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1.
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 ≤ ks ? 1, where sp. 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.  相似文献   

2.
We consider Hill's equation y″+(λq)y=0 where qL1[0,π]. We show that if ln—the length of the n-th instability interval—is of order O(n−(k+2)) then the real Fourier coefficients ank,bnk 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,….  相似文献   

3.
4.
Spectral property of the Bernoulli convolutions   总被引:1,自引:0,他引:1  
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.  相似文献   

5.
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(kn) 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(Fqn)→1 as q→∞ over the prime powers.  相似文献   

6.
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.  相似文献   

7.
Let p∈(1,N), ΩRN a bounded W1,p-extension domain and let μ be an upper d-Ahlfors measure on ∂Ω with d∈(Np,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.  相似文献   

8.
In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+up+εf(x), −Δu=μv+vq+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in RN (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.  相似文献   

9.
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 fY 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.  相似文献   

10.
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 ∈ Lp (1<p<∞),the sequence S nf converges weakly in Lp. (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 everyfL 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 forgL p (1<p<∞) Tng is weakly almost convergent. If forf ∈ Lp we have T n+1 f?T n f → 0weakly, then T nf converges weakly in Lp (1<p<∞).  相似文献   

11.
We consider the following boundary value problem: −Δny = F(k,y, Δy,…,Δn−1y), k ϵ Z[n − 1, N], Δiy(0) = 0, 0 ≤ in − 2, Δpy(N + n - p) = 0, where n ≥ 2 and p is a fixed integer satisfying 0 ≤ pn − 1. Using a fixed-point theorem for operators on a cone, we shall yield the existence of at least three positive solutions.  相似文献   

12.
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,∞).  相似文献   

13.
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 qk associated with a distribution dα on [−1, 1]. It is shown that if qn+1/qn max(qn+1(1)/qn(1), −qn+1(−1)/qn(−1)), then fH[f] fn + 1 · qn+1/qn + 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 qn+1/qn is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.  相似文献   

14.
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.  相似文献   

15.
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 ).  相似文献   

16.
For an integer k 1 and a geometric mesh (qi)−∞ with q ε (0, ∞), let Mi,k(x): = k[qi + k](· − x)+k − 1, Ni,k(x): = (qi + kqiMi,k(x)/k, and let Ak(q) be the Gram matrix (∝Mi,kNj,k)i,jεz. It is known that Ak(q)−1 is bounded independently of q. In this paper it is shown that Ak(q)−1 is strictly decreasing for q in [1, ∞). In particular, the sharp upper bound and lower bound for Ak (q)−1 are obtained: for all q ε (0, ∞).  相似文献   

17.
This article discusses linear differential boundary systems, which include nth-order differential boundary relations as a special case, in Lnp[0,1] × Lnp[0,1], 1 ? p < ∞. The adjoint relation in Lnq[0,1] × Lnq[0,1], 1p + 1q = 1, is derived. Green's formula is also found. Self-adjoint relations are found in Ln2[0,1] × Ln2[0,1], and their connection with Coddington's extensions of symmetric operators on subspaces of Lnp[0,1] × Ln2[0,1] is established.  相似文献   

18.
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?2k-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 = 2r-1 (q2?1), k = q+q1+?+qr, where qr is aprime power, qi = q2i+1 and q=q21.  相似文献   

19.
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.  相似文献   

20.
Let G be a simple graph with least eigenvalue λ and let S be a set of vertices in G which induce a subgraph with mean degree k. We use a quadratic programming technique in conjunction with the main angles of G to establish an upper bound of the form |S|?inf{(k+t)qG(t):t>-λ} where qG is a rational function determined by the spectra of G and its complement. In the case k=0 we obtain improved bounds for the independence number of various benchmark graphs.  相似文献   

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