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Let there be given a partition of the closed interval $$[-1,1]$$ by arbitrary nodes $$\{\eta_j\}_{j=0}^N$$ , where $$\lambda_N=\max_{0\le j \le N-1} (\eta_{j+1}-\eta_{j})$$ . For a continuous function $$f(t)$$ given on an arbitrary grid $$\Omega_N=\{t_j \mid \eta_{j} \le t_j \le \eta_{j+1}\}_{j=0}^{N-1}$$ , the approximation properties of the discrete Fourier sums $$\Lambda^{\alpha,\beta}_{n,N}(f,t)$$ in polynomials $$\widehat P^{\alpha,\beta}_{n, N} (t)$$ are investigated in the case of nonnegative integer parameters $$\alpha$$ , $$\beta$$ ; these polynomials are orthogonal to $$\Omega_N$$ with Jacobi weight $$\kappa^{\alpha,\beta}(t)=(1-t)^{\alpha}(1+t)^{\beta}$$ . Given the restriction $$n=O(\lambda_N^{-1/3})$$ on the order of the Fourier sums, a pointwise estimate of the Lebesgue function $$L^{\alpha,\beta}_{n, N}(t)$$ is obtained; it depends on $$n$$ and the position of the point $$t \in [-1,1]$$ : $$L^{\alpha,\beta}_{n,N}(t)=O\bigl[\ln{(n+1)}+ |\widehat P^{\alpha,\beta}_{n,N}(t)|+ |\widehat P^{\alpha,\beta}_{n+1,N}(t)|\bigr].$$

In this paper, we construct a new trivariate spline quasi-interpolation operator. It is expressed as blending sum of univariate and bivariate C 1 quadratic spline quasi-interpolants and it is of near-best type, i.e. it has a small infinity norm and the coefficients functionals defining it are determined by minimizing an upper bound of the operator infinity norm, derived from the Bernstein-Bezier coefficients of its Lebesgue function.

The first part of this paper surveys several results on the lattice structure of variable exponent Lebesgue function spaces (or Nakano spaces) L p ( ⋅ ) ( Ω ) . In the second part strictly singular and disjointly strictly singular operators between spaces L p ( ⋅ ) ( Ω ) are studied. New results on the disjoint strict singularity of the inclusions L p ( ⋅ ) ( Ω ) ↪ L q ( ⋅ ) ( Ω ) are given.

for arbitrary continuous function f(t) on the segment [−1,1] we construct discrete Fourier sums Sn,N(f,t) on a system of polynomials forming an orthonormal system on non-uniform grids $${T_N} = \left\{{{t_j}} \right\}\matrix{{N - 1} \cr {j = 0} \cr}$$ of N points from segment [−1,1] with weight Δtj=tj+1−tj. Approximation properties of the constructed partial sums Sn,N(f,t) of order n ≤ N−1 are investigated. A two-sided pointwise estimate is obtained for the Lebesgue function Ln,N(t) of considered discrete Fourier sums for $$n = O\left({\delta \matrix{{- 1/5} \cr N \cr}} \right),{\delta _N} = {\max _0} \le j \le N - 1\Delta {t_j}$$ . The question of the convergence of Sn,N(f,t) to f(t) is also investigated. In particular, we obtain the deflection estimation of a partial sum Sn,N(f,t) from f(t) for $$n = O\left({\delta \matrix{{- 1/5} \cr N \cr}} \right)$$ , which depends on n and the position of a point t ∊ [−1,1].

The main purpose of this note is to estimate the size of the set Tμλ of points, at which the Cantor function is not differentiable and we find that the Hausdorff dimension of Tμλ is [log2/log3]2. Moreover, the Packing dimension of Tμλ is log2/log3. The log2 = loge2 is that if ax = N (a >0, and a≠1), then the number x is called the logarithm of N with a base, recorded as x = logaN, read as the logarithm of N with a base, where a is called logarithm Base number, N is called true number.

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.

In this paper we characterize those exponents p ( ⋅ ) for which corresponding variable exponent Lebesgue space L p ( ⋅ ) ( [ 0 ; 1 ] ) has in common with L ∞ the property that the space of continuous functions is a closed linear subspace in it. In particular, we obtain necessary and sufficient condition on decreasing rearrangement of exponent p ( ⋅ ) for which exists equimeasurable exponent of p ( ⋅ ) which corresponding variable exponent Lebesgue space have the above mentioned property.

Let X be a random variable with independent ternary digits and let y = F ⁢ ( x ) {y=F(x)} be a classic singular Cantor function. For the distribution of the random variable Y = F ⁢ ( X ) {Y=F(X)} , the Lebesgue structure (i.e., the content of discrete, absolutely continuous and singular components), the structure of its point and the continuous spectra are exhaustively studied.

We consider the problem of approximation of discrete functions f = f(x) defined on the set Ω_δ = {0, δ, 2δ, . . .}, where δ =1/N, N > 0, using the Fourier sums in the modified Meixner polynomials M_(α;n,N)(x) = M(α;n)(Nx) (n = 0, 1, . . .), which for α > -1 constitute an orthogonal system on the grid Ωδ with the weight function w(x) = e^-(x)*Γ(Nx + α + 1)/Γ(Nx + 1). We study the approximative properties of partial sums of Fourier series in polynomials M(α_n,N)(x), with particular attention paid to estimating their Lebesgue function.

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

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 ∂ t u − div ( | u | m − 1 | D u | p − 2 D u ) = f with a non-negative Lebesgue function f on the right-hand side, where p > 2 n n + 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.

We investigate some conditions under which the Lebesgue constants or Lebesgue functions are bounded for the classical Lagrange polynomial interpolation on a compact subset of $\mathbb R$. In particular, relationships of such boundedness with uniform and pointwise convergence of Lagrange polynomials and with the existence of interpolating Faber bases are discussed., 17 pages

The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.

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, Martinez-Gimenez, 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.

In this note, we consider the metric theory of the dynamical covering problems on the triadic Cantor set K . More precisely, let T x = 3 x ( mod 1 ) be the natural map on K , μ the standard Cantor measure and x 0 ∈ K a given point. We consider the size of the set of points in K which can be well approximated by the orbit { T n x 0 } n ≥ 1 of x 0 , namely the set D ( x 0 , φ ) : = { y ∈ K : | T n x 0 − y | φ ( n ) for infinitely many n ∈ N } , where φ is a positive function defined on N . It is shown that for μ almost all x 0 ∈ K , the Hausdorff measure of D ( x 0 , φ ) is either zero or full depending upon the convergence or divergence of a certain series. Among the proof, as a byproduct, we obtain an inhomogeneous counterpart of Levesley, Salp and Velani's work on a Mahler's question about the Diophantine approximation on the Cantor set K .

We provide a generalization of the classical Cantor function. One characterization of the Cantor function is generated by a sequence of real numbers that starts with a seed value and at each step randomly applies one of two different linear functions. The Cantor function is defined as the probability that this sequence approaches infinity. We generalize the Cantor function to instead use a set of any number of linear functions with integer coefficients. We completely describe the resulting probability function and give a full explanation of which intervals of seed values lead to a constant probability function value.

We study the self-similar subsets of a class of Cantor sets with nice symmetric structure. Our result generalizes the result of Feng, Rao and Wang (2015) [3] , which characterizes all the self-similar subsets of the Middle-Third Cantor set (a question raised by P. Mattila in 1998). We simplify a number-theoretical argument in the paper of Feng, Rao and Wang, and make it applicable to a large class of self-similar sets.

Let ( Ω , Σ , μ ) be a σ-finite measure space and consider the Lebesgue function space L 1 ( μ ) endowed with its standard norm. We obtain a characterization of weak compactness for closed bounded convex subsets of L 1 ( μ ) in terms of the existence of fixed points for certain classes of eventually affine, uniformly Lipschitzian mappings.

El objetivo de este trabajo es construir el conjunto y la función de Cantor y estudiar sus propiedades fundamentales. The objetive of this work is construct the Cantor set and function and study their fundamental properties.

Urysohn's Lemma is a crucial property of normal spaces that deals with separation of closed sets by continuous functions. It is also a fundamental ingredient in proving the Tietze Extension Theorem, another property of normal spaces that deals with the existence of extensions of continuous functions. Using the Cantor function, we give alternative proofs for Urysohn's Lemma and the Tietze Extension Theorem., Comment: A slightly modified version has been accepted for publication in the Bulletin of the Australian Mathematical Society

We examine the chaotic behavior of certain continuous linear operators on infinite-dimensional Banach spaces, and provide several equivalent characterizations of when these operators have infinite topological entropy. For example, it is shown that infinite topological entropy is equivalent to non-zero topological entropy for translation operators on weighted Lebesgue function spaces. In particular, finite non-zero entropy is impossible for this class of operators, which answers a question raised by Yin and Wei.

Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the $$L^2$$ L 2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the $$L^2$$ L 2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.

Georg Cantor (1845-1918) introduced the notion of the cantor set, which consists of points along a single line segment with a number of remarkable and deep properties. This paper aims to emphasize a proceeding to obtain the Cantor (ternary) set, C by means of the so-called elimination of the open-middle third at each step using a general basic approach in constructing the set.

Given c > 0 a planar Cantor set X with a dim H (X) < 2 is constructed such that each c-monotone subspace of X has a smaller Hausdorff dimension than X.