The Riesz Basis Property of an Indefinite Sturm-Liouville Problem with a Non-Odd Weight Function

For the Sturm-Liouville eigenvalue problem − f′′ = λrf on [−1, 1] with Dirichlet boundary conditions and with an indefinite weight function r changing its sign at 0 we discuss the question whether the eigenfunctions form a Riesz basis of the Hilbert space L ² _|r|[− 1, 1]. So far a number of sufficient conditions on r for the Riesz basis property are known. However, a sufficient and necessary condition is only known in the special case of an odd weight function r. We shall here give a generalization of this sufficient and necessary condition for certain generally non-odd weight functions satisfying an additional assumption.      

On the Riesz transform associated with the ultraspherical polynomials

We define and investigate the Riesz transform associated with the differential operatorL _λ f(θ)=−f"(θ)−2λ cot’θ. We prove that it can be defined as a principal value and that it is bounded onL ^P ([0, π],dm _λ (θ)),dm _λ(θ)=sin^2λ θdθ, for every 1<p<∞ and of weak type (1,1). The same boundedness properties hold for the maximal operator of the truncated operators. The speed of convergence of the truncated operators is measured in terms of the boundedness inL ^P (dm _λ ), 1<p<∞, and weak type (1,1) of the oscillation and ρ-variation associated to them. Also, a multiplier theorem is proved to get the boundedness of the conjugate function studied by Muckenhoupt and Stein for 1<p<∞ as a corollary of the results for the Riesz transform. Moreover, we find a condition on the weightv which is necessary and sufficient for the existence of a weightu such that the Riesz transform is bounded fromL ^P (v dm _λ ) intoL ^P (u dm _λ ). The authors were partially supported by RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP. The first and fourth authors were supported in part by KBN grant 1-P93A 018 26. The second and third authors were partially supported by BFM grant 2002-04013-C02-02.     

The Ξ operator and its relation to Krein's spectral shift function

We explore connections between Krein's spectral shift function ζ(λ,H ₀, H) associated with the pair of self-adjoint operators (H ₀, H),H=H ₀+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K ^*(H ₀−λ−i0)⁻¹ K) associated with the operator-valued Herglotz functionJ+K ^*(H ₀−z)⁻¹ K, Im(z)>0 inH, whereV=KJK ^* andJ=sgn(V). Our principal results include a new representation for ζ(λ,H ₀,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (E J+A(λ)+tB(λ)(−∞, 0)),E _J((−∞, 0))), ℝ, whereA(λ)=Re(K ^*(H ₀−λ−i0⁻¹ K),B(λ)=Im(K ^*(H ₀−λ-i0)⁻¹ K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H ₀, H) coincides with the trindex associated with the pair (Ξ(J+K ^*(H ₀−λ−i0)K), Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ operators and the Fredholm determinant of the abstract scattering matrix. We also provide a generalization of the classical Birman—Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.     

A question of Gol’dberg concerning entire functions with prescribed zeros

Let (z_j) be a sequence of complex numbers satisfying |z_j|→ ∞ asj → ∞ and denote by n(r) the number of z_j satisfying |z_j|≤ r. Suppose that lim inf_{r → ⇈} log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫^∞ (ϕ(t)t logt)⁻¹ dt < ∞. It is proved that there exists an entire functionf whose zeros are the z_j such that log log M(r,f) = o((log n(r))²ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here. These results answer a question by A. A. Gol’dberg.     

A Reflexivity Result Concerning Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain Ω which is obtained by removing n closed disks which are centered at λ _j with radius r _j for j = 1, . . . , n from the unit disk. We assume that T is a bounded linear operator on a separable reflexive Banach space whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded. Then either T has a nontrivial hyperinvariant subspace or the WOT-closure of the algebra {f(T) : f is a rational function with poles off [`(W)]{\overline\Omega}} is reflexive.     

An identity for normal-like operators

Forλεσ(A) (A a bounded linear operator on a Hilbert space) withλ a boundary point of the numerical range, the ‘spectral theory’ forλ is ‘just as ifA were normal’. IfA isnormal-like (the smallest disk containingσ(A) has radiusr=inf _z ‖A − z‖), then also sup {‖Ax‖² − |〈x.Ax〉|²:‖x‖=1}=r ². This research was partially supported by Air Force Contract AF-AFOSR-62-414.     

A lower bound for the r -order of a matrix modulo N

For a positive integer N, we define the N-rank of a non singular integer d × d matrix A to be the maximum integer r such that there exists a minor of order r whose determinant is not divisible by N. Given a positive integer r, we study the growth of the minimum integer k, such that A ^k − I has N-rank at most r, as a function of N. We show that this integer k goes to infinity faster than log N if and only if for every eigenvalue λ which is not a root of unity, the sum of the dimensions of the eigenspaces relative to eigenvalues which are multiplicatively dependent with λ and are not roots of unity, plus the dimensions of the eigenspaces relative to eigenvalues which are roots of unity, does not exceed d − r − 1. This result will be applied to recover a recent theorem of Luca and Shparlinski [6] which states that the group of rational points of an ordinary elliptic curve E over a finite field with q ⁿ elements is almost cyclic, in a sense to be defined, when n goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that the orders of the groups of \Bbb F_qⁿ-{\Bbb F}_{q^n}- rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity.     

Estimation and detection of a function from tensor product spaces

We observe an unknown function of d variables ƒ(t), t ∈ [0, 1]^d, in the white Gaussian noise of level ε > 0. We assume that {ie4526-01}, where {ie4526-02} is a ball in the Hilbert space {ie4526-03} of tensor product structure. Under minimax setup, we consider two problems: estimate ƒ (for quadratic losses) and detect ƒ, i.e., test the null hypothesis H₀: ƒ = 0 against the alternatives {ie4526-04}. We are interested in the case {ie4526-05}. We study sharp, rate, and log-asymptotics (as ε → 0 and d → ∞) in the problems. In particular, we show that log-asymptotics are essentially different for d ≪ log ε⁻¹ and d ≫ log ε⁻¹. Bibliography: 19 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 180–218.     

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r _j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded, then there exists a nontrivial common invariant subspace for T ^* and (T − λ _j I)^*-1.     

零点有次线性项的椭圆问题的变号解

Abstract. It is proved that the semilinear elliptic problem with zero boundary value     

ON A CLASS OF BESICOVITCHFUNCTIONS TO HAVE EXACT BOX DIMENSION： A NECESSARY AND SUFFICIENT CONDITION

This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given byB(t) := ∞∑k=1 λs-2k sin(λkt),where 1 ＜ s ＜ 2, λk ＞ 0 tends to infinity as k →∞ and λk satisfies λk 1/λk ≥λ＞ 1. The results show thatlimk→∞ log λk 1/log λk = 1is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions.For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D-v,the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D-v(B)),0 ＜ v ＜ s - 1, to be s - v and box dimension of Graph(Du(B)),0 ＜ u ＜ 2 - s, to be s uis also lim k→∞logλk 1/log λk = 1.     

Sharp estimates for derivatives of functions in the Sobolev classes $$ \mathop {W_2^r }\limits^ \circ $$(−1,1)

Explicit formulas are obtained for the maximum possible values of the derivatives f ^(k)(x), x ∈ (−1, 1), k ∈ {0, 1, ..., r − 1}, for functions f that vanish together with their (absolutely continuous) derivatives of order up to ≤ r − 1 at the points ±1 and are such that $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 . As a corollary, it is shown that the first eigenvalue λ _1,r of the operator (−D ²) ^r with these boundary conditions is $ \sqrt 2 $ \sqrt 2 (2r)! (1 + O(1/r)), r → ∞.     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

Weak type estimates for maximal operators with a cylindric distance function

We establish sharp weak-type estimates for the maximal operators T^λ_* associated with cylindric Riesz means for functions on H^p(ℝ³) when 4/5 <p<1 and λ=3/p−5/2, and when p=4/5 and λ>3/p−5/2. The first author was supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD) No. R04-2002-000-20028-0. The third author was supported by a Korea University Grant.     

On the approximation of Tchebycheff-Fourier series in the C space andL P ω space

The degree of approximation to a function f(x)∈C[−1,1] by (U,λ) means and f(x)∈L _P ^ω by (J_r) means are discussed, some results in the literatures [1], [2], [3] have been improved.     

Rates of uniform convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions

Assume that {Xn} is a strictly stationary β-mixing random sequence with the β-mixing coefficient βk = O(k-r), 0 < r ≤1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in Lp, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((nr/(l+r)/logn)-1/2) or O((nr/(1+r)/ log n)-3/4) and that the Central Limit Theorem does not always hold for the empirical processes.``     

The asymptotic behavior of Chern-Simons Higgs model on a compact Riemann surface with boundary

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition.In the previous paper,we show that the Chern-Simons Higgs equation with parameter λ0 has at least two solutions(uλ1,uλ2) for λ sufficiently large,which satisfy that uλ1→u0 almost everywhere as λ→∞,and that uλ2→∞ almost everywhere as λ→∞,where u 0 is a(negative) Green function on M.In this paper,we study the asymptotic behavior of the solutions as λ→∞,and prove that uλ2-uλ2 converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary M is negative,or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.     

Comparison theorems for the volume of a complex submanifold of a Kaehler manifold

LetM be a Kaehler manifold of real dimension 2n with holomorphic sectional curvatureK _H≥4λ and antiholomorphic Ricci curvatureρ _A≥(2n−2)λ, andP is a complex hypersurface. We give a bound for the quotient (volume ofP)/(volume ofM) and prove that this bound is attained if and only ifP=C P ⁿ⁻¹(λ) andM=C P ⁿ(λ). Moreover, we give some results on the volume of of tubes aboutP inM. Work partially supported by a DGICYT Grant No. PS87-0115-CO3-01.     

On projective curves of maximal regularity

 Let 𝒞⊆ℙ ^r _K be a non-degenerate projective curve of degree d>r+1 of maximal regularity so that 𝒞 has an extremal secant line . We show that 𝒞∪ is arithmetically Cohen Macaulay if d<2r−1 and we study the Betti numbers and the Hartshorne-Rao module of the curve 𝒞. Received: 27 March 2002; in final form: 24 May 2002 / Published online: 1 April 2003 Mathematics Subject Classification (1991): 14H45, 13D02. The second author was partially supported by Swiss National Science Foundation (Projects No. 20-52762.97 and 20-59237.99).     

Harmonic Functions,Entropy, and a Characterization of the Hyperbolic Space

Let (M ⁿ ,g) be a compact Riemannian manifold with Ric ≥−(n−1). It is well known that the bottom of spectrum λ ₀ of its universal covering satisfies λ ₀≤(n−1)²/4. We prove that equality holds iff M is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy. The author was partially supported by NSF Grant 0505645.