Dr. Lalit Kumar Vashisht
Dr. Lalit Kumar Vashisht (Ph. D. University of Delhi)
Assistant Professor
Department of Mathematics
University of Delhi
Delhi -110 007.
Phone Office: +91-011-27666658
Email: lalit@maths.du.ac.in, lalitkvashisht@gmail.com
My Research Interests:
Frames for Banach Spaces; Hilbert Frames; Wavelet Analysis;
Fourier analysis; Functional Analysis.
Research Publications
- Unitary extension principle for non-uniform wavelet frames in L^2(R), ArXiv: 1806.00623 (with H.K. Malhotra).
- Vector-valued (Super) weaving frames, Journal of Geometry and Physics, 134 (2018), 48-57. (with Deepshikha).
- K-Matrix-valued wave packet frames in L^2(R^d, C^(s x r)), Mathematical Physics, Analysis and Geometry, 21 (3) (2018), Art 21, 19 pp. (With Jyoti). MR3835627, Zbl 1396.41020.
- Weaving K-frames in Hilbert spaces, Results in Mathematics, 73 (2) (2018), Art 81, 20 pp. (with Deepshikha). MR3805557, Zbl 06909700.
- On excess of retro Banach frames, Natsional'naya Akademiya Nauk Armenii. Izvestiya . Matematika, to appear. (with G. Verma).
- On weaving frames, Houston Journal of Mathematics, 44 (3) (2018), 887-915. (with Deepshikha).
- On generalized weaving frames in Hilbert spaces, Rocky Mountain Journal of Mathematics, 48 (2) (2018), 661-685. (with S. Garg, Deepshikha and P.K. Das). MR3810464, Zbl 1392.42030
- Sums of matrix-valued wave packet frames in L^2(R^d, C^(s x r)), Glasnik Matematicki , 53 (1) (2018) , 153-177. (with Jyoti, Deepshikha and G. Verma) MR3814235, Zbl 06912626.
- On WH-packets of matrix-valued wave packet frames in L^2(R^d, C^(s x r)), International Journal of Wavelets Multiresolution and Information Process, 16 (3) (2018) 1850022 (22 pages) (with Jyoti). MR3805708, Zbl 1396.42008.
- Traces of Hadamard and Kronecker products of matrices, Mathematics for Applications (Brno), 6 (2017) 143-150 (with P. K. Das) MR3742414, Zbl 1386.15042.
- Generalized weaving frames for operators in Hilbert spaces, Results in Mathematics, 72 (3) (2017), 1369-1391 (with Deepshikha and Geetika Verma). MR3721620, Zbl 1385.42028 .
- Frames in topological algebras, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math.Phys., 79 (3) (2017), 163-170. (with Saakshi Garg) MR3703289.
- Weaving K-fusion frames in Hilbert spaces, Ganita, 67 (1) (2017), 41-52. (with Saakshi Garg) MR3785319 .
- Extension of Bessel sequences to dual frames in Hilbert spaces, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 79 (2) (2017), 71-82. (with Deepshikha). MR3678067
- Necessary and sufficient conditions for discrete wavelet frames in C^N, Journal of Geometry and Physics, 117 (2017), 134-143. (with Deepshikha) MR3645837, Zbl 1364.42035.
- On Continuous weaving frames, Advances in Pure and Applied Mathematics, 8 (1) (2017), 15-31. (with Deepshikha) MR3592583, Zbl 1365.42028.
- Weaving properties of generalized continuous frames generated by an iterated function system, Journal of Geometry and Physics., 110 (2016), 282-295. (with Deepshikha) MR3566116, Zbl 06647253.
- On perturbation of frames in locally convex spaces, Jordan Journal of Mathematics and Statistics, 9 (4) (2016), 271-281 (with S. Garg and G. Khattar). MR3603193, Zbl 1362.42068.
- A note on discrete frames of translates in C^N, TWMS Journal of Applied and Engineering Mathematics, 6 (1) (2016), 143-149. (with Deepshikha) MR3594863, Zbl 1371.42038.
- Extension of Weyl-Heisenberg wave packet Bessel sequences to dual frames in L^2(R), J. Class. Anal., 8 (2) (2016), 131-145. (with Deepshikha). MR3535232
- Error locating codes by using blockwise-tensor product of blockwise detecting / correcting codes, Khayyam Journal of Mathematics, 2 (1) (2016), 6-17. (with P. K. Das) MR3577213 , Zbl 138094144.
- On exact frames in topological algebras, Palestine Journal of Mathematics, 5 (1) (2016), 131-134. (with Saakshi Garg ) MR3413765, Zbl 137742032
- On perturbation of binary linear codes, Mathematics for Applications (Brno), 4 (2015), 91-99 (with P. K. Das) MR3437508, Zbl 1370.94573 .
- Banach frames generated by compact operators associated with a boundary value problem, TWMS Journal of Pure and Applied Mathematics, 6 (2) (2015), 254-258. MR3689356, Zbl 1348.42041.
- Some types of convergence related to the reconstruction property in Banach spaces, Banach Journal of Mathematical Analysis, 9 (2) (2015), 253-275. (with G. Khattar) MR3296117, Zbl 1311.42082
- The reconstruction property in Banach spaces generated by matrices, Advances in Pure and Applied Mathematics, 5 (3) (2014), 151-160.(with G. Khattar) MR3259018, Zbl 1297.42044.
- Frames of eigenfunctions associated with a boundary value problem, International Journal of Analysis, Vol. 2014 Article Id. 590324. 6 pages, 2014. doi: 10.1155/2014/590324 (with Shalu Sharma) MR3219410, Zbl 06883059.
- Reconstruction property and frames in Banach spaces, Palestine Journal of Mathematics, 3(1) (2014), 11-26. (with S.K.Kaushik and G.Khattar) MR3109934.
- On I-reconstruction property, Advances in Pure Mathematics, 3 (3) (2013), 324 - 330. ( with G. Khattar) doi:10.4236/apm.2013.33046..
- Some results concerning frames associated with measureable spaces, TWMS Journal of Pure and Applied Mathematics, 4 (1) (2013), 52-60. (with S.K. Kaushik and S.K.Sharma) MR3097680, Zbl 1312.42034.
- On Φ-Schauder frames, TWMS Journal of Applied and Engineering Mathematics, 2 (1) (2012), 116-120. MR3068866, Zbl 1274.42080.
- On retro Banach frames of type P, Azerbaijan .Journal of Mathematics, 2 (1) (2012), 82-89. MR2967281, Zbl 1278.42041.
- On stability of Banach frames, Bulletin of the Korean Mathematical Society, 44 (2007), no. 1, 73-81. (with P.K.Jain and S.K. Kaushik) MR2297696, Zbl1190.42013.
- On Banach frames, Indian Journal of Pure and Applied Mathematics, 37 (2006), no. 5, 265-272 (with P.K.Jain and S.K. Kaushik) MR2271627, Zbl 1125.46013.
- On perturbation of Banach frames, International Journal of Wavelets Multiresolution and Information Process , 4 (2006), no. 3, 559-565. (with P.K.Jain and S.K. Kaushik) MR2260842, Zbl 1213.42113.
- Bessel sequences and Banach frames in Banach spaces, Bulletin of the Calcutta Mathematical Society, 98 (2006), no.1, 87-92.(with P.K.Jain and S.K.Kaushik) MR2218227.
- Banach frames for conjugate Banach spaces, Zeitschrift fur Analysis und ihre Anwendungen, 23 (2004), no. 4, 713-720. (with P.K.Jain and S. K. Kaushik) MR2110399, Zbl 1059.42024.
- Some remarks in the theory of frames in Banach spaces, Journal of Analysis and Applications, 2 (2004), no. 1, 39-50. (with P.K.Jain and S.K.Kaushik). MR2038296, Zbl 1090.42019.
Conference Papers
- Matrix-valued wave packet Bessel sequences and symmetric frames in L^2(R^d, C^(s x r)), In: Proceeding of the International Workshop on Wavelets, Frames and Applications ( IWWFA) III- 2017, organized by Department of Mathematics, Kirori Mal College, University of Delhi. Published by the Poincare Journal of Analysis and Applications (Special Issue) , 2018 (2) (with Jyoti, G. Verma, and Virender) .
- On weaving fusion frames for Hilbert spaces, In: Proceedings of Sampling Theory and Applications ( SampTA) (2017), 381-385 (with Deepshikha, S. Garg and G. Verma). Published by the IEEE. DOI: 10.1109/SAMPTA.2017.8024363.
- On perturbation of local atoms for subspaces, In: Proceeding of the International Workshop on Wavelets, Frames and Applications ( IWWFA) II- 2014, organized by Department of Mathematics, Kirori Mal College, University of Delhi. Published by the Poincare Journal of Analysis and Applications (Special Issue) , 2 (2015), 129-137. (with Deepshikha) MR3439947, Zbl 1352.42040 .
- On various types of frames in Banach spaces, Proceeding of the National Conference on Mathematics of soft computing, National Institute of Technology (NIT), Calicut, (2012), 109-112.
- On weighted Banach frames, In: Proceeding of the International Workshop on Wavelets, Frames and Applications ( IWWFA) I- 2011, organized by Department of Mathematics, Kirori Mal College, University of Delhi. Published in the Communications in Mathematics and Applications, (Special Issue) 3 (2012), 283-292 (with Shalu Sharma) .
- On frames in Banach spaces, In: Proceeding of the International Workshop on Wavelets, Frames and Applications ( IWWFA) I- 2011, organized by Department of Mathematics, Kirori Mal College, University of Delhi. Published in Communications in Mathematics and Applications, (Special Issue) 3 (2012), 313-332.
Research Guidance
(I) Ph. D.
The following scholars have completed and got the Ph. D. Degree
- Geetika Khattar. Thesis Title: ''On the Reconstruction Property and Frames in Banach Spaces'' . Currently, Research Associate, School of Information Technology and Mathematical Sciences, University of South Australia).
- Saakshi Garg. Thesis Title ''Weaving Generalized Frames in Hilbert Spaces and Frames in Locally Convex Spaces''. Currently, Assistant Professor in a college affiliated to the University of Delhi.
- Deepshikha. Thesis Title: ''Weaving Frames in Hilbert Spaces'' . Currently, Assistant Professor in a college affiliated to the University of Delhi.
The following scholars are currently doing Ph. D.
- Hari Krishan Malhotra (Under Progress)
- Jyoti (Under Progress)
- Uttam Kumar Sinha (Under Progress)
(II) M. Phil.
The following scholars have completed and got the M. Phil. Degree
- Sah Jahan. Dissertation Title: "Modular Frames and Riesz Bases in Hilbert C*-Modules".
- Salaj. Dissertation Title: "A Study of Finite Frames in Hilbert Spaces " . Currently, Assistant Professor in a college affiliated to the Maharishi Dayanand University, Rohtak.
- Sulbha Kumar Dissertation Title: '' Geometric Means of Positive Definite Matrices'' (jointly with Tanvi Jain)
- Soni. Dissertation Title: ''Wavelet Frames and Multiresolution Analysis on Local Fields '' . Currently, Assistant Professor in a college affiliated to the University of Delhi.
- Neha Sharma. Dissertation Title: ''Duality of Generalized Frames in Hilbert Spaces''.
- Rajni Gupta. Dissertation Title: ''Fusion Frames in Hilbert Spaces''. Currently, Assistant Professor in a college affiliated to the University of Delhi.
The following scholars are currently doing M. Phil.
- Ram Kishan (Under progress).
Conference Participation
Invited Talk
- Delivered a talk entitled ''Woven Fusion Frames in Separable Hilbert Spaces' in a National Conference on Recent Trends in Mathematics jointly organized by Bharata Ganita Parisad and Department of Mathematics & Astronomy, University of Lucknow, Lucknow during November 10-11 , 2018.
- Delivered an invited lecture entitled ''Hilbert Frames: A Signal Processing Prospective'' in International Conference on Applicable Mathematics organized by Department of Mathematics, Motilal Nehru College, University of Delhi, Delhi during 19-20 February, 2018.
- Delivered an invited lecture entitled "The Reconstruction Property of Fractals in L^p Spaces" in International workshop on wavelets, frames and applications III at Kirori Mal college, University of Delhi, Delhi during 14-20 December 2017.
- Delivered an invited talk entitled '' Weaving Properties of Generalized Frames in Hilbert Spaces'' in International Conference on Applicable Analysis, organized by Department of Mathematics, Shaheed Bhagat Singh College,University of Delhi, Delhi during 08-11 February, 2017.
- Delivered an invited talk entitled '' Discrete Frames of Translates in C^N'' in National workshop on Analysis, Differential equations & Applications, at Department of Mathematics & Statistics, M. L. Sukhadia University, Udaipur (Raj.) during 25-27 February, 2016.
- Delivered an invited lecture entitled " On Wave Packet Frames in L^2(R)" in International workshop on wavelets, frames and applications II at Kirori Mal college, University of Delhi, Delhi during 24-30 December 2014.
- Delivered an invited lecture entitled " On Retro Banach Frames in Banach spaces'' in National Conference on Recent Advances in Mathematics at Department of Mathematics & Astronomy, University of Lucknow, Lucknow during 2-4 February, 2012.
Paper Presentation
- Present a paper entitled '' Weaving Frames in Measure Spaces'' in Internaional conference on Recent Advances in Pure and Applied Mathematics & 28th Annual Conference of Rajastha Ganita Parishad, at Department of Mathematics & Statistics, M. L. Sukhadia Univesity, Udaipur (Raj.), during 13-14 February, 2017.
- Delivered a talk "Extension of Bessel Systems to Frames in L^2(R)" in Conference on Geometry of Banach spaces and Operator Theory (GBOT) at IIT Kanpur during 26 - 29 March, 2015.
- Present a paper entitled "Some types of linear combinations of wave packet system in L^2(R)" in National conference on emerging trends in mathematical sciences organised by Department of Mathematics, Chaudhary Devi Lal University, Sirsa (Haryana) during March 17-18, 2015.
- Present a paper entitled " Frames for subspaces in a discrete signal space" in National conference on Applications of Mathematics in Engineering and Sciences (AMES- 2014) organised by Department of Mathematics, Motilal Nehru National Institute of Technology (MNNIT) Allahabad, during November-29-30, 2014.
- Present a paper entitled " On (\omega, \mu)-frames in Hilbert spaces, in National conference on Mathematics (NCOM-2013) organised by department of Mathematics & Astronomy, University of Lucknow, Lucknow, during 29-11-2013 - 01-12-2013.
- Present a paper entitle "On duality of retro Banach frames" 11th Biennial conference of ISIAM at Gautam Buddha University, Greater Noida during Dec. 15-16, 2012.Present a paper entitled " The Dynamics of Banach frames " in National Conference on Advances in Mathematical Sciences (AMS 2012), at Motilal Nehru National Institute of Technology, Allahabad, during October 05-07, 2012.
- Present a paper entitled " On Various Types of Frames in Banach Spaces" in National Conference on Mathematics of Soft Computing at N I T Calicut, Kerla, during 05-07 July-2012.
- Present a paper entitled " On Banach Frames II " in International Workshop Wavelets, Frames and Applications at Kirori Mal College,University of Delhi during December 15-21, 2011.
Organized Conferences / Seminars
1 Organized a ''Research Scholar Seminar'', Department of Mathematics, University of Delhi during February 23-24, 2015.
2. Organized ''National Conference on Algebra, Analysis, Coding and Cryptography" in honor of Prof. Bal Kishan Dass on the occasion of his retirement in the Department of Mathematics, University of Delhi during October 14-15, 2016.
3. Organized ''Research Scholar Seminar & Annual Conference of the Society of Mathematical Sciences (Delhi)'' in the Department of Mathematics, University of Delhi during May 01-02, 2017.
4. Organized a Symposium on ''Frames and Wavelets '' in the 33rd Annual Conference of Ramanujan Mathematical Society in the Department of Mathematics, University of Delhi during June 01-03, 2018.
Courses Taught
U G Level: Real Analysis, Calculus, Metric Space, Linear Algebra, Group Theory, Ring Theory, Mechanics, Biomathematics, Differential Equations, Partial Differential Equations.
P G Level: Functional Analysis, Complex Analysis, Matrix Analysis, Measure & Integration, General Measure Theory, Frames & Wavelets, Theory of Bounded Operators.
M.Phil. / Ph.D. Coursework: Advanced Frame Theory.
Professional Activities
(I) Member of editorial board for the following journals
- International Journal of Wavelets Multiresolution and Information Processing (IJWMIP).
- Mathematical Journal of Interdisciplinary Sciences.