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TIGHT MATRIX-GENERATED GABOR FRAMES IN $L^2(\mathbb{R}^d)$ WITH DESIRED TIME-FREQUENCY LOCALIZATION
Christensen, Ole,Kim, Rae-Young Korean Society of Computational and Applied Mathem 2008 Journal of applied mathematics & informatics Vol.26 No.5
Based on two real and invertible $d{\times}d$ matrices Band C such that the norm $||C^T\;B||$ is sufficiently small, we provide a construction of tight Gabor frames $\{E_{Bm}T_{Cn}g\}_{m,n{\in}{\mathbb{Z}^d}$ with explicitly given and compactly supported generators. The generators can be chosen with arbitrary polynomial decay in the frequency domain.
Extensions of Bessel sequences to dual pairs of frames
Christensen, O.,Kim, H.O.,Kim, R.Y. Academic Press 2013 Applied and computational harmonic analysis Vol.34 No.2
Tight frames in Hilbert spaces have been studied intensively for the past years. In this paper we demonstrate that it often is an advantage to use pairs of dual frames rather than tight frames. We show that in any separable Hilbert space, any pairs of Bessel sequences can be extended to a pair of dual frames. If the given Bessel sequences are Gabor systems in L<SUP>2</SUP>®, the extension can be chosen to have Gabor structure as well. We also show that if the generators of the given Gabor Bessel sequences are compactly supported, we can choose the generators of the added Gabor systems to be compactly supported as well. This is a significant improvement compared to the extension of a Bessel sequence to a tight frame, where the added generator only can be compactly supported in some special cases. We also analyze the wavelet case, and find sufficient conditions under which a pair of wavelet systems can be extended to a pair of dual frames.
Monitoring $CO_2$ injection with cross-hole electrical resistivity tomography
Christensen, N.B.,Sherlock, D.,Dodds, K. Korean Society of Earth and Exploration Geophysici 2006 지구물리와 물리탐사 Vol.9 No.1
In this study, the resolution capabilities of electrical resistivity tomography (ERT) in the monitoring of $CO_2$ injection are investigated. The pole-pole and bipole-bipole electrode configuration types are used between two uncased boreholes straddling the $CO_2$ plume. Forward responses for an initial pre-injection model and three models for subsequent stages of $CO_2$ injection are calculated for the two different electrode configuration types, noise is added and the theoretical data are inverted with both L1- and L2-norm optimisation. The results show that $CO_2$ volumes over a certain threshold can be detected with confidence. The L1-norm proved superior to the L2-norm in most instances. Normalisation of the inverted models with the pre-injection inverse model gives good images of the regions of changing resistivity, and an integrated measure of the total change in resistivity proves to be a valid measure of the total injected volume.
International Repercussions of Source-based Capital Income Taxation
Christensen, Thomas Alslev,Nielsen, Søren Bo 세종대학교 국제경제연구소 1995 Journal of Economic Integration Vol.10 No.1
The paper is concerned with international effects of source-based capital income taxation in a large economy. We derive, within the context of a twocountry overlapping generations model in continuous time, the implications of such taxation for the world interest rate and for investment, consumption, saving and external balances at home and abroad. Furthermore, we argue that higher source-based taxes will hurt foreigners alive at the time of the policy change, whereas future citizens abroad stand to benefit. Finally, the effects of source-and residence-based taxed are compared.
Character Disorder:The Twentieth Century Neurosis
Christensen, Robert I 大韓神經精神醫學會 1962 신경정신의학 Vol.1 No.1
I have tried to define the adjustment problem generally called character disorder,partly by contrasting it against the better understood neurotic pattern. Implicit in this presentation is a plea for more accurate diagnosis-of the total personality picture, not merely the naming symptoms(often inaccurate and incomplete). I have also indicated the basic similarity between character disorder and psychoneurosis. Finally I have made some suggestions regarding the board field of compensation in psychiatry.
On Gabor frames generated by sign-changing windows and B-splines
Christensen, O.,Kim, H.O.,Kim, R.Y. Academic Press 2015 Applied and computational harmonic analysis Vol.39 No.3
For a class of compactly supported windows we characterize the frame property for a Gabor system {E<SUB>mb</SUB>T<SUB>na</SUB>g}<SUB>m,n@?Z</SUB>, for translation parameters a belonging to a certain range depending on the support size. We show that the obstructions to the frame property are located on a countable number of ''curves.'' For functions that are positive on the interior of the support these obstructions do not appear, and the considered region in the (a,b) plane is fully contained in the frame set. In particular this confirms a recent conjecture about B-splines by Grochenig in that particular region. We prove that the full conjecture is true if it can be proved in a certain ''hyperbolic strip.''
On entire functions restricted to intervals, partition of unities, and dual Gabor frames
Christensen, O.,Kim, H.O.,Kim, R.Y. Academic Press 2015 Applied and computational harmonic analysis Vol.38 No.1
Partition of unities appears in many places in analysis. Typically it is generated by compactly supported functions with a certain regularity. In this paper we consider partition of unities obtained as integer-translates of entire functions restricted to finite intervals. We characterize the entire functions that lead to a partition of unity in this way, and we provide characterizations of the ''cut-off'' entire functions, considered as functions of a real variable, to have desired regularity. In particular we obtain partition of unities generated by functions with small support and desired regularity. Applied to Gabor analysis this leads to constructions of dual pairs of Gabor frames with low redundancy, generated by trigonometric polynomials with small support and desired regularity.