共查询到20条相似文献,搜索用时 62 毫秒
1.
S. S. Platonov 《Siberian Mathematical Journal》2009,50(1):123-140
Approximation problems for functions on the half-line [0,+∞) in a weighted L p -metric are studied with the use of Bessel generalized translation. A direct theorem of Jackson type is proven for the modulus of smoothness of arbitrary order which is constructed on the basis of Bessel generalized translation. Equivalence is stated between the modulus of smoothness and the K-functional constructed by the Sobolev space corresponding to the Bessel differential operator. A particular class of entire functions of exponential type is used for approximation. The problems under consideration are studied mostly by means of Fourier-Bessel harmonic analysis. 相似文献
2.
E. Kh. Gimadi A. V. Kel’manov A. V. Pyatkin M. Yu. Khachai 《Proceedings of the Steklov Institute of Mathematics》2015,288(1):88-98
A generalized modulus of continuity is defined in the space L 2(? d ) with Dunkl weight by means of an arbitrary zero-sum sequence of complex numbers. A sharp generalized Jackson inequality is proved for this modulus and for the best approximations by entire functions of exponential spherical type. This inequality was earlier proved by S.N. Vasil’ev in the weightless case. 相似文献
3.
In the spaces L p on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the K-functional. We prove a direct and an inverse theorem of Jackson-Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the K-functional. 相似文献
4.
K. V. Runovski 《Mathematical Notes》2016,99(3-4):564-575
The quality of approximation by Fourier means generated by an arbitrary generator with compact support in the spaces Lp, 1 ≤ p ≤ +∞, of 2π-periodic pth integrable functions and in the space C of continuous 2π-periodic functions in terms of the generalized modulus of smoothness constructed froma 2π-periodic generator is studied. Natural sufficient conditions on the generator of the approximation method and values of smoothness ensuring the equivalence of the corresponding approximation error and modulus are obtained. As applications, Fourier means generated by classical kernels as well as the classical moduli of smoothness are considered. 相似文献
5.
A. A. Shkalikov 《Differential Equations》2009,45(4):580-590
We consider the operator function L(α, θ) = A(α) ? θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α 0 ∈ ? and ranging in the space of bounded operators in a Hilbert space ?. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α 0 and R is a bounded positive operator; moreover, the point α 0 is a normal eigenvalue of the operator function L(α, θ 0) for some θ 0 ∈ ?, and the number θ 0 is a normal eigenvalue of the operator function L(α 0 θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α 0 and θ 0, respectively, and on the representation of the corresponding eigenfunctions by series. 相似文献
6.
We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λk(f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L2. On the basis of this, for differentiable functions in the classes L2r, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ωk. For classes of functions defined by their smoothness characteristics Λk(f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths. 相似文献
7.
Given an indexing set I and a finite field Kα for each α ∈ I, let ? = {L2(Kα) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L2(P) (respectively SL2(P)) for a suitable locally finite field P. 相似文献
8.
S. I. Novikov 《Proceedings of the Steklov Institute of Mathematics》2018,300(1):136-144
We find exact values for the uniform Lebesgue constants of interpolating L-splines that are bounded on the real axis, have equidistant knots, and correspond to the linear thirdorder differential operator L3(D) = D(D2 + α2) with constant real coefficients, where α > 0. We compare the obtained result with the Lebesgue constants of other L-splines. 相似文献
9.
In this paper, we study the boundedness of the fractional integral operator I α on Carnot group G in the generalized Morrey spaces M p, φ (G). We shall give a characterization for the strong and weak type boundedness of I α on the generalized Morrey spaces, respectively. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two Sobolev–Stein embedding theorems on generalized Morrey spaces in the Carnot group setting. 相似文献
10.
The sharp inequality of different metrics (Nikol’skii’s inequality) for algebraic polynomials in the interval [?1, 1] between the uniform norm and the norm of the space L q (α,β) , 1 ≤ q < ∞, with Jacobi weight ?(α,β)(x) = (1 ? x)α(1 + x)β α ≥ β > ?1, is investigated. The study uses the generalized translation operator generated by the Jacobi weight. A set of functions is described for which the norm of this operator in the space L q (α,β) , 1 ≤ q < ∞, \(\alpha > \beta \geqslant - \frac{1}{2}\), is attained. 相似文献
11.
We study the Nikol’skii inequality for algebraic polynomials on the interval [?1, 1] between the uniform norm and the norm of the space L q (α,β) , 1 ≤ q < ∞, with the Jacobi weight ?(α,β)(x) = (1 ? x) α (1 + x) β , α ≥ β > ?1. We prove that, in the case α > β ≥ ?1/2, the polynomial with unit leading coefficient that deviates least from zero in the space L q (α+1,,β) with the Jacobi weight ? (α+1,β)(x) = (1?x) α+1(1+x) β is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation operator associated with the Jacobi weight. We describe the set of all functions at which the norm of this operator in the space L q (α,β) for 1 ≤ q < ∞ and α > β ≥ ?1/2 is attained. 相似文献
12.
A. A. Koshelev 《Proceedings of the Steklov Institute of Mathematics》2012,277(1):136-144
Close two-sided estimates are obtained for the best approximation in the space L p (? m ), m = 2 and 3, 1 ≤ p ≤ ∞, of the Laplace operator by linear bounded operators in the class of functions for which the second power of the Laplace operator belongs to the space L p (? m ). We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of values of the Laplace operator on functions from this class given with an error. We present an operator whose deviation from the Laplace operator is close to the best. 相似文献
13.
V. B. Korotkov 《Siberian Mathematical Journal》2009,50(1):96-99
We consider the class of the continuous L 2,1 linear operators in L 2 that are sums of the operators of multiplication by bounded measurable functions and the operators sending the unit ball of L 2 into a compact subset of L 1. We prove that a functional equation with an operator from L 2,1 is equivalent to an integral equation with kernel satisfying the Carleman condition. We also prove that if T ∈ L 2,1 and VTV ?1 ∈ L 2,1 for all unitary operators V in L 2 then T = α1 + C, where α is a scalar, 1 is the identity operator in L 2, and C is a compact operator in L 2. 相似文献
14.
In the L p -spaces, we study the complex powers of the operator where δ is the Laplace operator. The complex powers G λ ?α/2 , Reα > 0, are realized as potential type operators B λ α with a nonstandard metric. We obtain L p → L p + L s -estimates for the operator B λ α . By using the method of approximate inverse operators, we construct the inversion of the potentials B λ α φ with L p -densities and describe the range B λ α (L p ) in terms of the inversion constructions.
相似文献
$G_\lambda = m^2 I + \Delta - i\lambda \frac{{\partial ^2 }}{{\partial x_1^2 }},0 < \lambda < 1,m > 0,$
15.
V. I. Ivanov 《Mathematical Notes》2015,98(5-6):742-751
In the space L p , 1 ≤ p < 2, on the half-line with power weight, Jackson’s inequality between the value of the best approximation of a function by even entire functions of exponential type and its modulus of continuity defined by means of a generalized shift operator is well known. The question of the sharpness of the inequality remained open. For the constant in Jackson’s inequality, we obtain a lower bound, which proves its sharpness. 相似文献
16.
In this paper we consider the stochastic Dirichlet problem \(L\lozenge u=h+\nabla f\) in the framework of white noise analysis combined with Sobolev space and Colombeau algebra methods. The operator L is assumed to be strictly elliptic in divergence form \(L\lozenge u=\nabla(A\lozenge\nabla u+b\lozenge u)+c\lozenge\nabla u+d\lozenge u\). Its coefficients: the elements of the matrix A and of the vectors b, c and d are assumed to be generalized random processes, and the product of two generalized processes is interpreted as the Wick product. Generalized random processes are considered as linear bounded mappings from the Sobolev space \(W_0^{1,2}\) into the Kondratiev space (S)???1. In this paper we prove existence and uniqueness of the problem of this form in the case when the operator L generates a coercive bilinear form, and then extend this result to the general case. We also consider the case when the coefficients of L, the input data and the boundary condition are Colombeau-type generalized stochastic processes. 相似文献
17.
V. T. Shevaldin 《Proceedings of the Steklov Institute of Mathematics》2018,300(1):165-171
We construct an analog of two-scale relations for basis trigonometric splines with uniform knots corresponding to a linear differential operator of order 2r + 1 with constant coefficients L2r+1(D) = D(D2 + α12 )(D2 + α22 )... (D2 + α r 2 ), where α1, α2,..., α r are arbitrary positive numbers. The properties of nested subspaces of trigonometric splines are analyzed. 相似文献
18.
A. M. Savchuk 《Differential Equations》2016,52(4):431-446
We study the spectral properties of the Dirac operator LP,U generated in the space (L2[0, π])2 by the differential expression By′ + P(x)y and by Birkhoff regular boundary conditions U, where y = (y1, y2)t, \(B = \left( {\begin{array}{*{20}{c}} { - i}&0 \\ 0&i \end{array}} \right)\), and the entries of the matrix P are complexvalued Lebesgue measurable functions on [0, π]. We also study the asymptotic properties of the eigenvalues {λn}n∈Z of the operator LP,U as n → ∞ depending on the “smoothness” degree of the potential P; i.e., we consider the scale of Besov spaces B1,∞θ, θ ∈ (0, 1). In the case of strongly regular boundary conditions, we study the asymptotic behavior of the system of normalized eigenfunctions of the operator LP,U, and in the case of regular but not strongly regular boundary conditions, we find the asymptotics of two-dimensional spectral projections. 相似文献
19.
Let E and F be Banach spaces. A linear operator from E to F is said to be strictly singular if, for any subspace Q ? E, the restriction of A to Q is not an isomorphism. A compactness criterion for any strictly singular operator from Lp to Lq is found. There exists a strictly singular but not superstrictly singular operator on Lp, provided that p ≠ 2. 相似文献
20.
V. A. Aguilar-Arteaga M. Cruz-López S. Estala-Arias 《P-Adic Numbers, Ultrametric Analysis, and Applications》2017,9(3):165-182
For each finite set S of prime numbers there exists a unique completion ? S of ?, which is a second countable, locally compact and totally disconnected topological ring. This topological ring has a natural ultrametric that allows to define a pseudodifferential operator D α and to study an abstract heat equation on the Hilbert space L 2(? S ). The fundamental solution of this equation is a normal transition function of a Markov process on ? S . The techniques developed provides a general framework for these kind of problems on different ultrametric groups. 相似文献