1,721,027 research outputs found
Uncertainty principles on compact Riemannian manifolds
No full textAbstractBased on a result of Rösler and Voit for ultraspherical polynomials, we derive an uncertainty principle for compact Riemannian manifolds M. The frequency variance of a function in L2(M) is therein defined by means of the radial part of the Laplace–Beltrami operator. The proof of the uncertainty rests upon Dunkl theory. In particular, a special differential-difference operator is constructed which plays the role of a generalized root of the radial Laplacian. Subsequently, we prove with a family of Gaussian-like functions that the deduced uncertainty is asymptotically sharp. Finally, we specify in more detail the uncertainty principles for well-known manifolds like the d-dimensional unit sphere and the real projective space
A spectral interpolation scheme on the unit sphere based on the nodes of spherical Lissajous curves
No full textFor sampling values along spherical Lissajous curves we establish a spectral interpolation and quadrature scheme on the sphere. We provide a mathematical analysis of spherical Lissajous curves and study the characteristic properties of their intersection points. Based on a discrete orthogonality structure we are able to prove the unisolvence of the interpolation problem. As basis functions for the interpolation space we use a parity-modified double Fourier basis on the sphere that allows us to implement the interpolation scheme in an efficient way. We further show that the numerical condition number of the interpolation scheme displays a logarithmic growth. As an application, we use the developed interpolation algorithm to estimate the rotation of an object based on measurements at the spherical Lissajous nodes
Weak limits for weighted means of orthogonal polynomials
Full text linkThis article is a first attempt to obtain weak limit formulas for weighted means of orthogonal polynomials. For this, we introduce a new mean Nevai class that guarantees the existence of a limiting measure for the weighted means. We show that for a family of measures in this mean Nevai class also the means of the Christoffel-Darboux kernels and the asymptotic distribution of the roots converge weakly to the same limiting measure. As a main example, we study the mean Nevai classes in which the limiting measure is the orthogonality measure of the ultraspherical polynomials. The respective weak limit formula can be regarded as an asymptotic weak addition formula for the corresponding class of measures
Bivariate Lagrange interpolation at the node points of Lissajous curves-the degenerate case
No full textIn this article, we study bivariate polynomial interpolation on the node points of degenerate Lissajous figures. These node points form Chebyshev lattices of rank 1 and are generalizations of the well-known Padua points. We show that these node points allow unique interpolation in appropriately defined spaces of polynomials and give explicit formulas for the Lagrange basis polynomials. Further, we prove mean and uniform convergence of the interpolating schemes. For the uniform convergence the growth of the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature
Rhodonea Curves as Sampling Trajectories for Spectral Interpolation on the Unit Disk
No full textRhodonea curves are classical planar curves in the unit disk with the characteristic shape of a rose. In this work, we use these rose curves as sampling trajectories to create novel nodes for spectral interpolation on the disk. By generating the interpolation spaces with a parity-modified Chebyshev–Fourier basis, we will prove the unisolvence of the interpolation on the rhodonea nodes. Properties such as continuity, convergence, and numerical condition of the interpolation scheme depend on the spectral structure of the interpolation space. For rectangular spectral index sets, we show that the interpolant is continuous at the center, that the Lebesgue constant grows only logarithmically, and that the scheme converges fast if the interpolated function is smooth. Finally, we show that the scheme can be implemented efficiently using a fast Fourier method and that it can be applied to define a Clenshaw–Curtis quadrature on the disk
Optimally Space Localized Polynomials with Applications in Signal Processing
No full textFor the filtering of peaks in periodic signals, we specify polynomial filters that are optimally localized in space. The space localization of functions f having an expansion in terms of orthogonal polynomials is thereby measured by a generalized mean value e{open}(f). Solving an optimization problem including the functional e{open}(f), we determine those polynomials out of a polynomial space that are optimally localized. We give explicit formulas for these optimally space localized polynomials and determine in the case of the Jacobi polynomials the relation of the functional e{open}(f) to the position variance of a well-known uncertainty principle. Further, we will consider the Hermite polynomials as an example on how to get optimally space localized polynomials in a non-compact setting. Finally, we investigate how the obtained optimal polynomials can be applied as filters in signal processing. © 2011 Springer Science+Business Media, LLC
Approximation by positive definite functions on compact groups.
No full textWe consider approximation methods defined by translates of a positive definite function on a compact group. A characterization of the native space generated by a positive definite function on a compact group is presented. Starting from Bochner's theorem, we construct examples of well-localized positive definite central functions on the rotation group SO(3). Finally, the stability of the interpolation problem and the error analysis for the given examples are studied in detail
An alternative to Slepian functions on the unit sphere - A space-frequency analysis based on localized spherical polynomials
No full textIn this article, we present a space-frequency theory for spherical harmonics based on the spectral decomposition of a particular space-frequency operator. The presented theory is closely linked to the theory of ultraspherical polynomials on the one hand, and to the theory of Slepian functions on the 2-sphere on the other. Results from both theories are used to prove localization and approximation properties of the new band-limited yet space-localized basis. Moreover, particular weak limits related to the structure of the spherical harmonics provide information on the proportion of basis functions needed to approximate localized functions. Finally, a scheme for the fast computation of the coefficients in the new localized basis is provided
An orthogonal polynomial analogue of the Landau–Pollak–Slepian time–frequency analysis
No full textAbstractThe aim of this article is to present a time–frequency theory for orthogonal polynomials on the interval [−1,1] that runs parallel to the time–frequency analysis of bandlimited functions developed by Landau, Pollak and Slepian. For this purpose, the spectral decomposition of a particular compact time–frequency operator is studied. This decomposition and its eigenvalues are closely related to the theory of orthogonal polynomials. Results from both theories, the theory of orthogonal polynomials and the Landau–Pollak–Slepian theory, can be used to prove localization and approximation properties of the corresponding eigenfunctions. Finally, an uncertainty principle is proven that reflects the limitation of coupled time and frequency locatability
Applications of the monotonicity of extremal zeros of orthogonal polynomials in interlacing and optimization problems.
No full textWe investigate monotonicity properties of extremal zeros of orthogonal polynomials depending on a parameter. Using a functional analysis method we prove the monotonicity of extreme zeros of associated Jacobi, associated Gegenbauer and q-Meixner-Pollaczek polynomials. We show how these results can be applied to prove interlacing of zeros of orthogonal polynomials with shifted parameters and to determine optimally localized polynomials on the unit ball
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