Measure Dσ Research Articles

Estimates of the asymptotic Nikolskii constants for spherical polynomials

Let Πnd denote the space of spherical polynomials of degree at most n on the unit sphere Sd⊂Rd+1 that is equipped with the surface Lebesgue measure dσ normalized by ∫Sddσ(x)=1. This paper establishes a close connection between the asymptotic Nikolskii constant, L∗(d)≔limn→∞1dimΠndsupf∈Πnd‖f‖L∞(Sd)‖f‖L1(Sd), and the following extremal problem: Iα≔infak‖jα+1(t)−∑k=1∞akjα(qα+1,kt∕qα+1,1)‖L∞(R+) with the infimum being taken over all sequences {ak}k=1∞⊂R such that the infinite series converges absolutely a.e. on R+. Here jα denotes the Bessel function of the first kind normalized so that jα(0)=1, and {qα+1,k}k=1∞ denotes the strict increasing sequence of all positive zeros of jα+1. We prove that for α≥−0.272, Iα=∫0qα+1,1jα+1(t)t2α+1dt∫0qα+1,1t2α+1dt=1F2(α+1;α+2,α+2;−qα+1,124). As a result, we deduce that the constant L∗(d) goes to zero exponentially fast as d→∞: 0.5d≤L∗(d)≤(0.857⋯)d(1+εd)with εd=O(d−2∕3).

Norms of positive definite Toeplitz matrices and detection of almost periodic components in random signals

For positive definite Toeplitz matrices QN =( b(j −k)) N−1 j,k=0 generated by trigonomet- ric moments b(j) of a non-negative measure dσ(θ),θ ∈ (−π,π), we note that the Hilbert- Schmidt normQN � 2 and the maximal eigenvalue λm(N) satisfy the following relations

Semicircle law for random matrices of long-range percolation model

We study the normalized eigenvalue counting measure dσ of matrices of long-range percolation model. These are (2n + 1) × (2n + 1) random real symmetric matrices H = {H(i, j)}i,j whose elements are independent random variables taking zero value with probability 1 – ψ ((i – j)/b), b ∈ ℝ+, where ψ is an even positive function ψ(t) ≤ 1 vanishing at infinity. It is shown that if the third moment of , is uniformly bounded, then the measure dσ = dσn,b weakly converges in probability in the limit n, b → ∞, b = o(n), to the semicircle (or Wigner) distribution. The proof uses the resolvent technique combined with the cumulant expansions method. We show that the normalized trace of resolvent gn,b(z) converges in average and that the variance of gn,b(z) vanishes. In the second part of the paper, we estimate the rate of decreasing of the variance of gn,b(z) under further conditions on the moments of .

Extension of the Stieltjes moment sequence to the left and related problems of the spectral theory of inhomogeneous string

For an inhomogeneous string with known mass distribution (the total mass is assumed to be infinite), known finite length, and unknown spectral measure dσ(t), we construct an analogous string with spectral measure dσ(t)/t. This enables one to determine the moments of all non-negative orders for the measure dσ(t). The mechanical interpretation of Stieltjes’ investigation of the problem of moments proposed by Krein enables one to solve the problem of finding the moments of negative orders for the Stieltjes moment sequence that has a unique solution. This problem is equivalent to the problem of determining the asymptotic behavior of the associated Stieltjes function near zero on the basis of its known asymptotic behavior at infinity.

Damped Oscillatory Integrals and the Boundedness Problem for Maximal Operators

Let S ⊂ ℝRn +1 be a real-analytic hypersurface with surface measure dσ, and let ψ be a smooth nonnegative compactly supported cutoff function. Consider the surface measure dμ q = ψ|Λ(X)|qdσ, where Λ(X) is a damping factor determined by the matrices of the first and second fundamental forms of the surface. We show that its Fourier transform \(\widehat{d\mu _q }\left( \xi \right)\) decays for large |ξ| as O (|ξ|−(1/2+e)), e > 0, provided that q > 3/2. We also consider applications involving maximal operators associated with means of functions over hypersurfaces.

The method of layer potentials for electromagnetic waves in chiral media

Let Ω be a bounded domain in R, with surface measure dσ and outward unit normal n. Throughout the paper we shall assume that Ω is Lipschitz (i.e., ∂Ω can be locally described by means of graphs of Lipschitz continuous functions in appropriate systems of coordinates) and that ∂Ω is connected. Also, we set Ω+ := Ω and Ω− := R Ω. The classical Faraday-Ampere-Maxwell equations in Ω+ or Ω− read { curlE = iωB, curlH = −iωD, (1.1)

JACKSON TYPE THEOREMS FOR COMPACT RANK 1 SYMMETRIC SPACES

Since recently various problems of approximation theory have been actively studied for functions on the n-dimensional sphere Sn (see [1–3] and the bibliography therein). A broader but natural class of spaces for which we can pose such problems is that of all compact rank 1 symmetric spaces (CROSP in the terminology of the monograph [4]). Some results are already available for these spaces (see [5–9]); however, the main problems still remain open. This article is devoted to proving the direct Jackson-type theorems for an arbitrary CROSP M . The complete classification of all CROSPs is well known. It comprises the four series: the spheres Sn (n = 1, 2, . . . ) and the real, complex, and quaternionic projective spaces (Pn(R), n = 2, 3, . . . ; Pn(C), n = 4, 6, . . . ; and Pn(H), n = 8, 12, 16, . . . ) (everywhere the superscript n denotes the dimension of the space) and one special space, the Cayley elliptic plane P 16(Cay). Since the problems of harmonic analysis on Pn(R) are easily reduced to the corresponding problems on Sn, we assume that M 6= Pn(R). A CROSP M is always a Riemannian manifold. Let ∆ be the Laplace–Beltrami operator on M . The spectrum of ∆ is discrete, real, and nonpositive. We arrange it in decreasing order (0 = λ0 > λ1 > λ2 > . . . ) and denote the eigensubspace of ∆ corresponding to the eigenvalue λk by Hk (it is always finite-dimensional). Put PN (M) := H0 + H1 + · · · + HN . The functions in PN (M) are referred to as spherical polynomials on M of degree m (for M = Sn they coincide with the ordinary spherical polynomials). Given a set X with a measure dσ, we as usual denote by Lp(X, dσ) the Banach space of complexvalued measurable functions f(x) on X with the finite norm

Jacobi matrices for measures modified by a rational factor

This paper describes how, given the Jacobi matrixJ for the measure dσ(t), it is possible to produce the Jacobi matrix Ĵ for the measurer(t)dσ(t) wherer(t) is a quotient of polynomials. The method uses a new factoring algorithm to generate the Jacobi matrices associated with the partial fraction decomposition ofr(t) and then applies a previously developed summing technique to merge these Jacobi matrices. The factoring method performs best just where Gautschi's minimal solution method for this problem is weakest and vice versa. This suggests a hybrid strategy which is believed to be the most powerful yet for solving this problem. The method is demonstrated on a simple example and some numerical tests illustrate its performance characteristics.

A class of plastic constitutive equations with vertex effect—I. General theory

A class of plastic constitutive equations which shows one-to-one correspondence between plastic strain increment dϵp and stress increment do is proposed from the viewpoint of tensor algebra. It is shown that it inevitably represents the so-called vertex-hardening model. Several examples of plastic constitutive equation among this class are formulated. For plasticity, the stress time-like measure dσ¯=[(3/2)tr(dT2o)]1/2 and the strain time-like measure dε¯=[(2/3)tr(dep)2]1/2 are effectively used to represent loading or straining history, where d°T is the increment of deviatoric stress in the sense of Jaumann's rate, and dep denotes the plastic deviatoric strain increment. First, material is considered to be initially isotropic but to be anisotropic with deformation. For this case, Wang's representation theorem on isotropic tensor functions is effectively used. However, anisotropy in this case is limited. Therefore the theory is extended to the case where general initial and subsequent anisotropy plays an important role. Then it becomes possible that any anisotropic rule of yielding such as kinematic hardening, kinematic-isotropic hardening and other general anisotropic hardening without vertex formation is combined with vertex hardening. Introducing the natural time-measure dt, the theory is extended to express natural time-dependent inelastic constitutive equations such as creep and/ or viscoelasticity. Furthermore, introducing the natural and internal time-measures, which are combinations of dt and dσ¯, or dt and dεi¯, the theory is also extended to the case where natural and internal time-dependent inelastic constitutive equations such as viscoplasticity and/or dynamic plasticity are required to formulate. In some cases, temperature dependency is also considered.

Projections on Bergman Spaces Over Plane Domains

Let D be a bounded plane domain and let Lp(D) stand for the usual Lebesgue spaces of functions with domain D, relative to the area Lebesque measure dσ(z) = dxdy. The class of all holomorphic functions in D will be denoted by H(D) and we write Bp(D) = Lp(D) ∩ H(D). Bp(D) is called the Bergman p-space and its norm is given byLet be the Bergman kernel of D and consider the Bergman projection(1.1)It is well known that P is not bounded on Lp(D), p = 1, ∞, and moreover, it can be shown that there are no bounded projections of L∞(Δ) onto B∞(Δ).