47 research outputs found
A Gauss-Lucas type theorem on the location of the roots of a polynomial
AbstractIn this note, we prove a geometrical relationship between the zeros of a polynomial p of order m, say, and the zeros of another polynomial which is derived from p by multiplying each of p's coefficients, call them {αk}k = 0m, by a power of k or by k2 + 2kλ for λ > 0
On Quasi-interpolation with Radial Basis Functions
AbstractIt has been known since 1987 that quasi-interpolation with radial functions on the integer grid can be exact for certain order polynomials. If, however, we require that the basis functions of the quasi-interpolants be finite linear combinations of translates of the radial functions, then this can be done only in spaces whose dimension has a prescribed parity. In this paper we show how infinite linear combinations of translates of a given radial function can be found that provide polynomial exactness in spaces whose dimensions do not have this prescribed parity. These infinite linear combinations are of a simple form. They are, in particular, easier to find than the cardinal functions of radial basis function interpolation, which provide polynomial exactness in all dimensions. The techniques that are used in this work also give rise to some remarks about interpolation with radial functions both on the integers and on the nonnegative integers
On kernel engineering via PaleyâWiener
A radial basis function approximation takes the form
where the coefficients a 1,âŠ,a n are real numbers, the centres b 1,âŠ,b n are distinct points in â d , and the function Ï:â d ââ is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which Ï is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure ÎŒ for which the convolution Ï=ÎŒ Ï is a function of compact support, and when Ï is polyharmonic. The novelty of this construction is its use of the PaleyâWiener theorem to identify compact support via analysis of the Fourier transform of the new kernel Ï, so providing a new form of kernel engineering
Box spline prewavelets of small support
The purpose of this paper is the construction of bi- and trivariate prewavelets from box-spline spaces, \ie\ piecewise polynomials of fixed degree on a uniform mesh. They have especially small support and form Riesz bases of the wavelet spaces, so they are stable. In particular, the supports achieved are smaller than those of the prewavelets due to Riemenschneider and Shen in a recent, similar constructio
On spherical averages of radial basis functions
A radial basis function (RBF) has the general form
where the coefficients a 1,âŠ,a n are real numbers, the points, or centres, b 1,âŠ,b n lie in â d , and Ï:â d ââ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when Ï is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm Ï(x)=âxâ when d is an odd positive integer, the thin plate spline Ï(x)=âxâ2log ââxâ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243â264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the PaleyâWiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserlesâs study of geometric integration
Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators
In this paper we introduce a generalized Sobolev space by defining a
semi-inner product formulated in terms of a vector distributional operator
consisting of finitely or countably many distributional operators
, which are defined on the dual space of the Schwartz space. The types of
operators we consider include not only differential operators, but also more
general distributional operators such as pseudo-differential operators. We
deduce that a certain appropriate full-space Green function with respect to
now becomes a conditionally positive
definite function. In order to support this claim we ensure that the
distributional adjoint operator of is
well-defined in the distributional sense. Under sufficient conditions, the
native space (reproducing-kernel Hilbert space) associated with the Green
function can be isometrically embedded into or even be isometrically
equivalent to a generalized Sobolev space. As an application, we take linear
combinations of translates of the Green function with possibly added polynomial
terms and construct a multivariate minimum-norm interpolant to data
values sampled from an unknown generalized Sobolev function at data sites
located in some set . We provide several examples, such
as Mat\'ern kernels or Gaussian kernels, that illustrate how many
reproducing-kernel Hilbert spaces of well-known reproducing kernels are
isometrically equivalent to a generalized Sobolev space. These examples further
illustrate how we can rescale the Sobolev spaces by the vector distributional
operator . Introducing the notion of scale as part of the
definition of a generalized Sobolev space may help us to choose the "best"
kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D.
thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}
Extracting collective trends from Twitter using social-based data mining
The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-40495-5_62Proceedings 5th International Conference, ICCCI 2013, Craiova, Romania, September 11-13, 2013,Social Networks have become an important environment for Collective Trends extraction. The interactions amongst users provide information of their preferences and relationships. This information can be used to measure the influence of ideas, or opinions, and how they are spread within the Network. Currently, one of the most relevant and popular Social Network is Twitter. This Social Network was created to share comments and opinions. The information provided by users is specially useful in different fields and research areas such as marketing. This data is presented as short text strings containing different ideas expressed by real people. With this representation, different Data Mining and Text Mining techniques (such as classification and clustering) might be used for knowledge extraction trying to distinguish the meaning of the opinions. This work is focused on the analysis about how these techniques can interpret these opinions within the Social Network using information related to IKEAÂź company.The preparation of this manuscript has been supported
by the Spanish Ministry of Science and Innovation under the following projects:
TIN2010-19872, ECO2011-30105 (National Plan for Research, Development and
Innovation) and the Multidisciplinary Project of Universidad AutÂŽonoma de
Madrid (CEMU-2012-034