Faculty Publications

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Author: search.filters.author.Johnson, P.S.Subject: search.filters.subject.Frame

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    Construction of Mercedes–Benz Frame in R n
    (Springer, 2017) Parvathalu, B.; Johnson, P.S.
    In this article, Mercedes–Benz (MB) frame having 3 vectors in R 2 is generalized to the space R n with n+ 1 vectors through a complete concrete method. A necessary and sufficient condition for a normed tight frame to be an MB frame is given and MB frame is explored with the help of diagram vectors. In a new approach, it has been proved that there is no MB frame in R n with more than n+ 1 vectors and there is always an equiangular tight frame for every n? 2 , using MB frame. © 2017, Springer India Pvt. Ltd.
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    Towards characterizations of approximate Schauder frame and its duals for Banach spaces
    (Birkhauser, 2021) Mahesh Krishna, K.M.; Johnson, P.S.
    Characterizations for a frame and its duals are known for separable Hilbert spaces. In this paper, we characterize a class of approximate Schauder frame and its duals for separable Banach spaces. We also give an operator-theoretic characterization for similarity of ASFs. Our results encode the results of Holub, Li, Balan, Han, and Larson. We also address orthogonality of ASFs. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.
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    DILATION THEOREM FOR p-APPROXIMATE SCHAUDER FRAMES FOR SEPARABLE BANACH SPACES
    (Palestine Polytechnic University, 2022) Mahesh Krishna, K.M.; Johnson, P.S.
    Famous Naimark-Han-Larson dilation theorem for frames in Hilbert spaces states that every frame for a separable Hilbert space H is the image of a Riesz basis under an orthogonal projection from a separable Hilbert space H1 which contains H isometrically. In this paper, we derive dilation result for p-approximate Schauder frames for separable Banach spaces. Our result contains Naimark-Han-Larson dilation theorem as a particular case. © Palestine Polytechnic University-PPU 2022.
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    Frames for Metric Spaces
    (Birkhauser, 2022) Mahesh Krishna, K.M.; Johnson, P.S.
    We make a systematic study of frames for metric spaces. We prove that every separable metric space admits a metric Md-frame. Through Lipschitz-free Banach spaces we show that there is a correspondence between frames for metric spaces and frames for subsets of Banach spaces. We derive some characterizations of metric frames. We also derive stability results for metric frames. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
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    APPROXIMATELY DUAL p-APPROXIMATE SCHAUDER FRAMES
    (Austral Internet Publishing, 2023) Mahesh Krishna, K.M.; Johnson, P.S.
    Approximately dual frame in Hilbert spaces was introduced by Christensen and Laugesen to overcome difficulties in constructing dual frames for a given Hilbert space frame. It becomes even more difficult in Banach spaces to construct duals. For this purpose, we introduce approximately dual frames for a class of approximate Schauder frames for Banach spaces and develop basic theory. Approximate dual for this subclass is completely characterized and its perturbation is also studied. © 2023 Austral Internet Publishing. All rights reserved
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    ON THE CONSTRUCTION AND PROPERTIES OF FRAMES USING INCIDENCE MATRIX OF GRAPHS AND THEIR SPECTRA
    (Jangjeon Research Institute for Mathematical Sciences and Physics, 2024) Senthil Thilak, A.S.; Ayyanar, K.; Johnson, P.S.
    Frames are considered to be redundant counterparts of bases for vector spaces. This redundant structure favours frames to be rich in both theory and applications. In recent studies on frames, graph theory is one of the significant tools to analyze the properties of different types of frames. In graph theory, we associate a graph with different types of matrices, of which signless Laplacian matrix contributes significantly in exploring the properties of a graph. In this paper, given a graph G, we propose a method to construct a frame from its incidence matrix such that its frame graph is the line graph of a derived graph of G. We analyze various properties of the frame constructed as above, its dual, etc. Further, we investigate the existence of frames with constrained frame bounds, using the properties of the associated graph and its signless Laplacian spectrum. © 2024 Jangjeon Research Institute for Mathematical Sciences and Physics. All rights reserved.