共查询到20条相似文献,搜索用时 702 毫秒
1.
Let γn denote the length of the nth zone of instability of the Hill operator Ly=−y″−[4tαcos2x+2α2cos4x]y, where α≠0, and either both α, t are real, or both are pure imaginary numbers. For even n we prove: if t, n are fixed, then for α→0
2.
Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator LJf(x) = ∫(f(x) ? f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (?LJ, D) acts in L2(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel pJ (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (?LJ, D) extends in L2(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel pJ(t, x, y), and the function pJ(t, x, y) is uniformly comparable in t, x, y to the function pJ(t, x, y), the heat kernel related to the operator (?LJ,D). 相似文献
3.
A. M. Savchuk 《Proceedings of the Steklov Institute of Mathematics》2008,261(1):237-242
In the space L 2[0, π], the Sturm-Liouville operator L D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L 2[0, π], i.e., q ∈ W 2 ?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically. 相似文献
4.
Let L be a lattice of finite length, ξ = (x 1,…, x k )∈L k , and y ∈ L. The remoteness r(y, ξ) of y from ξ is d(y, x 1)+?+d(y, x k ), where d stands for the minimum path length distance in the covering graph of L. Assume, in addition, that L is a graded planar lattice. We prove that whenever r(y, ξ) ≤ r(z, ξ) for all z ∈ L, then y ≤ x 1∨?∨x k . In other words, L satisfies the so-called c 1 -median property. 相似文献
5.
I. V. Sadovnichaya 《Proceedings of the Steklov Institute of Mathematics》2008,261(1):243-252
We consider the Sturm-Liouville operator L = ?d 2/dx 2 + q(x) with the Dirichlet boundary conditions in the space L 2[0, π] under the assumption that the potential q(x) belongs to W 2 ?1 [0, π]. We study the problem of uniform equiconvergence on the interval [0, π] of the expansion of a function f(x) in the system of eigenfunctions and associated functions of the operator L and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function f(x) of class L 1. We also consider the case of potentials belonging to the scale of Sobolev spaces W 2 ?θ [0, π] with ½ < θ ≤ 1. We show that if the antiderivative u(x) of the potential belongs to some space W 2 θ [0, π] with 0 < θ < 1/2, then, for any function in the space L 2[0, π], the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing u(x). We also give an explicit estimate for the rate of equiconvergence. 相似文献
6.
I. V. Sadovnichaya 《Differential Equations》2008,44(5):675-684
We study the Sturm-Liouville operator L = ?d 2/dx 2 + q(x) in the space L 2[0, π] with the Dirichlet boundary conditions. We assume that the potential has the form q(x) = u′(x), u ∈ W 2 θ [0, π], 0 < θ < 1/2. We consider the problem on the uniform (on the entire interval [0, π]) equiconvergence of the expansion of a function f(x) in a series in the system of root functions of the operator L with its Fourier expansion in the system of sines. We show that if the antiderivative u(x) of the potential belongs to any of the spaces W 2 θ [0, π], 0 < θ < 1/2, then the equiconvergence rate can be estimated uniformly over the ball u(x) ∈ B R = {v(x) ∈ W 2 θ [0, π] | ∥v∥W 2 θ ≤ R} for any function f(x) ∈ L 2[0, π]. 相似文献
7.
Kil-Woung Jun 《Journal of Mathematical Analysis and Applications》2004,299(1):100-112
The purpose of this paper is to solve the stability problem of Ulam for an approximate mapping of the following generalized Pappus' equation:
n2Q(x+my)+mnQ(x−ny)=(m+n)[nQ(x)+mQ(ny)] 相似文献
8.
I. V. Sadovnichaya 《Differential Equations》2009,45(4):520-525
In the space L 2(?), we consider the self-adjoint extension \(\mathcal{L}\) of the Sturm-Liouville operator ly = ?y″ + q(x)y whose potential q is uniformly locally integrable on ?, i.e., satisfies the condition . We study the problem on the equiconvergence rate of the spectral expansion associated with \(\mathcal{L}\) of a function f ∈ L 1(?) with the Fourier integral on the entire real line. We obtain uniform estimates of the equiconvergence rate under some additional conditions on f or q.
相似文献
$\omega _q (h) = \mathop {\sup }\limits_{x \in \mathbb{R}} \int\limits_x^{x + h} {\left| {q(t)} \right|dt < + \infty ,h > 0.} $
9.
We consider the set S r,n of periodic (with period 1) splines of degree r with deficiency 1 whose nodes are at n equidistant points xi=i / n. For n-tuples y = (y0, ... , yn-1), we take splines s r,n (y, x) from S r,n solving the interpolation problem where t i = x i if r is odd and t i is the middle of the closed interval [x i , x i+1 ] if r is even. For the norms L r,n * of the operator y → s r,n (y, x) treated as an operator from l1 to L1 [0, 1] we establish the estimate with an absolute constant in the remainder. We study the relationship between the norms L r,n * and the norms of similar operators for nonperiodic splines.
相似文献
$$s_{r,n} (y,t_i ) = y_i,$$
$$L_{r,n}^ * = \frac{4}{{\pi ^2 n}}log min(r,n) + O\left( {\frac{1}{n}} \right)$$
10.
A. G. Kostyuchenko 《Differential Equations》2009,45(4):549-557
In the present paper, we compute the leading term of the asymptotics of the angular eigenvalue distribution function of the problem Au = λω(x)u(x) in a bounded domain Ω ? R n , where A is an elliptic differential operator of order 2m with domain D(A) ? W m 2m (Ω). The weight function ω(x) (x ∈ Ω) is indefinite and can also take zero values on a set of positive measure. 相似文献
11.
Ronald Begg 《Journal of Mathematical Analysis and Applications》2006,322(2):1168-1187
A class of nonlocal second-order ordinary differential equations of the form
y″(x)=f(x,y(x),(y○λ)(x),y′(x)) 相似文献
12.
V. G. Kurbatov 《Functional Analysis and Its Applications》2005,39(3):233-235
Let \(\mathbb{S}\) be a cone in ? n . A bounded linear operator T: L p (? n ) → L p (? n ) is said to be causal with respect to \(\mathbb{S}\) if the implication x(s) = 0 (s ε W ? \(\mathbb{S}\)) ? (Tx) (s) = 0 (s ε W ? \(\mathbb{S}\)) is valid for any x ε L p (? n ) and any open subset W\(\subseteq\) ? n . The set of all causal operators is a Banach algebra. We describe the spectrum of the operator in this algebra. Here x ranges in a Banach space \(\mathbb{E}\), the a n are bounded linear operators in \(\mathbb{E}\), and the function g ranges in the set of bounded operators in \(\mathbb{E}\).
相似文献
$(Tx)(t) = \sum\limits_{n = 1}^\infty {a_n x(t - t_n )} + \int {\mathbb{S}g(s)x(t - s)ds,} \quad t \in \mathbb{R}^n ,$
13.
Let ?1<α≤0 and let $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ be the generalizednth Laguerre polynomial,n=1,2,… Letx 1,x 2,…,x n andx*1,x*2,…,x* n?1 denote the roots ofL n (α) (x) andL n (α)′ (x) respectively and putx*0=0. In this paper we prove the following theorem: Ify 0,y 1,…,y n ?1 andy 1 ′ ,…,y n ′ are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n?1 satisfying the conditions $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered} $$ . 相似文献
14.
S. -M. Jung 《Acta Mathematica Hungarica》2009,124(1-2):155-163
We solve the inhomogeneous linear first order differential equations of the form y′(x) ? λy(x) = Σ m=0 ∞ a m (x ? c) m , and prove an approximation property of exponential functions. More precisely, we prove the local Hyers-Ulam stability of linear first order differential equations of the form y′(x) = λy(x) in a special class of analytic functions. 相似文献
15.
In this paper we investigate discrete spectrum of the non-selfadjoint matrix Sturm-Liouville operator L generated in L2(R+,S) by the differential expression
16.
Z. Kh. Rakhmonov F. Z. Rakhmonov 《Proceedings of the Steklov Institute of Mathematics》2017,296(1):211-233
For y ≥ x 4/5 L 8B+151 (where L = log(xq) and B is an absolute constant), a nontrivial estimate is obtained for short cubic exponential sums over primes of the form S 3(α; x, y) = ∑ x?y<n≤x Λ(n)e(αn 3), where α = a/q + θ/q 2, (a, q) = 1, L 32(B+20) < q ≤ y 5 x ?2 L ?32(B+20), |θ| ≤ 1, Λ is the von Mangoldt function, and e(t) = e 2πit. 相似文献
17.
Khachatur A. Khachatryan Emilya A. Khachatryan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2007,42(3):161-175
The paper looks for the solutions of integro-differential equations of the form in the class of functions which are absolutely continuous and of slow growth on ?. It is assumed that A and B are nonnegative parameters, 0 ≤ g ∈ L 1 (?), 0 ≤ k ∈ L 1 (?), ∫? k(x) dx = 1 and 0 ≤ λ(x) ≤ 1 is a measurable function in ?. The equation is solved by a special factorization of the corresponding integro-differential operator in combination with appropriately modified standard methods of the theory of convolution type integral equations.
相似文献
$ - \frac{{d\varphi }}{{dx}} + A\varphi (x) = g(x) + B\int_\mathbb{R} {k(x - t)\lambda (t)\varphi (t)dt, x \in \mathbb{R}} $
18.
V. B. Korotkov 《Siberian Mathematical Journal》2008,49(5):857-859
We construct a selfadjoint Akhiezer integral operator S on L 2(?1, 1) with the spectrum {?1; 0; 1} and the property that every function φ(S) that is not a multiple of S fails to be an Akhiezer integral operator. 相似文献
19.
Let R be a prime ring of char R ≠ 2, let d be a nonzero derivation of R, and let ρ be a nonzero right ideal of R such that [[d(x)x n , d(y)] m , [y, x] s ] t = 0 for all x, y ? ρ, where n ≥ 1, m ≥ 0, s ≥ 0, and t ≥ 1 are fixed integers. If [ρ, ρ]ρ ≠ 0 then d(ρ)ρ = 0. 相似文献
20.
A self-adjoint differential operator \(\mathbb{L}\) of order 2m is considered in L 2[0,∞) with the classic boundary conditions \(y^{(k_1 )} (0) = y^{(k_2 )} (0) = y^{(k_3 )} (0) = \ldots = y^{(k_m )} (0) = 0\), where 0 ≤ k 1 < k 2 < ... < k m ≤ 2m ? 1 and {k s } s=1 m ∪ {2m ? 1 ? k s } s=1 m = {0, 1, 2, ..., 2m ? 1}. The operator \(\mathbb{L}\) is perturbed by the operator of multiplication by a real measurable bounded function q(x) with a compact support: \(\mathbb{P}\) f(x) = q(x)f(x), f ∈ L 2[0,∞). The regularized trace of the operator \(\mathbb{L} + \mathbb{P}\) is calculated. 相似文献