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Journal articles on the topic 'Bilinear Hilbert transform'

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Published: 10 December 2022

Last updated: 29 January 2023

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1

Buchkovska, Aneta L., and Stevan PilipoviĆ. "Bilinear Hilbert Transform of Ultradistributions." Integral Transforms and Special Functions 13, no. 3 (January 2002): 211–21. http://dx.doi.org/10.1080/10652460213520.

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2

Shi, Zuoshunhua, and Dunyan Yan. "Criterion onLp1×Lp2→Lq-Boundedness for Oscillatory Bilinear Hilbert Transform." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/712051.

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We investigate the bilinear Hilbert transform with oscillatory factors and the truncated bilinear Hilbert transform. The main result is that theLp1×Lp2→Lq-boundedness of the two operators is equivalent with1≤p1,p2<∞, and1/q=1/p1+1/p2. In addition, we also discuss the boundedness of a variant operator of bilinear Hilbert transform with a nontrivial polynomial phase.

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3

Bučkovska, Aneta, Stevan Pilipović, and Mirjana Vuković. "Inversion theorem for bilinear Hilbert transform." Integral Transforms and Special Functions 19, no. 5 (May 2008): 317–25. http://dx.doi.org/10.1080/10652460701855948.

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4

Blasco, O., M. Carro, and T. A. Gillespie. "Bilinear Hilbert Transform on Measure Spaces." Journal of Fourier Analysis and Applications 11, no. 4 (August 2005): 459–70. http://dx.doi.org/10.1007/s00041-005-4074-1.

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5

Ciprian Demeter and Christoph Thiele. "On the two-dimensional bilinear Hilbert transform." American Journal of Mathematics 132, no. 1 (2010): 201–56. http://dx.doi.org/10.1353/ajm.0.0101.

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6

Bilyk, Dmitriy, and Loukas Grafakos. "Distributional estimates for the bilinear Hilbert transform." Journal of Geometric Analysis 16, no. 4 (December 2006): 563–84. http://dx.doi.org/10.1007/bf02922131.

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7

Di Plinio, Francesco, and Christoph Thiele. "Endpoint bounds for the bilinear Hilbert transform." Transactions of the American Mathematical Society 368, no. 6 (November 20, 2015): 3931–72. http://dx.doi.org/10.1090/tran/6548.

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8

Lacey, M., and C. Thiele. "Lp estimates for the bilinear Hilbert transform." Proceedings of the National Academy of Sciences 94, no. 1 (January 7, 1997): 33–35. http://dx.doi.org/10.1073/pnas.94.1.33.

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9

Amenta, Alex, and Gennady Uraltsev. "The bilinear Hilbert transform in UMD spaces." Mathematische Annalen 378, no. 3-4 (August 5, 2020): 1129–221. http://dx.doi.org/10.1007/s00208-020-02052-y.

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Abstract We prove $$L^p$$ L p -bounds for the bilinear Hilbert transform acting on functions valued in intermediate UMD spaces. Such bounds were previously unknown for UMD spaces that are not Banach lattices. Our proof relies on bounds on embeddings from Bochner spaces $$L^p(\mathbb {R};X)$$ L p ( R ; X ) into outer Lebesgue spaces on the time-frequency-scale space $$\mathbb {R}^3_+$$ R + 3 .

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10

Bucˇkovska, A. L., and S. Pilipovic´. "An Extension of Bilinear Hilbert Transform to Distributions." Integral Transforms and Special Functions 13, no. 1 (January 2002): 1–15. http://dx.doi.org/10.1080/10652460212891.

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11

Lacey, M. T., and C. M. Thiele. "On Calderon's conjecture for the bilinear Hilbert transform." Proceedings of the National Academy of Sciences 95, no. 9 (April 28, 1998): 4828–30. http://dx.doi.org/10.1073/pnas.95.9.4828.

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12

Demeter, Ciprian. "Pointwise convergence of the ergodic bilinear Hilbert transform." Illinois Journal of Mathematics 51, no. 4 (October 2007): 1123–58. http://dx.doi.org/10.1215/ijm/1258138536.

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13

Dong, Dong. "On the bilinear Hilbert transform along two polynomials." Proceedings of the American Mathematical Society 147, no. 10 (June 27, 2019): 4245–58. http://dx.doi.org/10.1090/proc/14518.

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14

Carro, María Jesús, Loukas Grafakos, José María Martell, and Fernando Soria. "Multilinear extrapolation and applications to the bilinear Hilbert transform." Journal of Mathematical Analysis and Applications 357, no. 2 (September 2009): 479–97. http://dx.doi.org/10.1016/j.jmaa.2009.04.021.

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15

Dong, Dong, and Xiao Chun Li. "On a hybrid of bilinear Hilbert transform and paraproduct." Acta Mathematica Sinica, English Series 34, no. 1 (February 6, 2017): 29–41. http://dx.doi.org/10.1007/s10114-017-6415-9.

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16

Duoandikoetxea, Javier. "The bilinear Hilbert transform acting on Hermite and Laguerre functions." Journal of Approximation Theory 162, no. 1 (January 2010): 131–40. http://dx.doi.org/10.1016/j.jat.2009.03.009.

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17

Dong, Dong. "Full range boundedness of bilinear Hilbert transform along certain polynomials." Mathematical Inequalities & Applications, no. 1 (2019): 151–56. http://dx.doi.org/10.7153/mia-2019-22-11.

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18

Dong, Dong. "Quasi Pieces of the Bilinear Hilbert Transform Incorporated into a Paraproduct." Journal of Geometric Analysis 29, no. 1 (February 1, 2018): 224–46. http://dx.doi.org/10.1007/s12220-018-9987-4.

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19

Do, Yen, Richard Oberlin, and Eyvindur Ari Palsson. "Variational bounds for a dyadic model of the bilinear Hilbert transform." Illinois Journal of Mathematics 57, no. 1 (2013): 105–19. http://dx.doi.org/10.1215/ijm/1403534488.

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20

Cruz-Uribe, David, and José María Martell. "Limited range multilinear extrapolation with applications to the bilinear Hilbert transform." Mathematische Annalen 371, no. 1-2 (February 2, 2018): 615–53. http://dx.doi.org/10.1007/s00208-018-1640-9.

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21

Fan, Dashan, and Shuichi Sato. "Transference on certain multilinear multiplier operators." Journal of the Australian Mathematical Society 70, no. 1 (February 2001): 37–55. http://dx.doi.org/10.1017/s1446788700002263.

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AbstractWe study DeLeeuw type theorems for certain multilinear operators on the Lebesgue spaces and on the Hardy spaces. As applications, on the torus we obtain an analog of Lacey—Thiele's theorem on the bilinear Hilbert transform, as well as analogies of some recent theorems on multilinear singular integrals by Kenig—Stein and by Grafakos—Torres.

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22

Oberlin, Richard, and Christoph Thiele. "New uniform bounds for a Walsh model of the bilinear Hilbert transform." Indiana University Mathematics Journal 60, no. 5 (2011): 1693–712. http://dx.doi.org/10.1512/iumj.2011.60.4445.

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23

Lie, Victor. "On the boundedness of the bilinear Hilbert transform along “non-flat” smooth curves." American Journal of Mathematics 137, no. 2 (2015): 313–63. http://dx.doi.org/10.1353/ajm.2015.0013.

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24

DURCIK, POLONA, VJEKOSLAV KOVAČ, KRISTINA ANA ŠKREB, and CHRISTOPH THIELE. "Norm variation of ergodic averages with respect to two commuting transformations." Ergodic Theory and Dynamical Systems 39, no. 3 (August 17, 2017): 658–88. http://dx.doi.org/10.1017/etds.2017.48.

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We study double ergodic averages with respect to two general commuting transformations and establish a sharp quantitative result on their convergence in the norm. We approach the problem via real harmonic analysis, using recently developed methods for bounding multilinear singular integrals with certain entangled structure. A byproduct of our proof is a bound for a two-dimensional bilinear square function related to the so-called triangular Hilbert transform.

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25

Lacey, Michael, and Christoph Thiele. "L p Estimates on the Bilinear Hilbert Transform for 2 < p < &#8734." Annals of Mathematics 146, no. 3 (November 1997): 693. http://dx.doi.org/10.2307/2952458.

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26

Lorist, Emiel, and Zoe Nieraeth. "Sparse domination implies vector-valued sparse domination." Mathematische Zeitschrift 301, no. 1 (January 12, 2022): 1–35. http://dx.doi.org/10.1007/s00209-021-02943-z.

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AbstractWe prove that scalar-valued sparse domination of a multilinear operator implies vector-valued sparse domination for tuples of quasi-Banach function spaces, for which we introduce a multilinear analogue of the $${{\,\mathrm{UMD}\,}}$$ UMD condition. This condition is characterized by the boundedness of the multisublinear Hardy-Littlewood maximal operator and goes beyond examples in which a $${{\,\mathrm{UMD}\,}}$$ UMD condition is assumed on each individual space and includes e.g. iterated Lebesgue, Lorentz, and Orlicz spaces. Our method allows us to obtain sharp vector-valued weighted bounds directly from scalar-valued sparse domination, without the use of a Rubio de Francia type extrapolation result. We apply our result to obtain new vector-valued bounds for multilinear Calderón-Zygmund operators as well as recover the old ones with a new sharp weighted bound. Moreover, in the Banach function space setting we improve upon recent vector-valued bounds for the bilinear Hilbert transform.

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27

Lie, Victor. "On the boundedness of the bilinear Hilbert transform along “non-flat” smooth curves. The Banach triangle case $(L^r, 1 ≤ r < \infty)$." Revista Matemática Iberoamericana 34, no. 1 (February 6, 2018): 331–53. http://dx.doi.org/10.4171/rmi/987.

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28

Yuan, Ping-Ping, Zuo-Cai Wang, Wei-Xin Ren, and Wen-Yu He. "Nonlinear joint model updating of strong column-weak beam type frames based on instantaneous characteristics of the responses under earthquake excitations." Advances in Structural Engineering 20, no. 5 (April 11, 2017): 682–93. http://dx.doi.org/10.1177/1369433217698343.

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In this article, a novel method based on the instantaneous frequencies and amplitudes of the principal response components is presented for nonlinear joint model updating. The instantaneous frequencies and amplitudes are first extracted by a low-pass filter with Hilbert transform. Then, limited point values of the extracted instantaneous frequencies and amplitudes are applied to represent the response of the nonlinear structure. Finally, an objective function based on the residuals of instantaneous frequencies and amplitudes between experimental structure and finite element model is established using the response surface method. The optimal values of the nonlinear joint model parameters are obtained by minimizing the objective function using simulated annealing algorithm. To verify the effectiveness of the proposed method, a three-story frame with bilinear moment–rotation relationship at the beam-column joints under earthquake excitations is simulated as a numerical example. The accuracy of the proposed nonlinear joint model updating procedure is quantified using the defined error indices. The effects of the selected data point number and the weight factors in the objective function are also discussed in the article. The results indicate that the proposed method can effectively update the nonlinear joint model with high accuracy even with noise effect.

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29

Dai, Wei, and Guozhen Lu. "Lpestimates for bilinear and multiparameter Hilbert transforms." Analysis & PDE 8, no. 3 (June 3, 2015): 675–712. http://dx.doi.org/10.2140/apde.2015.8.675.

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30

Blasco, Oscar, and Paco Villarroya. "Commutators of linear and bilinear Hilbert transforms." Proceedings of the American Mathematical Society 132, no. 7 (December 19, 2003): 1997–2004. http://dx.doi.org/10.1090/s0002-9939-03-07266-6.

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31

Guo, Jingwei, and Lechao Xiao. "Bilinear Hilbert Transforms Associated with Plane Curves." Journal of Geometric Analysis 26, no. 2 (February 3, 2015): 967–95. http://dx.doi.org/10.1007/s12220-015-9580-z.

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32

Li, Junfeng, and Haixia Yu. "Bilinear Hilbert transforms and (sub)bilinear maximal functions along convex curves." Pacific Journal of Mathematics 310, no. 2 (March 8, 2021): 375–446. http://dx.doi.org/10.2140/pjm.2021.310.375.

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33

Grafakos, Loukas, and Xiaochun Li. "Uniform bounds for the bilinear Hilbert transforms, I." Annals of Mathematics 159, no. 3 (May 1, 2004): 889–933. http://dx.doi.org/10.4007/annals.2004.159.889.

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34

Li, Xiaochun. "Bilinear Hilbert transforms along curves, I: the monomial case." Analysis & PDE 6, no. 1 (June 1, 2013): 197–220. http://dx.doi.org/10.2140/apde.2013.6.197.

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35

Li, Xiaochun, and Lechao Xiao. "Uniform estimates for bilinear Hilbert transforms and bilinear maximal functions associated to polynomials." American Journal of Mathematics 138, no. 4 (2016): 907–62. http://dx.doi.org/10.1353/ajm.2016.0030.

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36

Silva, Prabath. "Vector-valued inequalities for families of bilinear Hilbert transforms and applications to bi-parameter problems." Journal of the London Mathematical Society 90, no. 3 (September 4, 2014): 695–724. http://dx.doi.org/10.1112/jlms/jdu044.

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37

Gao, Wenxu, Zhengming Ma, Weichao Gan, and Shuyu Liu. "Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry." Entropy 23, no. 9 (August 27, 2021): 1117. http://dx.doi.org/10.3390/e23091117.

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Symmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD data, where bilinear transformation plays a vital role. Because linear operations are not supported in nonlinear spaces such as Riemannian manifolds, directly performing Euclidean DR methods on SPD matrices is inadequate and difficult in complex models and optimization. An SPD data DR method based on Riemannian manifold tangent spaces and global isometry (RMTSISOM-SPDDR) is proposed in this research. The main contributions are listed: (1) Any Riemannian manifold tangent space is a Hilbert space isomorphic to a Euclidean space. Particularly for SPD manifolds, tangent spaces consist of symmetric matrices, which can greatly preserve the form and attributes of original SPD data. For this reason, RMTSISOM-SPDDR transfers the bilinear transformation from manifolds to tangent spaces. (2) By log transformation, original SPD data are mapped to the tangent space at the identity matrix under the affine invariant Riemannian metric (AIRM). In this way, the geodesic distance between original data and the identity matrix is equal to the Euclidean distance between corresponding tangent vector and the origin. (3) The bilinear transformation is further determined by the isometric criterion guaranteeing the geodesic distance on high-dimensional SPD manifold as close as possible to the Euclidean distance in the tangent space of low-dimensional SPD manifold. Then, we use it for the DR of original SPD data. Experiments on five commonly used datasets show that RMTSISOM-SPDDR is superior to five advanced SPD data DR algorithms.

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38

Lacey, Michael. "The bilinear Hilbert transform is pointwise finite." Revista Matemática Iberoamericana, 1997, 411–69. http://dx.doi.org/10.4171/rmi/227.

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39

KOVAČ, VJEKOSLAV, CHRISTOPH THIELE, and PAVEL ZORIN-KRANICH. "DYADIC TRIANGULAR HILBERT TRANSFORM OF TWO GENERAL FUNCTIONS AND ONE NOT TOO GENERAL FUNCTION." Forum of Mathematics, Sigma 3 (November 1, 2015). http://dx.doi.org/10.1017/fms.2015.25.

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The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well-studied objects of harmonic analysis. We investigate $L^{p}$ bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.

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40

Di Plinio, Francesco, Kangwei Li, Henri Martikainen, and Emil Vuorinen. "Banach-Valued Multilinear Singular Integrals with Modulation Invariance." International Mathematics Research Notices, September 9, 2020. http://dx.doi.org/10.1093/imrn/rnaa234.

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Abstract We prove that the class of trilinear multiplier forms with singularity over a one-dimensional subspace, including the bilinear Hilbert transform, admits bounded $L^p$-extension to triples of intermediate $\operatorname{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\operatorname{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\textrm{UMD}$-valued setting. This is then employed to obtain appropriate single-tree estimates by appealing to the $\textrm{UMD}$-valued bound for bilinear Calderón–Zygmund operators recently obtained by the same authors.

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41

Kleiner, T., and R. Hilfer. "On extremal domains and codomains for convolution of distributions and fractional calculus." Monatshefte für Mathematik, March 2, 2022. http://dx.doi.org/10.1007/s00605-021-01646-1.

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AbstractIt is proved that the class of c-closed distribution spaces contains extremal domains and codomains to make convolution of distributions a well-defined bilinear mapping. The distribution spaces are systematically endowed with topologies and bornologies that make convolution hypocontinuous whenever defined. Largest modules and smallest algebras for convolution semigroups are constructed along the same lines. The fact that extremal domains and codomains for convolution exist within this class of spaces is fundamentally related to quantale theory. The quantale theoretic residual formed from two c-closed spaces is characterized as the largest c-closed subspace of the corresponding space of convolutors. The theory is applied to obtain maximal distributional domains for fractional integrals and derivatives, for fractional Laplacians, Riesz potentials and for the Hilbert transform. Further, maximal joint domains for families of these operators are obtained such that their composition laws are preserved.

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42

Li, Kaili, Fengshan Zhou, Anqi He, Ran Guo, Xiaopei Li, Yizhuang Xu, Isao Noda, Yukihiro Ozaki, and Jinguang Wu. "Intensity Enhancement of a Two-Dimensional Asynchronous Spectrum Without Noise Level Fluctuation Escalation Using a One-Dimensional Spectra Sequence Change." Applied Spectroscopy, November 26, 2020, 000370282097171. http://dx.doi.org/10.1177/0003702820971714.

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Previously, we demonstrated that the intensities of cross-peaks in a two-dimensional asynchronous spectrum could be enhanced using sequence change of the corresponding one-dimensional spectra. This unusual approach becomes useful when the determination of the sequential order of physicochemical events is not essential. However, it was not known whether the level of noise in the two-dimensional asynchronous spectrum was also escalated as the sequence of one-dimensional spectra changed. We first investigated the noise behavior in a two-dimensional asynchronous spectrum upon changing the sequence of the corresponding one-dimensional spectra on a model system. In the model system, bilinear data from a chromatographic–spectroscopic experiment on a mixture containing two components were analyzed using a two-dimensional asynchronous spectrum. The computer simulation results confirm that the cross-peak intensities in the resultant a two-dimensional asynchronous spectrum were indeed enhanced by more than 100 times as the sequence of one-dimensional spectra changed, whereas the fluctuation level of noise, reflected by the standard deviation of the value of a two-dimensional asynchronous spectrum at a given point, was almost invariant. Further analysis on the model system demonstrated that the special mathematical property of the Hilbert–Noda matrix (the modules of all column vectors of the Hilbert–Noda matrix being a near constant) accounts for the moderate variation of the noise level during the changes of the sequence of one-dimensional spectra. Next, a realistic example from a thermogravimetry–Fourier transform infrared spectroscopy experiment with added artificial noise in seven one-dimensional spectra was studied. As we altered the sequence of the seven FT-IR spectra, the variation of the cross-peak intensities covered four orders of magnitude in the two-dimensional asynchronous spectra. In contrast, the fluctuation of noise in the two-dimensional asynchronous spectra was within two times. The above results clearly demonstrate that a change in the sequence of one-dimensional spectra is an effective way to improve the signal-to-noise level of the two-dimensional asynchronous spectra.

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43

Li, Xiaochun. "Uniform Bounds for the Bilinear Hilbert Transforms, II." Revista Matemática Iberoamericana, 2006, 1069–126. http://dx.doi.org/10.4171/rmi/483.

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