abstract polynomials defined on vector spaces, in general, which includes the above result of the author [1] inK-inner product spaces. Our main theorem also deduces a well-known result due to Marden on linear combinations of polynomial and its derivative. At the end, we discuss some examples in support of certain claims.

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12

Ujević, Nenad. "A new generalization of Grüss inequality in inner product spaces." Mathematical Inequalities & Applications, no. 4 (2003): 617–23. http://dx.doi.org/10.7153/mia-06-57.

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13

Dragomir, S. S. "A generalization of J. Aczél's inequality in inner product spaces." Acta Mathematica Hungarica 65, no. 2 (June 1994): 141–48. http://dx.doi.org/10.1007/bf01874309.

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14

Sylviani, S., and H. Garminia. "The development of inner product spaces and its generalization: a survey." Journal of Physics: Conference Series 1722 (January 2021): 012031. http://dx.doi.org/10.1088/1742-6596/1722/1/012031.

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15

Dragomir, Sever Silvestru. "A Generalization of Grüss's Inequality in Inner Product Spaces and Applications." Journal of Mathematical Analysis and Applications 237, no. 1 (September 1999): 74–82. http://dx.doi.org/10.1006/jmaa.1999.6452.

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16

Molnár, Lajos. "Generalization of Wigner's Unitary-Antiunitary Theorem for Indefinite Inner Product Spaces." Communications in Mathematical Physics 210, no. 3 (April 1, 2000): 785–91. http://dx.doi.org/10.1007/s002200050799.

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17

Sylviani, S., and H. Garminia. "The development of inner product spaces and its generalization: a survey." Journal of Physics: Conference Series 1722 (January 2021): 012031. http://dx.doi.org/10.1088/1742-6596/1722/1/012031.

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18

Dragomir, Sever. "A generalization of Kurepa’s inequality." Filomat, no. 19 (2005): 7–17. http://dx.doi.org/10.2298/fil0519007d.

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A generalization of Kurepa?s inequality in inner product spaces that extends in its turn the de Bruijn refinement of the Cauchy-Buniakovsky-Schwarz inequality for sequences of real and complex numbers is given.

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19

Stanimirovic, Predrag, Xue-Zhong Wang, and Haifeng Ma. "Complex ZNN for computing time-varying weighted pseudo-inverses." Applicable Analysis and Discrete Mathematics 13, no. 1 (2019): 131–64. http://dx.doi.org/10.2298/aadm170628019s.

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We classify, extend and unify various generalizations of weighted Moore-Penrose inverses in indefinite inner product spaces. New kinds of generalized inverses are introduced for this purpose. These generalized inverses are included in the more general class called as the weighted indefinite pseudoinverses (WIPI), which represents an extension of the Minkowski inverse (MI), the weighted Minkowski inverse (WMI), and the generalized weighted Moore- Penrose (GWM-P) inverse. The WIPI generalized inverses are introduced on the basis of two Hermitian invertible matrices and two Hermitian involuntary matrices and represented as particular outer inverses with prescribed ranges and null spaces, in terms of appropriate full-rank and limiting representations. Application of introduced generalized inverses in solving some indefinite least squares problems is considered. New Zeroing Neural Network (ZNN) models for computing the WIPI are developed using derived full-rank and limiting representations. The convergence behavior of the proposed ZNN models is investigated. Numerical simulation results are presented.

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20

Molnár, Lajos. "Orthogonality Preserving Transformations on Indefinite Inner Product Spaces: Generalization of Uhlhorn's Version of Wigner's Theorem." Journal of Functional Analysis 194, no. 2 (October 2002): 248–62. http://dx.doi.org/10.1006/jfan.2002.3970.

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21

Yoon, Jin Hee, Taechang Byun, Ji Eun Lee, and Keun Young Lee. "Classification of Complex Fuzzy Numbers and Fuzzy Inner Products." Mathematics 8, no. 9 (September 20, 2020): 1626. http://dx.doi.org/10.3390/math8091626.

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The paper is concerned with complex fuzzy numbers and complex fuzzy inner product spaces. In the classical complex number set, a complex number can be expressed using the Cartesian form or polar form. Both expressions are needed because one expression is better than the other depending on the situation. Likewise, the Cartesian form and the polar form can be defined in a complex fuzzy number set. First, the complex fuzzy numbers (CFNs) are categorized into two types, the polar form and the Cartesian form, as type I and type II. The properties of the complex fuzzy number set of those two expressions are discussed, and how the expressions can be used practically is shown through an example. Second, we study the complex fuzzy inner product structure in each category and find the non-existence of an inner product on CFNs of type I. Several properties of the fuzzy inner product space for type II are proposed from the modulus that is newly defined. Specfically, the Cauchy-Schwartz inequality for type II is proven in a compact way, not only the one for fuzzy real numbers. In fact, it was already discussed by Hasanhani et al; however, they proved every case in a very complicated way. In this paper, we prove the Cauchy-Schwartz inequality in a much simpler way from a general point of view. Finally, we introduce a complex fuzzy scalar product for the generalization of a complex fuzzy inner product and propose to study the condition for its existence on CFNs of type I.

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22

Ahmad, Asif, Qi Liu, and Yongjin Li. "Geometric Constants in Banach Spaces Related to the Inscribed Quadrilateral of Unit Balls." Symmetry 13, no. 7 (July 19, 2021): 1294. http://dx.doi.org/10.3390/sym13071294.

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We introduce a new geometric constant Jin(X) based on a generalization of the parallelogram law, which is symmetric and related to the length of the inscribed quadrilateral side of the unit ball. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space, Jin(X) becomes 16 if and only if the norm is induced by an inner product. Moreover, its properties and some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented.

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23

Liu, Qi, Muhammad Sarfraz, and Yongjin Li. "Some aspects of generalized Zbăganu and James constant in Banach spaces." Demonstratio Mathematica 54, no. 1 (January 1, 2021): 299–310. http://dx.doi.org/10.1515/dema-2021-0033.

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Abstract We shall introduce a new geometric constant C Z ( λ , μ , X ) {C}_{Z}\left(\lambda ,\mu ,X) based on a generalization of the parallelogram law, which was proposed by Moslehian and Rassias. First, it is shown that, for a Banach space, C Z ( λ , μ , X ) {C}_{Z}\left(\lambda ,\mu ,X) is equal to 1 if and only if the norm is induced by an inner product. Next, a characterization of uniformly non-square is given, that is, X X has the fixed point property. Also, a sufficient condition which implies weak normal structure is presented. Moreover, a generalized James constant J ( λ , X ) J\left(\lambda ,X) is also introduced. Finally, some basic properties of this new coefficient are presented.

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24

Guo, Jiandong. "On a Generalization of a Result of Waldspurger." Canadian Journal of Mathematics 48, no. 1 (February 1, 1996): 105–42. http://dx.doi.org/10.4153/cjm-1996-005-3.

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AbstractWe consider a generalization of a trace formula identity of Jacquet, in the context of the symmetric spaces GL(2n)/GL(/n) × GL(n) and G′/H′. Here G′ is an inner form of GL(2n) over F with a subgroup H′ isomorphic to GL(n, E) where E/F is a quadratic extension of number field attached to a quadratic idele class character η of F. A consequence of this identity would be the following conjecture: Let π be an automorphic cuspidal representation of GL(2n). If there exists an automorphic representation π′ of G′ which is related to π by the Jacquet-Langlands correspondence, and a vector ø in the space of π′ whose integral over H′ is nonzero, then both L(1/2, π) and L(1/2,π ⊗ η) are nonvanishing. Moreover, we have L(1/2, π)L(1/2, π ⊗ η) > 0. Here the nonvanishing part of the conjecture is a generalization of a result of Waldspurger for GL(2) and the nonnegativity of the product is predicted from the generalized Riemann Hypothesis. In this article, we study the corresponding local orbital integrals for the symmetric spaces. We prove the "fundamental lemma for the unit Hecke functions" which says that unit Hecke functions have "matching" orbital integrals. This serves as the first step toward establishing the trace formula identity and in the same time it provides strong evidence for what we proposed.

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25

Kohli, J. K., and Rajesh Kumar. "On fuzzy inner product spaces and fuzzy co-inner product spaces." Fuzzy Sets and Systems 53, no. 2 (January 1993): 227–32. http://dx.doi.org/10.1016/0165-0114(93)90177-j.

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26

Misiak, Aleksander. "n-Inner Product Spaces." Mathematische Nachrichten 140, no. 1 (1989): 299–319. http://dx.doi.org/10.1002/mana.19891400121.

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27

Saheli, M., and S. Khajepour Gelousalar. "Fuzzy inner product spaces." Fuzzy Sets and Systems 303 (November 2016): 149–62. http://dx.doi.org/10.1016/j.fss.2015.11.008.

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28

El-Abyad, Abdelwahab M., and Hassan M. El-Hamouly. "Fuzzy inner product spaces." Fuzzy Sets and Systems 44, no. 2 (November 1991): 309–26. http://dx.doi.org/10.1016/0165-0114(91)90014-h.

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29

Falkowski, Bernd-Jürgen. "On certain generalizations of inner product similarity measures." Journal of the American Society for Information Science 49, no. 9 (1998): 854–58. http://dx.doi.org/10.1002/(sici)1097-4571(199807)49:9<854::aid-asi11>3.0.co;2-n.

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30

de Deus Marques, João. "On vectorial inner product spaces." Czechoslovak Mathematical Journal 50, no. 3 (September 2000): 539–50. http://dx.doi.org/10.1023/a:1022833626838.

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31

Sain, Debmalya, Kallol Paul, and Lokenath Debnath. "Characterization of Inner Product Spaces." International Journal of Applied and Computational Mathematics 1, no. 4 (February 26, 2015): 599–606. http://dx.doi.org/10.1007/s40819-015-0036-8.

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32

Mashadi, A. Hadi, S. Gemawati, and I. Nasfianti. "Fuzzy real Inner Product on Spaces of Fuzzy realn-Inner Product." Journal of Physics: Conference Series 1116 (December 2018): 022024. http://dx.doi.org/10.1088/1742-6596/1116/2/022024.

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33

Wójcik, Paweł. "Characterizations of inner product spaces by inequalities involving semi-inner product." Mathematical Inequalities & Applications, no. 3 (2015): 879–85. http://dx.doi.org/10.7153/mia-18-64.

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34

Bozkurt, Hacer, Sümeyye Çakan, and Yılmaz Yılmaz. "Quasilinear Inner Product Spaces and Hilbert Quasilinear Spaces." International Journal of Analysis 2014 (March 11, 2014): 1–7. http://dx.doi.org/10.1155/2014/258389.

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Aseev launched a new branch of functional analysis by introducing the theory of quasilinear spaces in the framework of the topics of norm, bounded quasilinear operators and functionals (Aseev (1986)). Furthermore, some quasilinear counterparts of classical nonlinear analysis that lead to such result as Frechet derivative and its applications were examined deal with. This pioneering work causes a lot of results in such applications such as (Rojas-Medar et al. (2005), Talo and Başar (2010), and Nikol'skiĭ (1993)). His work has motivated us to introduce the concept of quasilinear inner product spaces. Thanks to this new notion, we obtain some new theorems and definitions which are quasilinear counterparts of fundamental definitions and theorems in linear functional analysis. We claim that some new results related to this concept provide an important contribution to the improvement of quasilinear functional analysis.

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35

Dvurečenskij, Anatolij, and Sylvia Pulmannová. "Test spaces, dacey spaces, and completeness of inner product spaces." Letters in Mathematical Physics 32, no. 4 (December 1994): 299–306. http://dx.doi.org/10.1007/bf00761140.

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36

Misiak, Aleksander, and Alicja Ryż. "$n$-inner product spaces and projections." Mathematica Bohemica 125, no. 1 (2000): 87–97. http://dx.doi.org/10.21136/mb.2000.126265.

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37

Dey, Asit, and Madhumangal Pal. "Properties of Fuzzy Inner Product Spaces." International Journal of Fuzzy Logic Systems 4, no. 2 (April 30, 2014): 23–39. http://dx.doi.org/10.5121/ijfls.2014.4203.

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38

Falkner, Neil. "A Characterization of Inner Product Spaces." American Mathematical Monthly 100, no. 3 (March 1993): 246. http://dx.doi.org/10.2307/2324457.

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39

Das, Sujoy, and S. K. Samanta. "Operators on Soft Inner Product Spaces." Fuzzy Information and Engineering 6, no. 4 (December 2014): 435–50. http://dx.doi.org/10.1016/j.fiae.2015.01.003.

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40

Albahboh, Hussin, Harry Gingold, and Jocelyn Quaintance. "Trigonometry in complex inner product spaces." Linear Algebra and its Applications 575 (August 2019): 216–34. http://dx.doi.org/10.1016/j.laa.2019.04.014.

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41

Falkner, Neil. "A Characterization of Inner Product Spaces." American Mathematical Monthly 100, no. 3 (March 1993): 246–49. http://dx.doi.org/10.1080/00029890.1993.11990396.

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42

Shi-sheng, Zhang. "On the probabilistic inner product spaces." Applied Mathematics and Mechanics 7, no. 11 (November 1986): 1035–42. http://dx.doi.org/10.1007/bf01897206.

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43

Xiao, Jian-Zhong, and Feng-Qin Zhu. "Comments on “Fuzzy inner product spaces”." Fuzzy Sets and Systems 373 (October 2019): 180–83. http://dx.doi.org/10.1016/j.fss.2019.02.006.

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44

Dvurečenskij, Anatolij. "Regular measures and inner product spaces." International Journal of Theoretical Physics 31, no. 5 (May 1992): 889–905. http://dx.doi.org/10.1007/bf00678553.

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45

Alsina, C., P. Guijarro, and M. S. Tom�s. "Generalizations of characterizations of inner product structures under differentiability conditions." Archiv der Mathematik 66, no. 3 (March 1996): 228–32. http://dx.doi.org/10.1007/bf01195709.

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46

MOSNEGUTU, Bianca. "Ulam stability in real inner-product spaces." Constructive Mathematical Analysis 3, no. 3 (September 1, 2020): 113–15. http://dx.doi.org/10.33205/cma.758854.

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47

Sujoy Das and S. K. Samanta. "Operators on soft inner product spaces II." ANNALS OF FUZZY MATHEMATICS AND INFORMATICS 13, no. 3 (March 2017): 297–315. http://dx.doi.org/10.30948/afmi.2017.13.3.297.

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48

Sain, Debmalya, and Kallol Paul. "Operator norm attainment and inner product spaces." Linear Algebra and its Applications 439, no. 8 (October 2013): 2448–52. http://dx.doi.org/10.1016/j.laa.2013.07.008.

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49

Bovdi, Victor A., Tetiana Klymchuk, Tetiana Rybalkina, Mohamed A. Salim, and Vladimir V. Sergeichuk. "Operators on positive semidefinite inner product spaces." Linear Algebra and its Applications 596 (July 2020): 82–105. http://dx.doi.org/10.1016/j.laa.2020.03.004.

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50

Park, Choonkil, Won-Gil Park, and Abbas Najati. "Functional Equations Related to Inner Product Spaces." Abstract and Applied Analysis 2009 (2009): 1–11. http://dx.doi.org/10.1155/2009/907121.

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LetV,Wbe real vector spaces. It is shown that an odd mappingf:V→Wsatisfies∑i−12nf(xi−1/2n∑j=12nxj)=∑i=12nf(xi)−2nf(1/2n∑i=12nxi)for allx1,…,x2n∈Vif and only if the odd mappingf:V→Wis Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

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