Preface: symposium on progressive politics

Introduction to a special issue of Political Studies Review, based on papers presented at a conference on ‘Progressivism: Past and Present’ held at Senate House in London on 3 July 2012

Interpolation by multidimensional periodic splines

AbstractThis paper is concerned with interpolation by multidimensional periodic splines associated with certain elliptic differential operators. We develop an a priori error estimate for the solution obtained by spline interpolation. Finally we investigate the problem of uniform approximation by multidimensional periodic splines (as basis funnctions)

Splines and Wavelets on Geophysically Relevant Manifolds

Analysis on the unit sphere $\mathbb{S}^{2}$ found many applications in seismology, weather prediction, astrophysics, signal analysis, crystallography, computer vision, computerized tomography, neuroscience, and statistics. In the last two decades, the importance of these and other applications triggered the development of various tools such as splines and wavelet bases suitable for the unit spheres $\mathbb{S}^{2}$, $\>\>\mathbb{S}^{3}$ and the rotation group $SO(3)$. Present paper is a summary of some of results of the author and his collaborators on generalized (average) variational splines and localized frames (wavelets) on compact Riemannian manifolds. The results are illustrated by applications to Radon-type transforms on $\mathbb{S}^{d}$ and $SO(3)$.Comment: The final publication is available at http://www.springerlink.co

Combined Spherical Harmonic and Wavelet Expansion - a Future Concepts in Earth" s Gravitational Determination

The basic theory of spherical singular integrals is recapitulated. Criteria are given for measuring the space-frequency localization of functions on the sphere. The trade off between space localization on the sphere and frequency localization in terms of spherical harmonics is described in form of an uncertainty principle. A continuous version of spherical multiresolution is introduced, starting from continuous wavelet transform corresponding to spherical wavelets with vanishing moments up to a certain order. The wavelet transform is characterized by least-squares properties. Scale discretization enables us to construct spherical counterparts of wavelet packets and scale discrete Daubechies" wavelets. It is shown that singular integral operators forming a semigroup of contraction operators of class (Co) (like Abel-Poisson or Gauß-Weierstraß operators) lead in canonical way to pyramyd algorithms. Fully discretized wavelet transforms are obtained via approximate integration rules on the sphere. Finally applications to (geo-)physical reality are discussed in more detail. A combined method is proposed for approximating the low frequency parts of a physical quantity by spherical harmonics and the high frequency parts by spherical wavelets. The particular significance of this combined concept is motivated for the situation of today" s physical geodesy, viz. the determination of the high frequency parts of the earth" s gravitational potential under explicit knowledge of the lower order part in terms of a spherical harmonic expansion

Wavelet Deformation Analysis for Spherical Bodies

In this paper we introduce a multiscale technique for the analysis of deformation phenomena of the Earth. Classically, the basis functions under use are globally defined and show polynomial character. In consequence, only a global analysis of deformations is possible such that, for example, the water load of an artificial reservoir is hardly to model in that way. Up till now, the alternative to realize a local analysis can only be established by assuming the investigated region to be flat. In what follows we propose a local analysis based on tools (Navier scaling functions and wavelets) taking the (spherical) surface of the Earth into account. Our approach, in particular, enables us to perform a zooming-in procedure. In fact, the concept of Navier wavelets is formulated in such a way that subregions with larger or smaller data density can accordingly be modelled with a higher or lower resolution of the model, respectively

A Survey on Spherical Spline Approximation

Spline functions that approximate data given on the sphere are developed in a weighted Sobolev space setting. The flexibility of the weights makes possible the choice of the approximating function in a way which emphasizes attributes desirable for the particular application area. Examples show that certain choices of the weight sequences yield known methods. A convergence theorem containing explicit constants yields a usable error bound. Our survey ends with the discussion of spherical splines in geodetically relevant pseudodifferential equations

Wavelets, ridgelets and curvelets on the sphere

We present in this paper new multiscale transforms on the sphere, namely the isotropic undecimated wavelet transform, the pyramidal wavelet transform, the ridgelet transform and the curvelet transform. All of these transforms can be inverted i.e. we can exactly reconstruct the original data from its coefficients in either representation. Several applications are described. We show how these transforms can be used in denoising and especially in a Combined Filtering Method, which uses both the wavelet and the curvelet transforms, thus benefiting from the advantages of both transforms. An application to component separation from multichannel data mapped to the sphere is also described in which we take advantage of moving to a wavelet representation.Comment: Accepted for publication in A&A. Manuscript with all figures can be downloaded at http://jstarck.free.fr/aa_sphere05.pd

“Dancing with doxa”: A “Rhetorical Political Analysis” of David Cameron’s sense of Britishness

This chapter explains some of the background to political theorists' and political scientists' interest in language. It then introduces and explains Rhetorical Political Analysis (RPA) contrasting it with approaches to the study of political language, such as Critical Discourse Studies, found within Linguistics. It then briefly demonstrates the application of RPA through a discussion a study of David Cameron's Bloomberg speech on Europe. The analysis highlights in particular the way the speech echoes British conservative precedents and the the way in which "Britishness" emerges as a key 'catachrestical' term

Needatool: A Needlet Analysis Tool for Cosmological Data Processing

We introduce NeedATool (Needlet Analysis Tool), a software for data analysis based on needlets, a wavelet rendition which is powerful for the analysis of fields defined on a sphere. Needlets have been applied successfully to the treatment of astrophysical and cosmological observations, and in particular to the analysis of cosmic microwave background (CMB) data. Usually, such analyses are performed in real space as well as in its dual domain, the harmonic one. Both spaces have advantages and disadvantages: for example, in pixel space it is easier to deal with partial sky coverage and experimental noise; in harmonic domain, beam treatment and comparison with theoretical predictions are more effective. During the last decade, however, wavelets have emerged as a useful tool for CMB data analysis, since they allow to combine most of the advantages of the two spaces, one of the main reasons being their sharp localisation. In this paper, we outline the analytical properties of needlets and discuss the main features of the numerical code, which should be a valuable addition to the CMB analyst's toolbox.Comment: software available at: http://www.fisica.uniroma2.it/~pietrobon/dp_files/dp_NeedATool_download.htm

Simulating full-sky interferometric observations

Aperture array interferometers, such as that proposed for the Square Kilometre Array (SKA), will see the entire sky, hence the standard approach to simulating visibilities will not be applicable since it relies on a tangent plane approximation that is valid only for small fields of view. We derive interferometric formulations in real, spherical harmonic and wavelet space that include contributions over the entire sky and do not rely on any tangent plane approximations. A fast wavelet method is developed to simulate the visibilities observed by an interferometer in the full-sky setting. Computing visibilities using the fast wavelet method adapts to the sparse representation of the primary beam and sky intensity in the wavelet basis. Consequently, the fast wavelet method exhibits superior computational complexity to the real and spherical harmonic space methods and may be performed at substantially lower computational cost, while introducing only negligible error to simulated visibilities. Low-resolution interferometric observations are simulated using all of the methods to compare their performance, demonstrating that the fast wavelet method is approximately three times faster that the other methods for these low-resolution simulations. The computational burden of the real and spherical harmonic space methods renders these techniques computationally infeasible for higher resolution simulations. High-resolution interferometric observations are simulated using the fast wavelet method only, demonstrating and validating the application of this method to realistic simulations. The fast wavelet method is estimated to provide a greater than ten-fold reduction in execution time compared to the other methods for these high-resolution simulations.Comment: 16 pages, 9 figures, replaced to match version accepted by MNRAS (major additions to previous version including new fast wavelet method