The embedding theorems for anisotropic Nikol’skii-Besov spaces with generalized mixed smoothness

The theory of embedding of spaces of differentiable functions studies the important relations of differential (smoothness) properties of functions in various metrics and has a wide application in the theory of boundary value problems of mathematical physics, approximation theory, and other fields of mathematics. In this article, we prove the embedding theorems for anisotropic spaces Nikol’skii-Besov with a generalized mixed smoothness and mixed metric, and anisotropic Lorentz spaces. The proofs of the obtained results are based on the inequality of different metrics for trigonometric polynomials in Lebesgue spaces with mixed metrics and interpolation properties of the corresponding spaces.

On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey

The theory of Ulam stability was initiated by a problem raised in 1940 by S. Ulam and concerning approximate solutions to the equation of homomorphism in groups. It is somehow connected to various other areas of investigation such as, e.g., optimization and approximation theory. Its main issue is the error that we make when replacing functions satisfying the equation approximately with exact solutions of the equation. This article is a survey of the published so far results on Ulam stability for functional equations in 2-normed spaces. We present and discuss them, pointing to the various pitfalls they contain and showing possible simple generalizations. In this way, in particular, we demonstrate that the easily noticeable symmetry between them and the analogous results obtained for the classical metric or normed spaces is in fact only apparent.

Generalized proportional fractional integral Hermite–Hadamard’s inequalities

The theory of fractional integral inequalities plays an intrinsic role in approximation theory also it has been a key in establishing the uniqueness of solutions for some fractional differential equations. Fractional calculus has been found to be the best for modeling physical and engineering processes. More precisely, the proportional fractional operators are one of the recent important notions of fractional calculus. Our aim in this research paper is developing some novel ways of fractional integral Hermite–Hadamard inequalities in the frame of a proportional fractional integral with respect to another strictly increasing continuous function. The considered fractional integral is applied to establish some new fractional integral Hermite–Hadamard-type inequalities. Moreover, we present some special cases throughout discussing this work.

Abstract Fractional Monotone Approximation with Applications

Here we extended our earlier fractional monotone approximation theory to abstract fractional monotone approximation, with applications to Prabhakar fractional calculus and non-singular kernel fractional calculi. We cover both the left and right sides of this constrained approximation. Let f∈Cp−1,1, p≥0 and let L be a linear abstract left or right fractional differential operator such that Lf≥0 over 0,1 or −1,0, respectively. We can find a sequence of polynomials Qn of degree ≤n such that LQn≥0 over 0,1 or −1,0, respectively. Additionally f is approximated quantitatively with rates uniformly by Qn with the use of first modulus of continuity of fp.

On one generalization of the Hermite quadrature formula

In this paper we propose a new approach to the construction of quadrature formulas of interpolation rational type on an interval. In the introduction, a brief analysis of the results on the topic of the research is carried out. Most attention is paid to the works of mathematicians of the Belarusian school on approximation theory – Gauss, Lobatto, and Radau quadrature formulas with nodes at the zeros of the rational Chebyshev – Markov fractions. Rational fractions on the segment, generalizing the classical orthogonal Jacobi polynomials with one weight, are defined, and some of their properties are described. One of the main results of this paper consists in constructing quadrature formulas with nodes at zeros of the introduced rational fractions, calculating their coefficients in an explicit form, and estimating the remainder. This result is preceded by some auxiliary statements describing the properties of special rational functions. Classical methods of mathematical analysis, approximation theory, and the theory of functions of a complex variable are used for proof. In the conclusion a numerical analysis of the efficiency of the constructed quadrature formulas is carried out. Meanwhile, the choice of the parameters on which the nodes of the quadrature formulas depend is made in several standard ways. The obtained results can be applied for further research of rational quadrature formulas, as well as in numerical analysis.

How smooth is quantum complexity?

The “quantum complexity” of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates. While the notion of quantum complexity was first introduced as a quantum generalization of the classical computational complexity, it has since been argued to hold a fundamental significance in its own right, as a physical quantity analogous to the thermodynamic entropy. In this paper, we present a unified perspective on various notions of quantum complexity, viewed as functions on the space of unitary operators. One striking feature of these functions is that they can exhibit non-smooth and even fractal behaviour. We use ideas from Diophantine approximation theory and sub-Riemannian geometry to rigorously quantify this lack of smoothness. Implications for the physical meaning of quantum complexity are discussed.

On the boundedness of the partial sums operator for the Fourier series in the function classes families associated with harmonic intervals

The article is devoted to the study of some data from the theory of functions approximation by trigonometric polynomials with a spectrum from special sets called harmonic intervals. Due to the limited perception range of devices, the perception range of the senses of the person himself, when studying a mathematical model it is often enough to find an approximation of the object so that the error (noise, interference, distortion) is outside the interval of perception. Harmonic intervals model problems of this kind to some extent. In the article the main components of the approximation theory of functions by trigonometric polynomials with a spectrum from harmonic intervals are presented, the theorem on estimating the best approximation of a function by trigonometric polynomials through the best approximations of a function by trigonometric polynomials with a spectrum from harmonic intervals is proved. Theorems on the boundedness of the partial sums operator for the Fourier series in the function classes families associated with harmonic intervals are considered; such a theorem for the Lorentz space is generalized and proved. The article is mainly aimed at scientific researchers dealing with practical applications of the approximation theory of functions by trigonometric polynomials with a spectrum from special sets.