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Journal articles on the topic 'Orthonormal basis'

Author: Grafiati

Published: 4 June 2021

Last updated: 26 July 2025

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1

Toda, Hiroshi, and Zhong Zhang. "Orthonormal basis of wavelets with customizable frequency bands." International Journal of Wavelets, Multiresolution and Information Processing 14, no. 06 (2016): 1650050. http://dx.doi.org/10.1142/s0219691316500508.

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We already proved the existence of an orthonormal basis of wavelets having an irrational dilation factor with an infinite number of wavelet shapes, and based on its theory, we proposed an orthonormal basis of wavelets with an arbitrary real dilation factor. In this paper, with the development of these fundamentals, we propose a new type of orthonormal basis of wavelets with customizable frequency bands. Its frequency bands can be freely designed with arbitrary bounds in the frequency domain. For example, we show two types of orthonormal bases of wavelets. One of them has an irrational dilation

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2

Toda, Hiroshi, and Zhong Zhang. "Orthonormal wavelet basis with arbitrary real dilation factor." International Journal of Wavelets, Multiresolution and Information Processing 14, no. 03 (2016): 1650010. http://dx.doi.org/10.1142/s0219691316500107.

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Daubechies posed the following problem in Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992): “It is an open question whether there exist orthonormal wavelet bases (not necessarily associated with a multiresolution analysis), with good time-frequency localization, and with irrational [Formula: see text]” (that is, for an arbitrary irrational dilation factor [Formula: see text], with appropriate wavelet function [Formula: see text] and constant [Formula: see text], whether can [Formula: see text] construct an orthonormal wavelet basis with good time-frequency localization?). Our answer is

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3

Tipton, James. "Classification of polynomials and an orthonormal basis construction on the associated basin of attraction." Gulf Journal of Mathematics 20 (June 14, 2025): 81–95. https://doi.org/10.56947/gjom.v20i.2813.

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We study a construction that associates certain Fatou subsets with reproducing kernel Hilbert spaces and a method for constructing an orthonormal basis for said Hilbert space, which depends on the map of the given Fatou set. We provide a classification of those polynomials the method applies to, and we extend a previous result concerning the dynamics of these orthonormal bases. Essential to the proof of this classification are the Newton--Girard identities and Vieta's formulas.

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4

Hecht, K. T., R. Le Blanc, and D. J. Rowe. "Canonical orthonormal Wigner supermultiplet basis." Journal of Physics A: Mathematical and General 20, no. 2 (1987): 257–75. http://dx.doi.org/10.1088/0305-4470/20/2/013.

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5

Guo, Xunxiang. "g-Bases in Hilbert Spaces." Abstract and Applied Analysis 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/923729.

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The concept ofg-basis in Hilbert spaces is introduced, which generalizes Schauder basis in Hilbert spaces. Some results aboutg-bases are proved. In particular, we characterize theg-bases andg-orthonormal bases. And the dualg-bases are also discussed. We also consider the equivalent relations ofg-bases andg-orthonormal bases. And the property ofg-minimal ofg-bases is studied as well. Our results show that, in some cases,g-bases share many useful properties of Schauder bases in Hilbert spaces.

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6

El Amrani, Abdelkhalek, Mohamed Rossafi, and Tahar El krouk. "K-Riesz bases and K-g-Riesz bases in Hilbert C∗-module." Proyecciones (Antofagasta) 42, no. 5 (2023): 1241–60. http://dx.doi.org/10.22199/issn.0717-6279-5713.

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This paper is devoted to studying the K-Riesz bases and the K-g-Riesz bases in Hilbert C∗-modules; we characterize the concept of K-Riesz bases by a bounded below operator and the standard orthonormal basis for Hilbert C∗-modules H. Also We give some properties and characterization of K-g-Riesz bases by a bounded surjective operator and g-orthonormal basis for H. Finally we consider the relationships between K-Riesz bases and K-g-Riesz bases.

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7

Zelditch, Steve. "Quantum ergodicity of random orthonormal bases of spaces of high dimension." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, no. 2007 (2014): 20120511. http://dx.doi.org/10.1098/rsta.2012.0511.

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We consider a sequence of finite-dimensional Hilbert spaces of dimensions . Motivating examples are eigenspaces, or spaces of quasi-modes, for a Laplace or Schrödinger operator on a compact Riemannian manifold. The set of Hermitian orthonormal bases of may be identified with U ( d N ), and a random orthonormal basis of is a choice of a random sequence U N ∈ U ( d N ) from the product of normalized Haar measures. We prove that if and if tends to a unique limit state ω ( A ), then almost surely an orthonormal basis is quantum ergodic with limit state ω ( A ). This generalizes an earlier result o

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8

WOJDYŁŁO, PIOTR. "CHARACTERIZATION OF WILSON SYSTEMS FOR GENERAL LATTICES." International Journal of Wavelets, Multiresolution and Information Processing 06, no. 02 (2008): 305–14. http://dx.doi.org/10.1142/s0219691308002367.

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The Wilson orthonormal basis was constructed in 1991 by Daubechies, Jaffard and Journé using combinations of elements of Gabor tight frame with redundancy 2. In 1994, Auscher gave a characterization of the atoms for which the Wilson system is an orthonormal basis. Recently, Kutyniok and Strohmer generalized the notion of the Wilson system to the lattices whose generator matrix is in Hermite normal form.We extend their result to the full characterization of Wilson orthonormal bases on the general lattice of volume 1/2. Moreover, we generalize this result to other forms of Wilson systems differi

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9

GHOBBER, SAIFALLAH. "SHAPIRO’S UNCERTAINTY PRINCIPLE IN THE DUNKL SETTING." Bulletin of the Australian Mathematical Society 92, no. 1 (2015): 98–110. http://dx.doi.org/10.1017/s000497271500026x.

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The Dunkl transform ${\mathcal{F}}_{k}$ is a generalisation of the usual Fourier transform to an integral transform invariant under a finite reflection group. The goal of this paper is to prove a strong uncertainty principle for orthonormal bases in the Dunkl setting which states that the product of generalised dispersions cannot be bounded for an orthonormal basis. Moreover, we obtain a quantitative version of Shapiro’s uncertainty principle on the time–frequency concentration of orthonormal sequences and show, in particular, that if the elements of an orthonormal sequence and their Dunkl tra

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10

Lemma, D. T., M. Ramasamy, and M. Shuhaimi. "System Identification using Orthonormal Basis Filters." Journal of Applied Sciences 10, no. 21 (2010): 2516–22. http://dx.doi.org/10.3923/jas.2010.2516.2522.

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11

Chung, Young-Bok, and Heui-Geong Na. "ORTHONORMAL BASIS FOR THE BERGMAN SPACE." Honam Mathematical Journal 36, no. 4 (2014): 777–86. http://dx.doi.org/10.5831/hmj.2014.36.4.777.

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12

Bodin, Per, Lars F. Villemoes, and Bo Wahlberg. "Selection of Best Orthonormal Rational Basis." SIAM Journal on Control and Optimization 38, no. 4 (2000): 995–1032. http://dx.doi.org/10.1137/s036301299732818x.

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13

KAMADA, M., K. TORAICHI, Y. IKEBE, and R. MORI. "Orthonormal basis for spline signal spaces." International Journal of Systems Science 20, no. 1 (1989): 157–70. http://dx.doi.org/10.1080/00207728908910113.

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14

Agora, Elona, Jorge Antezana, and Mihail N. Kolountzakis. "Tiling functions and Gabor orthonormal basis." Applied and Computational Harmonic Analysis 48, no. 1 (2020): 96–122. http://dx.doi.org/10.1016/j.acha.2018.02.005.

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15

Fricain, Emmanuel, and Rishika Rupam. "Asymptotically orthonormal basis and Toeplitz operators." Journal of Mathematical Analysis and Applications 474, no. 2 (2019): 944–60. http://dx.doi.org/10.1016/j.jmaa.2019.01.081.

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16

WOJDYŁŁO, PIOTR. "WILSON SYSTEM FOR TRIPLE REDUNDANCY." International Journal of Wavelets, Multiresolution and Information Processing 09, no. 01 (2011): 151–67. http://dx.doi.org/10.1142/s0219691311003980.

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A Wilson orthonormal basis was constructed in 1991 by Daubechies, Jaffard and Journé from Gabor tight frame elements, when the redundancy of the Gabor system is 2. In 1994, Auscher gave a characterization of the atoms for which the Wilson system is an orthonormal basis. Afterwards, Gröchenig posed a question whether the construction of an orthonormal Wilson basis is possible for a Gabor tight frame of redundancy 3. We give a partial positive answer to this question constructing in this case a Wilson system being a tight frame with bound 1.

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17

Wojdyłło, Piotr. "Symmetric Wilson systems." International Journal of Wavelets, Multiresolution and Information Processing 17, no. 05 (2019): 1950036. http://dx.doi.org/10.1142/s021969131950036x.

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The Wilson orthonormal basis was constructed in 1991 by Daubechies, Jaffard and Journé using combinations of elements of Gabor tight frame with redundancy 2. In 1994, Auscher gave a characterization of the atoms for which the Wilson system is an orthonormal basis. Then, Kutyniok and Strohmer generalized the notion of Wilson system to the lattices whose generator matrix is in Hermite normal form.In the present publication, we introduce Wilson systems where the time-frequency shifts are combined symmetrically with respect to the origin. Using the arguments from previous papers that work also in

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18

Fricain, Emmanuel, and Javad Mashreghi. "Orthonormal polynomial basis in local Dirichlet spaces." Acta Scientiarum Mathematicarum 87, no. 34 (2021): 595–613. http://dx.doi.org/10.14232/actasm-021-465-4.

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19

Vasileiou, Panagiotis N., Konstantinos Maliatsos, Evangelos D. Thomatos, and Athanasios G. Kanatas. "Reconfigurable Orthonormal Basis Patterns Using ESPAR Antennas." IEEE Antennas and Wireless Propagation Letters 12 (2013): 448–51. http://dx.doi.org/10.1109/lawp.2013.2255254.

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20

Wahlberg, Bo. "Orthonormal basis functions models: A transformation analysis." IFAC Proceedings Volumes 32, no. 2 (1999): 4123–28. http://dx.doi.org/10.1016/s1474-6670(17)56703-8.

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21

Van Den Hof, Paul M. J., Peter S.C. Heuberger, and József Bokor. "System identification with generalized orthonormal basis functions." Automatica 31, no. 12 (1995): 1821–34. http://dx.doi.org/10.1016/0005-1098(95)00074-4.

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22

Liao, Jian Quan, Xing Min Li, and Jin Xun Wang. "Orthonormal basis of the octonionic analytic functions." Journal of Mathematical Analysis and Applications 366, no. 1 (2010): 335–44. http://dx.doi.org/10.1016/j.jmaa.2009.10.002.

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23

Wardani, Irma Budi, and Hartanto Sunardi. "ORTONORMALISASI VEKTOR BASIS DENGAN PROSES GRAM SCHMIDT." Buana Matematika : Jurnal Ilmiah Matematika dan Pendidikan Matematika 5, no. 2: (2016): 1–8. http://dx.doi.org/10.36456/buanamatematika.v5i2:.391.

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Gram schmidt process is one of linear algebra roles that associated by basis vector. This thesis aims to determine theoretically step by step in the process of gram schmidt. Gram schmidt process is a method that used to convert an arbitrary basis vector into an orthogonal basis vector. After orthogonal basis vector had been obtained, the orthogonal basis vector was compiled into an orthonormal basis through step by step. A vector on will be expressed as a basis vector if the vector if the vectors in are linear independently and spinning against . And a basis vector can be expressed

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24

Leleury, Zeth A. "SISTEM ORTONORMAL DALAM RUANG HILBERT." BAREKENG: Jurnal Ilmu Matematika dan Terapan 8, no. 2 (2014): 19–26. http://dx.doi.org/10.30598/barekengvol8iss2pp19-26.

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Hilbert space is one of the important inventions in mathematics. Historically, the theory of Hilbert space originated from David Hilbert’s work on quadratic form in infinitely many variables with their applications to integral equations. This paper contains some definitions such as vector space, normed space and inner product space (also called pre-Hilbert space), and which is important to construct the Hilbert space. The fundamental ideas and results are discussed with special attention given to finite dimensional pre-Hilbert space and some basic propositions of orthonormal systems in Hilbert

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25

Marino, Guiseppe, and Paolamaria Pietramala. "An Unconventional Orthonormal Basis Provides an Unexpected Counterexample." Mathematics Magazine 66, no. 5 (1993): 309. http://dx.doi.org/10.2307/2690507.

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26

Namiki, Ryo. "Non-Gaussian Entangled States and Entangled Orthonormal Basis." Journal of the Physical Society of Japan 79, no. 1 (2010): 013001. http://dx.doi.org/10.1143/jpsj.79.013001.

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27

Hughes, John F., and Tomas Moller. "Building an Orthonormal Basis from a Unit Vector." Journal of Graphics Tools 4, no. 4 (1999): 33–35. http://dx.doi.org/10.1080/10867651.1999.10487513.

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28

Marino, Giuseppe, and Paolamaria Pietramala. "An Unconventional Orthonormal Basis Provides an Unexpected Counterexample." Mathematics Magazine 66, no. 5 (1993): 309–11. http://dx.doi.org/10.1080/0025570x.1993.11996151.

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29

Akçay, Hüseyin, and Brett Ninness. "Orthonormal basis functions for modelling continuous-time systems." Signal Processing 77, no. 3 (1999): 261–74. http://dx.doi.org/10.1016/s0165-1684(99)00039-0.

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30

Bodin, Per. "Selection of local discriminant generalized orthonormal rational basis." IFAC Proceedings Volumes 32, no. 2 (1999): 4147–52. http://dx.doi.org/10.1016/s1474-6670(17)56707-5.

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31

Hon, Y. C., and T. Wei. "An orthonormal basis functions method for moment problems." Engineering Analysis with Boundary Elements 26, no. 10 (2002): 855–60. http://dx.doi.org/10.1016/s0955-7997(02)00032-2.

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32

Tiels, Koen, and Johan Schoukens. "Wiener system identification with generalized orthonormal basis functions." Automatica 50, no. 12 (2014): 3147–54. http://dx.doi.org/10.1016/j.automatica.2014.10.010.

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33

de Vries, D. K., and P. M. J. Van den Hof. "Frequency domain identification with generalized orthonormal basis functions." IEEE Transactions on Automatic Control 43, no. 5 (1998): 656–69. http://dx.doi.org/10.1109/9.668831.

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34

Heuberger, P. S. C., P. M. J. Van den Hof, and O. H. Bosgra. "A generalized orthonormal basis for linear dynamical systems." IEEE Transactions on Automatic Control 40, no. 3 (1995): 451–65. http://dx.doi.org/10.1109/9.376057.

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35

Daubechies, Ingrid, Stéphane Jaffard, and Jean-Lin Journé. "A Simple Wilson Orthonormal Basis with Exponential Decay." SIAM Journal on Mathematical Analysis 22, no. 2 (1991): 554–73. http://dx.doi.org/10.1137/0522035.

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36

Dhanuk, B. B., K. Pudasainee, H. P. Lamichhane, and R. P. Adhikari. "Dirac Delta Function from Closure Relation of Orthonormal Basis and its Use in Expanding Analytic Functions." Journal of Nepal Physical Society 6, no. 2 (2020): 158–63. http://dx.doi.org/10.3126/jnphyssoc.v6i2.34872.

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One of revealing and widely used concepts in Physics and mathematics is the Dirac delta function. The Dirac delta function is a distribution on real lines which is zero everywhere except at a single point, where it is infinite. Dirac delta function has vital role in solving inhomogeneous differential equations. In addition, the Dirac delta functions can be used to explore harmonic information’s imbedded in the physical signals, various forms of Dirac delta function and can be constructed from the closure relation of orthonormal basis functions of functional space. Among many special functions,

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37

Xin, Jianguo, and Wei Cai. "A Well-Conditioned Hierarchical Basis for Triangular H(curl)-Conforming Elements." Communications in Computational Physics 9, no. 3 (2011): 780–806. http://dx.doi.org/10.4208/cicp.220310.030610s.

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AbstractWe construct a well-conditioned hierarchical basis for triangular H(curl)-conforming elements with selected orthogonality. The basis functions are grouped into edge and interior functions, and the later is further grouped into normal and bubble functions. In our construction, the trace of the edge shape functions are orthonormal on the associated edge. The interior normal functions, which are perpendicular to an edge, and the bubble functions are both orthonormal among themselves over the reference element. The construction is made possible with classic orthogonal polynomials, viz., Le

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38

Zhang, Yingchao, Yuntao Jia, and Yingzhen Lin. "An $ {\varepsilon} $-approximate solution of BVPs based on improved multiscale orthonormal basis." AIMS Mathematics 9, no. 3 (2024): 5810–26. http://dx.doi.org/10.3934/math.2024282.

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<abstract><p>In the present paper, we construct a set of multiscale orthonormal basis based on Legendre polynomials. Using this orthonormal basis, a new algorithm is designed for solving the second-order boundary value problems. This algorithm is to find numerical solution by seeking $ {\varepsilon} $-approximate solution. Moreover, we prove that the order of convergence depends on the boundedness of $ u^{(m)}(x) $. In addition, third numerical examples are provided to validate the efciency and accuracy of the proposed method. Numerical results reveal that the present method yields

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39

Saito, Naoki, and Yiqun Shao. "eGHWT: The Extended Generalized Haar–Walsh Transform." Journal of Mathematical Imaging and Vision 64, no. 3 (2022): 261–83. http://dx.doi.org/10.1007/s10851-021-01064-w.

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AbstractExtending computational harmonic analysis tools from the classical setting of regular lattices to the more general setting of graphs and networks is very important, and much research has been done recently. The generalized Haar–Walsh transform (GHWT) developed by Irion and Saito (2014) is a multiscale transform for signals on graphs, which is a generalization of the classical Haar and Walsh–Hadamard transforms. We propose the extended generalized Haar–Walsh transform (eGHWT), which is a generalization of the adapted time–frequency tilings of Thiele and Villemoes (1996). The eGHWT exami

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40

Yu, Miao, Youyi Wang, Wanli Wang, and Yongtao Wei. "Continuous-Time Subspace Identification with Prior Information Using Generalized Orthonormal Basis Functions." Mathematics 11, no. 23 (2023): 4765. http://dx.doi.org/10.3390/math11234765.

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This paper presents a continuous-time subspace identification method utilizing prior information and generalized orthonormal basis functions. A generalized orthonormal basis is constructed by a rational inner function, and the transformed noises have ergodic properties. The lifting approach and the Hambo system transform are used to establish the equivalent nature of continuous and transformed discrete-time stochastic systems. The constrained least squares method is adopted to investigate the incorporation of prior knowledge in order to further increase the subspace identification algorithm’s

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41

Tóth, R., P. S. C. Heuberger, and P. M. J. Van den Hof. "An LPV identification Framework Based on Orthonormal Basis Functions." IFAC Proceedings Volumes 42, no. 10 (2009): 1328–33. http://dx.doi.org/10.3182/20090706-3-fr-2004.00221.

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42

Nouri, B., R. Achar, and M. S. Nakhla. "$z$-Domain Orthonormal Basis Functions for Physical System Identifications." IEEE Transactions on Advanced Packaging 33, no. 1 (2010): 293–307. http://dx.doi.org/10.1109/tadvp.2009.2019965.

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43

Heuberger, P. S. C., P. M. J. Van den Hof, and O. H. Bosgra. "Modelling Linear Dynamical Systems through Generalized Orthonormal Basis Functions." IFAC Proceedings Volumes 26, no. 2 (1993): 19–22. http://dx.doi.org/10.1016/s1474-6670(17)48214-0.

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44

de Hoog, Thomas J., Zoltán Szabó, Peter S. C. Heuberger, Paul M. J. Van den Hof, and József Bokor. "Minimal partial realization from generalized orthonormal basis function expansions." Automatica 38, no. 4 (2002): 655–69. http://dx.doi.org/10.1016/s0005-1098(01)00247-3.

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45

Akçay, Hüseyin. "Discrete-time system modelling in with orthonormal basis functions." Systems & Control Letters 39, no. 5 (2000): 365–76. http://dx.doi.org/10.1016/s0167-6911(99)00116-4.

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46

Kibangou, Alain Y., Gérard Favier, and Moha M. Hassan. "Generalized orthonormal basis selection for expanding quadratic volterra filters." IFAC Proceedings Volumes 36, no. 16 (2003): 1077–82. http://dx.doi.org/10.1016/s1474-6670(17)34902-9.

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47

Blomqvist, A., and G. Fanizza. "Identification of rational spectral densities using orthonormal basis functions." IFAC Proceedings Volumes 36, no. 16 (2003): 1327–32. http://dx.doi.org/10.1016/s1474-6670(17)34944-3.

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48

Heuberger, P. S. C., Z. Szabó, T. J. de Hoog, P. M. J. Van den Hof, and J. Bokor. "Realization algorithms for expansions in generalized orthonormal basis functions." IFAC Proceedings Volumes 32, no. 2 (1999): 4153–58. http://dx.doi.org/10.1016/s1474-6670(17)56708-7.

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49

Krommweh, Jens. "An Orthonormal Basis of Directional Haar Wavelets on Triangles." Results in Mathematics 53, no. 3-4 (2009): 323–31. http://dx.doi.org/10.1007/s00025-008-0343-z.

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50

Daubechies, Ingrid, Stéphane Jaffard, and Jean-Lin Journé. "Erratum: A Simple Wilson Orthonormal Basis with Exponential Decay." SIAM Journal on Mathematical Analysis 22, no. 3 (1991): 878. http://dx.doi.org/10.1137/0522056.

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