Asymptotic Representation of L<Subscript>p</Subscript>-Minimal Polynomials, 1 < p < ∞

While the theory of asymptotics for L₂-minimal polynomials is highly developed, so far not much is known about L_p-minimal polynomials, Indeed, Bernstein gave asymptotics for the minimum deviation, Fekete and Walsh gave nth root asymptotics and, recently, Lubinsky and Saff came up with asymptotics outside the support [-1,1]. But the main point of interest, the asymptotic representation on the support, still remains open. Here we settle it for weight functions of the form where w is positive and on [-1,1] with and 相似文献   

Global optimality condition and fixed point continuation algorithm for non-Lipschitz ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> regularized matrix minimization

Regularized minimization problems with nonconvex, nonsmooth, even non-Lipschitz penalty functions have attracted much attention in recent years, owing to their wide applications in statistics, control, system identification and machine learning. In this paper, the non-Lipschitz ? _p (0 < p < 1) regularized matrix minimization problem is studied. A global necessary optimality condition for this non-Lipschitz optimization problem is firstly obtained, specifically, the global optimal solutions for the problem are fixed points of the so-called p-thresholding operator which is matrix-valued and set-valued. Then a fixed point iterative scheme for the non-Lipschitz model is proposed, and the convergence analysis is also addressed in detail. Moreover, some acceleration techniques are adopted to improve the performance of this algorithm. The effectiveness of the proposed p-thresholding fixed point continuation (p-FPC) algorithm is demonstrated by numerical experiments on randomly generated and real matrix completion problems.     

Sur la rigidité des groupes de triangles (<Emphasis Type=Italic>r</Emphasis>, <Emphasis Type=Italic>p</Emphasis>, <Emphasis Type=Italic>q</Emphasis>)

We show the length spectral rigidity for the family of fuchsian triangular groups.     

The Lagrange trigonometric interpolation polynomial with the minimal norm considered as an operator from <Emphasis Type="Italic">C</Emphasis><Subscript>2π</Subscript> to <Emphasis Type="Italic">C</Emphasis><Subscript>2π</Subscript>

In this paper we perform a comparative analysis of Lebesgue functions and constants of a family of Lagrange polynomials. We prove that if a polynomial from the family has the minimal norm in the space of square summable functions, then it also has the minimal norm as an operator which maps the space of continuous functions into itself.     

A new characterization of Anderson’s inequality in <Emphasis Type="Italic">C</Emphasis><Subscript>1</Subscript>-classes

Let ℋ be a separable infinite dimensional complex Hilbert space, and let ℒ(H) denote the algebra of all bounded linear operators on ℋ into itself. Let A = (A ₁, A ₂,..., A _n), B = (B ₁, B ₂,..., B _n) be n-tuples of operators in ℒ(H); we define the elementary operators Δ _A,B : ℒ(H) ↦ ℒ(H) by

Convergence Theorems for ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Norm Minimizers with Respect to <Emphasis Type="Italic">p</Emphasis>

Let C be a fixed compact convex subset of and let x_p be the unique minimal _p-norm element in C for any . In this paper, we study the convergence of x_p as p or , respectively. We characterize also the limit point as the minimal element of C with respect to the lexical minimax order relation or the lexical minitotal order relation, respectively.Communicated by D. G. LuenbergerThe author thanks Professor H. Komiya for valuable advice and an anonymous referee for kind comments.     

Smolyak's algorithm for weighted <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-approximation of multivariate functions with bounded <Emphasis Type="Italic">r</Emphasis>th mixed derivatives over ℝ<Superscript>d</Superscript>

We are interested in numerical algorithms for weighted L₁ approximation of functions defined on . We consider the space ℱ_r,d which consists of multivariate functions whose all mixed derivatives of order r are bounded in L₁-norm. We approximate f∈ℱ_r,d by an algorithm which uses evaluations of the function. The error is measured in the weighted L₁-norm with a weight function ρ. We construct and analyze Smolyak's algorithm for solving this problem. The algorithm is based on one-dimensional piecewise polynomial interpolation of degree at most r−1, where the interpolation points are specially chosen dependently on the smoothness parameter r and the weight ρ. We show that, under some condition on the rate of decay of ρ, the error of the proposed algorithm asymptotically behaves as , where n denotes the number of function evaluations used. The asymptotic constant is known and it decreases to zero exponentially fast as d→∞.     

A volume formula for ℤ<Subscript>2</Subscript>-symmetric spherical tetrahedra

We obtain formulas for the volume of a spherical tetrahedron with ℤ₂-symmetry realized as rotation about the axis passing through the midpoints of a pair of skew edges. We show the dependence of the volume formula on the edge lengths and dihedral angles of the tetrahedron. Several different formulas result whose scopes are determined by the geometric characteristics of the tetrahedron.     

A closed-form expression for the Drinfeld modular polynomial Φ<Subscript><Emphasis Type="Italic">T</Emphasis></Subscript>(<Emphasis Type="Italic">X</Emphasis>, <Emphasis Type="Italic">Y</Emphasis>)

In this paper we give a closed-form expression for the Drinfeld modular polynomial \({\Phi_T(X,Y) \in \mathbb{F}_q(T)[X,Y]}\) for arbitrary q and prove a conjecture of Schweizer. A new identity involving the Catalan numbers plays a central role.     

Containment of c<Subscript>0</Subscript> and ℓ<Subscript>1</Subscript> in π<Subscript>1</Subscript>(<Emphasis Type="Italic">E,F</Emphasis>)

Suppose π₁(E, F) is the space of all absolutely 1-summing operators between two Banach spacesE andF. We show that ifF has a copy of c₀, then π₁ (E, F) will have a copy of c₀, and under some conditions ifE has a copy of ℓ₁ then π₁ (E, F) would have a complemented copy of ℓ₁.     

An Improved Bound for the Crossing Number of C<Subscript><Emphasis Type="Italic">m</Emphasis></Subscript>C<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>: a Self–Contained Proof Using Mostly Combinatorial Arguments

We present some surprisingly elementary arguments to prove that for every >0, if m is sufficiently large, then the crossing number of the Cartesian product C_m×C_n is at least (0.8–)mn, for every nm. The self–contained proof we give involves only one (rather elementary) geometrical result. The rest of the proof involves purely combinatorial arguments.Supported by CONA-CYT (Grant J32168E) and by FAI-UASLPFinal version received: July 14, 2003     

On automorphisms of distance-regular graphs with intersection arrays {2<Emphasis Type="Italic">r</Emphasis> + 1, 2<Emphasis Type="Italic">r</Emphasis> − 2, 1; 1, 2, 2<Emphasis Type="Italic">r</Emphasis> + 1}

Let Γ be an antipodal graph with intersection array {2r+1, 2r?2, 1; 1, 2, 2r+1}, where 2r(r + 1) ≤ 4096. If 2r + 1 is a prime power, then Mathon’s scheme provides the existence of an arc-transitive graph with this intersection array. Note that 2r + 1 is not a prime power only for r ∈ {7, 17, 19, 22, 25, 27, 31, 32, 37, 38, 42, 43}. We study automorphisms of hypothetical distance-regular graphs with the specified values of r. The cases r ∈ {7, 17, 19} were considered earlier. We prove that, if Γ is a vertex-symmetric graph with intersection array {2r + 1, 2r ? 2, 1; 1, 2, 2r +1}, 2r + 1 is not a prime power, and r ≤ 43, then r = 25, 27, or 31.     

Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for <Emphasis Type="Italic">p</Emphasis> ≥ 1

We prove that the mixed problem for the Klein–Gordon–Fock equation u _tt (x, t) ? u _xx (x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q _T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L _p [0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L _p (Q _T ) for p ≥ 1. We construct the solution in explicit analytic form.     

On the basis property of eigenelements of ordinary linear differential operators in a space of the type <Emphasis Type=Italic>L</Emphasis><Subscript><Emphasis Type=Italic>p</Emphasis></Subscript>, <Emphasis Type=Italic>p</Emphasis> ≥ 2

For a linear differential expression with matrix coefficients in the class L _p , p ≥ 2, and with a parameter λ, we consider a boundary value problem with boundary conditions at the endpoints of the interval [a, b]. Under the condition that the problem is regular, we obtain a formula for the Fourier series expansion of an arbitrary vector function of the class L _p in the root functions of the problem.     

Trace operators and theta series for Γ<Subscript>0,n</Subscript>[q] and Γ<Subscript>n</Subscript>[q]

We consider the action of suitable trace operators on non homogeneous theta series that are Siegel modular forms for the principal congruence subgroups of the symplectic group of level q, _n[q]. Then, we prove that modular forms for the Hecke subgroup of level q, _0,n[q], that are linear combination of such theta series, can also be expressed as combination of (homogeneous) theta series that are modular forms with respect to _0,n[q].     

On <Emphasis Type="Italic">L</Emphasis> <Subscript>p</Subscript>-convergence of Cesàro means for Fourier series with monotonic coefficients

For sine and cosine Fourier serieswithmonotonic coefficients we study L_p-convergence (1 < p < ∞) of their Cesàro means of a negative order.     

A Hilbert-type theorem for spacelike surfaces with constant Gaussian curvature in ℍ<Superscript>2</Superscript> × ℝ<Subscript>1</Subscript>

There are examples of complete spacelike surfaces in the Lorentzian product ℍ² × ℝ₁ with constant Gaussian curvature K ≤ −1. In this paper, we show that there exists no complete spacelike surface in ℍ² × ℝ₁ with constant Gaussian curvature K > −1.