Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces

Direct and inverse approximation theorems are proved in the Besicovitch-Stepanets spaces $B{\mathcal S}^{p}$ of almost periodic functions in terms of the best approximations of functions and their generalized moduli of smoothness.

Strong Embeddings for Transitory Queueing Models

In this paper, we establish strong embedding theorems, in the sense of the Komlós-Major-Tusnády framework, for the performance metrics of a general class of transitory queueing models of nonstationary queueing systems. The nonstationary and non-Markovian nature of these models makes the computation of performance metrics hard. The strong embeddings yield error bounds on sample path approximations by diffusion processes in the form of functional strong approximation theorems.

Complex Shepard Operators and Their Summability

In this paper, we construct the complex Shepard operators to approximate continuous and complex-valued functions on the unit square. We also examine the effects of regular summability methods on the approximation by these operators. Some applications verifying our results are provided. To illustrate the approximation theorems graphically we consider the real or imaginary part of the complex function being approximated and also use the contour lines of the modulus of the function.

Korovkin type approximation via statistical e-convergence on two dimensional weighted spaces

In the present work, we prove a Korovkin theorem for statistical e-convergence on two dimensional weighted spaces. We show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. We also study the rate of statistical e-convergence by using the weighted modulus of continuity and afterwards we present an application in support of our result.

Statistical Riemann and Lebesgue Integrable Sequence of Functions with Korovkin-Type Approximation Theorems

In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fundamental limit theorems connecting these beautiful and potentially useful notions. Furthermore, based upon our proposed techniques, we establish the Korovkin-type approximation theorems with algebraic test functions. Finally, we present two illustrative examples under the consideration of positive linear operators in association with the Bernstein polynomials to exhibit the effectiveness of our findings.

Approximation Theorems for the Schrödinger Equation and Quantum Vortex Reconnection

We prove the existence of smooth solutions to the Gross–Pitaevskii equation on $$\mathbb {R}^3$$ R 3 that feature arbitrarily complex quantum vortex reconnections. We can track the evolution of the vortices during the whole process. This permits to describe the reconnection events in detail and verify that this scenario exhibits the properties observed in experiments and numerics, such as the $$t^{1/2}$$ t 1 / 2 and change of parity laws. We are mostly interested in solutions tending to 1 at infinity, which have finite Ginzburg–Landau energy and physically correspond to the presence of a background chemical potential, but we also consider the cases of Schwartz initial data and of the Gross–Pitaevskii equation on the torus. In the proof, the Gross–Pitaevskii equation operates in a nearly linear regime, so the result applies to a wide range of nonlinear Schrödinger equations. Indeed, an essential ingredient in the proofs is the development of novel global approximation theorems for the Schrödinger equation on $$\mathbb {R}^n$$ R n . Specifically, we prove a qualitative approximation result that applies for solutions defined on very general spacetime sets and also a quantitative result for solutions on product sets in spacetime $$D\times \mathbb {R}$$ D × R . This hinges on frequency-dependent estimates for the Helmholtz–Yukawa equation that are of independent interest.

Approximation Theorem for New Modification of q -Bernstein Operators on (0,1)

In this work, we extend the works of F. Usta and construct new modified q -Bernstein operators using the second central moment of the q -Bernstein operators defined by G. M. Phillips. The moments and central moment computation formulas and their quantitative properties are discussed. Also, the Korovkin-type approximation theorem of these operators and the Voronovskaja-type asymptotic formula are investigated. Then, two local approximation theorems using Peetre’s K -functional and Steklov mean and in terms of modulus of smoothness are obtained. Finally, the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators are studied, and some graphs and numerical examples are shown by using Matlab algorithms.