Associate Professor Jeffrey Hogan
Associate Professor
School of Information and Physical Sciences (Mathematics)
- Email:jeff.hogan@newcastle.edu.au
- Phone:(02) 4921 7235
Career Summary
Biography
Biography
I was awarded the BSc (Hons) in 1985 and the PhD in 1990 (under the supervision of Professors John Price and Michael Cowling) from the University of New South Wales. After completing my study, I held postdoctoral positions at the University of Texas at Austin (1989-91) with Professor John Gilbert, Flinders University (1991-93) with Professor Garth Gaudry and Macquarie University (1993-2000) with Professor Alan McIntosh. I then moved to the University of Arkansas in Fayetteville, and gained tenure and promotion to Associate Professor in 2006. In 2009 I accepted a position at the University of Newcastle.
Research
Broadly speaking, I work in pure, applied and computational harmonic analysis. My past and present projects include:
- Uncertainty Principles for the Fourier transform on locally compact groups
- Singular integrals and wavelet frames
- Sampling theory in Fourier and wavelet analysis
- Clifford-Fourier theory
- Bandlimited and bandpass prolate functions in one- and higher dimensions
- Applications of optimisation in the construction of one- and multi-dimensional wavelets
- Fractionation of minerals according to size and denisty
Research Collaboration
I have active collaborations with:
- Laureate Professor Kevin Galvin (University of Newcastle)
- Professor Joseph Lakey (New Mexico State University)
- Dr Matthew Tam (University of Goettingen)
- Dr Scott Lindstrom (Curtin University)
- Dr Peter Massopust (Technical University of Munich)
- Professor Brigitte Forster (University of Passau)
- Dr Mark Craddock (University of Technology Sydney)
Teaching
Since arrival at UON, I have taught across the undergraduate mathematics curriculum, including courses in first-year calculus, multivariable calculus, linear algebra, mathematical modelling, differential equations, complex analysis, numerical analysis and my favourite -- Fourier analysis.
Administration
Administrative duties at UON have included:
- Head of Mathematics Discipline: July 2023 -- present
- Deputy Head of School of Mathematical and Physical Sciences: 2015 -- 2022
- Acting Head of Mathematics Discipline: July 2017 - July 2018
- MAPS HDR Coordinator: January 2017 -- present
- Bachelor of Mathematics Convenor: 2009 -- 2012
Qualifications
- PhD, University of New South Wales
- Bachelor of Science, University of New South Wales
Keywords
- Clifford analysis
- Fourier analysis
- signal processing
- singular integrals
- wavelets
Fields of Research
Code | Description | Percentage |
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490406 | Lie groups, harmonic and Fourier analysis | 60 |
490101 | Approximation theory and asymptotic methods | 30 |
400607 | Signal processing | 10 |
Professional Experience
UON Appointment
Title | Organisation / Department |
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Associate Professor | University of Newcastle School of Mathematical and Physical Sciences Australia |
Academic appointment
Dates | Title | Organisation / Department |
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1/7/2006 - 1/12/2008 | Associate Professor | University of Arkansas Department of Mathematical Sciences United States |
1/8/2000 - 1/7/2006 | Assistant Professor | University of Arkansas Department of Mathematical Sciences United States |
15/7/1993 - 31/8/2000 | Research Associate | MACCS, Macquarie University Australia |
1/9/1991 - 1/6/1993 | Research Associate | Flinders University Australia |
1/8/1989 - 1/8/1991 | Lecturer | University of Texas At Austin Department of Mathematics United States |
1/3/1985 - 1/7/1989 | Tutor | The University of New South Wales School of Mathematical Sciences Australia |
Publications
For publications that are currently unpublished or in-press, details are shown in italics.
Book (7 outputs)
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2013 |
Hogan J, AMSI International Conference on Harmonic Analysis and Applications (Macquarie University, February 2011), Centre for Mathematics and its Applications, Mathematical Sciences Institute, the Australian National University, Canberra, 147 (2013)
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2013 |
Hogan J, AMSI International Conference on Harmonic Analysis and Applications (Macquarie University, February 2011), Centre for Mathematics and its Applications, Mathematical Sciences Institute, the Australian National University, Canberra, 147 (2013)
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2012 |
Hogan JA, Lakey JD, Duration and Bandwidth Limiting: Prolate Functions, Sampling, and Applications, Birkhauser, New York, 258 (2012) [A1]
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2012 | Hogan JA, Lakey JD, Preface (2012) | ||||
2005 |
Hogan JA, Lakey JD, Time-frequency and time-scale methods: Adaptive decompositions, uncertainty principles, and sampling (2005) [A1]
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2005 | Hogan JA, Lakey JD, Preface (2005) | ||||
2002 |
Gilbert JE, Han YS, Hogan JA, Lakey JD, Weiland D, Weiss G, Smooth molecular decompositions of functions and singular integral operators (2002) [A1] Under minimal assumptions on a function ¿ we obtain wavelettype frames of the form ¿j,k(x) = r1/2nj¿(rjx - sk) j ¿ Z k ¿ Zn for some r > 1 and s > 0. This collection is show... [more] Under minimal assumptions on a function ¿ we obtain wavelettype frames of the form ¿j,k(x) = r1/2nj¿(rjx - sk) j ¿ Z k ¿ Zn for some r > 1 and s > 0. This collection is shown to be a frame for a scale of Triebel-Lizorkin spaces (which includes Lebesgue, Sobolev and Hardy spaces) and the reproducing formula converges in norm as well as pointwise a.e. The construction follows from a characterization of those operators which are bounded on a space of smooth molecules. This characterization also allows us to decompose a broad range of singular integral operators in terms of smooth molecules.
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Chapter (19 outputs)
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2023 |
Ghaffari HB, Hogan JA, Lakey JD, 'Clifford Prolate Spheroidal Wavefunctions and Associated Shift Frames', Applied and Numerical Harmonic Analysis 393-411 (2023) [B1] Prolate spheroidal wave functions have long been used in mathematical physics as a basis in which to expand solutions of the Helmholtz equation in prolate spheroidal coordinates. ... [more] Prolate spheroidal wave functions have long been used in mathematical physics as a basis in which to expand solutions of the Helmholtz equation in prolate spheroidal coordinates. They are simultaneous eigenfunctions of a Sturm-Liouville differential operator and a truncation of the Fourier transform to an interval. In 1961, Slepian and Pollak exploited the connection to the truncated Fourier transform to show that the prolates provide the solution to the spectral concentration problem. Shortly after, Slepian constructed eigenfunctions of a higher-dimensional truncation of the Fourier transform (in which the truncations are performed with respect to balls in n-dimensional Euclidean space) by a separation of variables approach. The radial parts of these solutions were shown to be eigenfunctions of a Sturm-Liouville differential operator with a singularity at the origin. In this chapter we first review a Clifford analysis-based approach to the construction of higher-dimensional prolates associated with the ball-truncated Fourier transform. A non-singular Clifford differential operator acting on multidimensional Clifford-valued functions is shown to commute with the ball-truncated Fourier transform, and the associated eigenfunctions (Clifford prolate spheroidal wave functions, or CPSWFs) are constructed numerically. Properties of these functions and their associated eigenvalues are explored. The advantages of this approach are that the singular Sturm-Liouville operator plays no role, replaced by a better-behaved operator, and the CPSWFs take values in a multi-channel algebra, allowing for analysis of vector-valued signals such as those which arise in electromagnetic theory. Finally, we apply this theory to the construction of frames for the space of Clifford-valued bandlimited functions generated from the translates of a finite collection of CPSWFs. This generalizes the one-dimensional scalar-valued construction of Hogan and Lakey.
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2022 |
Dizon ND, Hogan JA, Lindstrom SB, 'Centering Projection Methods for Wavelet Feasibility Problems', Trends in Mathematics 661-669 (2022) [B1] We revisit the feasibility approach to the construction of compactly supported smooth orthogonal wavelets on the line. We highlight its flexibility and illustrate how symmetry and... [more] We revisit the feasibility approach to the construction of compactly supported smooth orthogonal wavelets on the line. We highlight its flexibility and illustrate how symmetry and cardinality properties are easily embedded in the design criteria. We solve the resulting wavelet feasibility problems using recently introduced centering methods, and we compare performance. Solutions admit real-valued compactly supported smooth orthogonal scaling functions and wavelets with near symmetry and near cardinality properties.
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2021 |
Hogan JA, Lakey JD, 'Spatio spectral limiting on redundant cubes: a case study', Excursions in Harmonic Analysis, Springer, Switzerland 97-115 (2021) [B1]
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2020 |
Hogan JA, Lakey JD, 'Prolate Shift Frames and Sampling of Bandlimited Functions', Sampling: Theory and Applications A Centennial Celebration of Claude Shannon, Birkhäuser, Cham, Switzerland 141-167 (2020) [B1]
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2017 |
Hogan JA, Lakey JD, 'Frame properties of shifts of prolate and bandpass prolate functions', Frames and other bases in abstract and function spaces. Novel Methods in Harmonic Analysis, Birkhauser Basel, New York 215-235 (2017) [B1]
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2012 |
Hogan JA, Lakey JD, 'Thomson s multitaper method and applications to channel modeling', Applied and Numerical Harmonic Analysis 91-127 (2012) One of the most basic applications of Fourier analysis is power spectrum estimation. Some historical comments on this age-old problem can be found in review articles by Robinson [... [more] One of the most basic applications of Fourier analysis is power spectrum estimation. Some historical comments on this age-old problem can be found in review articles by Robinson [279] and Benedetto [23], and in Percival and Walden¿s book [264]. We refer to Appendix A for basic definitions and properties, and references for stochastic processes. In the problem of estimating a wide-sense stationary random process from data, one has only finitely many samples, often of a single realization of the process, with which to work.
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2012 |
Hogan JA, Lakey JD, 'Time-localized sampling approximations', Applied and Numerical Harmonic Analysis 199-221 (2012) In this last chapter we explore briefly some connections among sampling and time and band limiting. The chapter begins by pointing out a general connection between the samples of ... [more] In this last chapter we explore briefly some connections among sampling and time and band limiting. The chapter begins by pointing out a general connection between the samples of eigenfunctions of time and band limiting and the eigenvectors of a certain matrix whose entries are, in essence, the samples of time-localized images of functions that interpolate the samples in the given Paley¿Wiener space. Next, a discrete method is considered for generating eigenfunctions of time¿frequency localizations to unions of sets from their separate localizations. We then reconsider the connection between eigenfunctions and their samples in the concrete context of localization to intervals of the real line, outlining work ofWalter and Shen [347] and of Khare and George [177].Walter and Shen provided L2-estimates for approximate prolate spheroidal wave functions (PSWFs) constructed from interpolation of their sample values within the time-localization interval. We provide a partial sharpening of their estimates by using a slightly enlarged set of samples.
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2012 |
Hogan JA, Lakey JD, 'Time and band limiting of multiband signals', Applied and Numerical Harmonic Analysis 129-151 (2012) When a = 2OT, the operator POQT corresponding to single time and frequency intervals has an eigenvalue ¿¿a¿¿1/2, as Theorem 4.1.2 belowwill show. The norm ¿0(a = 1) of the operato... [more] When a = 2OT, the operator POQT corresponding to single time and frequency intervals has an eigenvalue ¿¿a¿¿1/2, as Theorem 4.1.2 belowwill show. The norm ¿0(a = 1) of the operator PQ1/2 satisfies ¿0(a = 1) = ¿sinc 1[1/2,1/2]¿ > 0.88. The trace of PQ1/2 is equal to a = 1, on the one hand and to S¿n on the other, so ¿1(a = 1) = 1¿0(a = 1) < 1/2. Suppose that T = 1 and S is a finite, pairwise disjoint union of a frequency intervals I1, ¿, Ia each of unit length. Then PSQ should have on the order of a eigenvalues of magnitude at least 1/2. Consider now the limiting case in which the frequency intervals become separated at infinity. Any function ¿j that is concentrated in frequency on Ij will be almost orthogonal over [T,T], in the separation limit, to any function ¿k that is frequency-concentrated on Ik when j ¿ k.
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2012 |
Hogan JA, Lakey JD, 'Sampling of band-limited and multiband signals', Applied and Numerical Harmonic Analysis 153-198 (2012) We provide here an overview of sampling theory that emphasizes real-variable aspects and functional analytic methods rather than analytic function-theoretic ones. While this appro... [more] We provide here an overview of sampling theory that emphasizes real-variable aspects and functional analytic methods rather than analytic function-theoretic ones. While this approach does not justify the most powerful mathematical results, it does provide the basis for practical sampling techniques for band-limited and multiband signals.
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2012 |
Hogan JA, Lakey JD, 'Numerical aspects of time and band limiting', Applied and Numerical Harmonic Analysis 45-90 (2012) This chapter is concerned with the role of the prolates in numerical analysis¿ particularly their approximation properties and application in the numerical solution of differentia... [more] This chapter is concerned with the role of the prolates in numerical analysis¿ particularly their approximation properties and application in the numerical solution of differential equations. The utility of the prolates in these contexts is due principally to the fact that they form a Markov system (see Defn. 2.1.6) of functions on [-11], a property that stems from their status as eigenfunctions of the differential operator P of (1.6), and allows the full force of the Sturm¿Liouville theory to be applied. The Markov property immediately gives the orthogonality of the prolates on [-11] (previously observed in Sect. 1.2 as the double orthogonality property) and also a remarkable collection of results regarding the zeros of the prolates as well as quadrature properties that are central to applications in numerical analysis.
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2012 |
Hogan JA, Lakey JD, 'The bell labs theory', Applied and Numerical Harmonic Analysis 1-43 (2012) Duration limiting, or time limiting, refers to restricting a signal by setting its values equal to zero outside of a finite time interval or, more generally, outside of a compact ... [more] Duration limiting, or time limiting, refers to restricting a signal by setting its values equal to zero outside of a finite time interval or, more generally, outside of a compact set. Bandwidth limiting, or band limiting, refers to restricting a signal by setting its amplitudes equal to zero outside of a finite frequency interval or again, more generally, outside of a compact set. This book addresses primarily the theory of time and band limiting whose core was developed by Landau, Pollak, and Slepian in a series of papers [195, 196, 303, 309] appearing in the Bell System Technical Journal in the early to middle part of the 1960s, and a broader body of work that grew slowly but steadily out of that core up until around 1980, with a resurgence since 2000, due in large part to the importance of time and band limiting in wireless communications. The 1960s Bell Labs theory of time and band limiting is but one aspect of the Bell Labs information theory. The foundations of this encompassing theory were laid, in large part, in Nyquist¿s fundamental papers ¿Certain Topics in Telegraph Transmission Theory¿ [247], which appeared in the Transactions of the American Institute of Electrical Engineers in 1928, and ¿Certain Factors Affecting Telegraph Speed,¿ published in April 1924 in the Bell System Technical Journal, along with Hartley¿s paper ¿Transmission of Information,¿ which also appeared in the Bell System Technical Journal in 1928 [137]. These papers quantified general ideas that were in the air, though certain specific versions were attributed to Kelvin and Wiener among others. Of course, Claude Shannon¿s seminal work, ¿A Mathematical Theory of Communication,¿ which appeared in the Bell System Technical Journal in July and October 1948 [293], is often cited as providing the basis for much of modern communications theory. His sampling theory plays a central role in Chap. 5 of this monograph. The works of Nyquist and Hartley however remain, in some ways, more germane to the study at hand.
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2008 |
Hogan JA, Lakey JD, 'Sampling and time-frequency localization of band-limited and multiband signals', Applied and Numerical Harmonic Analysis 275-291 (2008) [B1] This chapter develops aspects of previous work ([10] and [11]) that pertain distinctly to band-limited functions. In particular, we discuss some representation formulas for band-l... [more] This chapter develops aspects of previous work ([10] and [11]) that pertain distinctly to band-limited functions. In particular, we discuss some representation formulas for band-limited functions in terms of periodic nonuniform samples. In the case of multiband signals, periodic nonuniform sampling is often valid at a lower sampling rate than is uniform sampling, as will be discussed. Finally, we will consider some related questions about optimally time- and multiband-limited signals.
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2006 |
Hogan JA, Lakey JD, 'Periodic Nonuniform Sampling in Shift-Invariant Spaces', Applied and Numerical Harmonic Analysis 253-287 (2006) [B1] This chapter reviews several ideas that grew out of observations of Djokovic and Vaidyanathan to the effect that a generalized sampling method for bandlimited functions, due to Pa... [more] This chapter reviews several ideas that grew out of observations of Djokovic and Vaidyanathan to the effect that a generalized sampling method for bandlimited functions, due to Papoulis, could be carried over in many cases to the spline spaces and other shift-invariant spaces. Papoulis¿ method is based on the sampling output of linear, time-invariant systems. Unser and Zerubia formalized Papoulis¿ approach in the context of shift-invariant spaces. However, it is not easy to provide useful conditions under which the Unser-Zerubia criterion provides convergent and stable sampling expansions. Here we review several methods for validating the Unser-Zerubia approach for periodic nonuniform sampling, which is a very special case of generalized sampling. The Zak transform plays an important role.
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Show 16 more chapters |
Journal article (41 outputs)
Year | Citation | Altmetrics | Link | ||||||||
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2023 |
Hogan JA, Lakey JD, 'Spatio Spectral Limiting on Replacements of Tori by Cubes', Mathematics, 11 (2023) [C1] A class of graphs is defined in which each vertex of a discrete torus is replaced by a Boolean hypercube in such a way that vertices in a fixed subset of each replacement cube are... [more] A class of graphs is defined in which each vertex of a discrete torus is replaced by a Boolean hypercube in such a way that vertices in a fixed subset of each replacement cube are adjacent to corresponding vertices of a neighboring replacement cube. Bases of eigenvectors of the Laplacians of the resulting graphs are described in a manner suitable for quantifying the concentration of a low-spectrum vertex function on a single vertex replacement. Functions that optimize this concentration on these graphs can be regarded as analogues of Slepian prolate functions that optimize concentration of a bandlimited signal on an interval in the classical setting of the real line. Comparison to the case of a simple discrete cycle shows that replacement allows for higher concentration.
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2022 |
Ghaffari HB, Hogan JA, Lakey JD, 'Properties of Clifford-Legendre Polynomials', ADVANCES IN APPLIED CLIFFORD ALGEBRAS, 32 (2022) [C1]
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2022 |
Dizon ND, Hogan JA, Lindstrom SB, 'Circumcentering Reflection Methods for Nonconvex Feasibility Problems', SET-VALUED AND VARIATIONAL ANALYSIS, 30 943-973 (2022) [C1]
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2022 |
Dizon ND, Hogan JA, Lakey JD, 'Optimization in the construction of cardinal and symmetric wavelets on the line', International Journal of Wavelets, Multiresolution and Information Processing, 20 (2022) [C1] We present an optimization approach to wavelet architecture that hinges on the Zak transform to formulate the construction as a minimization problem. The transform warrants parame... [more] We present an optimization approach to wavelet architecture that hinges on the Zak transform to formulate the construction as a minimization problem. The transform warrants parametrization of the quadrature mirror filter in terms of the possible integer sample values of the scaling function and the associated wavelet. The parameters are predicated to satisfy constraints derived from the conditions of regularity, compact support and orthonormality. This approach allows for the construction of nearly cardinal scaling functions when an objective function that measures deviation from cardinality is minimized. A similar objective function based on a measure of symmetry is also proposed to facilitate the construction of nearly symmetric scaling functions on the line.
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2021 |
Dao MN, Dizon ND, Hogan JA, Tam MK, 'Constraint Reduction Reformulations for Projection Algorithms with Applications to Wavelet Construction', Journal of Optimization Theory and Applications, 190 201-233 (2021) [C1] We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibil... [more] We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibility problem by replacing a pair of its constraint sets with their intersection, before applying Pierra¿s classical product space reformulation. The step of combining the two constraint sets reduces the dimension of the product spaces. We refer to this technique as the constraint reduction reformulation and use it to obtain constraint-reduced variants of well-known projection algorithms such as the Douglas¿Rachford algorithm and the method of alternating projections, among others. We prove global convergence of constraint-reduced algorithms in the presence of convexity and local convergence in a nonconvex setting. In order to analyze convergence of the constraint-reduced Douglas¿Rachford method, we generalize a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. Finally, we apply the constraint-reduced versions of Douglas¿Rachford and alternating projections to solve the wavelet feasibility problems and then compare their performance with their usual product variants.
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2021 |
Hogan JA, Lakey JD, 'Spatio-Spectral Limiting on Boolean Cubes', Journal of Fourier Analysis and Applications, 27 (2021) [C1] The operator that first truncates to a neighborhood of the origin in the spatial domain then truncates to a neighborhood of the origin in the spectral domain is investigated in th... [more] The operator that first truncates to a neighborhood of the origin in the spatial domain then truncates to a neighborhood of the origin in the spectral domain is investigated in the case of Boolean cubes. This operator is self adjoint on the space of functions spanned by the Laplacian eigenvectors corresponding to small eigenvalues. The eigenspaces of this iterated projection operator are studied through reduced matrices based on certain invariant subspaces. They are shown to depend fundamentally on the neighborhood structure of the cube defined in terms of Hamming distance.
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2021 |
Hogan JA, Lakey JD, 'Spatio-spectral limiting on discrete tori: adjacency invariant spaces', Sampling Theory, Signal Processing, and Data Analysis, 19 (2021) [C1] Discrete tori are ZmN thought of as vertices of graphs CmN whose adjacencies encode the Cartesian product structure. Space-limiting refers to truncation to a symmetric path neighb... [more] Discrete tori are ZmN thought of as vertices of graphs CmN whose adjacencies encode the Cartesian product structure. Space-limiting refers to truncation to a symmetric path neighborhood of the zero element and spectrum-limiting in this case refers to corresponding truncation in the isomorphic Fourier domain. Composition spatio-spectral limiting (SSL) operators are analogues of classical time and band limiting operators. Certain adjacency-invariant spaces of vectors defined on ZmN are shown to have bases consisting of Fourier transforms of eigenvectors of SSL operators. We show that when m= 3 or m= 4 , all eigenvectors of SSL arise in this way. We study the structure of corresponding invariant spaces when m= 5 and give an example to indicate that the relationship between eigenvectors of SSL and the corresponding adjacency-invariant spaces should extend to m= 5.
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2019 |
Hogan JA, Lakey JD, 'An Analogue of Slepian Vectors on Boolean Hypercubes', Journal of Fourier Analysis and Applications, 25 2004-2020 (2019) [C1]
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2019 |
Hogan JA, Massopust PR, 'Quaternionic Fundamental Cardinal Splines: Interpolation and Sampling', Complex Analysis and Operator Theory, 13 3373-3403 (2019) [C1]
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2017 |
Hogan JA, Massopust P, 'Quaternionic B-splines', Journal of Approximation Theory, 224 43-65 (2017) [C1]
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2017 |
Franklin DJ, Hogan JA, Larkin KG, 'Hardy, Paley-Wiener and Bernstein spaces in Clifford analysis', COMPLEX VARIABLES AND ELLIPTIC EQUATIONS, 62 1314-1328 (2017) [C1]
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2017 |
Hogan JA, Lakey JD, 'On the Numerical Evaluation of Bandpass Prolates II', JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, 23 125-140 (2017) [C1]
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2016 |
Hogan JA, Lakey JD, 'Frame expansions of bandlimited signals using prolate spheroidal wave functions', Sampling Theory in Signal and Image Processing, 15 139-153 (2016) [C1]
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2015 |
Hogan JA, Lakey JD, 'Frame properties of shifts of prolate spheroidal wave functions', Applied and Computational Harmonic Analysis, 39 21-32 (2015) [C1] © 2014 Elsevier Inc. Abstract We provide conditions on a shift parameter and number of basic prolate spheroidal wave functions with a fixed bandwidth and time concentrated to a fi... [more] © 2014 Elsevier Inc. Abstract We provide conditions on a shift parameter and number of basic prolate spheroidal wave functions with a fixed bandwidth and time concentrated to a fixed duration such that the shifts of the basic prolates form a frame or a Riesz basis for the Paley-Wiener space consisting of all square integrable functions with the given bandlimit.
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2014 |
Ogburn DX, Waters CL, Sciffer MD, Hogan JA, Abbott PC, 'A finite difference construction of the spheroidal wave functions', COMPUTER PHYSICS COMMUNICATIONS, 185 244-253 (2014) [C1]
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2014 |
Hogan JA, Kroger J, Lakey JD, 'Time and bandpass limiting and an application to EEG', Sampling Theory in Signal and Image Processing, 13 295-313 (2014) [C1]
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2013 |
Craddock MJ, Hogan JA, 'The Fractional Clifford-Fourier Kernel', JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, 19 683-711 (2013) [C1]
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2013 |
Hogan JA, Lakey JD, 'Letter to the Editor: On the Numerical Evaluation of Bandpass Prolates', Journal of Fourier Analysis and Applications, 19 439-446 (2013) [C1] This letter provides a technique for numerical evaluation of certain eigenfunctions of the integral kernel operator corresponding to time truncation of a square-integrable functio... [more] This letter provides a technique for numerical evaluation of certain eigenfunctions of the integral kernel operator corresponding to time truncation of a square-integrable function to a finite interval, followed by frequency limiting to frequencies in an annular band. © 2013 Springer Science+Business Media New York.
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2012 |
Hogan JA, Morris AJ, 'Quaternionic wavelets', Numerical Functional Analysis and Optimization, 33 1031-1062 (2012) [C1]
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2012 |
Lakey JD, Hogan JA, 'On the numerical computation of certain eigenfunctions of time and multiband limiting', Numerical Functional Analysis and Optimization, 33 1095-1111 (2012) [C1]
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2010 |
Hogan JA, Izu S, Lakey JD, 'Sampling approximations for time- and bandlimiting', Sampling Theory in Signal and Image Processing, 9 91-117 (2010) [C1]
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2009 |
Hogan JA, Lakey J, 'Non-translation-invariance and the synchronization problem in wavelet sampling', Acta Applicandae Mathematicae, 107 373-398 (2009) [C1]
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2006 |
Hogan JA, Lakey JD, 'On uncertainty bounds and growth estimates for fractional Fourier transforms', Applicable Analysis, 85 891-899 (2006) [C1] Gelfand¿Shilov spaces are spaces of entire functions defined in terms of a rate of growth in one direction and a concomitant rate of decay in an orthogonal direction. Gelfand and ... [more] Gelfand¿Shilov spaces are spaces of entire functions defined in terms of a rate of growth in one direction and a concomitant rate of decay in an orthogonal direction. Gelfand and Shilov proved that the Fourier transform is an isomorphism among certain of these spaces. In this article we consider mapping properties of fractional Fourier transforms on Gelfand¿Shilov spaces. Just as the Fourier transform corresponds to rotation through a right angle in the phase plane, fractional Fourier transforms correspond to rotations through intermediate angles. Therefore, the aim of fractional Fourier estimates is to set up a correspondence between growth properties in the complex plane versus decay properties in phase space. © 2006, Taylor & Francis Group, LLC.
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2006 |
Hogan JA, Lakey JD, 'Hardy's theorem and rotations', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 134 1459-1466 (2006) [C1]
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2005 |
Hogan JA, Lakey JD, 'Sampling and oversampling in shift-invariant and multiresolution spaces 1: Validation of sampling schemes', INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING, 3 257-281 (2005) [C1]
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2005 |
Gilbert JE, Hogan JA, Lakey JD, 'BMO, boundedness of affine operators, and frames', APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS, 18 3-24 (2005) [C1]
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2002 |
Gilbert JE, Han YS, Hogan JA, Lakey JD, Weiland D, Weiss G, 'Smooth molecular decompositions of functions and singular integral operators', Memoirs of the American Mathematical Society, (2002) Under minimal assumptions on a function ¿ we obtain wavelettype frames of the form ¿j,k(x) = r1/2nj¿(rjx - sk) j ¿ Z k ¿ Zn for some r > 1 and s > 0. This collection is show... [more] Under minimal assumptions on a function ¿ we obtain wavelettype frames of the form ¿j,k(x) = r1/2nj¿(rjx - sk) j ¿ Z k ¿ Zn for some r > 1 and s > 0. This collection is shown to be a frame for a scale of Triebel-Lizorkin spaces (which includes Lebesgue, Sobolev and Hardy spaces) and the reproducing formula converges in norm as well as pointwise a.e. The construction follows from a characterization of those operators which are bounded on a space of smooth molecules. This characterization also allows us to decompose a broad range of singular integral operators in terms of smooth molecules.
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2000 |
Gilbert JE, Hogan JA, Lakey JD, 'Characterization of Hardy spaces by singular integrals and 'divergence-free' wavelets', PACIFIC JOURNAL OF MATHEMATICS, 193 79-105 (2000)
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2000 |
Hogan J, Li C, McIntosh A, Zhang K, 'Global higher integrability of Jacobians on bounded domains', Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, 17 193-217 (2000) We give conditions for a vector-valued function u ¿ W1,n (O, Rn), satisfying det Du (x) 0 on a bounded domain O, which imply that det Du (x) is globally higher integrable on O. We... [more] We give conditions for a vector-valued function u ¿ W1,n (O, Rn), satisfying det Du (x) 0 on a bounded domain O, which imply that det Du (x) is globally higher integrable on O. We also give conditions for u ¿ W1,n (O, Rn) such that det Du belongs to the Hardy space h1z (O) and exhibit some examples which show that our conditions are in some sense optimal. Applications to the weak convergence of Jacobians follow. Div-curl type extensions of these results to forms are also considered. © 2000 Editions scientifiques et médicales Elsevier SAS.
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Show 38 more journal articles |
Conference (22 outputs)
Year | Citation | Altmetrics | Link | |||||
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2023 |
Ghaffari HB, Hogan JA, Lakey JD, 'New Properties of Clifford Prolate Spheroidal Wave Functions', 2023 International Conference on Sampling Theory and Applications, SampTA 2023, New Haven, CT (2023) [E1]
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Nova | ||||||
2023 |
Hogan JA, Lakey JD, 'Pseudo Clifford Bandpass Prolates', 2023 International Conference on Sampling Theory and Applications, SampTA 2023, New Haven, CT (2023) [E1]
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Nova | ||||||
2019 |
Ghaffari HB, Hogan JA, Lakey JD, 'A Clifford Construction of Multidimensional Prolate Spheroidal Wave Functions', 2019 13th International Conference on Sampling Theory and Applications, SampTA 2019, Bordeaux, France (2019) [E1]
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Nova | ||||||
2019 |
Franklin D, Hogan JA, Tam M, 'Higher-dimensional wavelets and the Douglas-Rachford algorithm', 2019 13th International Conference on Sampling Theory and Applications, SampTA 2019, Bordeaux, France (2019) [E1]
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Nova | ||||||
2019 |
Hogan JA, Lakey JD, 'Numerical computation of eigenspaces of spatio-spectral limiting on hypercubes', 2019 13th International Conference on Sampling Theory and Applications, SampTA 2019, Bordeaux, France (2019) [E1]
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Nova | ||||||
2019 |
Dizon ND, Hogan JA, Lakey JD, 'Optimization in the construction of nearly cardinal and nearly symmetric wavelets', 2019 13th International Conference on Sampling Theory and Applications, SampTA 2019, Bordeaux, France (2019) [E1]
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Nova | ||||||
2017 | Hogan JA, Lakey J, 'Sampling in Paley-Wiener spaces, uncertainty and the prolate spheroidal wavefunctions', Proceedings of the AMSI/AustMS 2014 Workshop in Harmonic Analysis and its Applications, Sydney, Australia (2017) [E1] | Nova | ||||||
2017 |
Hogan JA, Lakey JD, 'Riesz bounds for prolate shifts', 2017 INTERNATIONAL CONFERENCE ON SAMPLING THEORY AND APPLICATIONS (SAMPTA), Tallinn, ESTONIA (2017) [E1]
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Nova | ||||||
2017 | Hogan JA, Lakey JD, 'Bandpass pseudo prolate shift frames and Riesz bases', 2017 INTERNATIONAL CONFERENCE ON SAMPLING THEORY AND APPLICATIONS (SAMPTA), Tallinn, ESTONIA (2017) [E1] | Nova | ||||||
2015 |
Hogan JA, Lakey JD, 'Prolate shift frames and their duals', 2015 International Conference on Sampling Theory and Applications, SampTA 2015, Washington, DC (2015) [E1]
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Nova | ||||||
2015 |
Hogan JA, Lakey JD, 'Wavelet frames generated by bandpass prolate functions', 2015 International Conference on Sampling Theory and Applications, SampTA 2015 (2015) [E1] We refer to eigenfunctions of the kernel corresponding to truncation in a time interval followed by truncation in a frequency band as bandpass prelates (BPPs). We prove frame boun... [more] We refer to eigenfunctions of the kernel corresponding to truncation in a time interval followed by truncation in a frequency band as bandpass prelates (BPPs). We prove frame bounds for certain families of shifts of bandpass prolates, and we numerically construct dual frames for finite dimensional analogues. In the continuous case, the corresponding families produce wavelet frames for the space of square-integrable functions.
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Nova | ||||||
2013 | Hogan JA, Lakey JD, 'Sampling aspects of approximately time-limited multiband and bandpass signals', Proceedings of the 10th International Conference on Sampling Theory and Applications (SampTA 2013), Bremen, Germany (2013) [E1] | Nova | ||||||
2000 |
Hogan JA, Lakey J, 'Sampling for shift-invariant and wavelet subspaces', WAVELET APPLICATIONS IN SIGNAL AND IMAGE PROCESSING VIII PTS 1 AND 2, SAN DIEGO, CA (2000)
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Show 19 more conferences |
Grants and Funding
Summary
Number of grants | 6 |
---|---|
Total funding | $1,163,034 |
Click on a grant title below to expand the full details for that specific grant.
20201 grants / $546,913
Enhanced Fractionation of Mineral Particles According to Density$546,913
Funding body: ARC (Australian Research Council)
Funding body | ARC (Australian Research Council) |
---|---|
Project Team | Laureate Professor Kevin Galvin, Associate Professor Jeffrey Hogan, Mr Jason Mackellar |
Scheme | Discovery Projects |
Role | Investigator |
Funding Start | 2020 |
Funding Finish | 2022 |
GNo | G1900136 |
Type Of Funding | C1200 - Aust Competitive - ARC |
Category | 1200 |
UON | Y |
20171 grants / $13,500
Multichannel Image and Signal Analysis: Phase and Geometry$13,500
Funding body: Bayerische Forschunsallianz
Funding body | Bayerische Forschunsallianz |
---|---|
Project Team | Prof. Brigitte Forster, Dr Jeffrey Hogan, Dr Peter Massopust |
Scheme | Bayerischen Hochshulforderprogramms zur Ahnbahnung und Vertiefung internationaler Forschnungskooperationen |
Role | Investigator |
Funding Start | 2017 |
Funding Finish | 2017 |
GNo | |
Type Of Funding | International - Competitive |
Category | 3IFA |
UON | N |
20161 grants / $593,621
Relaxed reflection methods for feasibility and matrix completion problems$593,621
Funding body: ARC (Australian Research Council)
Funding body | ARC (Australian Research Council) |
---|---|
Project Team | Associate Professor Jeffrey Hogan, Laureate Professor Jon Borwein, Associate Professor Russell Luke, Associate Professor Brailey Sims |
Scheme | Discovery Projects |
Role | Lead |
Funding Start | 2016 |
Funding Finish | 2018 |
GNo | G1500027 |
Type Of Funding | C1200 - Aust Competitive - ARC |
Category | 1200 |
UON | Y |
20141 grants / $2,000
Faculty PVC Conference Assistance Grant 2014$2,000
Funding body: University of Newcastle - Faculty of Science & IT
Funding body | University of Newcastle - Faculty of Science & IT |
---|---|
Project Team | Associate Professor Jeffrey Hogan |
Scheme | PVC Conference Assistance Grant |
Role | Lead |
Funding Start | 2014 |
Funding Finish | 2014 |
GNo | G1401195 |
Type Of Funding | Internal |
Category | INTE |
UON | Y |
20131 grants / $2,000
Faculty PVC Conference Assistance Grant 2013$2,000
Funding body: University of Newcastle - Faculty of Science & IT
Funding body | University of Newcastle - Faculty of Science & IT |
---|---|
Project Team | Associate Professor Jeffrey Hogan |
Scheme | PVC Conference Assistance Grant |
Role | Lead |
Funding Start | 2013 |
Funding Finish | 2013 |
GNo | G1401161 |
Type Of Funding | Internal |
Category | INTE |
UON | Y |
20101 grants / $5,000
Hypercomplex Signal Processing$5,000
Funding body: University of Newcastle
Funding body | University of Newcastle |
---|---|
Project Team | Associate Professor Jeffrey Hogan |
Scheme | New Staff Grant |
Role | Lead |
Funding Start | 2010 |
Funding Finish | 2010 |
GNo | G1000074 |
Type Of Funding | Internal |
Category | INTE |
UON | Y |
Research Supervision
Number of supervisions
Current Supervision
Commenced | Level of Study | Research Title | Program | Supervisor Type |
---|---|---|---|---|
2024 | PhD | Cross Curvature Flow: Existence, Uniqueness, and Negative Sectional Curvature | PhD (Mathematics), College of Engineering, Science and Environment, The University of Newcastle | Co-Supervisor |
2023 | PhD | Number Theory Results With A Focus On Elliptic Curves | PhD (Mathematics), College of Engineering, Science and Environment, The University of Newcastle | Co-Supervisor |
2023 | Masters | Towards an Algorithm for the De-Convolution of Fractionation Data | M Philosophy (Mathematics), College of Engineering, Science and Environment, The University of Newcastle | Principal Supervisor |
Past Supervision
Year | Level of Study | Research Title | Program | Supervisor Type |
---|---|---|---|---|
2022 | PhD | Higher-dimensional Prolate Spheroidal Wave Functions | PhD (Mathematics), College of Engineering, Science and Environment, The University of Newcastle | Principal Supervisor |
2022 | PhD | Metric Projections in Inner Product Spaces | PhD (Mathematics), College of Engineering, Science and Environment, The University of Newcastle | Principal Supervisor |
2021 | PhD | Optimisation in the Construction of Multidimensional Wavelets | PhD (Mathematics), College of Engineering, Science and Environment, The University of Newcastle | Principal Supervisor |
2019 | PhD | Forecasting in Supply Chains: The Impact of Demand Volatility in the Presence of Promotions | PhD (Statistics), College of Engineering, Science and Environment, The University of Newcastle | Co-Supervisor |
2018 | PhD | Projection Algorithms for Non-separable Wavelets and Clifford Fourier Analysis | PhD (Mathematics), College of Engineering, Science and Environment, The University of Newcastle | Principal Supervisor |
2018 | PhD | Random Walks On Groups | PhD (Mathematics), College of Engineering, Science and Environment, The University of Newcastle | Co-Supervisor |
2014 | PhD | Fourier and Wavelet Analysis of Clifford-Valued Functions | PhD (Mathematics), College of Engineering, Science and Environment, The University of Newcastle | Principal Supervisor |
Research Projects
Enhanced fractionation of minerals according to density 2021 - 2023
This project is focused on solving a long-standing problem in minerals processing -- the quantification of the density distribution of a system of particles. We pursue major advances in the experimental fractionation of particles on the basis of density and simultaneously develop an algorithm for the de-convolution of fractionation data to produce an accurate measure of the true washability data set. A successful solution to this problem will create low cost opportunities for efficient recovery of mineral resources without the need for the use of toxic heavy liquids.
The key theoretical aims of this study are to:
- establish an algorithm to de-convolve differential fractionation data to produce an accurate approximation of the true washability data set, and investigate the de-convolution as a function of the quality of the fractionator;
- extend the differential fractionation to the discrete case, and examine the impact of random noise on the error in restoring the washability data;
- validate the combined use of the algorithm and experimental fractionation in quantifying the washability data of the flow streams of a separator to obtain the overall partition curve of the separator.
Grants
Enhanced Fractionation of Mineral Particles According to Density
Funding body: ARC (Australian Research Council)
Funding body | ARC (Australian Research Council) |
---|---|
Project Team | Mr Jason Mackellar, Laureate Professor Kevin Galvin, Associate Professor Jeffrey Hogan |
Scheme | Discovery Projects |
Collaborators
Name | Organisation |
---|---|
Laureate Professor Kevin Patrick Galvin | University of Newcastle |
Prolate functions : bandlimited and bandpass signal processing 2009 -
Prolate functions have maximal energy concentration among all signals of a fixed bandlimit and have found many applications in signal processing, especially as taper functions for spectral estimation algorithms. This is due to their ``spectral accumulation'' properties. In this project, we
- construct bandpass prolates whose spectra are supported on a union of intervals on the line.
- construct prolate functions on graphs.
- develop the required aspects of Clifford analysis to enable the construction of bandlimited and bandpass prolates in higher dimensions.
Publications
Hogan JA, Lakey JD, Time-frequency and time-scale methods: Adaptive decompositions, uncertainty principles, and sampling (2005) [A1]
Hogan JA, Lakey JD, 'Sampling and time-frequency localization of band-limited and multiband signals', Applied and Numerical Harmonic Analysis 275-291 (2008) [B1]
Hogan JA, Lakey JD, Duration and Bandwidth Limiting: Prolate Functions, Sampling, and Applications, Birkhauser, New York, 258 (2012) [A1]
Hogan JA, Lakey JD, 'Numerical aspects of time and band limiting', Applied and Numerical Harmonic Analysis 45-90 (2012)
Lakey JD, Hogan JA, 'On the numerical computation of certain eigenfunctions of time and multiband limiting', Numerical Functional Analysis and Optimization, 33 1095-1111 (2012) [C1]
Hogan JA, Lakey JD, 'Time-localized sampling approximations', Applied and Numerical Harmonic Analysis 199-221 (2012)
Hogan JA, Lakey JD, 'Letter to the Editor: On the Numerical Evaluation of Bandpass Prolates', Journal of Fourier Analysis and Applications, 19 439-446 (2013) [C1]
Hogan JA, Kroger J, Lakey JD, 'Time and bandpass limiting and an application to EEG', Sampling Theory in Signal and Image Processing, 13 295-313 (2014) [C1]
Hogan JA, Lakey JD, 'Frame properties of shifts of prolate spheroidal wave functions', Applied and Computational Harmonic Analysis, 39 21-32 (2015) [C1]
Hogan JA, Lakey JD, 'Prolate shift frames and their duals', 2015 International Conference on Sampling Theory and Applications, SampTA 2015, Washington, DC (2015) [E1]
Hogan JA, Lakey JD, 'Wavelet frames generated by bandpass prolate functions', 2015 International Conference on Sampling Theory and Applications, SampTA 2015 (2015) [E1]
Hogan JA, Lakey JD, 'Bandpass pseudo prolate shift frames and Riesz bases', 2017 INTERNATIONAL CONFERENCE ON SAMPLING THEORY AND APPLICATIONS (SAMPTA), Tallinn, ESTONIA (2017) [E1]
Hogan JA, Lakey JD, 'Frame properties of shifts of prolate and bandpass prolate functions', Frames and other bases in abstract and function spaces. Novel Methods in Harmonic Analysis, Birkhauser Basel, New York 215-235 (2017) [B1]
Hogan JA, Lakey JD, 'On the Numerical Evaluation of Bandpass Prolates II', JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, 23 125-140 (2017) [C1]
Hogan JA, Lakey JD, 'Riesz bounds for prolate shifts', 2017 INTERNATIONAL CONFERENCE ON SAMPLING THEORY AND APPLICATIONS (SAMPTA), Tallinn, ESTONIA (2017) [E1]
Hogan JA, Lakey JD, 'An Analogue of Slepian Vectors on Boolean Hypercubes', Journal of Fourier Analysis and Applications, 25 2004-2020 (2019) [C1]
Hogan JA, Lakey JD, 'Prolate Shift Frames and Sampling of Bandlimited Functions', Sampling: Theory and Applications A Centennial Celebration of Claude Shannon, Birkhäuser, Cham, Switzerland 141-167 (2020) [B1]
Hogan JA, Lakey JD, 'Spatio-Spectral Limiting on Boolean Cubes', Journal of Fourier Analysis and Applications, 27 (2021) [C1]
Students
Program | Research Title |
---|---|
PhD College of Engineering, Science and Environment |
Higher-dimensional Prolate Spheroidal Wave Functions |
Collaborators
Name | Organisation |
---|---|
Joseph Lakey | New Mexico State University, Las Cruces |
Multidimensional wavelets 2016 -
Among modern signal processing techniques, wavelets have provided the most important theoretical challenges and technological advances. The by now classical construction methods on the line require techniques from complex analysis that are not available in higher dimensions. In this project, we apply modern techniques of optimisation to the notoriously difficult construction of wavelets in two and higher dimensions. Outputs include:
- Wavelets on the line with improved properties (near-cardinality, near-symmetry, etc).
- Wavelets on the plane with prescribed regularity.
With a view to applications in the processing of colour images, the team is working towards the development of quaternion-valued wavelets on the plane.
Grants
Relaxed reflection methods for feasibility and matrix completion problems
Funding body: ARC (Australian Research Council)
Funding body | ARC (Australian Research Council) |
---|---|
Project Team | Associate Professor Jeffrey Hogan, Laureate Professor Jon Borwein, Associate Professor Russell Luke, Associate Professor Brailey Sims |
Scheme | Discovery Projects |
Publications
Hogan JA, Morris AJ, 'Quaternionic wavelets', Numerical Functional Analysis and Optimization, 33 1031-1062 (2012) [C1]
Hogan JA, Morris AJ, 'Translation-invariant Clifford operators', , Australian National University 48 (2012) [E3]
Franklin D, Hogan JA, Tam M, 'Higher-dimensional wavelets and the Douglas-Rachford algorithm', 2019 13th International Conference on Sampling Theory and Applications, SampTA 2019, Bordeaux, France (2019) [E1]
Dizon ND, Hogan JA, Lakey JD, 'Optimization in the construction of nearly cardinal and nearly symmetric wavelets', 2019 13th International Conference on Sampling Theory and Applications, SampTA 2019, Bordeaux, France (2019) [E1]
Students
Program | Research Title |
---|---|
PhD College of Engineering, Science and Environment |
Optimisation in the Construction of Multidimensional Wavelets |
PhD College of Engineering, Science and Environment |
Fourier and Wavelet Analysis of Clifford-Valued Functions |
Clifford splines 2012 -
Spline functions are important components in signal processing algorithms -- they are often used in finite element algorithms for the solution of differential equations and are also used in the smoothing of discrete data in computer graphics. In this project we investigate the application of Clifford-analytic techniques to the construction of multi-channel, multi-variate, rotation-covariant splines appropriate to the treatment of multi-channel signals such as colour images and video.
Grants
Multichannel Image and Signal Analysis: Phase and Geometry
Funding body: Bayerische Forschunsallianz
Funding body | Bayerische Forschunsallianz |
---|---|
Description | For research visits between the participants to work on current research in which methods of Clifford analysis are applied to problems of multidimensional, multichannel signal and image analysis. |
Scheme | Bayerischen Hochshulforderprogramms zur Ahnbahnung und Vertiefung internationaler Forschnungskooperationen |
Publications
Hogan JA, Massopust P, 'Quaternionic B-splines', Journal of Approximation Theory, 224 43-65 (2017) [C1]
Hogan JA, Massopust PR, 'Quaternionic Fundamental Cardinal Splines: Interpolation and Sampling', Complex Analysis and Operator Theory, 13 3373-3403 (2019) [C1]
Edit
Research Collaborations
The map is a representation of a researchers co-authorship with collaborators across the globe. The map displays the number of publications against a country, where there is at least one co-author based in that country. Data is sourced from the University of Newcastle research publication management system (NURO) and may not fully represent the authors complete body of work.
Country | Count of Publications | |
---|---|---|
United States | 56 | |
Australia | 55 | |
Germany | 3 | |
Hong Kong | 2 | |
Canada | 1 |
News
News • 19 Dec 2019
ARC Discovery projects funding success
Laureate Professor Kevin Galvin and Associate Professor Jeffrey Hogan have been awarded $520,000 in ARC Discovery Project funding commencing in 2020 for their research project “Enhanced fractionation of mineral particles according to density”.
News • 4 Nov 2015
ARC Discovery Projects funding success
Laureate Professor Jon Borwein, Dr Jeffrey Hogan and Professor Dr Russell Luke have been awarded more than $560,000 in ARC Discovery Project funding commencing in 2016 for their research project Relaxed reflection methods for feasibility and matrix completion problems.
Associate Professor Jeffrey Hogan
Position
Associate Professor
School of Information and Physical Sciences
College of Engineering, Science and Environment
Focus area
Mathematics
Contact Details
jeff.hogan@newcastle.edu.au | |
Phone | (02) 4921 7235 |
Mobile | 0487444128 |
Fax | (02) 4921 6898 |
Office
Room | SR-243 |
---|---|
Building | Mathematics Building |
Location | Callaghan University Drive Callaghan, NSW 2308 Australia |