Construction of multiwavelets with high approximation order and symmetry

In this paper, based on existing symmetric multiwavelets, we give an explicit algorithm for constructing multiwavelets with high approximation order and symmetry. Concretely, suppose Φ(x) := (φ1(x), . . . , φr(x))T is a symmetric refinable function vectors with approximation order m. For an arbitrary nonnegative integer n, a new symmetric refinable function vector Φnew(x) := (φn1 ew(x), . . . , φrn ew(x))T with approximation order m + n can be constructed through the algorithm mentioned above. Additionally,...     

Orthogonality criteria for compactly supported refinable functions and refinable function vectors

A refinable function φ(x):ℝⁿ→ℝ or, more generally, a refinable function vector Φ(x)=[φ₁(x),...,φ_r(x)]^T is an L¹ solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φ_j(x−α):α∈ℤⁿ, 1≤j≤r form an orthogonal set of functions in L²(ℝⁿ). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and multiwavelet bases of L²(ℝⁿ). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported refinable functions and refinable function vectors.     

Pseudo-dimension and entropy of manifolds formed by affine-invariant dictionary

We consider the manifolds H _n(φ) formed by all possible linear combinations of n functions from the set {φ(A⋅+b)}, where x→Ax+b is arbitrary affine mapping in the space ℝ^d. For example, neural networks and radial basis functions are the manifolds of type H _n(φ). We obtain estimates for pseudo-dimension of the manifold H _n(φ) for wide collection of the generator function φ. The estimates have the order O(d ² n) in degree scale, that is the order is proportional to number of parameters of the manifold H _n(φ). Moreover the estimates for ɛ-entropy of the manifold H _n(φ) are obtained. Mathematics subject classifications (2000) 41A46, 41A50, 42A61, 42C10 V. Maiorov: Supported by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt Foundation.     

Additive Maps Preserving Similarity of Operators on Banach Spaces

Let X be an infinite-dimensional complex Banach space and denote by B（X） the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B（X） onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B（X） such that either Φ（T） = cATA^-1 ＋ ψ（T）I for all T, or Φ（T） = cAT＊A^-1 ＋ ψ（T）I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero.     

Equations fonctionnelles et estimations de normes de matrices

We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(x−y)/θ(x−y) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(x−y)+τ(x)τ(y), τ(0)=0. In both cases we find θ²=aθ⁴+bθ²+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x _r −x _s )) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex _r are equidistant.

Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux     

Ordering Trees with Nearly Perfect Matchings by Algebraic Connectivity

Let T2k＋1 be the set of trees on 2k＋1 vertices with nearly perfect matchings and α（T） be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T2k＋1. Specifically, 10 trees T2,T3,... ,T11 and two classes of trees T（1） and T（12） in T2k＋1 are introduced. It is shown in this paper that for each tree T^′1,T^″1∈T（1）and T^′12,T^″12∈T（12） and each i,j with 2≤i〈j≤11,α（T^′1）=α（T^″1）〉α（Tj）〉α（T^′12）=α（T^″12）.It is also shown that for each tree T with T∈T2k＋1／（T（1）∪{T2,T3,…,T11}∪T（12））,α（T^′12）〉α（T）.     

Augmented group systems and shifts of finite type

Let (G, χ, x) be a triple consisting of a finitely presented groupG, epimorphism χ:G →Z, and distinguished elementx ∈G such that χ(x)=1. Given a finite symmetric groupS _r, we construct a finite directed graph Γ that describes the set Φ _r of representations π: Ker χ →S _r as well as the mapping σ _x :Φ _r →Φ _r defined by (σ _x ϱ)(a) = ϱ(x ⁻¹ ax) for alla ∈ Ker χ. The pair (Φ _r ,σ _x has the structure of a shift of finite type, a well-known type of compact 0-dimensional dynamical system. We discuss basic properties and applications of therepresentation shift (Φ _r ,σ _x ), including applications to knot theory.     

On some properties of Musielak-Orlicz spaces and their subspaces of order continuous elements

Some criterions in order thatl ¹ embeds complementably inE ^Φ(μ) and inL ^Φ(μ) are given. It is also proved that every idealL inL ^Φ(μ) such thatI _Φ(x/‖x‖Φ)=1 for anyxεL/{0} is contained inE ^Φ(μ).     

Orthogonal two-direction multiscaling functions

The concept of a two-direction multiscaling functions is introduced. We investigate the existence of solutions of the two-direction matrix refinable equation

Convergence of Lagrange Interpolation Polynomials for Piecewise Smooth Functions

The present paper first establishes a decomposition result for f(x)∈ C ^r C ^r+1. By using this decomposition we thus can obtain an estimate of ∣f(x) - L _n (f,x)∣ which reflects the influence of the position of the x's and ω(f ^(r+1),δ)_j, j = 0,1,...,s, on the error of approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Positive linear operators generated by analytic functions

Let φ be a power series with positive Taylor coefficients {a _k } _k=0^∞ and non-zero radius of convergence r ≤ ∞. Let ξ _x , 0 ≤ x < r be a random variable whose values α _k , k = 0, 1, …, are independent of x and taken with probabilities a _k x ^k /φ(x), k = 0, 1, …. The positive linear operator (A _φ f)(x):= E[f(ξ _x )] is studied. It is proved that if E(ξ _x ) = x, E(ξ _x ²) = qx ² + bx + c, q, b, c ∈ R, q > 0, then A _φ reduces to the Szász-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupaş operator in the case q > 1.     

Linear ordinary differential equations of the second order with unbounded solutions

We consider three families of equations of the form y″ + (1 + φ(x))y = 0, where the coefficient φ(x) satisfies the condition lim _x→+∞ φ(x) = 0. We obtain solutions of these equations in closed form. We show that the maximum absolute values of solutions grow at the rate of a logarithmic function, a power-law function, and even an exponential function as x → ∞.     

Nearly flat almost monotone measures are big pieces of lipschitz graphs

A concentrated (ξ, m) almost monotone measure inR ⁿ is a Radon measure Φ satisfying the two following conditions: (1) Θ ^m (Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R ⁿ the ratioexp [ξ(r)]r^−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim _r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R ⁿ and a λ-Lipschitz function f from x+W into x+W^‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ.     

Maximal subspaces for solutions of the second order abstract cauchy problem

For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family.     

Characterising the Accuracy of Multivariable Refinable Functions via the Symbol Function

Suppose that f(x) = (f ₁(x),...,f _r (x)) ^T , x∈R ^d is a vector-valued function satisfying the refinement equation f(x) = ∑_Λ c _κ f(2x−κ) with finite set Λ of Z ^d and some r×r matricex c _κ. The requirements for f to have accuracy p are given in terms of the symbol function m(ξ). Supported by NSFC     

Explicit evaluations of Turán expressions

Summary Positive representations for [P _n ^(λ) (x)]²−P _n ^−1/(λ) (x)P _n ^+1/(λ) (x) and for analogous expressions involving orthogonal polynomials are obtained. This is an excerpt from the author's doctoral dissertation, written under the direction of ProfessorW. Seidel, to whom the author is grateful for his encouragement and assistance.     

Construction of orthonormal multi-wavelets with additional vanishing moments

An iterative scheme for constructing compactly supported orthonormal (o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established. Precisely, let φ=[φ₁,. . .,φ_r]^⊤ be an r-dimensional o.n. scaling function vector with polynomial preservation of order (p.p.o.) m, and ψ=[ψ₁,. . .,ψ_r]^⊤ an o.n. multi-wavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a new (r+1)-dimensional o.n. scaling function vector φ^♯:=[φ^⊤,φ_r+1]^⊤ and some corresponding o.n. multi-wavelet ψ^♯ are constructed in such a way that φ^♯ has p.p.o.=n>m and their two-scale symbols P^♯ and Q^♯ are lower and upper triangular block matrices, respectively, without increasing the size of the supports. For instance, for r=1, if we consider the mth order Daubechies o.n. scaling function φ_m^D, then φ^♯:=[φ_m^D,φ₂]^⊤ is a scaling function vector with p.p.o. >m. As another example, for r=2, if we use the symmetric o.n. scaling function vector φ in our earlier work, then we obtain a new pair of scaling function vector φ^♯=[φ^⊤,φ₃]^⊤ and multi-wavelet ψ^♯ that not only increase the order of vanishing moments but also preserve symmetry. Dedicated to Charles A. Micchelli in Honor of His Sixtieth Birthday Mathematics subject classifications (2000) 42C15, 42C40. Charles K. Chui: Supported in part by NSF grants CCR-9988289 and CCR-0098331 and Army Research Office under grant DAAD 19-00-1-0512. Jian-ao Lian: Supported in part by Army Research Office under grant DAAD 19-01-1-0739.     

Some inverse theorems for approximation of functions by the Fourier-Laguerre sums

In this paper we prove two inverse theorems for approximation of functions of two variables by Fourier-Laguerre sums in the space L ₂(ℝ₊²;x ^α y ^β e ^−x−y ).     

A global volume lemma and applications

Letf ^t be aC ² Axiom A dynamical system on a compact manifold satisfying the transversality condition. We prove that ifB _x (ε,t)=[y: dist (f ^s x,f ^s y)≤ε for all 0≤s≤t], then volB _x (ε,t) has the order exp(∫ ₀ ^t φ (f ^s x)ds) in the continuous time case and exp (Σ _s ^t−1 φ (f ^s x)) in the discrete time case, whereφ is a Holder continuous extension from basic hyperbolic sets of the negative of the differential expansion coefficient in the unstable direction. An application to the theory of large deviations is given. Partially supported by US-Israel BSF. Partially supported by a Darpa grant.