, where \(\mathcal{L}_2 (D)\) is a linear differential operator of the second order whose characteristic polynomial has only real roots, we construct a noninterpolating linear positive method of exponential spline approximation possessing extremal and smoothing properties and locally inheriting the monotonicity of the initial data (the values of a function
We consider the problem of optimal boundary control of string vibrations by a displacement at one endpoint with the other endpoint being fixed; the problem is considered in the space
, p ≥ 1. The control brings the vibration process from the quiescent state to an arbitrarily prescribed state in a time that is a multiple of the double string length.
Let a function f : \(\Pi ^{ * ^m } \) → ? be Lebesgue integrable on \(\Pi ^{ * ^m } \) and Riemann-Stieltjes integrable with respect to a function G : \(\Pi ^{ * ^m } \) → ? on \(\Pi ^{ * ^m } \). Then the Parseval equality
χk(x) dG(x) are Fourier coefficients of the function f and Fourier-Stieltjes coefficients of the function G with respect to the Haar system, respectively; the integrals in the equality and in the definition of the coefficients of the function G are the Riemann-Stieltjes integrals; the series in the right-hand side of the equality converges in the sense of rectangular partial sums; and the overline indicates the complex conjugation. If f : Πm → ? is a complex-valued Lebesgue integrable function, G is a complex-valued function of bounded variation on Πm,
holds for almost all x ∈ \(\Pi ^{ * ^m } \) in the sense of any summation method with respect to which the Fourier series of Lebesgue integrable functions are summable to these functions almost everywhere (the integral here is interpreted in the sense of Lebesgue-Stieltjes).
over the Grassmannian manifolds G(n, p) as noncompact symmetric affine spaces together with their Cartan model in the group of the Euclidean motions SE(n).
, and the aαjk and pjk are constants, x ∈ Ω, and Ω is a bounded open set. The boundary conditions correspond to the Dirichlet problem. Let N±(μ) be the positive and negative spectral counting functions. We establish the asymptotics N±(μ) ～ (mesmΩ)φ±(μ) as μ → +0. The functions φ±(μ) are independent of Ω. In the nonelliptic case, these asymptotics are in general different from the classical (Weyl) asymptotics.
, this coincides with the notion of a Jordan ideal.) We study conditions under which Jordan submodules are subbimodules. General criteria are given in the purely algebraic situation as well as for the case of Banach bimodules over Banach algebras. We also consider symmetrically normed Jordan submodules over C*-algebras. It turns out that there exist C*-algebras in which not all Jordan ideals are ideals.
This addendum to [1] completely characterizes the boundedness and compactness of a recently introduced integral type operator from the space of bounded holomorphic functions H∞(\(\mathbb{D}^n \)) on the unit polydisk \(\mathbb{D}^n \) to the mixed norm space
with p, q ∈ [1,∞) and α = (α1, ..., αn) such that αj > ?1 for every j = 1, ..., n. We show that the operator is bounded if and only if it is compact and if and only if g ∈
of step two to another such group satisfies a Beltrami-type system of partial differential equations which is usually not elliptic but subelliptic when the group
is strongly 2-pseudoconcave. We derive an integral representation formula for CR-mappings from a strongly 2-pseudoconcave nilpotent Lie group of step two to another such group and establish the Hölder continuity of ε-quasi-CR-mappings and the stability of CR-mappings between such groups.
A theorem of Baker says that a function F entire on ?d such that F(?d) ? ? and increasing slower (in a precise sense) than \(2^{z_1 + \cdots + z_d } \) is necessarily a polynomial. This is a multivariate generalisation of the celebrated theorem of Pólya (case d = 1). Using the theory of analytic functionals with non-compact carrier, Yoshino proved a general theorem dealing with the growth of arithmetic analytic functions, which implies that the conclusion of Baker’s theorem holds if F is only assumed to be holomorphic on the domain
The case d = 1 was also treated in a different way by Gel’fond and Pólya by means of the characteristic function of Carlson-Nörlund. This function was introduced to bound in a nearly optimal way the growth of holomorphic functions of one variable that can be expanded in a Newton interpolation series in the half-plane
that can be expanded in multiple Newton series. These considerations enable us to improve Gel’fond-Pólya’s and Yoshino’s theorems, in particular, to remove or to weaken certain of their technical conditions.
It is well-known that Morgan-Voyce polynomials Bn(x) and bn(x) satisfy both a Sturm-Liouville equation of second order and a three-term recurrence equation ([SWAMY, M.: Further properties of Morgan-Voyce polynomials, Fibonacci Quart. 6 (1968), 167–175]). We study Diophantine equations involving these polynomials as well as other modified classical orthogonal polynomials with this property. Let A, B, C ∈ ? and {pk(x)} be a sequence of polynomials defined by
In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s＋t（Lp0（Ω））→Bq1s（Lp1（Ω））, t〉max{d（1/p0-1/p1）, 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s＋t（Lp0（Ω）） is a Besov space defined on the bounded Lipschitz domain Ω ？ Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions. 相似文献
Let Open image in new window be the class of radial real-valued functions of m variables with support in the unit ball \(\mathbb{B}\) of the space ?m that are continuous on the whole space ?m and have a nonnegative Fourier transform. For m ≥ 3, it is proved that a function f from the class Open image in new window can be presented as the sum \(\sum {f_k \tilde *f_k } \) of at most countably many self-convolutions of real-valued functions fk with support in the ball of radius 1/2. This result generalizes the theorem proved by Rudin under the assumptions that the function f is infinitely differentiable and the functions fk are complex-valued. 相似文献
Results on the solvability of boundary integral equations on a plane contour with a peak obtained in collaboration with V.G. Maz’ya are developed. Earlier, it was proved that, on a contour Γ with an outward peak, the operator of the boundary equation of the Dirichlet boundary value problem maps the space ?p, β + 1 (Γ) continuously onto \(\mathcal{N}_{p,\beta } (\Gamma )\). The norm of a function in ?p, β (Γ) is defined as
, where q± are the intersection points of Γ with the circle {z: |z| = |q|} and δ > 0 is a fixed small number. On a contour with an inward peak, the operator of the boundary equation of the Dirichlet problem continuously maps ?p, β + 1 (Γ) onto ?p, β(Γ), where ?p, β(Γ) is the direct sum of \(\mathcal{N}_{p,\beta }^ + (\Gamma )\) (Γ) and the space
(Γ) of functions on Γ of the form p(z) = Σk = 0mt(k)Rezk with the parameter m = [μ ? β ? p?1]. The operator I ? 2W of the boundary integral equation of plane elasticity theory, where W is the elastic double-layer potential, is considered. The main result is that the operator I ? 2W continuously maps the space ?p, β + 1 × ?p, β + 1(Γ) to the space \(\mathcal{N}_{p,\beta }^ - \times \mathcal{N}_{p,\beta }^ - (\Gamma )\).
On a contour with an inward peak, the obtained representation of the operator I ? 2W and theorems on the boundedness of auxiliary integral operators imply that the images of vector-valued functions from ?p, β + 1 × ?p, β + 1(Γ) have components representable as sums of functions from the spaces \(\mathcal{N}_{p,\beta }^ - (\Gamma )\)(Γ) and ?p, β(Γ). 相似文献
We define an R?-generalized solution of the first boundary value problem for a second-order elliptic equation with degeneration of the input data on the entire boundary of the two-dimensional domain and prove the existence and uniqueness of the solution in the weighted set Open image in new window. 相似文献
