Lebesgue Function Research Articles

Estimate of the Lebesgue Function of Fourier Sums in Terms of Modified Meixner Polynomials

The paper is devoted to the study of the approximation properties of Fourier sums in terms of the modified Meixner polynomials m (x), n = 0,1,..., which generate, for α > -1, an orthonormal system on the grid Ωδ = {0, δ, 2δ,...} with weight $${\rho _N}(x) = {e^{ - x}}\frac{{\Gamma (Nx + \alpha + 1)}}{{\Gamma (Nx + 1)}}{(1 - {e^{ - \delta }})^{\alpha + 1}},\;\;\;\;\text{where}\;\;\delta = \frac{1}{N},\;N \geq 1.$$ The main attention is paid to the derivation of a pointwise estimate for the Lebesgue function λ (x) of Fourier sums in terms of the modified Meixner polynomials for x ∈ [θn/2, ∞) and θn = 4n + 2α + 2.

Some Examples of Non-Measurable Lebesgue Functions

Abstract The structuralism is a powerful toll for ordering and classifying knowledge of fundamental mathematical objects. Such an important structure is the Lebesgue measurable sets or Lebesgue non-measurable sets (such a set exists, according to Vitali construction), as well as Lebesgue measurable functions or Lebesgue non-measurable functions. In the following we show that f : (ℝ, £,) → ℝ Lebesgue measurable ⇎ {x ∈ X | f(x) = c} ∈ £, for any c ∈ ℝ, and we will give some important examples of Lebegue non-measurable functions.

A robust error estimator and a residual-free error indicator for reduced basis methods

The Reduced Basis Method (RBM) is a rigorous model reduction approach for solving parameterizedpartial differential equations. It identifies a low-dimensional subspace for approximation of the parametric solution manifold that is embedded in high-dimensional space. A reduced order model is subsequently constructed in this subspace. RBM relies on residual-based error indicators or a posteriori error bounds to guide construction of the reduced solution subspace, to serve as a stopping criteria, and to certify the resulting surrogate solutions. Unfortunately, it is well-known that the standard algorithm for residual norm computation suffers from premature stagnation at the level of the square root of machine precision.In this paper, we develop two alternatives to the standard offline phase of reduced basis algorithms. First, we design a robust strategy for computation of residual error indicators that allows RBM algorithms to enrich the solution subspace with accuracy beyond root machine precision. Secondly, we propose a new error indicator based on the Lebesgue function in interpolation theory. This error indicator does not require computation of residual norms, and instead only requires the ability to compute the RBM solution. This residual-free indicator is rigorous in that it bounds the error committed by the RBM approximation, but up to an uncomputable multiplicative constant. Because of this, the residual-free indicator is effective in choosing snapshots during the offline RBM phase, but cannot currently be used to certify error that the approximation commits. However, it circumvents the need for a posteriori analysis of numerical methods, and therefore can be effective on problems where such a rigorous estimate is hard to derive.

Existence of Ψ-bounded Solutions for System of Linear Dynamic equations on Time Scales

In this work, we develop the criteria for existence of Ψ- bounded solutions of system of linear dynamic equations on time scales. The advantage of results in this dynamical system is it unifies discrete as well as continuous systems. Initially, we develop if and only if conditions for the existence of at least one Ψ-bounded solution for linear dynamic equation y∆(τ ) = P (τ )y +g(τ ), for each Ψ- delta integrable Lebesgue function g, on time scale T +. Later, we obtain asymptotic nature of Ψ-bounded solutions of dynamical system. Also we provided the examples for supporting the results.AMS Subject Classification: 74H20, 34N05, 34C11

Two-Sided Estimates of Fourier Sums Lebesgue Functions with Respect to Polynomials Orthogonal on Nonuniform Grids

Let Ω = {t0, t1, …, tN} and ΩN = {x0, x1, …, xN–1}, where xj = (tj + tj + 1)/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums Sn, N( f, x) with respect to the system of polynomials {pk,N(x)} =0 –1 , forming an orthonormal system on nonuniform point systems ΩN consisting of finite number N of points from the segment [–1, 1] with weight Δtj = tj + 1–tj. We find the growth order for the Lebesgue function Ln,N (x) of the considered partial discrete Fourier sums Sn,N ( f, x) as n = O(δ −2/7 ), δN = max0≤ j≤N−1 Δtj More exactly, we have a two-sided pointwise estimate for the Lebesgue function Ln, N(x), depending on n and the position of the point x from [–1, 1].

Some new results on orthogonal polynomials for Laguerre type exponential weights

We prove some results on the root-distances and the weighted Lebesgue function corresponding to orthogonal polynomials for Laguerre type exponential weights.

Double-sided to Lebesgue function of Fourier sums by polynomials orthogonal on non-uniform grids

Double-sided to Lebesgue function of Fourier sums by polynomials orthogonal on non-uniform grids

Asymptotic formulas for Lebesgue functions corresponding to the family of Lagrange interpolation polynomials

The asymptotic behavior of Lebesgue functions of trigonometric Lagrange interpolation polynomials constructed on an even number of nodes is studied. For these functions, asymptotic formulas involving concrete simplest trigonometric and algebraic-trigonometric polynomials were first obtained.

Existence of weak solutions of doubly nonlinear parabolic equations

We deal with a Cauchy–Dirichlet problem with homogeneous boundary conditions on the parabolic boundary of a space–time cylinder for doubly nonlinear parabolic equations, whose prototype is∂tu−div(|u|m−1|Du|p−2Du)=f with a non-negative Lebesgue function f on the right-hand side, where p>2nn+2 and m>0. The central objective is to establish the existence of weak solutions under the optimal integrability assumption on the inhomogeneity f. The constructed solution is obtained by a limit of approximations, i.e. we use solutions of regularized Cauchy–Dirichlet problems and pass to the limit to receive a solution for the original Cauchy–Dirichlet problem.

The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions

For kernels ν which are positive and integrable we show that the operator g↦Jνg=∫0xν(x−s)g(s)ds on a finite time interval enjoys a regularizing effect when applied to Hölder continuous and Lebesgue functions and a “contractive” effect when applied to Sobolev functions. For Hölder continuous functions, we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel, namely by the factor N(x)=∫0xν(s)ds. For functions in Lebesgue spaces, we prove that an improvement always exists, and it can be expressed in terms of Orlicz integrability. Finally, for functions in Sobolev spaces, we show that the operator Jν “shrinks” the norm of the argument by a factor that, as in the Hölder case, depends on the function N (whereas no regularization result can be obtained).These results can be applied, for instance, to Abel kernels and to the Volterra function I(x)=μ(x,0,−1)=∫0∞xs−1/Γ(s)ds, the latter being relevant for instance in the analysis of the Schrödinger equation with concentrated nonlinearities in R2.

The specification property and infinite entropy for certain classes of linear operators

We study the specification property and infinite topological entropy for two specific types of linear operators: translation operators on weighted Lebesgue function spaces and weighted backward shift operators on sequence F-spaces. It is known, from the work of Bartoll, Martínez-Giménez, Murillo-Arcila, and Peris, that for weighted backward shift operators, the existence of a single non-trivial periodic point is sufficient for specification. We show this also holds for translation operators on weighted Lebesgue function spaces. This implies, in particular, that for these operators, the specification property is equivalent to Devaney chaos. We show that these forms of chaos (unsurprisingly) imply infinite topological entropy, but that (surprisingly) the converse does not hold.

The overlapped radial basis function-finite difference (RBF-FD) method: A generalization of RBF-FD

We present a generalization of the RBF-FD method that computes RBF-FD weights in finite-sized neighborhoods around the centers of RBF-FD stencils by introducing an overlap parameter δ∈(0,1] such that δ=1 recovers the standard RBF-FD method and δ=0 results in a full decoupling of stencils. We provide experimental evidence to support this generalization, and develop an automatic stabilization procedure based on local Lebesgue functions for the stable selection of stencil weights over a wide range of δ values. We provide an a priori estimate for the speedup of our method over RBF-FD that serves as a good predictor for the true speedup. We apply our method to parabolic partial differential equations with time-dependent inhomogeneous boundary conditions – Neumann in 2D, and Dirichlet in 3D. Our results show that our method can achieve as high as a 60× speedup in 3D over existing RBF-FD methods in the task of forming differentiation matrices.

A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps

We consider a stochastic functional delay differential equation, namely an equation whose evolution depends on its past history as well as on its present state, driven by a pure diffusive component plus a pure jump Poisson compensated measure. We lift the problem in the infinite dimensional space of square integrable Lebesgue functions in order to show that its solution is an \(L^2\)-valued Markov process whose uniqueness can be shown under standard assumptions of locally Lipschitzianity and linear growth for the coefficients. Coupling the aforementioned equation with a standard backward differential equation, and deriving some ad hoc results concerning the Malliavin derivative for systems with memory, we are able to derive a non-linear Feynman–Kac representation theorem under mild assumptions of differentiability.

Special series in Laguerre polynomials and their approximation properties

We consider special series in the classical Laguerre polynomials coinciding in particular cases with the mixed series associated to Laguerre polynomials, introduced by the author previously, as well as Fourier–Sobolev series in Sobolev orthogonal Laguerre polynomials. We address the questions of uniform convergence of these series on a finite segment of the positive half-axis. We study the approximation properties of partial sums of special series on the positive half-axis, with particular attention paid to estimating their Lebesgue function.

Asymptotic Formulas for Lebesgue Functions Corresponding to the Family of Lagrange Interpolation Polynomials

В работе исследуются асимптотическое поведение функций Лебега тригонометрических интерполяционных полиномов Лагранжа, построенных в четном числе узлов. Для рассматриваемых функций впервые получены асимптотические формулы, выраженные через вполне определенные простейшие тригонометрические и алгебро-тригонометрические полиномы. Библиография: 23 названия.

Existence of solutions to a non‐variational singular elliptic system with unbounded weights

In this paper we prove an existence result for the following singular elliptic system urn:x-wiley:0025584X:media:mana201600038:mana201600038-math-0001where Ω is a bounded open set in (), is the p‐laplacian operator, and are suitable Lebesgue functions and , , are positive parameters satisfying suitable assumptions.

Lebesgue functions and Lebesgue constants in polynomial interpolation

The Lebesgue constant is a valuable numerical instrument for linear interpolation because it provides a measure of how close the interpolant of a function is to the best polynomial approximant of the function. Moreover, if the interpolant is computed by using the Lagrange basis, then the Lebesgue constant also expresses the conditioning of the interpolation problem. In addition, many publications have been devoted to the search for optimal interpolation points in the sense that these points lead to a minimal Lebesgue constant for the interpolation problems on the interval $[-1, 1]$ . In Section 1 we introduce the univariate polynomial interpolation problem, for which we give two useful error formulas. The conditioning of polynomial interpolation is discussed in Section 2. A review of some results for the Lebesgue constants and the behavior of the Lebesgue functions in view of the optimal interpolation points is given in Section 3.

On barycentric interpolation. I. (On the T-Lebesgue function and T-Lebesgue constant)

We prove theorems on the Lebesgue function and Lebesgue constant of barycentric Lagrange interpolation based on arbitrary node-sytem in [-1, 1]. It turns out that the results are very similar to the ones known for the classical Lagrange interpolation.

Limit ultraspherical series and their approximation properties

We study new series of the form \(\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} \) in which the general term \(f_k^{ - 1} \hat P_k^{ - 1} (x)\), k = 0, 1, …, is obtained by passing to the limit as α→−1 from the general term \(\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)\) of the Fourier series \(\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} \) in Jacobi ultraspherical polynomials \(\hat P_k^{\alpha ,\alpha } (x)\) generating, for α> −1, an orthonormal system with weight (1 − x2)α on [−1, 1]. We study the properties of the partial sums \(S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} \) of the limit ultraspherical series \(\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} \). In particular, it is shown that the operator Sn−1(f) = Sn−1 (f, x) is the projection onto the subspace of algebraic polynomials pn = pn(x) of degree at most n, i.e., Sn(pn) = pn; in addition, Sn−1(f, x) coincides with f(x) at the endpoints ±1, i.e., Sn−1 (f,±1) = f(±1). It is proved that the Lebesgue function Λn(x) of the partial sums Sn−1(f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that \(\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1\).

Lebesgue functions corresponding to a family of Lagrange interpolation polynomials

In this paper we obtain various explicit forms of the Lebesgue function corresponding to a family of Lagrange interpolation polynomials defined at an even number of nodes. We study these forms by using the derivatives up to the second order inclusive. We estimate exact values of Lebesgue constants for this family from below and above in terms of known parameters. In a particular case we obtain new convenient formulas for calculating these estimates.