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## Spatiospectral concentration of vector fields on a sphere (2014)

### Citations

7677 |
Matrix Analysis
- Horn, Johnson
- 1985
(Show Context)
Citation Context ...′ Plm,l′m′Ul′m′ L∑ lm U2lm . (37) Here we have used orthonormality (22) and the quantities Plm,l′m′ = ∫ R Plm ·Pl′m′ dΩ = ∫ R YlmYl′m′ dΩ. (38) We can reformulate (35) as a matrix variational problem =-=[60]-=-: λ = gTPg gTg = maximum (39) using the (L+ 1)2 × (L+ 1)2 matrix P = P00,00 · · · P00,LL ... ... PLL,00 · · · PLL,LL , (40) The stationary solutions of the Rayleigh quotient λ in (39) ... |

519 |
Angular Momentum and Quantum Mechanics
- Edmonds
- 1960
(Show Context)
Citation Context ...in (1)–(3) to guarantee orthonormality:∫ Ω YlmYl′m′ dΩ = δll′δmm′ . (4) 3 A. Plattner and F. J. Simons / Appl. Comput. Harmon. Anal. XX (2012) XXX-XXX The product of two normalized Legendre functions =-=[51,53]-=- is the linear combination Xlm(θ)Xl′m′(θ) = (−1)m+m′ l+l′∑ n=|l−l′| √ (2n+ 1)(2l + 1)(2l′ + 1) 4pi × l n l′ 0 0 0 l n l′ m −m−m′ m′ Xnm+m′(θ). (5) The index arrays in (5) are Wigner 3-j... |

513 |
Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. Extensions to many dimensions; generalized prolate spheroidal functions
- Slepian
- 1964
(Show Context)
Citation Context ...t to their spatial energy, inside a target region. Slepian, Landau, and Pollak presented the solution for the problem of optimally concentrating a signal in time and frequency in their seminal papers =-=[1,2,3,4]-=-. Their construction leads to a family of orthogonal tapers or data windows that have been widely applied as windows to regularize the quadratic inverse problem of power spectral estimation from time-... |

445 |
Quantum Mechanics
- Messiah
- 1958
(Show Context)
Citation Context ...inear combination Xlm(θ)Xl′m′(θ) = (−1)m+m′ l+l′∑ n=|l−l′| √ (2n+ 1)(2l + 1)(2l′ + 1) 4pi × l n l′ 0 0 0 l n l′ m −m−m′ m′ Xnm+m′(θ). (5) The index arrays in (5) are Wigner 3-j symbols =-=[53,54]-=-. We will use the following two recursion relations [36,55] for the derivatives of the Xlm(θ) and their divisions by sin θ. Define X ′lm = dXlm dθ . (6) Then, from the work by Ilk [55] follows that, f... |

196 |
Necessary density conditions for sampling and interpolation of certain entire functions,”
- Landau
- 1967
(Show Context)
Citation Context ...of vector fields u(r̂) that are approximately limited in both the spectral domain to vector spherical-harmonic degrees 0 ≤ l ≤ L, and in the spatial domain to an arbitrarily shaped region R of area A =-=[62,63]-=-. Instead of constructing a bandlimited field g(r̂) ∈ SL that is optimally energy-concentrated within a spatial region R, we could have sought to construct one that is optimally excluded from R, i.e.,... |

194 |
Prolate spheroidal wave functions, Fourier analysis and uncertainty
- Landau, Pollak
- 1961
(Show Context)
Citation Context ...t to their spatial energy, inside a target region. Slepian, Landau, and Pollak presented the solution for the problem of optimally concentrating a signal in time and frequency in their seminal papers =-=[1,2,3,4]-=-. Their construction leads to a family of orthogonal tapers or data windows that have been widely applied as windows to regularize the quadratic inverse problem of power spectral estimation from time-... |

167 |
Time–Frequency/Time–Scale Analysis.
- Flandrin
- 1999
(Show Context)
Citation Context ...f the eigenvectors is orthogonal. The transformed 13 A. Plattner and F. J. Simons / Appl. Comput. Harmon. Anal. XX (2012) XXX-XXX representation (72) is a vector spherical version of Mercer’s theorem =-=[61,64,65]-=-. Upon setting r̂′ = r̂ in (72) and applying the trace [51], we deduce that the sum of the squares of the 3(L + 1)2 − 2 bandlimited eigenfields gα(r̂) is a constant that is independent of position r̂ ... |

165 |
Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques
- Percival, Walden
- 1993
(Show Context)
Citation Context ...bandlimit is L = 18. Each symbol stands for two eigenvalues, arising from the plus-minus-degeneracy for m > 0 and from the block-diagonal shape for m = 0. The spectra have a characteristic step shape =-=[62,70,71]-=-, showing significant (λ ≈ 1) and insignificant (λ ≈ 0) eigenvalues separated by a transition band. In all four cases the reasonably well concentrated eigensolutions (λ ≥ 0.5) and the more poorly conc... |

155 |
Theoretical Global Seismology
- Dahlen, Tromp
- 1998
(Show Context)
Citation Context ...e constructed from their scalar counterparts. Each scalar spherical harmonic Ylm has a degree 0 ≤ l and, for each degree, an order −l ≤ m ≤ l. Our spherical harmonics are unit-normalized in the sense =-=[51]-=- Ylm(θ, φ) = √ 2Xl|m|(θ) cosmφ if − l ≤ m < 0, Xl0(θ) if m = 0,√ 2Xlm(θ) sinmφ if 0 < m ≤ l, (1) Xlm(θ) = (−1)m ( 2l + 1 4pi )1/2 [ (l −m)! (l +m)! ]1/2 Plm(cos θ), (2) Plm(µ) = 1 2ll! (1− µ2)... |

102 |
Some comments on Fourier analysis, uncertainty and modeling
- Slepian
- 1983
(Show Context)
Citation Context ...t to their spatial energy, inside a target region. Slepian, Landau, and Pollak presented the solution for the problem of optimally concentrating a signal in time and frequency in their seminal papers =-=[1,2,3,4]-=-. Their construction leads to a family of orthogonal tapers or data windows that have been widely applied as windows to regularize the quadratic inverse problem of power spectral estimation from time-... |

47 |
Integral Equations, Interscience,
- Tricomi
- 1957
(Show Context)
Citation Context ...ial-domain eigenvalue problem∫ R K(r̂, r̂′) · g(r̂′) dΩ = λg(r̂), r̂ ∈ Ω, (60) 10 A. Plattner and F. J. Simons / Appl. Comput. Harmon. Anal. XX (2012) XXX-XXX a homogeneous Fredholm integral equation =-=[61]-=- with a finite-rank, symmetric, separable, bandlimited vector Dirac delta function kernel, K(r̂, r̂′) = L∑ lm Plm(r̂)Plm(r̂ ′) +Blm(r̂)Blm(r̂′) +Clm(r̂)Clm(r̂′), (61) a reproducing kernel [5,7] in the... |

35 |
Spatiospectral concentration on a sphere
- Simons, Dahlen, et al.
(Show Context)
Citation Context ...entration, R = R1 ∪R2 ∪ . . ., is denoted by A in the text. in a wide range of disciplines. Several authors have studied the time-scale and time-frequency concentration for more general settings (see =-=[5,6,7]-=- and references therein for a review). More specifically, spherical scalar Slepian functions, spatially concentrated while bandlimited or spectrally concentrated while spacelimited, have been applied ... |

30 |
Linear Integral Equations: Theory & Technique, 2nd ed.; Modern Birkhauser Classic; Birkhauser:
- Kanwal
- 2013
(Show Context)
Citation Context ...f the eigenvectors is orthogonal. The transformed 13 A. Plattner and F. J. Simons / Appl. Comput. Harmon. Anal. XX (2012) XXX-XXX representation (72) is a vector spherical version of Mercer’s theorem =-=[61,64,65]-=-. Upon setting r̂′ = r̂ in (72) and applying the trace [51], we deduce that the sum of the squares of the 3(L + 1)2 − 2 bandlimited eigenfields gα(r̂) is a constant that is independent of position r̂ ... |

27 |
Evaluation of the rotation matrices in the basis of real spherical harmonics,
- Blanco, Florez, et al.
- 1997
(Show Context)
Citation Context ...the optimally concentrated eigenfields of the polar cap R = {θ : 0 ≤ θ ≤ Θ} can be rotated to anywhere on the unit sphere using the same transformations that apply in the rotation of scalar functions =-=[66,67,68]-=-. 4.1. Decomposition of the Spectral-Domain Eigenvalue Problem In the axisymmetric case the matrix elements (38) and (47)–(49) reduce to Plm,l′m′ = 2piδmm′ ∫ Θ 0 XlmXl′m sin θ dθ, (77) Blm,l′m′ = 2piδ... |

23 |
Swarm: A constellation to study the Earth’s magnetic field.
- Friis-Christensen, Lühr, et al.
- 2006
(Show Context)
Citation Context ...ted spherical vectorial multitapers that we will construct in the following should be useful in many scientific applications. In particular in geomagnetism, one of the objectives of the SWARM mission =-=[39]-=- is to model the lithospheric magnetic field with maximal resolution and accuracy, even in the presence of contaminating signals from secondary sources. In addition, and more generally, lithospheric2 ... |

21 |
Eigenvalues associated with prolate spheroidal wave functions of zero order
- Slepian, Sonnenblick
- 1965
(Show Context)
Citation Context ...bandlimit is L = 18. Each symbol stands for two eigenvalues, arising from the plus-minus-degeneracy for m > 0 and from the block-diagonal shape for m = 0. The spectra have a characteristic step shape =-=[62,70,71]-=-, showing significant (λ ≈ 1) and insignificant (λ ≈ 0) eigenvalues separated by a transition band. In all four cases the reasonably well concentrated eigensolutions (λ ≥ 0.5) and the more poorly conc... |

20 |
From global to regional analysis of the magnetic field on the sphere using wavelet frames, Phys. Earth Planet
- Holschneider, Chambodut, et al.
(Show Context)
Citation Context ...e. In the past decade or so, a variety of global-to-regional modeling techniques have come of age including harmonic splines [40,41,42], stitching together local models [43,44,45,46,47], and wavelets =-=[48,49,50]-=-. Due to their optimal combination of spatial locality and spectral bandlimitation the multitapers constructed in this paper should be well suited to combine global and local data while respecting the... |

19 | Spherical Slepian functions and the polar gap in geodesy, Geophys
- Simons, Dahlen
- 2006
(Show Context)
Citation Context ...l scalar Slepian functions, spatially concentrated while bandlimited or spectrally concentrated while spacelimited, have been applied in physical, computational, and biomedical fields such as geodesy =-=[8,9,10,11]-=- and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing t... |

19 |
The Main Field
- Langel
- 1987
(Show Context)
Citation Context ...have to successfully merge information from the global to the regional scale. In the past decade or so, a variety of global-to-regional modeling techniques have come of age including harmonic splines =-=[40,41,42]-=-, stitching together local models [43,44,45,46,47], and wavelets [48,49,50]. Due to their optimal combination of spatial locality and spectral bandlimitation the multitapers constructed in this paper ... |

19 |
Wavelet frames: an alternative to spherical harmonic representation of potential fields, Geophys
- Chambodut, Panet, et al.
(Show Context)
Citation Context ...e. In the past decade or so, a variety of global-to-regional modeling techniques have come of age including harmonic splines [40,41,42], stitching together local models [43,44,45,46,47], and wavelets =-=[48,49,50]-=-. Due to their optimal combination of spatial locality and spectral bandlimitation the multitapers constructed in this paper should be well suited to combine global and local data while respecting the... |

17 | On the construction of invertible filterbank on the 2-sphere,"
- Yeo, Ou, et al.
- 2006
(Show Context)
Citation Context ...such as geodesy [8,9,10,11] and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science =-=[28,29]-=-, while continuing to be of interest in information and communication theory [30], signal processing [31,32], and mathematics [33,34]. To date only a few attempts have been made to bring the advantage... |

16 |
Localized spectral analysis on the sphere, Geophys
- Wieczorek, Simons
- 2005
(Show Context)
Citation Context ...set Xlm = 0 for m > l. Since this solution to (41) is identical to that of the scalar spherical concentration problem for the polar cap, alternate expressions and special cases can be found elsewhere =-=[6,69]-=-. For the Bmll′ at positive orders m ≥ 0, as first noted by Eshagh [36], we first need to transform the derivative products X ′lmX ′l′m, and m2(sin θ)−2XlmXl′m, into products of Xlm using the lemmas (... |

13 |
The eigenvalue behavior of certain convolution equations
- Landau
- 1965
(Show Context)
Citation Context ...of vector fields u(r̂) that are approximately limited in both the spectral domain to vector spherical-harmonic degrees 0 ≤ l ≤ L, and in the spatial domain to an arbitrarily shaped region R of area A =-=[62,63]-=-. Instead of constructing a bandlimited field g(r̂) ∈ SL that is optimally energy-concentrated within a spatial region R, we could have sought to construct one that is optimally excluded from R, i.e.,... |

12 |
Spatiospectral localization of global geopotential fields from the Gravity Recovery and Climate Experiment GRACE reveals the coseismic gravity change owing to the 2004 Sumatra-Andaman earthquake
- Han, Simons
- 2008
(Show Context)
Citation Context ... spatially concentrated while bandlimited or spectrally concentrated while spacelimited, have been applied in physical, computational, and biomedical fields such as geodesy [8,9,10,11] and gravimetry =-=[12,13,14,15,16]-=-, geomagnetism [17,18] and geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information ... |

11 | G.: Representations of SO(3) and angular polyspectra
- Marinucci, Peccati
- 2010
(Show Context)
Citation Context ...iomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theory [30], signal processing [31,32], and mathematics =-=[33,34]-=-. To date only a few attempts have been made to bring the advantages of spherical Slepian functions into the realm of spherical vector fields. The first successful construction of spatially concentrat... |

11 |
Revised spherical cap harmonic analysis (R-SCHA): validation and properties
- Thébault, Schott, et al.
(Show Context)
Citation Context ...the global to the regional scale. In the past decade or so, a variety of global-to-regional modeling techniques have come of age including harmonic splines [40,41,42], stitching together local models =-=[43,44,45,46,47]-=-, and wavelets [48,49,50]. Due to their optimal combination of spatial locality and spectral bandlimitation the multitapers constructed in this paper should be well suited to combine global and local ... |

10 |
Spectral estimation on a sphere in geophysics and cosmology, Geophys
- Dahlen, Simons
- 2008
(Show Context)
Citation Context ...tional, and biomedical fields such as geodesy [8,9,10,11] and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology =-=[26,27]-=-, and computer science [28,29], while continuing to be of interest in information and communication theory [30], signal processing [31,32], and mathematics [33,34]. To date only a few attempts have be... |

9 |
Mapping Greenland’s mass loss in space and time
- Harig, Simons
- 2012
(Show Context)
Citation Context ... spatially concentrated while bandlimited or spectrally concentrated while spacelimited, have been applied in physical, computational, and biomedical fields such as geodesy [8,9,10,11] and gravimetry =-=[12,13,14,15,16]-=-, geomagnetism [17,18] and geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information ... |

9 |
Spherical Functions of Mathematical Geosciences,
- Freeden, Schreiner
- 2009
(Show Context)
Citation Context ...the sense ∫ Ω Plm ·Pl′m′ dΩ = ∫ Ω Blm ·Bl′m′ dΩ = ∫ Ω Clm ·Cl′m′ dΩ = δll′δmm′ , (22)∫ Ω Plm ·Bl′m′ dΩ = ∫ Ω Plm ·Cl′m′ dΩ = ∫ Ω Blm ·Cl′m′ dΩ = 0. (23) The vector spherical-harmonic addition theorem =-=[57]-=- comprises the identities 5 A. Plattner and F. J. Simons / Appl. Comput. Harmon. Anal. XX (2012) XXX-XXX l∑ m=−l Plm(r̂) ·Plm(r̂) = ( 2l + 1 4pi ) (24) = l∑ m=−l Blm(r̂) ·Blm(r̂) = l∑ m=−l Clm(r̂) ·Cl... |

8 | Slepian functions and their use in signal estimation and spectral analysis
- Simons
- 2010
(Show Context)
Citation Context ...entration, R = R1 ∪R2 ∪ . . ., is denoted by A in the text. in a wide range of disciplines. Several authors have studied the time-scale and time-frequency concentration for more general settings (see =-=[5,6,7]-=- and references therein for a review). More specifically, spherical scalar Slepian functions, spatially concentrated while bandlimited or spectrally concentrated while spacelimited, have been applied ... |

8 |
Postseismic gravity change following the great 2004 Sumatra‐Andaman earthquake from the regional harmonic analysis of GRACE inter‐satellite tracking data: Implication for the regional viscoelastic response,
- Han, Sauber, et al.
- 2008
(Show Context)
Citation Context ... spatially concentrated while bandlimited or spectrally concentrated while spacelimited, have been applied in physical, computational, and biomedical fields such as geodesy [8,9,10,11] and gravimetry =-=[12,13,14,15,16]-=-, geomagnetism [17,18] and geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information ... |

8 |
Efficient analysis and representation of geophysical processes using localized spherical basis functions
- Simons, Hawthorne, et al.
- 2009
(Show Context)
Citation Context ...andlimited or spectrally concentrated while spacelimited, have been applied in physical, computational, and biomedical fields such as geodesy [8,9,10,11] and gravimetry [12,13,14,15,16], geomagnetism =-=[17,18]-=- and geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theo... |

7 |
Recurrence relations for integrals of associated Legendre functions
- Paul
- 1978
(Show Context)
Citation Context ... m ≤ l, (sin θ)−1mXlm = b−lmXl−1m−1 + b + lmXl−1m+1, (9) where b±lm = − √ 2l + 1 2l − 1 √ (l ∓m)(l ∓m− 1) 2 . (10) In both (8) and (10), for l < 0 or m < 0, we have Xlm = 0. Finally, as shown by Paul =-=[56]-=-, integrals of the type Ilm(Θ) = ∫ Θ 0 Xlm(θ) sin θ dθ (11) can be exactly evaluated recursively. When l ≥ 2 and 0 ≤ m < l, we have Ilm(Θ) = l − 2 l + 1 √√√√(2l + 1)[(l − 1)2 −m2] (2l − 3)(l2 −m2) Il−... |

7 |
Enhancing comprehensive inversions using the Swarm constellation
- Sabaka, Olsen
- 2006
(Show Context)
Citation Context ...hat we will be showing should be well suited to the localized analysis of global vector-valued satellite-magnetic data such as measured to study the magnetization of the terrestrial lithosphere (e.g. =-=[39,72,73,74]-=-), or more generally, planetary magnetic fields [17,75,76,77]. 20 A. Plattner and F. J. Simons / Appl. Comput. Harmon. Anal. XX (2012) XXX-XXX Table 1 Fractional areas, tangential Shannon numbers, and... |

7 |
An ellipsoidal harmonic representation of Earth’s lithospheric magnetic field to degree and order 720, Geochem. Geophys
- Maus
- 2010
(Show Context)
Citation Context ...n error within the region R while simultaneously keeping the outside leakage bias small. Figure 11 shows the outcome of such an experiment conducted on the terrestrial crustal-field model NGDC-720 V3 =-=[78]-=-. We multiply the spherical-harmonic coefficients with the corresponding Blm vector harmonics up to bandlimit L = 72. The left panel of Figure 11 shows the tangential vector field that results: this i... |

6 |
Global lithospheric magnetic field modelling by successive regional analysis
- Thébault
- 2006
(Show Context)
Citation Context ...the global to the regional scale. In the past decade or so, a variety of global-to-regional modeling techniques have come of age including harmonic splines [40,41,42], stitching together local models =-=[43,44,45,46,47]-=-, and wavelets [48,49,50]. Due to their optimal combination of spatial locality and spectral bandlimitation the multitapers constructed in this paper should be well suited to combine global and local ... |

5 |
The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid
- Slobbe, Simons, et al.
- 2012
(Show Context)
Citation Context ...l scalar Slepian functions, spatially concentrated while bandlimited or spectrally concentrated while spacelimited, have been applied in physical, computational, and biomedical fields such as geodesy =-=[8,9,10,11]-=- and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing t... |

5 |
Hydrological estimates for small basins: Evaluating processing approaches on
- Longuevergne, Scanlon, et al.
(Show Context)
Citation Context |

5 |
On the effective dimension of light transport
- Lessig, Fiume
- 2010
(Show Context)
Citation Context ...such as geodesy [8,9,10,11] and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science =-=[28,29]-=-, while continuing to be of interest in information and communication theory [30], signal processing [31,32], and mathematics [33,34]. To date only a few attempts have been made to bring the advantage... |

5 |
2003), Regularization of spherical cap harmonics, Geophys
- Korte, Holme
(Show Context)
Citation Context ...the global to the regional scale. In the past decade or so, a variety of global-to-regional modeling techniques have come of age including harmonic splines [40,41,42], stitching together local models =-=[43,44,45,46,47]-=-, and wavelets [48,49,50]. Due to their optimal combination of spatial locality and spectral bandlimitation the multitapers constructed in this paper should be well suited to combine global and local ... |

5 | Simulation of the high-degree lithospheric field recovery for the Swarm constellation of satellites, Earth Planets Space, 58, this issue
- Maus, Lühr, et al.
- 2006
(Show Context)
Citation Context ...hat we will be showing should be well suited to the localized analysis of global vector-valued satellite-magnetic data such as measured to study the magnetization of the terrestrial lithosphere (e.g. =-=[39,72,73,74]-=-), or more generally, planetary magnetic fields [17,75,76,77]. 20 A. Plattner and F. J. Simons / Appl. Comput. Harmon. Anal. XX (2012) XXX-XXX Table 1 Fractional areas, tangential Shannon numbers, and... |

4 | Spatiospectral concentration in the Cartesian plane 33 - Simons, Dahlen - 2007 |

4 |
Localized spectral analysis of global satellite gravity fields for recovering timevariable mass redistributions
- Han, Ditmar
- 2007
(Show Context)
Citation Context |

4 |
Constraints on the composition of the martian south polar cap from gravity and topography.
- Wieczorek
- 2008
(Show Context)
Citation Context ...ile spacelimited, have been applied in physical, computational, and biomedical fields such as geodesy [8,9,10,11] and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics [19], planetary =-=[20,21,22,23]-=- and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theory [30], signal processing [31,32], and mathem... |

4 |
The concentration problem for vector fields
- Maniar, Mitra
- 2005
(Show Context)
Citation Context ...n physical, computational, and biomedical fields such as geodesy [8,9,10,11] and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics [19], planetary [20,21,22,23] and biomedical science =-=[24,25]-=-, cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theory [30], signal processing [31,32], and mathematics [33,34]. To date only a f... |

4 |
Spatio-spectral analysis on the sphere using spatially localized spherical harmonics transform
- Khalid, Durrani, et al.
- 1109
(Show Context)
Citation Context ...etary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theory [30], signal processing =-=[31,32]-=-, and mathematics [33,34]. To date only a few attempts have been made to bring the advantages of spherical Slepian functions into the realm of spherical vector fields. The first successful constructio... |

4 |
Optimally localized approximate identities on the 2-sphere
- Michel
(Show Context)
Citation Context ...iomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theory [30], signal processing [31,32], and mathematics =-=[33,34]-=-. To date only a few attempts have been made to bring the advantages of spherical Slepian functions into the realm of spherical vector fields. The first successful construction of spatially concentrat... |

4 |
Harmonic splines for geomagnetic modeling,” Phys
- Shure, Parker, et al.
- 1982
(Show Context)
Citation Context ...have to successfully merge information from the global to the regional scale. In the past decade or so, a variety of global-to-regional modeling techniques have come of age including harmonic splines =-=[40,41,42]-=-, stitching together local models [43,44,45,46,47], and wavelets [48,49,50]. Due to their optimal combination of spatial locality and spectral bandlimitation the multitapers constructed in this paper ... |

4 |
Translated origin spherical cap harmonics analysis, Geophy
- Santis
- 1991
(Show Context)
Citation Context |

4 |
Separating inner and outer Earth’s magnetic field from CHAMP satellite measurements by means of vector scaling functions and wavelets,” Geophys
- Mayer, Maier
- 2006
(Show Context)
Citation Context ...e. In the past decade or so, a variety of global-to-regional modeling techniques have come of age including harmonic splines [40,41,42], stitching together local models [43,44,45,46,47], and wavelets =-=[48,49,50]-=-. Due to their optimal combination of spatial locality and spectral bandlimitation the multitapers constructed in this paper should be well suited to combine global and local data while respecting the... |

4 |
Vector Slepian basis functions with optimal energy concentration in high numerical aperture focusing
- Jahn, Bokor
- 2012
(Show Context)
Citation Context ...tics, where prior work has previously considered 28 A. Plattner and F. J. Simons / Appl. Comput. Harmon. Anal. XX (2012) XXX-XXX a number of special cases of the vectorial concentration on the sphere =-=[25,81]-=- that we have treated more completely here. The ability of scalar Slepian functions on the sphere to perform localized bandlimited analysis has led to observations made from global data that remain ob... |

3 |
Improved nearside gravity field of the Moon by localizing the power law constraint, Geophys
- Han, Mazarico, et al.
(Show Context)
Citation Context ...ile spacelimited, have been applied in physical, computational, and biomedical fields such as geodesy [8,9,10,11] and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics [19], planetary =-=[20,21,22,23]-=- and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theory [30], signal processing [31,32], and mathem... |

3 |
Concentration maximization and local basis expansions (LBEX) for linear inverse problems
- Mitra, Maniar
(Show Context)
Citation Context ...n physical, computational, and biomedical fields such as geodesy [8,9,10,11] and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics [19], planetary [20,21,22,23] and biomedical science =-=[24,25]-=-, cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theory [30], signal processing [31,32], and mathematics [33,34]. To date only a f... |

3 |
Analysis of lithospheric magnetization in vector spherical harmonics,” Geophys
- Gubbins, Ivers, et al.
- 1111
(Show Context)
Citation Context ...t a complete extension of Slepian’s spatiospectral concentration problem to vector fields on the sphere, and give suggestions and examples as to their usage for problems of a geomagnetic nature (e.g. =-=[37,38]-=-). The family of optimally concentrated spherical vectorial multitapers that we will construct in the following should be useful in many scientific applications. In particular in geomagnetism, one of ... |

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Local spectral variability and the origin of the Martian crustal magnetic field,” Geophys
- Lewis, Simons
- 1029
(Show Context)
Citation Context ... analysis of global vector-valued satellite-magnetic data such as measured to study the magnetization of the terrestrial lithosphere (e.g. [39,72,73,74]), or more generally, planetary magnetic fields =-=[17,75,76,77]-=-. 20 A. Plattner and F. J. Simons / Appl. Comput. Harmon. Anal. XX (2012) XXX-XXX Table 1 Fractional areas, tangential Shannon numbers, and bandwidths for the vectorial concentration problem to contin... |

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Using Slepian functions for local geodetic computations
- Albertella, Sacerdote
(Show Context)
Citation Context ...l scalar Slepian functions, spatially concentrated while bandlimited or spectrally concentrated while spacelimited, have been applied in physical, computational, and biomedical fields such as geodesy =-=[8,9,10,11]-=- and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing t... |

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Localized analysis of satellite tracking data for studying time-variable Earth’s gravity fields
- Han, Rowlands, et al.
(Show Context)
Citation Context |

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Modelling the earths magnetic field from global to regional scales
- Schott, Thébault
- 2011
(Show Context)
Citation Context ...andlimited or spectrally concentrated while spacelimited, have been applied in physical, computational, and biomedical fields such as geodesy [8,9,10,11] and gravimetry [12,13,14,15,16], geomagnetism =-=[17,18]-=- and geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theo... |

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Constraints on upper-mantle viscosity inferred from the flowinduced pressure gradient across a continental keel, Geochem. Geophys
- Harig, Zhong, et al.
- 2010
(Show Context)
Citation Context ... concentrated while spacelimited, have been applied in physical, computational, and biomedical fields such as geodesy [8,9,10,11] and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics =-=[19]-=-, planetary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theory [30], signal proce... |

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Geophysical limitations on the erosion history within
- Evans, Andrews-Hanna, et al.
- 2010
(Show Context)
Citation Context ...ile spacelimited, have been applied in physical, computational, and biomedical fields such as geodesy [8,9,10,11] and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics [19], planetary =-=[20,21,22,23]-=- and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theory [30], signal processing [31,32], and mathem... |

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2012), Local lunar gravity field analysis over the South Pole-Aitken basin from SELENE farside tracking data
- Goossens, Ishihara, et al.
(Show Context)
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Efficient power spectrum estimation for high resolution CMB maps,” Phys
- Das, Hajian, et al.
- 2009
(Show Context)
Citation Context ...tional, and biomedical fields such as geodesy [8,9,10,11] and gravimetry [12,13,14,15,16], geomagnetism [17,18] and geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology =-=[26,27]-=-, and computer science [28,29], while continuing to be of interest in information and communication theory [30], signal processing [31,32], and mathematics [33,34]. To date only a few attempts have be... |

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Spatially restricted integrals in gradiometric boundary value problems
- Eshagh
- 2009
(Show Context)
Citation Context ...elds. The first successful construction of spatially concentrated bandlimited tangential spherical vector fields was reported for applications in magnetoencephalography [24,25,35]. In geodesy, Eshagh =-=[36]-=- has developed methods to explicitly evaluate the product integrals arising in the concentration problem whose solutions are the vectorial Slepian functions. In this paper we present a complete extens... |

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Ein Beitrag zur Dynamik ausgedehnter Körper: Gravitationswechselwirkung,” Deutsche Geodätische Kommission, vol
- Ilk
- 1983
(Show Context)
Citation Context ...√ (2n+ 1)(2l + 1)(2l′ + 1) 4pi × l n l′ 0 0 0 l n l′ m −m−m′ m′ Xnm+m′(θ). (5) The index arrays in (5) are Wigner 3-j symbols [53,54]. We will use the following two recursion relations =-=[36,55]-=- for the derivatives of the Xlm(θ) and their divisions by sin θ. Define X ′lm = dXlm dθ . (6) Then, from the work by Ilk [55] follows that, for 0 ≤ l and 0 ≤ m ≤ l, X ′lm = a − lmXl m−1 + a + lmXl m+1... |

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Spectral and spatial decomposition of lithospheric magnetic field models using spherical Slepian functions,” Geophys
- Beggan, Saarimäki, et al.
- 1093
(Show Context)
Citation Context ...hat we will be showing should be well suited to the localized analysis of global vector-valued satellite-magnetic data such as measured to study the magnetization of the terrestrial lithosphere (e.g. =-=[39,72,73,74]-=-), or more generally, planetary magnetic fields [17,75,76,77]. 20 A. Plattner and F. J. Simons / Appl. Comput. Harmon. Anal. XX (2012) XXX-XXX Table 1 Fractional areas, tangential Shannon numbers, and... |

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Crustal magnetic fields of terrestrial planets
- Langlais, Lesur, et al.
- 2010
(Show Context)
Citation Context ... analysis of global vector-valued satellite-magnetic data such as measured to study the magnetization of the terrestrial lithosphere (e.g. [39,72,73,74]), or more generally, planetary magnetic fields =-=[17,75,76,77]-=-. 20 A. Plattner and F. J. Simons / Appl. Comput. Harmon. Anal. XX (2012) XXX-XXX Table 1 Fractional areas, tangential Shannon numbers, and bandwidths for the vectorial concentration problem to contin... |

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Error distribution in regional inversion of potential field data,” Geophys
- Schachtschneider, Holschneider, et al.
- 1111
(Show Context)
Citation Context ...dlimit L = 72. While we chose 1.5N here for convenience, in real-world applications the optimal choice for the Slepian truncation would depend on the behavior of the signal-to-noise ratio of the data =-=[9,79,80]-=-. The relative error and bias of the reconstruction over Africa, as defined by (91), are 0.4% and 14%, respectively. Figure 12 shows the evolution of error and bias for reconstructions using different... |

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Error distribution in regional modelling of the geomagnetic field,” Geophys
- Schachtschneider, Holschneider, et al.
- 1111
(Show Context)
Citation Context ...dlimit L = 72. While we chose 1.5N here for convenience, in real-world applications the optimal choice for the Slepian truncation would depend on the behavior of the signal-to-noise ratio of the data =-=[9,79,80]-=-. The relative error and bias of the reconstruction over Africa, as defined by (91), are 0.4% and 14%, respectively. Figure 12 shows the evolution of error and bias for reconstructions using different... |

1 |
Concentration problems for bandpass filters in communication theory over disjoint frequency intervals and numerical solutions
- SenGupta, Sun, et al.
- 1007
(Show Context)
Citation Context ...d geodynamics [19], planetary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theory =-=[30]-=-, signal processing [31,32], and mathematics [33,34]. To date only a few attempts have been made to bring the advantages of spherical Slepian functions into the realm of spherical vector fields. The f... |

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Quadratic variational framework for signal design on the 2-sphere
- Wei, Kennedy, et al.
- 1109
(Show Context)
Citation Context ...etary [20,21,22,23] and biomedical science [24,25], cosmology [26,27], and computer science [28,29], while continuing to be of interest in information and communication theory [30], signal processing =-=[31,32]-=-, and mathematics [33,34]. To date only a few attempts have been made to bring the advantages of spherical Slepian functions into the realm of spherical vector fields. The first successful constructio... |

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Local basis expansions for MEG source localization
- Mitra, Maniar
- 2005
(Show Context)
Citation Context ...he realm of spherical vector fields. The first successful construction of spatially concentrated bandlimited tangential spherical vector fields was reported for applications in magnetoencephalography =-=[24,25,35]-=-. In geodesy, Eshagh [36] has developed methods to explicitly evaluate the product integrals arising in the concentration problem whose solutions are the vectorial Slepian functions. In this paper we ... |

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Imaging Earth’s crustal magnetic field with satellite data: a regularized spherical triangle tessellation approach,” Geophys
- Stockmann, Finlay, et al.
- 1111
(Show Context)
Citation Context ...t a complete extension of Slepian’s spatiospectral concentration problem to vector fields on the sphere, and give suggestions and examples as to their usage for problems of a geomagnetic nature (e.g. =-=[37,38]-=-). The family of optimally concentrated spherical vectorial multitapers that we will construct in the following should be useful in many scientific applications. In particular in geomagnetism, one of ... |

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Efficient modeling of the Earth’s magnetic-field with harmonic splines,” Geophys
- Parker, Shure
- 1982
(Show Context)
Citation Context ...have to successfully merge information from the global to the regional scale. In the past decade or so, a variety of global-to-regional modeling techniques have come of age including harmonic splines =-=[40,41,42]-=-, stitching together local models [43,44,45,46,47], and wavelets [48,49,50]. Due to their optimal combination of spatial locality and spectral bandlimitation the multitapers constructed in this paper ... |

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Spherical cap harmonic-analysis
- Haines
- 1985
(Show Context)
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The propagation of earthquake waves,” Philos
- Jeans
- 1923
(Show Context)
Citation Context ...0(θ) if m = 0,√ 2Xlm(θ) sinmφ if 0 < m ≤ l, (1) Xlm(θ) = (−1)m ( 2l + 1 4pi )1/2 [ (l −m)! (l +m)! ]1/2 Plm(cos θ), (2) Plm(µ) = 1 2ll! (1− µ2)m/2 ( d dµ )l+m (µ2 − 1)l. (3) The asymptotic wavenumber =-=[52]-=- associated with a harmonic degree l is √ l(l + 1). The function Plm(µ) in (3) is called the associated Legendre function of integer degree l and order m. The spherical harmonics Ylm(r̂) are eigenfunc... |

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Application of Slepian basis functions to magnetic field analysis of Saturn
- Sterenborg, Bloxham
- 2007
(Show Context)
Citation Context ... analysis of global vector-valued satellite-magnetic data such as measured to study the magnetization of the terrestrial lithosphere (e.g. [39,72,73,74]), or more generally, planetary magnetic fields =-=[17,75,76,77]-=-. 20 A. Plattner and F. J. Simons / Appl. Comput. Harmon. Anal. XX (2012) XXX-XXX Table 1 Fractional areas, tangential Shannon numbers, and bandwidths for the vectorial concentration problem to contin... |