Unicity of meromorphic functions and their derivatives

In this paper, we prove that if a transcendental meromorphic function f shares two distinct small functions CM with its kth derivative f^(k) (k>1), then f=f^(k). We also resolve the same question for the case k=1. These results generalize a result due to Frank and Weissenborn.     

Supports of fourier transforms of refinable frame functions and their applications to FMRA

Let M be a d × d expansive matrix, and FL ²(??) be a reducing subspace of L ²(? ^d ). This paper characterizes bounded measurable sets in ? ^d which are the supports of Fourier transforms of M-refinable frame functions. As applications, we derive the characterization of bounded measurable sets as the supports of Fourier transforms of FMRA (W-type FMRA) frame scaling functions and MRA (W-type MRA) scaling functions for FL ²(??), respectively. Some examples are also provided.     

Construction of bent functions via Niho power functions

A Boolean function with an even number n=2k of variables is called bent if it is maximally nonlinear. We present here a new construction of bent functions. Boolean functions of the form f(x)=tr(α₁x^d₁+α₂x^d₂), α₁,α₂,x∈F_2ⁿ, are considered, where the exponents d_i (i=1,2) are of Niho type, i.e. the restriction of x^d_i on F_2^k is linear. We prove for several pairs of (d₁,d₂) that f is a bent function, when α₁ and α₂ fulfill certain conditions. To derive these results we develop a new method to prove that certain rational mappings on F_2ⁿ are bijective.     

Minimum-energy wavelet frame on the interval with arbitrary integer dilation factor

In this paper, we study minimum-energy frame Ψ={ψ¹,ψ²,…,ψ^M} on the interval with arbitrary factor d for L²[0,1], Ψ corresponding to some refinable functions with compact support. We give the constructive proof as well as the necessary and sufficient conditions of minimum-energy frames for L²[0,1], present the decomposition and reconstruction formulas of minimum-energy frame on the interval [0,1], and some examples. The experimental results show that the proposed minimum-energy frame on the interval improves the performance in the application of image denoising significantly.     

Admissible majorants for model subspaces, and arguments of inner functions

Let Θ be an inner function in the upper half-plane ?⁺ and let K _Θ denote the model subspace H ² ? Θ H ² of the Hardy space H ² = H ²(?⁺). A nonnegative function w on the real line is said to be an admissible majorant for K _Θ if there exists a nonzero function f ∈ K _Θ such that {f} ? w a.e. on ?. We prove a refined version of the parametrization formula for K _Θ-admissible majorants and simplify the admissibility criterion (in terms of arg Θ) obtained in [8]. We show that, for every inner function Θ, there exist minimal K _Θ-admissible majorants. The relationship between admissibility and some weighted approximation problems is considered.     

Meromorphic functions sharing two small functions with its derivative

In this paper, we find all the forms of meromorphic functions f(z) that share the value 0 CM^∗, and share b(z)IM^∗ with g(z)=a₁(z)f(z)+a₂(z)f^′(z). And a₁(z), a₂(z) and b(z) (a₂(z),b(z)?0) be small functions with respect to f(z). As an application, we show that some of nonlinear differential equations have no transcendental meromorphic solution.     

A note on the Riccati differential equation

In this note, we obtain some results for the Riccati differential equations u′=A(z)+u² with nonentire meromorphic functions A(z). Some examples are given to illustrate our some results are sharp.     

Sum of graphs of continuous functions and boundedness of additive operators

Assume that and are uniformly continuous functions, where D₁,D₂⊂X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x)=x^∗(x)+a and g(x)=x^∗(x)+b with some x^∗∈X^∗ and a,b∈R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X×R treated as a normed space with a norm .     

Properties of delta functions of a class of observables on white noise functionals

Let δa be the Dirac delta function at a∈R and (E)⊂(L²)⊂^∗(E) the canonical framework of white noise analysis over white noise space (E^∗,μ), where E^∗=S^∗(R). For h∈H=L²(R) with h≠0, denote by Mh the operator of multiplication by Wh=〈⋅,h〉 in (L²). In this paper, we first show that Mh is δa-composable. Thus the delta function δa(Mh) makes sense as a generalized operator, i.e. a continuous linear operator from (E) to ^∗(E). We then establish a formula showing an intimate connection between δa(Mh) as a generalized operator and δa(Wh) as a generalized functional. We also obtain the representation of δa(Mh) as a series of integral kernel operators. Finally we prove that δa(Mh) depends continuously on a∈R.     

Inequalities for the Gaussian hypergeometric function

we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(-1/2,1/2;1;1- xc) and F(-1/2- δ,1/2 + δ;1;1- xd) on(0,1) for given 0 c 5d/6 ∞ andδ∈(-1/2,1/2),and find the largest value δ1 = δ1(c,d) such that inequality F(-1/2,1/2;1;1- xc) F(-1/2- δ,1/2 + δ;1;1- xd) holds for all x ∈(0,1). Besides,we also consider the Gaussian hypergeometric functions F(a- 1- δ,1- a + δ;1;1- x3) and F(a- 1,1- a;1;1- x2) for given a ∈ [1/29,1) and δ∈(a- 1,a),and obtain the analogous results.     

Multiplier operators and extremal functions related to the dual dunkl-sonine operator

We study the dual Dunkl-Sonine operator ^tS_k,? on ?^d, and give expression of ^tS_k,?, using Dunkl multiplier operators on ?^d. Next, we study the extremal functions f^*_λ, λ >0 related to the Dunkl multiplier operators, and more precisely show that {f^*_λ} λ >0 converges uniformly to ^tS_k,?(f) as λ → 0+. Certain examples based on Dunkl-heat and Dunkl-Poisson kernels are provided to illustrate the results.     

The improper infinite derivatives of Takagi's nowhere-differentiable function

Let T be Takagi's continuous but nowhere-differentiable function. Using a representation in terms of Rademacher series due to N. Kono [Acta Math. Hungar. 49 (1987) 315-324], we give a complete characterization of those points where T has a left-sided, right-sided, or two-sided infinite derivative. This characterization is illustrated by several examples. A consequence of the main result is that the sets of points where T^′(x)=±∞ have Hausdorff dimension one. As a byproduct of the method of proof, some exact results concerning the modulus of continuity of T are also obtained.     

Short note on Hilbert’s inequality

Considering five different parameters, we obtain some new Hilbert-type integral inequalities for functions f(x), g(x) in L²[0, ∞). Then, we extract from our results some special cases which have been proved before.     

An improvement of Montel's criterion

Let F be a family of holomorphic functions in a domain D, and let a(z), b(z) be two holomorphic functions in D such that a(z)?b(z), and a(z)?a^′(z) or b(z)?b^′(z). In this paper, we prove that: if, for each f∈F, f(z)−a(z) and f(z)−b(z) have no common zeros, f^′(z)=a(z) whenever f(z)=a(z), and f^′(z)=b(z) whenever f(z)=b(z) in D, then F is normal in D. This result improves and generalizes the classical Montel's normality criterion, and the related results of Pang, Fang and the first author. Some examples are given to show the sharpness of our result.     

Applications of utility functions defined on quasi-metric spaces

A quasi-metric space (X,d) is called sup-separable if (X,d^s) is a separable metric space, where d^s(x,y)=max{d(x,y),d(y,x)} for all x,y∈X. We characterize those preferences, defined on a sup-separable quasi-metric space, for which there is a semi-Lipschitz utility function. We deduce from our results that several interesting examples of quasi-metric spaces which appear in different fields of theoretical computer science admit semi-Lipschitz utility functions. We also apply our methods to the study of certain kinds of dynamical systems defined on quasi-metric spaces.     

Absolutely (completely) monotonic functions and Jordan-type inequalities

Detailed analysis shows that a function f admits the double Jordan-type inequality if and only if f is analytic and even. Associated with f is the function g with f(x)=g(x²). In this short note, based on this association, and using properties of absolutely and/or (completely) monotonic functions, we propose a concise method to derive the inequality from the coefficients in the Taylor’s series of f. The results include some existing ones as special cases.     

Differential polynomials sharing one value

In this paper, we investigate uniqueness problems of differential polynomials of meromorphic functions. Let a, b be non-zero constants and let n, k be positive integers satisfying n ≥ 3k+12. If fⁿ + af^(k) and gⁿ + ag^(k) share b CM and the b-points of fⁿ + af^(k) are not the zeros of f and g, then f and g are either equal or closely related.     

Two versions of the Nikodym maximal function on the Heisenberg group

The classical Nikodym maximal function on the Euclidean plane R² is defined as the supremum over averages over rectangles of eccentricity N; its operator norm in L²(R²) is known to be O(logN). We consider two variants, one on the standard Heisenberg group H¹ and the other on the polarized Heisenberg group . The latter has logarithmic L² operator norm, while the former has the L² operator norm which grows essentially of order O(N1/4). We shall imbed these two maximal operators in the family of operators associated to the hypersurfaces {(x₁,x₂,αx₁x₂)} in the Heisenberg group H¹ where the exceptional blow up in N occurs when α=0.     

Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation

In this work we study the period function T of solutions to the conservative equation x^″(t)+f(x(t))=0. We present conditions on f that imply the monotonicity and convexity of T. As a consequence we obtain the criterium established by C. Chicone and find conditions easier to apply. We also get a condition obtained by Cima, Gasull and Mañosas about monotonicity and, following some of their calculations, present results on the period function of Hamiltonian systems where H(x,y)=F(x)+n^-1|y|ⁿ. Using the monotonicity of T, we count the homogeneous solutions to the semilinear elliptic equation Δu=γu^γ-1 in two dimensions.