Increasing functions and inverse Santaló inequality for unconditional functions

Let be a convex function and be its Legendre tranform. It is proved that if is invariant by changes of signs, then . This is a functional version of the inverse Santaló inequality for unconditional convex bodies due to J. Saint Raymond. The proof involves a general result on increasing functions on together with a functional form of Lozanovskii’s lemma. In the last section, we prove that for some c > 0, one has always . This generalizes a result of B. Klartag and V. Milman.      

On α-quasi-irresolute functions

Joseph and Kwack [9] introduced the notion of (θ,s)-continuous functions in order to investigateS-closed spaces due to Thompson [32]. In [26], the present authors investigated further properties of (θ,s)-continuous functions. In this paper, we introduce a new class of functions called α-quasi-irresolute functions which is weaker than (θ,s)-continuous and improve some results established in [26].     

On α-close-to-convex functions

LetP(α) denote the class of functionsf analytic in the unit discE, withf(0)=0,f(z)≠0 (0<|z|<1) andf′(z)≠0 inE, satisfying the condition $$\int\limits_{\theta _1 }^{\theta _2 } {\operatorname{Re} } \left\{ {a\left( {1 + \frac{{zf''\left( z \right)}}{{f'\left( z \right)}}} \right) + \left( {1 - a} \right)\frac{{zf'\left( z \right)}}{{f\left( z \right)}}} \right\}d\theta > - \pi $$ whenever 0≤θ₁≤θ₂≤θ₁+2π,z=re^iθ r<1 and α is any positive real number. The functions inP(α) unify the classes of close-to-starlike (α=0) and close-to-convex (α=1) functions. We callf∈P(α) and α-close-to-convex function. In this paper we investigate certain properties of the classP(α).     

Elementarily λ-homogeneous binary functions

Characterizations of compact Hausdorff topological MV-algebras, StoneMV-algebras, and MV-algebras that are isomorphic to their profinite completionsare established. It is proved that compact Hausdorff topological MV-algebras areproducts (both topological and algebraic) of copies [0, 1] with the interval topologyand finite ?ukasiewicz chains with the discrete topology. Going one step further, wealso prove that Stone MV-algebras are products (both topological and algebraic) of finite ?ukasiewicz chains with the discrete topology. Finally, it is proved that an MV-algebra is isomorphic to its profinite completion if and only if it is profinite andeach of its maximal ideals of finite rank is principal.     

Approximation of partially smooth functions

In this paper we discuss approximation of partially smooth functions by smooth functions. This problem arises naturally in the study of laminated currents.     

On conditionally δ-convex functions

Let X be a real vector space, V a subset of X and δ ≧ 0 a given number. We say that f: V → ? is a conditionally δ-convex function if for each convex combination t ₁ υ ₁ + … + t _n υ _n of elements of V such that t ₁ υ ₁ + … + t _n υ _n ∈ V the following inequality holds true: $$ f(t_1 v_1 + \cdots + t_n v_n ) \leqq t_1 f(v_1 ) + \cdots + t_n f(v_n ) + \delta . $$ We prove that f: V → ? is conditionally δ-convex if and only if there exists a convex function $ \tilde f $ : conv V → [?∞, ∞) such that $$ \tilde f(v) \leqq f(v) \leqq \tilde f(v) + \delta for v \in V. $$ In case X = ? ⁿ some conditions equivalent to conditional δ-convexity are also presented.     

Generalized Büchi’s problem for algebraic functions and meromorphic functions

Büchi’s nth power problem asks is there a positive integer M such that any sequence ${(x_1^n,\ldots ,x_M^n)}$ of nth powers of integers with nth difference equal to n! is necessarily a sequence of nth powers of successive integers. In this paper, we study an analogue of this problem for meromorphic functions and algebraic functions.     

Log-convexity properties of Schur functions and generalized hypergeometric functions of matrix argument

We establish a positivity property for the difference of products of certain Schur functions, s _λ (x), where λ varies over a fundamental Weyl chamber in ? ⁿ and x belongs to the positive orthant in ? ⁿ . Further, we generalize that result to the difference of certain products of arbitrary numbers of Schur functions. We also derive a log-convexity property of the generalized hypergeometric functions of two Hermitian matrix arguments, and we show how that result may be extended to derive higher-order log-convexity properties.     

Positivstellensätze for differentiable functions

We present a canonical proof of both the strict and weak Positivstellensatz for rings of differentiable and smooth functions. Our construction is explicit, preserves definability in expansions of the real field, and it works in definably complete expansions of real closed fields as well as for real-valued functions on Banach spaces.     

Padé approximants of special functions

Let $ {f_{\gamma }}(x) = \sum\nolimits_{{k = 0}}^{\infty } {{{{T_k (x)}} \left/ {{{{\left( \gamma \right)}_k}}} \right.}} $ , where (??) _k =??(??+1) ? (??+k?1) and T _k (x)=cos (k arccos x) are Padé?CChebyshev polynomials. For such functions and their Padé?CChebyshev approximations $ \pi_{n,m}^{ch}\left( {x;{f_{\gamma }}} \right) $ , we find the asymptotics of decreasing the difference $ {f_{\gamma }}(x) - \pi_{n,m}^{ch}\left( {x;{f_{\gamma }}} \right) $ in the case where 0 ? m ? m(n), m(n) = o (n), as n???? for all x ?? [?1, 1]. Particularly, we determine that, under the same assumption, the Padé?CChebyshev approximations converge to f _?? uniformly on the segment [?1, 1] with the asymptotically best rate.     

Zero sets for functions from Λω

The following result is proved: Theorem. Let S be an inner function ¦S¦1, Spec SE, and assume that the set E satisfies the conditions and where is an arbitrary modulus of continuity. Then there exists a function such that.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 178–188, 1982.I am very grateful to V. P. Khavin, S. A. Vinogradov, and S. V. Khrushchev for useful discussions regarding the topic of this paper.     

Two-dimensional Coulomb scattering of a quantum particle: Wave functions and Green’s functions

We solve the problem of the propagation of a charged quantum particle in a two-dimensional plane embedded in the three-dimensional coordinate space. We consider scattering of this particle by a stable Coulomb center situated in the same plane. We study the wave function of this particle, its Green’s function, and all radial components of these functions. We derive uniform majorant bounds on absolute values of these functions and find the wave function representation in terms of regular radial Coulomb functions and the scattering amplitude representation via partial phases. We obtain integral representations of the Greens’s function and all its radial components.     

Normal functions and α-Normal Functions

This paper has studied two open questions about normal functions due to Lappan, and obtained two corresponding results for α-normal functions. Received April 5, 1999, Accepted June 9, 1999     

Best φ-approximation by n-convex functions

A n-convex function defined on a bounded open interval J ₀ n ≥2 is the (n?l)-st indefinite integral of a nondecreasing function. This fact and the simple structure of the latter enable to obtain concrete results about a n-convex best φ approximation g to a function f ? L _φ on J ₀, where φ: [0, ∞) → [0, ∞) is a convex function that generaJizes the p^th -power functions, 1 ≤ p < ∞. It is shown that g may also be a best generalized spline φ approximation to the restriction of f on the maximal subintervals of J₀ where g is a generalized spline. This is the situation in some cases, among which the L_p -approximation is includedp ≥ 1. For n = 2 it is proven that g is a polynomial of best φ-approximation to f ? L _φ on any maximal interval where g is a polynomial. If f is in addition continuous, then this fact implies the uniqueness of g Under the same assumption, it is shown that the best 3-convex L ₁-approximation is also unique whenever its derivative is bounded.     

Estimating functions for jump–diffusions

Asymptotic theory for approximate martingale estimating functions is generalised to diffusions with finite-activity jumps, when the sampling frequency and terminal sampling time go to infinity. Rate-optimality and efficiency are of particular concern. Under mild assumptions, it is shown that estimators of drift, diffusion, and jump parameters are consistent and asymptotically normal, as well as rate-optimal for the drift and jump parameters. Additional conditions are derived, which ensure rate-optimality for the diffusion parameter as well as efficiency for all parameters. The findings indicate a potentially fruitful direction for the further development of estimation for jump–diffusions.     

The ranks of Maiorana-McFarland bent functions

In this paper, the ranks of a special family of Maiorana-McFarland bent functions are discussed. The upper and lower bounds of the ranks are given and those bent functions whose ranks achieve these bounds are determined. As a consequence, the inequivalence of some bent functions are derived. Furthermore, the ranks of the functions of this family are calculated when t 6.     

Asymptotic analysis of Θ-hypergeometric functions

We define the -hypergeometric functions as a generalization of the hypergeometric functions associated with root systems of Heckman and Opdam. In the geometric setting, the -hypergeometric functions can be specialized to Harish-Chandras spherical functions on Riemannian symmetric spaces of noncompact type, and also to the spherical functions on noncompactly causal symmetric spaces. After describing their regularity properties, we prove estimates for the -hypergeometric functions which are uniform in the space parameter and locally uniform in the spectral parameter. Particular cases are sharp uniform estimates for the Harish-Chandra series up to the walls of the positive Weyl chamber. New estimates for the spherical functions on noncompactly causal symmetric spaces are deduced. Mathematics Subject Classification (2000) 33C67, 43A90, 43A85