Functions of normal operators under perturbations

In Peller (1980) [27], Peller (1985) [28], Aleksandrov and Peller (2009) [2], Aleksandrov and Peller (2010) [3], and Aleksandrov and Peller (2010) [4] sharp estimates for f(A)−f(B) were obtained for self-adjoint operators A and B and for various classes of functions f on the real line R. In this paper we extend those results to the case of functions of normal operators. We show that if a function f belongs to the Hölder class Λα(R²), 0<α<1, of functions of two variables, and N₁ and N₂ are normal operators, then ‖f(N₁)−f(N₂)‖?const‖fΛα_‖‖N₁−N₂α^‖. We obtain a more general result for functions in the space for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class , then it is operator Lipschitz, i.e., . We also study properties of f(N₁)−f(N₂) in the case when f∈Λα(R²) and N₁−N₂ belongs to the Schatten–von Neumann class Sp.     

Coquet-type formulas for the rarefied weighted Thue–Morse sequence

Newman proved for the classical Thue–Morse sequence, ((−1)^s(n))_n≥0, that for all N∈N with real constants satisfying c₂>c₁>0 and λ=log3/log4. Coquet improved this result and deduced , where F(x) is a nowhere-differentiable, continuous function with period 1 and η(N)∈{−1,0,1}. In this paper we obtain for the weighted version of the Thue–Morse sequence that for the sum a Coquet-type formula exists for every r∈{0,1,2} if and only if the sequence of weights is eventually periodic. From the specific Coquet-type formulas we derive parts of the weak Newman-type results that were recently obtained by Larcher and Zellinger.     

Augmented truncation approximations of discrete-time Markov chains

Let P be a positive recurrent infinite transition matrix with invariant distribution π and be a truncated and arbitrarily augmented stochastic matrix with invariant distribution _(n)π. We investigate the convergence ‖_(n)π−π‖→0, as n→∞, and derive a widely applicable sufficient criterion. Moreover, computable bounds on the error ‖_(n)π−π‖ are obtained for polynomially and geometrically ergodic chains. The bounds become rather explicit when the chains are stochastically monotone.     

Best Approximation of Functions like |x| exp(−A|x|)

We determine the exact order of best approximation by polynomials and entire functions of exponential type of functions like?_λ, α(x)=|x|^λ exp(−A|x|^−α). In particular, it is shown thatE(?_λ, α, _n, L_p(−1, 1))∼n^{−(2λp+αp+2)/2p(1+α)}×exp(−(1+α⁻¹)(Aα)^1/(1+α) cos απ/2(1+α) n^α/(1+α)), whereE(?_λ, α, _n, L_p(−1, 1)) denotes best polynomial approximation of?_λ, αinL_p(−1, 1),λ∈,α∈(0, 2],A>0, 1?p?∞. The problem, concerning the exact order of decrease ofE(?_0, 2, _n, L_∞(−1, 1)), has been posed by S. N. Bernstein.     

Asymptotic Enumeration of Convex Polygons

A polygon is an elementary (self-avoiding) cycle in the hypercubic lattice ^dtaking at least one step in every dimension. A polygon on ^dis said to be convex if its length is exactly twice the sum of the side lengths of the smallest hypercube containing it. The number ofd-dimensional convex polygonsp_{n, d}of length 2nwithd(n)→∞ is asymptoticallywherer=r(n, d) is the unique solution ofr coth r=2n/d−1andb(r)=d(r coth r−r²/sinh² r). The convergence is uniform over alld?ω(n) for any functionω(n)→∞. Whendis constant the exponential is replaced with (1−d⁻¹)^2d. These results are proved by asymptotically enumerating a larger class of combinatorial objects calledconvex proto-polygonsby the saddle-point method and then finding the asymptotic probability a randomly chosen convex proto-polygon is a convex polygon.     

Bounded Functions in Möbius Invariant Dirichlet Spaces

Forp∈(0, 1), letQ_p(Q_{p, 0}) be the space of analytic functionsfon the unit diskΔwith sup_w∈Δ ‖f°?_w‖_p<∞ (lim_|w|→1 ‖f°?_w‖_p=0), where ‖·‖_pmeans the weighted Dirichlet norm and?_wis the Möbius map ofΔonto itself with?_w(0)=w. In this paper, we prove the Corona theorem for the algebraQ_p∩H^∞(Q_{p, 0}∩H^∞); then we provide a Fefferman–Stein type decomposition forQ_p(Q_{p, 0}), and finally we describe the interpolating sequences forQ_p∩H^∞(Q_{p, 0}∩H^∞)).     

Unconditional constants and polynomial inequalities

If P is a polynomial on R^m of degree at most n, given by , and P_n(R^m) is the space of such polynomials, then we define the polynomial |P| by . Now if is a convex set, we define the norm on P_n(R^m), and then we investigate the inequality providing sharp estimates on for some specific spaces of polynomials. These ’s happen to be the unconditional constants of the canonical bases of the considered spaces.     

The index formula and the spectral shift function for relatively trace class perturbations

We compute the Fredholm index, index(DA), of the operator DA=(d/dt)+A on L²(R;H) associated with the operator path , where (Af)(t)=A(t)f(t) for a.e. t∈R, and appropriate f∈L²(R;H), via the spectral shift function ξ(⋅;A₊,A₋) associated with the pair (A₊,A₋) of asymptotic operators A_±=A(±∞) on the separable complex Hilbert space H in the case when A(t) is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator A₋.We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function ξ(⋅;A₊,A₋) for the pair (A₊,A₋), and the corresponding spectral shift function ξ(⋅;H₂,H₁) for the pair of operators in this relative trace class context,This formula is then used to identify the Fredholm index of DA with ξ(0;A₊,A₋). In addition, we prove that index(DA) coincides with the spectral flow of the family {A(t)}_t∈R and also relate it to the (Fredholm) perturbation determinant for the pair (A₊,A₋): with the choice of the branch of ln(detH(⋅)) on C₊ such thatWe also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant.     

Growth rate of the functions in Bergman type spaces

For 0<p,α<∞, let ‖f‖_p,α be the L^p-norm with respect the weighted measure . We define the weighted Bergman space A_α^p(D) consisting of holomorphic functions f with ‖f‖_p,α<∞. For any σ>0, let A^−σ(D) be the space consisting of holomorphic functions f in D with . If D has C² boundary, then we have the embedding A_α^p(D)⊂A^−(n+α)/p(D). We show that the condition of C²-smoothness of the boundary of D is necessary by giving a counter-example of a convex domain with C^1,λ-smooth boundary for 0<λ<1 which does not satisfy the embedding.     

A pattern sequence approach to Stern’s sequence

Suppose that w∈1{0,1}^∗ and let a_w(n) be the number of occurrences of the word w in the binary expansion of n. Let {s(n)}_n?0 denote the Stern sequence, defined by s(0)=0, s(1)=1, and for n?1, In this note, we show that where denotes the complement of w (obtained by sending 0?1 and 1?0) and [w]₂ denotes the integer specified by the word w∈{0,1}^∗ interpreted in base 2.     

Logarithmic Sobolev Inequalities for Pinned Loop Groups

LetGbe a connected compact type Lie group equipped with anAd_G-invariant inner product on the Lie algebra ofG. Given this data there is a well known left invariant “H¹-Riemannian structure” on =(G)—the infinite dimensional group of continuous based loops inG. Using this Riemannian structure, we define and construct a “heat kernel”ν_T(g₀, ·) associated to the Laplace–Beltrami operator on (G). HereT>0,g₀∈(G), andν_T(g₀,·) is a certain probability measure on (G). For fixedg₀∈(G) andT>0, we use the measureν_T(g₀,·) and the Riemannian structure on (G) to construct a “classical” pre-Dirichlet form. The main theorem of this paper asserts that this pre-Dirichlet form admits a logarithmic Sobolev inequality.     

On spacelike hypersurfaces with constant scalar curvature in the anti-de Sitter space

In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space (n?3) with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we also obtain several rigidity theorems for such hypersurfaces.     

Convergence of Modified Lagrange Interpolation toLp-Functions Based on the Zeros of Orthonormal Polynomials with Freud Weights

We consider the “Freud weight”W²_Q(x)=exp(−Q(x)). let 1<p<∞, and letL*_n(f) be a modified Lagrange interpolation polynomial to a measurable functionf∈{f; ess sup_x∈ |f(x)| W_Q(x)(1+|x|)^α<∞},α>0. Then we have lim_n→∞ ∫^∞_−∞ [|f(x)−L*_n(f; x)| W_Q(x)(1+|x|)^−Δ]^p dx=0, whereΔis a constant depending onpandα.     

The Baer-invariant of a semidirect product

In 1972 K.I. Tahara [7,2, Theorem 2.2.5], using cohomological methods, showed that if a finite group is the semidirect product of a normal subgroup N and a subgroup T, then M(T) is a direct factor of M(G), where M(G) is the Schur-multiplicator of G and in the finite case, is the second cohomology group of G. In 1977 W. Haebich [1, Theorem 1.7] gave another proof using a different method for an arbitrary group G.In this paper we generalize the above theorem. We will show that scN_cM(T) is a direct factor of _cM(G), where _c[3, p. 102] is the variety of nilpotent groups of class at most c ≥ 1 and _cM(G) is the Baer-invariant of the group G with respect to the variety _c [3, p. 107].     

Problems arising from jackknifing the estimate of a Kaplan-Meier integral

Stute and Wang (1994) considered the problem of estimating the integral S^θ = ∫ θ dF, based on a possibly censored sample from a distribution F, where θ is an F-integrable function. They proposed a Kaplan-Meier integral to approximate S^θ and derived an explicit formula for the delete-1 jackknife estimate . differs from only when the largest observation, X_(n), is not censored (δ_(n) = 1 and next-to-the-largest observation, X_(n-1), is censored (δ_(n-1) = 0). In this note, it will pointed out that when X_(n) is censored is based on a defective distribution, and therefore can badly underestimate . We derive an explicit formula for the delete-2 jackknife estimate . However, on comparing the expressions of and , their difference is negligible. To improve the performance of and , we propose a modified estimator according to Efron (1980). Simulation results demonstrate that is much less biased than and and .     

Sequences that omit a box (modulo 1)

Let be a strictly increasing sequence of real numbers satisfying(0.1)aj+1−aj?σ>0. For an open box I in [0,1d⁾, we write It is shown that the Hausdorff dimension of is d−1 whenever The case d=1 is due to Boshernitzan. The proof builds on his approach.Now let S₁,…,Sd be strictly increasing in N. Define to be the set of x in [0, 1) for which A sequence S is said to fulfill condition D(C) if it containsBr=[ur,vr]∩S for which vr−ur→∞ and1+vr−ur?C#(Br). Kaufman has shown that is countable whenever S₁,…,Sd fulfill condition D(C). Here it is shown that is finite under this hypothesis. An upper bound for is provided.     

An endpoint space–time estimate for the Schrödinger equation

We obtain endpoint estimates for the Schrödinger operator f→eitΔf in with initial data f in the homogeneous Sobolev space . The exponents and regularity index satisfy and . For n=2 we prove the estimates in the range q>16/5, and for n?3 in the range q>2+4/(n+1).     

Existence Theorems for Certain Elliptic and Parabolic Semilinear Equations

Existence theorems for the nonlinear parabolic differential equation −∂u/∂t + Δu + |u|^p + f(x, t) = 0 in ⁿ × [0, ∞) with zero initial value are established given explicit conditions on the nonhomogeneous termf(x, t). An existence theorem is also demonstrated for the corresponding elliptic equation.     

Backward Shift Invariant Operator Ranges

Results on first order Ext groups for Hilbert modules over the disk algebra are used to study certain backward shift invariant operator ranges, namely de Branges–Rovnyak spaces and a more general class called (W; B) spaces. Necessary and sufficient conditions are given for the groups Ext¹_A()(, (W; B)) to vanish whereis thedualof the vector-valued Hardy module, H². One condition involves an extension problem for the Hankel operator with symbolB,Γ_B, but viewed as a module map from H²into (W; B). The group Ext¹_A()(, (W; B))=(0) precisely whenΓ_Bextends to a module map from L²into (W; B) and this in turn is equivalent to the injectivity of (W; B) in the category of contractive HilbertA()-modules. This result applied to the de Branges–Rovnyak spaces yields a connection between the extension problem for the HankelΓ_B and the operator corona problem.     

The Diametric Theorem in Hamming Spaces—Optimal Anticodes

For a Hamming space (ⁿ_α, d_H), the set ofn-length words over the alphabet _α = {0, 1,…,α − 1} endowed with the distanced_H, which for two wordsxⁿ = (x₁,…,x_n),yⁿ = (y₁,…,y_n) ∈ ⁿ_αcounts the number of different components, we determine the maximal cardinality of subsets with a prescribed diameterdor, in another language, anticodes with distanced. We refer to the result as the diametric theorem.In a sense anticodes are dual to codes, which have a prescribedlowerbound on the pairwise distance. It is a hopeless task to determine their maximal sizes exactly.We find it remarkable that the diametric theorem (for arbitrary α) can be derived from our recent complete intersection theorem, which can be viewed as a diametric theorem (for α = 2) in the restricted case, where alln-length words considered have exactlykones.