Asymptotic expansions for Riesz fractional derivatives of Airy functions and applications

Riesz fractional derivatives of a function, (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions, and , via and . Here Bi(x) is the Airy function of the second type. Integral representations are presented for the function A₂(a,b;x)=Ai(x−a)Ai(x−b) with a,b∈R and its Hilbert transform. Combined with the above asymptotic expansions they can be used for computing asymptotics of the Hankel transform of . These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero).     

Fractional derivatives of products of Airy functions

Fractional derivatives of the products of Airy functions are investigated, and Dα{Ai(x)×Bi(x)}, where Ai(x) and Bi(x) are the Airy functions of the first and second type, respectively. They turn out to be linear combinations of Dα{Ai(x)} and Dα{Gi(x)}, where Gi(x) is the Scorer function. It is also proved that the Wronskian W(x) of the system of half integrals {D−1/2Ai(x),D−1/2Gi(x)} and its Hilbert transform can be considered special functions in their own right since they are expressed in terms of and Ai(x)Bi(x), respectively. Various integral relations are established. Integral representations for Dα{Ai(x−a)Ai(x+a)} and its Hilbert transform −HDα{Ai(x−a)Ai(x+a)} are derived.     

Differential and integral relations involving fractional derivatives of Airy functions and applications

Various differential and integral relations are deduced that involve fractional derivatives of the Airy function Ai(x) and the Scorer function Gi(x). Several new Wronskian relations are obtained that lead to the calculation of a number of indefinite integrals containing fractional derivatives of the Airy functions. New fractional derivative conservation laws are derived for equations of the Korteweg-de Vries type.     

On normal families and differential polynomials for meromorphic functions

We consider the normality criterion for a families F meromorphic in the unit disc Δ, and show that if there exist functions a(z) holomorphic in Δ, a(z)≠1, for each z∈Δ, such that there not only exists a positive number ε₀ such that |an(a(z)−1)−1|?ε₀ for arbitrary sequence of integers an(n∈N) and for any z∈Δ, but also exists a positive number B>0 such that for every f(z)∈F, B|f^′(z)|?|f(z)| whenever f(z)f^″(z)−a(z)(f^′2(z))=0 in Δ. Then is normal in Δ.     

On Bernstein's inequality for entire functions of exponential type

It was shown by S.N. Bernstein that if f is an entire function of exponential type τ such that |f(x)|?M for −∞<x<∞, then |f^′(x)|?Mτ for −∞<x<∞. If p is a polynomial of degree at most n with |p(z)|?M for |z|=1, then f(z):=p(eiz) is an entire function of exponential type n with |f(x)|?M on the real axis. Hence, by the just mentioned inequality for functions of exponential type, |p^′(z)|?Mn for |z|=1. Lately, many papers have been written on polynomials p that satisfy the condition znp(1/z)≡p(z). They do form an intriguing class. If a polynomial p satisfies this condition, then f(z):=p(eiz) is an entire function of exponential type n that satisfies the condition f(z)≡einzf(−z). This led Govil [N.K. Govil, Lp inequalities for entire functions of exponential type, Math. Inequal. Appl. 6 (2003) 445-452] to consider entire functions f of exponential type satisfying f(z)≡eiτzf(−z) and find estimates for their derivatives. In the present paper we present some additional observations about such functions.     

Complete monotonicity of some functions involving polygamma functions

In the present paper, we establish necessary and sufficient conditions for the functions x^α|ψ⁽ⁱ⁾(x+β)| and α|ψ⁽ⁱ⁾(x+β)|−x|ψ⁽ⁱ⁺¹⁾(x+β)| respectively to be monotonic and completely monotonic on (0,∞), where i∈N, α>0 and β≥0 are scalars, and ψ⁽ⁱ⁾(x) are polygamma functions.     

Normal families of meromorphic functions with multiple zeros

Let F be a family of meromorphic functions defined in a domain D such that for each f∈F, all zeros of f(z) are of multiplicity at least 3, and all zeros of f^′(z) are of multiplicity at least 2 in D. If for each f∈F, f^′(z)−1 has at most 1 zero in D, ignoring multiplicity, then F is normal in D.     

Porosity of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty closed subset of X. Let be a lower semicontinuous function bounded from below. This paper is concerned with the perturbed optimization problem infz∈Z{J(z)+‖x−z‖}, denoted by (x,J)-inf for x∈X. In the case when X is compactly fully 2-convex, it is proved in the present paper that the set of all points x in X for which there does not exist z₀∈Z such that J(z₀)+‖x−z₀‖=infz∈Z{J(z)+‖x−z‖} is a σ-porous set in X. Furthermore, if X is assumed additionally to be compactly locally uniformly convex, we verify that the set of all points x∈X?Z⁰ such that the problem (x,J)-inf fails to be approximately compact, is a σ-porous set in X?Z⁰, where Z⁰ denotes the set of all z∈Z such that z∈PZ(z). Moreover, a counterexample to which some results of Ni [R.X. Ni, Generic solutions for some perturbed optimization problem in nonreflexive Banach space, J. Math. Anal. Appl. 302 (2005) 417-424] fail is provided.     

A localization inequality for set functions

We prove the following theorem, which is an analog for discrete set functions of a geometric result of Lovász and Simonovits. Given two real-valued set functions f₁,f₂ defined on the subsets of a finite set S, satisfying for i∈{1,2}, there exists a positive multiplicative set function μ over S and two subsets A,B⊆S such that for i∈{1,2}μ(A)f_i(A)+μ(B)f_i(B)+μ(A∪B)f_i(A∪B)+μ(A∩B)f_i(A∩B)?0. The Ahlswede-Daykin four function theorem can be deduced easily from this.     

Generic solutions for some perturbed optimization problem in non-reflexive Banach spaces

Let Z be a closed, boundedly relatively weakly compact, nonempty subset of a Banach space X, and J:Z→R a lower semicontinuous function bounded from below. If X₀ is a convex subset in X and X₀ has approximatively Z-property (K), then the set of all points x in X₀?Z for which there exists z₀∈Z such that J(z₀)+‖x−z₀‖=?(x) and every sequence {z_n}⊂Z satisfying lim_n→∞[J(z_n)+‖x−z_n‖]=?(x) for x contains a subsequence strongly convergent to an element of Z is a dense G_δ-subset of X₀?Z. Moreover, under the assumption that X₀ is approximatively Z-strictly convex, we show more, namely that the set of all points x in X₀?Z for which there exists a unique point z₀∈Z such that J(z₀)+‖x−z₀‖=?(x) and every sequence {z_n}⊂Z satisfying lim_n→∞[J(z_n)+‖x−z_n‖=?(x) for x converges strongly to z₀ is a dense G_δ-subset of X₀?Z. Here . These extend S. Cobzas's result [J. Math. Anal. Appl. 243 (2000) 344-356].     

Existence and porosity for a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty closed subset of X. Let be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem supz∈Z{J(z)+‖x−z‖}, which is denoted by (x,J)-sup. We shall prove in the present paper that if Z is a closed boundedly relatively weakly compact nonempty subset, then the set of all x∈X for which the problem (x,J)-sup has a solution is a dense Gδ-subset of X. In the case when X is uniformly convex and J is bounded, we will show that the set of all points x in X for which there does not exist z₀∈Z such that J(z₀)+‖x−z₀‖=supz∈Z{J(z)+‖x−z‖} is a σ-porous subset of X and the set of all points x∈X?Z⁰ such that there exists a maximizing sequence of the problem (x,J)-sup which has no convergent subsequence is a σ-porous subset of X?Z⁰, where Z⁰ denotes the set of all z∈Z such that z is in the solution set of (z,J)-sup.     

On friendly index sets of 2-regular graphs

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f:V→A induces an edge labeling f^∗:E→A defined by f^∗(xy)=f(x)+f(y). For i∈A, let v_f(i)=card{v∈V:f(v)=i} and e_f(i)=card{e∈E:f^∗(e)=i}. A labeling f is said to be A-friendly if |v_f(i)−v_f(j)|≤1 for all (i,j)∈A×A, and A-cordial if we also have |e_f(i)−e_f(j)|≤1 for all (i,j)∈A×A. When A=Z₂, the friendly index set of the graph G is defined as {|e_f(1)−e_f(0)|:the vertex labelingf is Z₂-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if n?2 (mod 4).     

Integral representations of harmonic functions in half spaces

In this paper, using a modified Poisson kernel in an upper half-space, we prove that a harmonic function u(z) in a upper half space with its positive part u⁺(x)=max{u(x),0} satisfying a slowly growing condition can be represented by its integral in the boundary of the upper half space, the integral representation is unique up to the addition of a harmonic polynomial, vanishing in the boundary of the upper half space and that its negative part u⁻(x)=max{−u(x),0} can be dominated by a similar slowly growing condition, this improves some classical result about harmonic functions in the upper half space.     

Inclusion relations and convolution properties of a certain class of analytic functions associated with the Ruscheweyh derivatives

Let A denote the class of functions f(z) with

f(0)=f^′(0)−1=0,     

Boundedness of rough oscillatory singular integral on Triebel-Lizorkin spaces

We give the boundedness on Triebel-Lizorkin spaces for oscillatory singular integral operators with polynomial phases and rough kernels of the form eiP(x)Ω(x)|x|⁻ⁿ, where Ω∈Llog⁺L(Sn−1) is homogeneous of degree zero and satisfies certain cancellation condition.     

On the Weyl-Heisenberg frames generated by simple functions

Let with , and let (?,a,1), 0<a?1 be a Weyl-Heisenberg system {e2πimx?(x−na):m,n∈Z}. We show that if E=[0,1] (and some modulo extension of E), then (?,a,1) is a frame for each 0<a?1 (for certain a, respectively) if and only if the analytic function has no zero on the unit circle {z:|z|=1}. These results extend the case of Casazza and Kalton (2002) [6] that and a=1, which brought together the frame theory and the function theory on the closed unit disk. Our techniques of proofs are based on the Zak transform and the distribution of fractional parts of {na}_n∈Z.     

Divisibility of matrices associated with multiplicative functions

Let S={x₁,…,x_n} be a set of n distinct positive integers. For x,y∈S and y<x, we say the y is a greatest-type divisor of x in S if y∣x and it can be deduced that z=y from y∣z,z∣x,z<x and z∈S. For x∈S, let G_S(x) denote the set of all greatest-type divisors of x in S. For any arithmetic function f, let (f(x_i,x_j)) denote the n×n matrix having f evaluated at the greatest common divisor (x_i,x_j) of x_i and x_j as its i,j-entry and let (f[x_i,x_j]) denote the n×n matrix having f evaluated at the least common multiple [x_i,x_j] of x_i and x_j as its i,j-entry. In this paper, we assume that S is a gcd-closed set and . We show that if f is a multiplicative function such that (f∗μ)(d)∈Z whenever and f(a)|f(b) whenever a|b and a,b∈S and (f(x_i,x_j)) is nonsingular, then the matrix (f(x_i,x_j)) divides the matrix (f[x_i,x_j]) in the ring M_n(Z) of n×n matrices over the integers. As a consequence, we show that (f(x_i,x_j)) divides (f[x_i,x_j]) in the ring M_n(Z) if (f∗μ)(d)∈Z whenever and f is a completely multiplicative function such that (f(x_i,x_j)) is nonsingular. This confirms a conjecture of Hong raised in 2004.     

Well-posedness of a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty subset of X. Let J:Z→R be a lower semicontinuous function bounded from below and p?1. This paper is concerned with the perturbed optimization problem of finding z₀∈Z such that ‖x−z₀p^‖+J(z₀)=infz∈Z{‖x−zp^‖+J(z)}, which is denoted by minJ(x,Z). The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x∈X for which every minimizing sequence of the problem minJ(x,Z) has a converging subsequence is a dense Gδ-subset of X?Z₀, where Z₀ is the set of all points z∈Z such that z is a solution of the problem minJ(z,Z). If additionally p>1 and X is J-strictly convex with respect to Z, then the set of all x∈X for which the problem minJ(x,Z) is well-posed is a dense Gδ-subset of X?Z₀.     

On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry

We consider families (Yn) of degenerating hyperbolic surfaces. The surfaces are geometrically finite of fixed topological type. Let Zn be the Selberg Zeta function of Yn, and let zn be the contribution of the pinched geodesics to Zn. Extending a result of Wolpert's, we prove that Zn(s)/zn(s) converges to the Zeta function of the limit surface if Re(s)>1/2. The technique is an examination of resolvent of the Laplacian, which is composed from that for elementary surfaces via meromorphic Fredholm theory. The resolvent ⁻¹(Δn−t) is shown to converge for all t∉[1/4,∞). We also use this property to define approximate Eisenstein functions and scattering matrices.     

Rose window graphs underlying rotary maps

Given natural numbers n≥3 and 1≤a,r≤n−1, the rose window graph R_n(a,r) is a quartic graph with vertex set {x_i∣i∈Z_n}∪{y_i∣i∈Z_n} and edge set {{x_i,x_i+1}∣i∈Z_n}∪{{y_i,y_i+r}∣i∈Z_n}∪{{x_i,y_i}∣i∈Z_n}∪{{x_i+a,y_i}∣i∈Z_n}. In this paper rotary maps on rose window graphs are considered. In particular, we answer the question posed in [S. Wilson, Rose window graphs, Ars Math. Contemp. 1 (2008), 7-19. http://amc.imfm.si/index.php/amc/issue/view/5] concerning which of these graphs underlie a rotary map.