Analysis Mathematica最新文献

In this paper we consider the discrete Laplacian acting on1-forms and we study its spectrum relative to the spectrum of the 0-form Laplacian.We show that the nonzero spectrum can coincide for these Laplacians withthe same nature. We examine the characteristics of 0-spectrum of the 1-formLaplacian compared to the cycles of graphs.

We consider random Dirichlet series (f(s)=sum_{n=1}^{infty} varepsilon_n a_n e^{-lambda_{n} s}), with (a_n) complex numbers, (lambda_n geq 0), increasing to (infty) , and otherwise arbitrary; and with ((varepsilon_n)) a Rademacher sequence of random variables. We study their almost sure convergence on the critical line of convergence({ text{Re},, s=sigma_{c}(f)}.)When (lambda_n=n) (periodic case), a well-known sufficient condition on the coefficients a_n ensuring almost sure uniform convergence on ([0,2pi] ) (equivalently uniform convergence on (mathbb{R})) has been given by Salem and Zygmund, who made strong use of Bernstein's inequality. When ((lambda_n)) is arbitrary (non-periodic case), one must distinguish between uniform convergence on compact subsets of (mathbb{R}) (local convergence) and uniform convergence on (mathbb{R}). We extend Salem–Zygmund's theorem to general random Dirichlet series in this non-periodic case. Our main tools are a simple “local” Bernstein's inequality, and P. Lévy's symmetry principle.

We summarize some results as well as we prove some new results about the Orlicz–Lorentz–Karamata spaces and martingale Hardy Orlicz–Lorentz–Karamata spaces. More precisely, Doob's maximal inequality for submartingales and Burkholder–Davis–Gundy inequality are presented. We also show some fundamental martingale inequalities and modular inequalities. Additionally, based on atomic decompositions, duality theorems and fractional integral operators are discussed. As applications in Fourier analysis, we consider the Walsh–Fourier series on Orlicz–Lorentz–Karamata spaces. The dyadic maximal operators on martingale Hardy Orlicz–Lorentz–Karamata spaces are presented. The boundedness of maximal Fejér operator is proved, which further implies some convergence results of the Fejér means.

The well-known inequality (lvert {rm supp}(f) rvert lvert {rm supp}( widehat f) rvert geq |G|) gives a lower estimation for each support. In this paper we consider the case where there exists a slowly increasing function (F) such that (lvert {rm supp}(f) rvert leq F(lvert {rm supp}( widehat f) rvert )). We will show that this can be done under some arithmetic constraint.The two links that help us come from additive combinatorics and theoretical computer science. The first is the additive energy which plays a central role in additive combinatorics. The second is the influence of Boolean functions. Our main tool is the spectral analysis of Boolean functions. We prove an uncertainty inequality in which the influence of a function and its Fourier spectrum play a role.

We show that the sum function of the Möbius function of a Beurling number system must satisfy the asymptotic bound (M(x)=o(x)) if it satisfies the prime number theorem and its prime distribution function arises from a monotone perturbation of either the classical prime numbers or the logarithmic integral.

Optimal quasiconformal dimension distortions bounds for subsetsof the complex plane have been established by Astala. We show that theseestimates can be improved when one considers subsets of the real line of arbitraryHausdorff dimension. We present some explicit numerical bounds.

Let (A) be a Hilbert-Schmidt operator, whose eigenvalues are (lambda_k(A)(k=1,2 , ldots )).We derivea new inequality for the series (sum^{infty}_{k=1}|lambda_k(A)-z_k|^2), where ({z_k}) is a sequence of numberssatisfying the condition(sum_k |z_k|^2<{infty}). That inequality is expressedvia the self-commutator (AA^*-A^*A). If (A) is a nuclear operator, we obtain an inequality for the eigenvalues via the trace and self-commutator.

Our results are based on the generalization of the theorem of R. Bhatia andL. Elsner [1] which is an infinite-dimensional analog of the Hoffman–Wielandttheorem on perturbations of normal matrices.

A classical problem posed in 1992 by Huijsmans and de Pagter asks whether, for every positive operator (T) on a Banach lattice with spectrum (sigma(T) = {1}), the inequality (T ge operatorname{id}) holds true. While the problem remains unsolved in its entirety, a positive solution is known in finite dimensions. In the broader context of ordered Banach spaces, Drnovšek provided an infinite-dimensional counterexample. In this note, we demonstrate the existence of finite-dimensional counterexamples, specifically on the ice cream cone and on a polyhedral cone in (mathbb{R}^3). On the other hand, taking inspiration from the notion of (m)-isometries, we establish that each counterexample must contain a Jordan block of size at least (3).

We prove that any finite union P of interior-disjoint polytopes in (mathbb R^d) has the Pompeiu property, a result first proved by Williams [15]. This means that if a continuous function f on (mathbb R^d) integrates to 0 on any congruent copy of (P) then (f) is identically 0. By a fundamental result of Brown, Schreiber and Taylor [4], this is equivalent to showing that the Fourier–Laplace transform of the indicator function of P does not vanish identically on any 0-centered complex sphere in (mathbb C^d). Our proof initially follows the recent one of Machado and Robins [12] who are using the Brion–Barvinok formula for the Fourier–Laplace transform of a polytope. But we simplify this method considerably by removing the use of properties of Bessel function zeros. Instead we use some elementary arguments on the growth of linear combinations of exponentials with rational functions as coefficients. Our approach allows us to prove the non-existence of complex spheres of any center in the zero-set of the Fourier–Laplace transform. The planar case is even simpler in that we do not even need the Brion–Barvinok formula. We then go further in the question of which sets can be contained in the null set of the Fourier–Laplace transform of a polytope by extending results of Engel [7] who showed that rationally parametrized hypersurfaces, under some mild conditions, cannot be contained in this null-set. We show that a rationally parametrized curve which is not contained in an affine hyperplane in (mathbb C^d) cannot be contained in this null-set. Results about curves parametrized by meromorphic functions are also given.