Density of the Set of Generators of Wavelet Systems

Given a function ψ in the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions In this paper we prove that the set of functions generating affine systems that are a Riesz basis of ${\cal L}^2({\Bbb R}^d)$ is dense in We also prove that a stronger result is true for affine systems that are a frame of In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems, that are compactly supported in frequency, are dense in the unit sphere of with the induced metric. As a byproduct we introduce the p-Grammian of a function and prove a convergence result of this Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems.     

Harmonic Functions for a Class of Integro-differential Operators

We consider the operator defined on functions by

Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space

We establish a close link between the amenability property of a unitary representation p \pi of a group G (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system (\Bbb S_p, G) ({\Bbb S}_{\pi}, G) , where \Bbb S_H {\Bbb S}_{\cal H} is the unit sphere the Hilbert space of representation. We prove that p \pi is amenable if and only if either p \pi contains a finite-dimensional subrepresentation or the maximal uniform compactification of (\Bbb S_p ({\Bbb S}_{\pi} has a G-fixed point. Equivalently, the latter means that the G-space (\Bbb S_p, G) ({\Bbb S}_{\pi}, G) has the concentration property: every finite cover of the sphere \Bbb S_p {\Bbb S}_{\pi} contains a set A such that for every e > 0 \epsilon > 0 the e \epsilon -neighbourhoods of the translations of A by finitely many elements of G always intersect. As a corollary, amenability of p \pi is equivalent to the existence of a G-invariant mean on the uniformly continuous bounded functions on \Bbb S_p {\Bbb S}_{\pi} . As another corollary, a locally compact group G is amenable if and only if for every strongly continuous unitary representation of G in an infinite-dimensional Hilbert space H {\cal H} the system (\Bbb S_H, G) ({\Bbb S}_{\cal H}, G) has the property of concentration.     

Large Deviations for Brownian Intersection Measures

We consider p independent Brownian motions in \input amssym ${\Bbb R}^d$ . We assume that p ≥ 2 and p (d ? 2) < d. Let ?_t denote the intersection measure of the p paths by time t, i.e., the random measure on \input amssym ${\Bbb R}^d$ that assigns to any measurable set \input amssym $A \subset {\Bbb R}^d$ the amount of intersection local time of the motions spent in A by time t. Earlier results of X. Chen derived the logarithmic asymptotics of the upper tails of the total mass \input amssym $\ell _t \left({{\Bbb R}^d } \right)$ as t → ∞. In this paper, we derive a large‐deviation principle for the normalized intersection measure t^?p?_t on the set of positive measures on some open bounded set \input amssym $B \subset {\Bbb R}^d$ as t → ∞ before exiting B. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalized occupation times measures of the p motions. Our proof makes the classical Donsker‐Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from B, conditional on a large total mass in some compact set . This extends earlier studies on the intersection measure by König and Mörters. © 2012 Wiley Periodicals, Inc.     

Periodic and solitary traveling wave solutions for the generalized Kadomtsev–Petviashvili equation

This paper is concerned with traveling waves for the generalized Kadomtsev–Petviashvili equation \input amssym.tex $(w_{t}+w_{\xi\xi\xi}+f(w)_{\xi})_{\xi}=w_{yy},(\xi,y)\in{\Bbb R}^{2}, t\in{\Bbb R}$ , i.e. solutions of the form . We study both, solutions periodic in and solitary waves, which are decaying in x, and their interrelations. In particular, we prove the existence of a sequence of k‐periodic solutions, \input amssym.def $k\in{\Bbb N}$ , which is uniformly bounded in norm and converges to a solitary wave in a suitable topology. This result also holds for the corresponding ground states, i.e. solutions with minimal energy. Copyright © 1999 John Wiley & Sons, Ltd.     

Idempotent multipliers for LP (\Bbb R)(\Bbb R)

The algebra B_p(\Bbb R){\cal B}_p({\Bbb R}), p ? (1,￥)\{2}p\in (1,\infty )\setminus \{2\}, consisting of all measurable sets in \Bbb R{\Bbb R} whose characteristic function is a Fourier p-multiplier, forms an algebra of sets containing many interesting and non-trivial elements (e.g. all intervals and their finite unions, certain periodic sets, arbitrary countable unions of dyadic intervals, etc.). However, B_p(\Bbb R){\cal B}_p({\Bbb R}) fails to be a s\sigma -algebra. It has been shown by V. Lebedev and A. Olevskii [4] that if E ? B_p(\Bbb R)E\in {\cal B}_p({\Bbb R}), then E must coincide a.e. with an open set, a remarkable topological constraint on E. In this note we show if $2 < p < \infty $2 < p < \infty , then there exists E ? B_p(\Bbb R)E\in {\cal B}_p({\Bbb R}) which is not in B_q(\Bbb R){\cal B}_q({\Bbb R}) for any q > pq>p.     

Optimal recovery of anisotropic Besov--Wiener classes

This paper deals with the problem of optimal recovery of some anisotropic Besov--Wiener classes $S^{\bf r}_{pq\theta} B({\bf R}^d)$ in $L_q({\bf R}^d)$ and the dual case $S^{\bf r}_{p\theta} B({\bf R}^d)$ in $L_{qp} ({\bf R}^d)$ $(1相似文献   

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Let denote the linear space over spanned by . Define the (real) inner product , where V satisfies: (i) V is real analytic on ; (ii) ; and (iii) . Orthogonalisation of the (ordered) base with respect to yields the even degree and odd degree orthonormal Laurent polynomials , and . Define the even degree and odd degree monic orthogonal Laurent polynomials: and . Asymptotics in the double-scaling limit such that of (in the entire complex plane), , and (in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on , and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].     

A sharp equivalence between H ∞ functional calculus and square function estimates

Let (T _t ) _{t?≥ 0} be a bounded analytic semigroup on L ^p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A ^* both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ .     

Area‐minimizing projective planes in 3‐manifolds

Let (M,g) be a compact Riemannian manifold of dimension 3, and let ? denote the collection of all embedded surfaces homeomorphic to \input amssym ${\Bbb R}{ \Bbb P}^2$ . We study the infimum of the areas of all surfaces in ?. This quantity is related to the systole of (M,g). It makes sense whenever ? is nonempty. In this paper, we give an upper bound for this quantity in terms of the minimum of the scalar curvature of (M,g). Moreover, we show that equality holds if and only if (M,g) is isometric to \input amssym ${\Bbb R}{ \Bbb P}^3$ up to scaling. The proof uses the formula for the second variation of area and Hamilton's Ricci flow. © 2010 Wiley Periodicals, Inc.     

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Let \input amssym $S\subset{\Bbb R}^2$ be a bounded domain with boundary of class C^∞, and let g_ij = δ_ij denote the flat metric on \input amssym ${\Bbb R}^2$ . Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of ∂S) of all W^2,2 isometric immersions of the Riemannian manifold (S, g) into \input amssym ${\Bbb R}^3$ . In this article we derive the Euler‐Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C³ away from a certain singular set Σ and C^∞ away from a larger singular set Σ ∪ Σ₀. We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Σ₀. Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates. © 2010 Wiley Periodicals, Inc.     

Probability measures on [SIN] groups and some related ideals in group algebras

In 1921, Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that , whenever is a convex body containing n + 1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area F(K), namely . The proof is based on a slight improvement of Blichfeldt’s bound in the case when K is a non-lattice translate of a lattice polytope, i.e., K = t + P, where and P is an n-dimensional polytope with integral vertices. Then we have . Moreover, in the 3-dimensional case we prove a stronger inequality, namely . Authors’ addresses: Martin Henk, Institut für Algebra und Geometrie, Universit?t Magdeburg, Universit?tsplatz 2, D-39106 Magdeburg, Germany; J?rg M. Wills, Mathematisches Institut, Universit?t Siegen, ENC, D-57068 Siegen, Germany     

Almost Optimal Estimates for Best Approximation by Translates on a Torus

We investigate the best approximation of some classes of functions defined on a $d$-dimensional torus ${\mbox{\smallbf T}}^d$ by the transfer manifold $H_n(\varphi)=\{\xxsum_{i=1}^n b_i\varphi(\cdot -a_i)\}$ in the Hilbert space $L_2({\mbox{\smallbf T}}^d)$. For Sobolev classes of functions $\tilde{W}_2^r$ we obtain two-sided estimates of the deviation ${\rm dist}(\tilde{W}_2^r,H_n(\varphi))$. These results are applied to obtain estimates for radial basis approximation.     

Conjugates of Rational Equivariant Holomorphic Maps of Symmetric Domains

Let t: D ?D￠\tau: {\cal D} \rightarrow{\cal D}^\prime be an equivariant holomorphic map of symmetric domains associated to a homomorphism r: \Bbb G ?\Bbb G￠{\bf\rho}: {\Bbb G} \rightarrow{\Bbb G}^\prime of semisimple algebraic groups defined over \Bbb Q{\Bbb Q} . If G ì \Bbb G (\Bbb Q)\Gamma\subset {\Bbb G} ({\Bbb Q}) and G￠ ì \Bbb G￠(\Bbb Q)\Gamma^\prime \subset {\Bbb G}^\prime ({\Bbb Q}) are torsion-free arithmetic subgroups with r (G) ì G￠{\bf\rho} (\Gamma) \subset \Gamma^\prime , the map G\D ?G￠\D￠\Gamma\backslash {\cal D} \rightarrow\Gamma^\prime \backslash {\cal D}^\prime of arithmetic varieties and the rationality of D{\cal D} and D￠{\cal D}^\prime as well as the commensurability groups of s ? Aut (\Bbb C)\sigma \in {\rm Aut} ({\Bbb C}) determines a conjugate equivariant holomorphic map t^s: D^s ?D^￠s\tau^\sigma: {\cal D}^\sigma \rightarrow{\cal D}^{\prime\sigma} of f^s: (G\D)^s ?(G￠\D￠)^s\phi^\sigma: (\Gamma\backslash {\cal D})^\sigma \rightarrow(\Gamma^\prime \backslash {\cal D}^\prime)^\sigma of . We prove that is rational if is rational.     

Conjugates of Rational Equivariant Holomorphic Maps of Symmetric Domains

Let be an equivariant holomorphic map of symmetric domains associated to a homomorphism of semisimple algebraic groups defined over . If and are torsion-free arithmetic subgroups with , the map induces a morphism : of arithmetic varieties and the rationality of is defined by using symmetries on and as well as the commensurability groups of and . An element determines a conjugate equivariant holomorphic map of which induces the conjugate morphism of . We prove that is rational if is rational.     

Weighted estimates for multilinear commutators of Marcinkiewicz integral

Let μ be the n-dimensional Marcinkiewicz integral and μb the multilinear commutator of μ. In this paper, the following weighted inequalities are proved for ω ∈ A∞ and 0 〈 p 〈 ∞,
｜｜μ（f）｜｜LP（ω）≤C｜Mf｜LP（ω） and ｜｜μb（f）｜｜LP（ω）≤C｜｜ML（log L）^1/r f｜｜LP（ω）.
The weighted weak L（log L）^1/r -type estimate is also established when p=1 and ω∈A1.     

The affine preservers of non-singular matrices

When \mathbbK{\mathbb{K}} is an arbitrary field, we study the affine automorphisms of M_n(\mathbbK){{\rm M}_n(\mathbb{K})} that stabilize GL_n(\mathbbK){{\rm GL}_n(\mathbb{K})}. Using a theorem of Dieudonné on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n > 2 or # ${\mathbb{K} > 2}${\mathbb{K} > 2}. We include a short new proof of the more general Flanders theorem for affine subspaces of M_p,q(\mathbbK){{\rm M}_{p,q}(\mathbb{K})} with bounded rank. We also find that the group of affine transformations of M₂(\mathbbF₂){{\rm M}_2(\mathbb{F}_2)} that stabilize GL₂(\mathbbF₂){{\rm GL}_2(\mathbb{F}_2)} does not consist solely of linear maps. Using the theory of quadratic forms over \mathbbF₂{\mathbb{F}_2}, we construct explicit isomorphisms between it, the symplectic group Sp₄(\mathbbF₂){{\rm Sp}_4(\mathbb{F}_2)} and the symmetric group \mathfrakS₆{\mathfrak{S}_6}.     

Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums , as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form , where is a continuous function with , runs over , the set of Farey fractions of order Q in the unit interval [0,1] and are consecutive elements of . We show that the limit lim _Q→∞ A _h (Q) exists and is independent of h.     

Inverse scattering up to smooth functions for the Dirac-ZS-AKNS system

We consider the Dirac-ZS-AKNS system (1) where (the space of functions with n derivatives in L ¹), (2) We consider for (1) the transition matrix and, in addition, for the case of the Dirac system (i.e. for the selfadjoint case the scattering matrix We can divide main results of the present work into three parts. I. We show that the inverse scattering transform and the inverse Fourier transform give the same solution, up to smooth functions, of the inverse scattering problem for (1). More preciseley, we show that, under condition (2) with , the following formulas are valid: (3) and, in addition, for the case of the Dirac system (4) where denotes the factor space. II. Using (3), (4), we give the characterization of the transition matrix and the scattering matrix for the case of the Dirac system under condition (2) with III. As applications of the results mentioned above, we show that 1) for any real-valued initial data , the Cauchy problem for the sh-Gordon equation has a unique solution such that and for any t > 0, 2) in addition, for , for such a solution the following formula is valid: where denotes the space of functions locally integrable with n derivatives. We give also a review of preceding results.