Analysis Mathematica最新文献

A Blaschke product has no radial limits on a subset E of the unit circle T but has unrestricted limit at each point of T E if and only if E is a closed set of measure zero.

In this article, we mainly obtain the measure of Julia limiting directions and transcendental directions of Jackson difference operators of non-trivial transcendental entire solutions for differential-difference equation ({f^n}(z) + sumlimits_{k = 0}^n {{a_{{lambda _k}}}(z){p_{{lambda _k}}}(z,f) = h(z),} ) where ({p_{{lambda _k}}}(z,f),,,(lambda in mathbb{N})) are distinct differential-difference monomials, ({a_{{lambda _k}}}(z)) are entire functions of growth smaller than that of the transcendental entire h(z). For non-trivial entire solutions f of differential-difference equation ({P_2}(z,f) + {A_1}(z){P_1}(z,f) + {A_0}(z) = 0,) where P_λ(z,f)(λ = 1, 2) are differential-difference polynomials. By considering the entire coefficient associated with Petrenko’s deviation, the measure of common transcendental directions of classical difference operators and Jackson difference operators of f was studied.

In this paper we apply ideas from the theory of Uniform Distribution of sequences to Functional Analysis and then drawing inspiration from the consequent results, we study concepts and results in Uniform Distribution itself. so let E be a Banach space. then we prove:

1. (a)

   If F is a bounded subset of E and (x in overline {{rm{co}}} (F)) (= the closed convex hull of F), then there is a sequence (x_n) ⊆ F which is Cesàro summable to x.

2. (b)

   If E is separable, F ⊆ E* bounded and (f in {overline {{rm{co}}} ^{{w^ ast}}},(F)), then there is a sequence (f_n) ⊆ F whose sequence of arithmetic means ({{{f_1} + cdots +{f_N}} over N}), N ≥ 1 weak*-converges to f.

By the aid of the Krein-Milman theorem, both (a) and (b) have interesting implications for closed, convex and bounded subsets Ω of E such that (Omega = overline {{rm{co}}} ({rm{ex}},Omega)) and for weak* compact and convex subsets of E*. Of particular interest is the case when Ω = B_C(K)*, where K is a compact metric space.

By further expanding the previous ideas and results, we are able to generalize a classical theorem of Uniform Distribution which is valid for increasing functions φ: I =[0,1] → ℝ with φ(0) = 0 and φ(1) = 1, for functions φ of bounded variation on I with φ(0) = 0 and total variation V₀¹φ = 1.

Let ψ be the digamma function and let L(a,b) = (b − a)/log(b/a) be the logarithmic mean of a and b. We prove that the inequality

holds for all real numbers a and b with b > a ≥ α₀. Here, α₀ ≈ 0.56155 is the only positive solution of

The constant lower bound α₀ is best possible. This refines a result of Chu, Zhang and Tang, who showed that (*) is valid for b > a ≥ 2. Moreover, we prove that the following companion to (*) holds for all a and b with b > a > 0,

We consider the spectral eigenmatrix problem of the planar self-similar spectral measures μ_Q,D generated by

where q ≥ 2 is an integer. For matrix R ∈ M₂(ℤ), we prove that there exist some spectrum Λ such that Λ and RΛ are both the spectra of μ_Q,D if and only if det R ∈ 2ℤ + 1.

We consider, for a class of functions φ: ℝ² {0} → ℝ² satisfying a nonisotropic homogeneity condition, the Fourier transform û of the Borel measure on ℝ⁴ defined by

where E is a Borel set of ℝ⁴ and (U = left{{left({{t^{{alpha _1}}},{t^{{alpha _2}}}s} right):c < s < d,,,0 < t < 1} right}). The aim of this article is to give a decay estimate for û for the case where the set of nonelliptic points of φ is a curve in (overline U backslash left{{bf{0}} right}). From this estimate we obtain a restriction theorem for the usual Fourier transform to the graph of φ∣_U: U → ℝ². We also give L^p-improving properties for the convolution operator T_μf = μ * f.

It is shown that the map z ↦ log(1 − c⁻¹ log(1 − z)) is absolutely monotone on [0, 1) if and only if c ≥ 1. The proof is based on an integral representation for the associated Taylor coefficients and on one of Gautschi’s double inequalities for the quotient of two gamma functions. The result is used to verify that, for every c ≥ 1 and α ∈ (0, 1], the map z ↦ 1 − exp(c − c(1 − c⁻¹ log(1 − z))^α) is absolutely monotone on [0, 1). The proof exploits a continuous-time discrete state space branching process.

In this short article we present some properties regarding the order and the type of an entire function.

Let X be a Banach space and T be a bounded linear operator acting in l_p(ℤ^c,X), 1 ≤ p ≤ ∞. The operator T is called locally nuclear if it can be represented in the form

where b_km: X → X are nuclear,

(left|cdotright|{_{{mathfrak{S}_1}}}) is the nuclear norm, β ∈ l₁(ℤ^c,ℂ) or β ∈ l_1,g(ℤ^c,ℂ), and g is an appropriate weight on ℤ^c. It is established that if T is locally nuclear and the operator 1 + T is invertible, then the inverse operator (1 + T)⁻¹ has the form 1 + T₁, where T₁ is also locally nuclear. This result is refined for the case of operators acting in L_p (ℝ^c,ℂ).

Many studies have been done for one-dimensional Cauchy integral operator. We consider n-dimensional Cauchy integral operator for hypersurface, or we say, the double layer potential operator, and obtain the boundedness from H^p(Rⁿ) to h^p(Rⁿ) (local Hardy space). For the proof we introduce Clifford valued Hardy spaces.