161 research outputs found
Polyharmonic functions of infinite order on annular regions
Polyharmonic functions f of infinite order and type {\tau} on annular regions
are systematically studied. The first main result states that the
Fourier-Laplace coefficients f_{k,l}(r) of a polyharmonic function f of
infinite order and type 0 can be extended to analytic functions on the complex
plane cut along the negative semiaxis. The second main result gives a
constructive procedure via Fourier-Laplace series for the analytic extension of
a polyharmonic function on annular region A(r_{0},r_{1}) of infinite order and
type less than 1/2r_{1} to the kernel of the harmonicity hull of the annular
region. The methods of proof depend on an extensive investigation of Taylor
series with respect to linear differential operators with constant
coefficients.Comment: 32 page
Cardinal interpolation with polysplines on annuli
AbstractCardinal polysplines of order p on annuli are functions in C2p-2Rn⧹0 which are piecewise polyharmonic of order p such that Δp-1S may have discontinuities on spheres in Rn, centered at the origin and having radii of the form ej, j∈Z. The main result is an interpolation theorem for cardinal polysplines where the data are given by sufficiently smooth functions on the spheres of radius ej and center 0 obeying a certain growth condition in j. This result can be considered as an analogue of the famous interpolation theorem of Schoenberg for cardinal splines
Cardinal interpolation and spline fucntions V. The B-splines for cardinal Hermite interpolation
AbstractIn the third paper of this series on cardinal spline interpolation [4] Lipow and Schoenberg study the problem of Hermite interpolation S(v) = Yv, S′(v) = Yv′,…,S(r−1)(v) = Yv(r−1) for allv. The B-splines are there conspicuous by their absence, although they were found very useful for the case γ = 1 of ordinary (or Lagrange) interpolation (see [5–10]). The purpose of the present paper is to investigate the B-splines for the case of Hermite interpolation (γ > 1). In this sense the present paper is a supplement to [4] and is based on its results. This is done in Part I. Part II is devoted to the special case when we want to solve the problem S(v) = Yv, S′(v) = Yv′ for all v by quintic spline functions of the class C‴(– ∞, ∞). This is the simplest nontrivial example for the general theory. In Part II we derive an explicit solution for the problem (1), where v = 0, 1,…, n
Bernstein operators for exponential polynomials
Let be a linear differential operator with constant coefficients of order
and complex eigenvalues . Assume that the set
of all solutions of the equation is closed under complex
conjugation. If the length of the interval is smaller than , where M_{n}:=\max \left\{| \text{Im}% \lambda_{j}| :j=0,...,n\right\}
, then there exists a basis %, , of the space with
the property that each has a zero of order at and a zero of
order at and each is positive on the open interval
Under the additional assumption that and
are real and distinct, our first main result states that there exist points and positive numbers %,
such that the operator \begin{equation*}
B_{n}f:=\sum_{k=0}^{n}\alpha_{k}f(t_{k}) p_{n,k}(x) \end{equation*} satisfies
, for The second main result
gives a sufficient condition guaranteeing the uniform convergence of
to for each .Comment: A very similar version is to appear in Constructive Approximatio
Complex B-Splines
We propose a complex generalization of Schoenberg's cardinal splines. To this end, we go back to the Fourier domain definition of the B-splines and extend it to complex-valued degrees. We show that the resulting complex B-splines are piecewise modulated polynomials, and that they retain most of the important properties of the classical ones: smoothness, recurrence, and two-scale relations, Riesz basis generator, explicit formulae for derivatives, including fractional orders, etc. We also show that they generate multiresolution analyses of (R) and that they can yield wavelet bases. We characterize the decay of these functions which are no-longer compactly supported when the degree is not an integer. Finally, we prove that the complex B-splines converge to modulated Gaussians as their degree increases, and that they are asymptotically optimally localized in the time-frequency plane in the sense of Heisenberg's uncertainty principle
Spectrophotometric calibration of low-resolution spectra
Low-resolution spectroscopy is a frequently used technique. Aperture prism
spectroscopy in particular is an important tool for large-scale survey
observations. The ongoing ESA space mission Gaia is the currently most relevant
example. In this work we analyse the fundamental limitations of the calibration
of low-resolution spectrophotometric observations and introduce a calibration
method that avoids simplifying assumptions on the smearing effects of the line
spread functions. To this aim, we developed a functional analytic mathematical
formulation of the problem of spectrophotometric calibration. In this
formulation, the calibration process can be described as a linear mapping
between two suitably constructed Hilbert spaces, independently of the
resolution of the spectrophotometric instrument. The presented calibration
method can provide a formally unusual but precise calibration of low-resolution
spectrophotometry with non-negligible widths of line spread functions. We used
the Gaia spectrophotometric instruments to demonstrate that the calibration
method of this work can potentially provide a significantly better calibration
than methods neglecting the smearing effects of the line spread functions.Comment: Final versio
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