Analysis Mathematica最新文献

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.

In this paper we connect the finiteness property and the periodicityin the study of the generalized Yang’s conjecture and its variations, whichinvolve the inverse question of whether f(z) is still periodic when some differentialpolynomial in f is periodic. The finiteness property can be dated back toWeierstrass in the characterization of addition law for meromorphic functions. Tothe best of our knowledge, it seems the first time that the finiteness property isused to investigate generalized Yang’s conjecture, which gives a partial affirmativeanswer for the meromorphic functions with at least one pole.

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.

Our main goal is to track down an algebraic basis of Hilbert space ( ell^2) which is a connected and locally connected subset of the unit sphere.

We study the value distributions of the random analytic functions on the unit disk of the form

where (a_jinmathbb{C}) and (chi_j(omega)) are independent and identically distributed random variables defined on a probability space ((Omega, mathcal{F}, mu)). Some of the theorems complement the work in [6], which deals with random entire functions.We first define a family of random analytic functions in the above form, which includes Gaussian, Rademacher, and Steinhaus analytic functions. Then we prove the relationship between the integrated counting function (N(r, a, f_omega))and the (L_2) norm of (f) on the circle (|z|=r) as (r) is close to (1). As a by-product, we obtain Nevanlinna's second main theorem on the unit disk. Finally, we show theorems on the maximum modulus of (f) and (f_omega) on the unit disk.

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).

Let ((X,d,mu)) be a space of homogeneous type and (p(cdot) colon X to[1,infty]) be a variable exponent. We show that if the measure (mu) is Borel-semiregular and reverse doubling, then the condition ({ess,inf}_{xin X}p(x)>1) is necessary for the boundedness of the Hardy–Littlewood maximal operator (M) on the variable Lebesgue space (L^{p(cdot)}(X,d,mu)).

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.