On the Riesz basis property of the eigen- and associated functions of periodic and antiperiodic Sturm-Liouville problems

The paper deals with the Sturm-Liouville operator $$ Ly = - y'' + q(x)y, x \in [0,1], $$ generated in the space L ₂ = L ₂[0, 1] by periodic or antiperiodic boundary conditions. Several theorems on the Riesz basis property of the root functions of the operator L are proved. One of the main results is the following. Let q belong to the Sobolev spaceW ₁ ^p [0, 1] for some integer p ≥ 0 and satisfy the conditions q ^(k)(0) = q ^(k)(1) = 0 for 0 ≤ k ≤ s ? 1, where s ≤ p. Let the functions Q and S be defined by the equalities $$ Q(x) = \int_0^x {q(t)dt, S(x) = Q^2 (x)} $$ and let q _n , Q _n , and S _n be the Fourier coefficients of q, Q, and S with respect to the trigonometric system $ \{ e^{2\pi inx} \} _{ - \infty }^\infty $ . Assume that the sequence q _2n ? S _2n + 2Q ₀ Q _2n decreases not faster than the powers n ^?s?2. Then the system of eigenfunctions and associated functions of the operator L generated by periodic boundary conditions forms a Riesz basis in the space L ₂[0, 1] (provided that the eigenfunctions are normalized) if and only if the condition $$ q_{2n} - s_{2n} + 2Q_0 Q_{2n} \asymp q_{ - 2n} - s_{2n} + 2Q_0 Q_{ - 2n} , n > 1, $$ holds.     

Some inverse results for Hill's equation

We consider Hill's equation y″+(λ−q)y=0 where q∈L¹[0,π]. We show that if l_n—the length of the n-th instability interval—is of order O(n^−(k+2)) then the real Fourier coefficients a_n^k,b_n^k of q^(k)—k-th derivative of q—are of order O(n⁻²), which implies that q^(k) is absolutely continuous almost everywhere for k=0,1,2,….     

Spectral property of the Bernoulli convolutions

For 0<ρ<1, let μρ be the Bernoulli convolution associated with ρ. Jorgensen and Pedersen [P. Jorgensen, S. Pedersen, Dense analytic subspaces in fractal L²-spaces, J. Anal. Math. 75 (1998) 185-228] proved that if ρ=1/q where q is an even integer, then L²(μρ) has an exponential orthonormal basis. We show that for any 0<ρ<1, L²(μρ) contains an infinite orthonormal set of exponential functions if and only if ρ is the nth root of a fraction p/q where p is an odd integer and q is an even integer.     

On the Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field

We prove that for every field k and every positive integer n there exists an absolutely simple n-dimensional abelian variety over k. We also prove an asymptotic result for finite fields: For every finite field k and positive integer n, we let S(k, n) denote the fraction of the isogeny classes of n-dimensional abelian varieties over k that consist of absolutely simple ordinary abelian varieties. Then for every n we have S(F_q, n)→1 as q→∞ over the prime powers.     

Non-orientable fundamental surfaces in lens spaces

We give a concrete example of an infinite sequence of (pn,qn)-lens spaces L(pn,qn) with natural triangulations T(pn,qn) with pn tetrahedra such that L(pn,qn) contains a certain non-orientable closed surface which is fundamental with respect to T(pn,qn) and of minimal crosscap number among all closed non-orientable surfaces in L(pn,qn) and has n−2 parallel sheets of normal disks of a quadrilateral type disjoint from the pair of core circles of L(pn,qn). Actually, we can set p₀=0, q₀=1, pk+1=3pk+2qk and qk+1=pk+qk.     

A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions

Let p∈(1,N), Ω⊂RN a bounded W1,p-extension domain and let μ be an upper d-Ahlfors measure on ∂Ω with d∈(N−p,N). We show in the first part that for every p∈[2N/(N+2),N)∩(1,N), a realization of the p-Laplace operator with (nonlinear) generalized nonlocal Robin boundary conditions generates a (nonlinear) strongly continuous submarkovian semigroup on L²(Ω), and hence, the associated first order Cauchy problem is well posed on Lq(Ω) for every q∈[1,∞). In the second part we investigate existence, uniqueness and regularity of weak solutions to the associated quasi-linear elliptic equation. More precisely, global a priori estimates of weak solutions are obtained.     

Enumerating pairs of permutations with the same up-down form

For each sequence q = {q_i} = ± 1, i = 1, …, n−1 let N_q = the number of permutations σ of 1, 2, …, n with up-down sequence sgn(σ_i+1 − σ_i) = q_i, i = 1,…, n−1. Clearly Σ_q (N_q/n!) =1 but what is the probability p_n = Σ_q (N_q/n!)² that two random permutations have the same up-down sequence? We show that p_n = (Kⁿ⁻¹1,1) where 1 = 1(x, y) ≡ 1 and Kⁿ⁻¹ is the iterated integral operator with Kφ(x, y) = ∫₀¹∫₀¹ K(x, y; x′, y′)φ(x′, y′) dx′ dy′ on L²[0, 1] × [0, 1] where K(x, y; x′, y′) is 1 if (x− x′)(y − y′) > 0 otherwise, and (f, g) = ∫₀¹∫₀¹fg. The eigenexpression of K yeilds p_n ∼ cαⁿ as n → ∞, where c ≈ 1.6, α ≈ 0.55.We also give a recursion formula for a polynomial whose coefficients are the frequencies of all the possible forms.     

Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+u^p+εf(x), −Δu=μv+v^q+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in R^N (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ₁) and ε,δ∈(0,δ₀), where λ₁ is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ₀ is a positive number.     

Nikol’skii’s inequality for different metrics and properties of the sequence of norms of the Fourier sums of a function in the Lorentz space

Let (X, Y) be a pair of normed spaces such that X ? Y ? L ₁[0, 1] ⁿ and {e _k } _k be an expanding sequence of finite sets in ? ⁿ with respect to a scalar or vector parameter k, k ∈ ? or k ∈ ? ⁿ . The properties of the sequence of norms $\{ \left\| {S_{e_k } (f)} \right\|x\} _k $ of the Fourier sums of a fixed function f ∈ Y are studied. As the spaces X and Y, the Lebesgue spaces L _p [0, 1], the Lorentz spaces L _p,q [0, 1], L _p,q [0, 1] ⁿ , and the anisotropic Lorentz spaces L _p,q*[0, 1] ⁿ are considered. In the one-dimensional case, the sequence {e _k } _k consists of segments, and in the multidimensional case, it is a sequence of hyperbolic crosses or parallelepipeds in ? ⁿ . For trigonometric polynomials with the spectrum given by step hyperbolic crosses and parallelepipeds, various types of inequalities for different metrics in the Lorentz spaces L _p,q [0, 1] ⁿ and L _p,q*[0, 1] ⁿ are obtained.     

A limit theorem for order preserving nonexpansive operators inL 1

We prove the following theorem:Let T be an order preserving nonexpansive operator on L ₁ (μ) (or L ₁ ⁺ ) of a σ-finite measure, which also decreases theL _∞-norm, and let S=tI+(1?t)T for 0<t<1. Then for everyf ∈ L_p (1<p<∞),the sequence S ⁿf converges weakly in L_p. (The assumptions do not imply thatT is nonexpansive inL _p for anyp>1, even ifμ is finite.) For the proof we show that ∥S ⁿ⁺¹ f?S ⁿf∥ _p → 0 for everyf ∈L _p, 1<p<∞, and apply toS the following theorem:Let T be order preserving and nonexpansive in L ₁ ⁺ , and assume that T decreases theL _∞-norm. Then forg ∈L _p (1<p<∞) Tⁿg is weakly almost convergent. If forf ∈ L_p we have T ⁿ⁺¹ f?T ⁿ f → 0weakly, then T ⁿf converges weakly in L_p (1<p<∞).     

Existence of triple solutions of discrete (n,p) boundary value problems

We consider the following boundary value problem: −Δⁿy = F(k,y, Δy,…,Δⁿ⁻¹y), k ϵ Z[n − 1, N], Δⁱy(0) = 0, 0 ≤ i ≤ n − 2, Δ^py(N + n - p) = 0, where n ≥ 2 and p is a fixed integer satisfying 0 ≤ p ≤ n − 1. Using a fixed-point theorem for operators on a cone, we shall yield the existence of at least three positive solutions.     

On the completeness of the system of root vectors of the Sturm-Liouville operator with general boundary conditions

We study general boundary value problems with nondegenerate characteristic determinant Δ(λ) for the Sturm-Liouville equation on the interval [0, 1]. Necessary and sufficient conditions for the completeness of root vectors are obtained in terms of the potential. In particular, it is shown that if Δ(λ) ≠ const, q(·) ∈ C ^k [0, 1] for some k ? 0, and q ^(k)(0) ≠ (?1) ^k q ^(k)(1), then the system of root vectors is complete and minimal in L ^p [0, 1] for p ∈ [1,∞).     

Error bounds for least squares approximation by polynomials

Let f ε Cⁿ⁺¹[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials q_k associated with a distribution dα on [−1, 1]. It is shown that if q_n+1/q_n max(q_n+1(1)/q_n(1), −q_n+1(−1)/q_n(−1)), then f − H[f] f^{n + 1} · q_n+1/q_{n + 1}^{(n + 1)}, where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)^α (1 + t)^β dt, α, β > −1, the condition on q_n+1/q_n is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.     

The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

In this paper, we study the Fu?ik spectrum of the problem: (^*) ?+(λ₊+q₊(t))x₊+(λ₋+q₋(t))x₋=0 with the 2π-periodic boundary condition, where q_±(t) are 2π-periodic. After introducing a rotation number function ρ(λ₊, λ₋) for (^*), we prove using the Hamiltonian structure and the positive homogeneity of (^*) that for any positive integer n, the two boundary curves of the domain ρ⁻¹(n/2) in the (λ₊, λ₋)-plane are Fu?ik curves of (^*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fu?ik spectrum with the Dirichlet condition. In particular, it remains open if the Fu?ik spectrum of (^*) is composed of only these curves.     

On Abel—Poisson type and Riesz means

В работе исследуются ядра методов суммиро вания типа Абеля—Пуассона и Рис са, применяемых к кратны м интегралам Фурье. Вы ясняются условия на параметры, определяющие эти методы, при которы х их ядра неотрицател ьны. Полученные результа ты можно сформулировать в тер минах положительной определенности неко торых функций. Наприм ер, функция exp(? ¦x¦^α) при 0<а ≦2 является, а при α>2 не является положитель но определенной в евклидовом простра нствеE _N размерностиN (N=1, 2, ...). Далее, еслиt ₊=max (t, 0), то при любом натураль номN на интервале 0<λ<2 существует неубываю щая непрерывная функцияk _N (λ) такая, что функция (1 ? ¦х¦^λ) ₊ ^k приk≧k _N (λ) является, а приk<k _N (λ) не является положительно опреде ленной в пространств еE _N . При этом $$k_N (1) = \frac{{N + 1}}{2}, k_N (2 - 0) = + \infty , k_N (\lambda ) \geqq \lambda + \frac{{N - 1}}{2}.$$ Если же λ≧2, то функция (1?¦x¦^λ) ₊ ^k ни при каком значении параметраk не является положите льно определенной в прост ранствеE _N ,N=1, 2, .... Кроме того, исследует ся порядок приближен ия функцийN переменных класса Н икольскогоk _P ^α , 1≦р<∞, 0<а<2, операторам и типа Абеля—Пуассон а в метрикеL _p (E _N ).     

L∞-upper bound of L2-projections onto splines at a geometric mesh

For an integer k 1 and a geometric mesh (qⁱ)_−∞^∞ with q ε (0, ∞), let M_i,k(x): = k[q^{i + k}](· − x)₊^{k − 1}, N_i,k(x): = (q^{i + k} − qⁱM_i,k(x)/k, and let A_k(q) be the Gram matrix (∝M_i,kN_j,k)_i,jεz. It is known that A_k(q)⁻¹_∞ is bounded independently of q. In this paper it is shown that A_k(q)⁻¹_∞ is strictly decreasing for q in [1, ∞). In particular, the sharp upper bound and lower bound for A_k (q)⁻¹ are obtained: for all q ε (0, ∞).     

nth order Stieltjes differential boundary operators and Stieltjes differential boundary systems

This article discusses linear differential boundary systems, which include nth-order differential boundary relations as a special case, in $\text{L}$_n^p[0,1] × $\text{L}$_n^p[0,1], 1 ? p < ∞. The adjoint relation in $\text{L}$_n^q[0,1] × $\text{L}$_n^q[0,1], $\text{1}\text{p}\text{+}\text{1}\text{q}\text{= 1}$, is derived. Green's formula is also found. Self-adjoint relations are found in $\text{L}$_n²[0,1] × $\text{L}$_n²[0,1], and their connection with Coddington's extensions of symmetric operators on subspaces of $\text{L}$_n^p[0,1] × $\text{L}$_n²[0,1] is established.     

Incidence graphs and subdesigns of semisymmetric designs

A semisymmetric design is a connected incidence structure satisfying; two points (blocks) are on 0 or λ blocks (points). Every block (point) is incident with k points (blocks). Properties of the incidence graph of these structures are investigated, leading to bounds on its diameter (d?k if λ = 2, d?[2k/(λ + 1)]+ 1 if λ > 2), and the number of points of these structures (υ?2^k-1 if λ = 2, υ?k2^{[2k/(λ + 1)]} if λ > 2). Bounds are also found for semisymmetric designs containing a subdesign. We give characterizations of semisymmetric designs with λ = 2 (semibiplanes) which contain a subdesign and achieve the bounds. This leads to a construction for a semibiplane with parameters υ = 2^r-1 (q²?1), k = q+q₁+?+q_r, where q_r is aprime power, q_i = q²_i+1 and q=q²₁.     

Strong Sperner property of the subgroup lattice of an Abelianp-group

Letn andk be arbitrary positive integers,p a prime number and L_(k ⁿ)(p) the subgroup lattice of the Abelianp-group (Z/p ^k ) ⁿ . Then there is a positive integerN(n,k) such that whenp N(n,k),L _(k ^N )(p) has the strong Sperner property.