An application of the sampling theorem

According a classical Weierstrass-Hadamard-Lindelf theorem, for each distribution of points with a finite upper density in the complex plane, there exists a non-zero entire function of exponential type vanishing on the these points with the multiplicity taken into account. In the beginning of 1960s, in a joint work by P. Malliavin and L.A. Rubel, the following problem was completely solved. Given two distributions of points on the positive half-line with finite upper densities, find relations between these distributions under which for each non-zero entire function of exponential type vanishing on one of the distributions, there exists a non-zero entire function of exponential type vanishing on the other distribution and having the absolute value not exceeding that of the first function. A complete solution of this problem going back to works by F. Carlson, T. Carleman, M. Cartwright, L. Schwartz, J.-P. Kahane and many others, was given in terms of so-called logarithmic characteristics of distributions of points, which are expressed via reciprocals to points in these distributions. In this paper we extend these results on complex distributions of the points separated from the imaginary axis by a pair of vertical angles of an arbitrary small opening; here we develop logarithmic characteristics for complex distributions of points. We consider three types of possible restrictions on the growth along the imaginary axis, very strict ones, as by P. Malliavin and L.A. Rubel, and less restrictive as in previous works by the second co-author. The main results are of a completed form and are formulated as criterions.

A classical result of Laguerre says that if P is a polynomial of degree n such that P(z) ≠ 0 for | z | < 1 then (ξ - z)P' (z) + nP(z) ≠ 0 for | z | < 1 and | ξ | < 1. Rahman and Schmeisser have obtained an extension of that result to entire functions of exponential type: if f is an entire function of exponential type τ, bounded on ℝ, such that hf(π/2) = 0 then (ξ- l)f'(z) + iτ(z) ≠ 0 for Im(z) > 0 and | ξ | < 1, whenever f(z) ≠ 0 if Im(z) > 0. We obtain a new proof of that result. We also obtain a generalization, to entire functions of exponential type, of a result of Szegö according to which the inequality | P(Rz) — P(z) | < Rn - 1, | z | ≤ 1, R ≥ 1, holds for all polynomials P, of degree ≤ n, such that | P(z) | ≤ 1 for | z | ≤ 1.

In this paper we generalize a Liouville type theorem obtained by Gay in [7] and apply this result to several problems in analysis and mathematical physics such as the non-existence of a smallest carrier for an analytic functional in several dimensions, the determination of the discriminant of Hill's equation and the eigenvalue problem of trace class operators on Hilbert spaces. Furthermore Lerch's theorem obtained in [20] is applied to arithmetic entire functions of exponential type and some interpolation formulas for entire functions of exponential type are established. Finally a characterization is given of analytic functionals with carrier at the origin.

This paper is concerned with a class of linear operators acting in the space of the trigonometric polynomials and preserving the inequalities of the form \S(8)\ < \T(8)\ in the half plane Im 8 > 0. Some inequalities for entire functions of exponential type and some theorems concerning the distribution of the zeros of the trigonometric polynomials, including an analogue to the Gauss-Lucas theorem, are derived.1. Introduction.Using interpolation series, R. P. Boas [1] obtains the following interesting extension of the classic S. Bernstein inequality: Theorem.Letf(z) be an entire function of exponential type a with \f(z)\ ê M on the real axis.Then the inequality (1) I/O + iy)e~iw +f(x -iy)eiw\ â 2 M(cosh2 ay -sin2w)1/2, w real, holds.This theorem, as shown by Boas himself [1], has a number of important consequences.Our purpose is to give a new proof and some extension of (1).At the same time our method, which is based on a principle suggested by a paper of De Bruijn [2], and on a theorem of Obreshkov [3] concerning the zeros of the rational polynomials, allows us to prove some theorems about the zeros of the trigonometric polynomials, including a theorem analogous to the classic Gauss-Lucas theorem.2. The principle mentioned above is given by Theorem 1.Let % be a closed subset of the complex plane C and let 911 be a complex linear space of meromorphic functions with poles in %.Further, let L: 911 -> 911 be a linear operator and 91 the subset of 9H consisting of the functions having no zeros in C\% Then the inequality \fProof.Let us suppose that L(9t) C 91 and |/(z)| < |g(z)| in C\% but nevertheless, there exists z0 G C\9Csuch that |L(/)(z0)| è |L(g)(z0)|.Intro--

Sampling in spaces of entire functions of exponential type in [formula omitted

This work includes some new aspects of motion estimation by the optic flow method in scale spaces. The usual techniques for motion estimation are limited to the application of coarse to fine strategies. The coarse to fine strategies can be successful only if there is enough information in every scale. In this work we investigate the motion estimation in the scale space more basically. The wavelet choice for scale space decomposition of image sequences is discussed in the first part of this work. We make use of the continuous wavelet transform with rotationally symmetric wavelets. Bandpass decomposed sequences allow the replacement of the structure tensor by the phase invariant energy operator. The structure tensor is computationally more expensive because of its spatial or spatio-temporal averaging. The energy operator needs in general no further averaging. The numerical accuracy of the motion estimation with the energy operator is compared to the results of usual techniques, based on the structure tensor. The comparison tests are performed on synthetic and real life sequences. Another practical contribution is the accuracy measurement for motion estimation by adaptive smoothed tensor fields. The adaptive smoothing relies on nonlinear anisotropic diffusion with discontinuity and curvature preservation. We reached an accuracy gain under properly chosen parameters for the diffusion filter. A theoretical contribution from mathematical point of view is a new discontinuity and curvature preserving regularization for motion estimation. The convergence of solutions for the isotropic case of the nonlocal partial differential equation is shown. For large displacements between two consecutive frames the optic flow method is systematically corrupted because of the violence of the sampling theorem. We developed a new method for motion analysis by scale decomposition, which allows to circumvent the systematic corruption without using the coarse to fine strategy. The underlying assumption is, that in a certain neighborhood the grey value undergoes the same displacement. If this is fulfilled, then the same optic flow should be measured in all scales. If there arise inconsistencies in a pixel across the scale space, so they can be detected and the scales containing this inconsistencies are not taken into account.

In this paper, we study the sharp Jackson inequality for the best approximation of f ∈ L2,κ(ℝd) by a subspace Eκ2(σ) (SEκ2(σ)), which is a subspace of entire functions of exponential type (spherical exponential type) at most σ. Here L2,κ(ℝd) denotes the space of all d-variate functions f endowed with the L2-norm with the weight \(v_\kappa (x) = \prod\nolimits_{\xi \in R_ + } {|(\xi ,x)|^{2\kappa (\xi )} } \), which is defined by a positive subsystem R+ of a finite root system R ⊂ ℝd and a function κ(ξ): R → ℝ+ invariant under the reflection group G(R) generated by R. In the case G(R) = ℤ2d, we get some exact results. Moreover, the deviation of best approximation by the subspace Eκ2(σ) (SEκ2(σ)) of some class of the smooth functions in the space L2,κ(ℝd) is obtained.

We generalize a classical Bernstein Theorem about approximation of functions on the real line by entire functions of the exponential type, i.e. for any function f continuous and bounded on an unbounded quasi-smooth (in the sense of Lavrentiev) curve L in the complex plane we construct entire functions of exponential type σ > 0 which in some sense converge optimally to f on L as σ → ∞.

Tail risk of hedge funds: an extreme value application

In this paper, we first establish some new types of fixed point theorem which generalize and improve Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem and many results in [W.-S. Du, Some new results and generalizations in metric fixed point theory, Nonlinear Anal. 73 (2010), 1439-1446] and references therein. Applying those new results, we also present some existence theorems of coincidence point and others.

We obtain some explicit formulae for series of the typewhere f is an entire function of exponential type τ, bounded on the real exis (and satisfying in the first case). These series are expressed in terms of the derivatives of f and Bernoulli numbers. We examine the case where f is a trigonometric polynomial which lead us, in particular, to a new representation of the associated Fejér mean.

We describe closed invariant eigenspaces of the Pommmiez operator in the (LF)-space of entire functions of exponential type. This space is topologically equivalent (by means of the Laplace transform) to the strong dual space of all germs of functions that are analytic on a convex, locally closed subset of the complex plane.

We obtain sufficient conditions for the basis property of the family of exponentials $$\{ e^{i\lambda _n t} /(1 + |\lambda _n |^s )\} _{n \in \mathbb{Z}} $$ in the Sobolev spaceH s (0,a) in terms of the behavior of the generating function, which is an entire function of exponential type with zeros λ n . This result is a generalization of the Levin-Golovin theorem on the basis property of the family of exponentials generated by a function of sine type inL 2(0,a). We apply the theorem obtained to the interpolation of entire functions of exponential type; this application is a generalization of the Kotel’nikov-Shannon theorem in signal theory.

Gauss and Markov quadrature formulae with nodes at zeros of eigenfunctions of a Sturm-Liouville problem, which are exact for entire functions of exponential type, are established. They generalize quadrature formulae involving zeros of Bessel functions, which were first designed by Frappier and Olivier. Bessel quadratures correspond to the Fourier-Hankel integral transform. Some other examples, connected with the Jacobi integral transform, Fourier series in Jacobi orthogonal polynomials and the general Sturm-Liouville problem with regular weight are also given. Bibliography: 39 titles.

Nikol’skii-type inequalities for entire functions of exponential type on \({\mathbb{R}}^{n}\) for the Lorentz–Zygmund spaces are obtained. Some new limiting cases are examined. Application to Besov–type spaces of logarithmic smoothness is given.