scholarly journals Estimation of deviation of continuous $2\pi$-periodic functions from corresponding trigonometric polynomials of S.N. Bernstein's type

2021 ◽  
pp. 43
Author(s):  
N.Ya. Yatsenko
Keyword(s):  
Periodic Function ◽  
Trigonometric Polynomial ◽  
Modulus Of Continuity ◽  
Trigonometric Polynomials ◽  
Periodic Functions

We have established the estimation of deviation of continuous $2\pi$-periodic function $f(x)$ from the trigonometric polynomial of S.N. Bernstein's type that corresponds to it, by the modulus of continuity of the function $f(x)$.

scholarly journals Distribution of Small Values of Bohr Almost Periodic Functions with Bounded Spectrum

2019 ◽  
pp. 571-578
Author(s):  
Wayne M. Lawton
Keyword(s):  
Periodic Function ◽  
Riemann Hypothesis ◽  
Mahler Measure ◽  
Trigonometric Polynomials ◽  
Almost Periodic Function ◽  
Almost Periodic ◽  
Periodic Functions ◽  
The Mean ◽  
Relationship Of ◽  
The Relationship

For f a nonzero Bohr almost periodic function on R with a bounded spectrum we proved there exist Cf > 0 and integer n > 0 such that for every u > 0 the mean measure of the set f x : jf(x)j < u g is less than Cf u1=n: For trigonometric polynomials with n + 1 frequencies we showed that Cf can be chosen to depend only on n and the modulus of the largest coefficient of f: We showed this bound implies that the Mahler measure M(h); of the lift h of f to a compactification G of R; is positive and discussed the relationship of Mahler measure to the Riemann Hypothesis

scholarly journals Inequalities of Jackson type for functions with values in Hilbert space

2021 ◽  
Vol 16 ◽  
pp. 10
Author(s):  
V.F. Babenko ◽  
S.V. Savela
Keyword(s):  
Hilbert Space ◽  
Periodic Function ◽  
Modulus Of Continuity ◽  
Trigonometric Polynomials ◽  
Jackson Type

We present the generalization of M.I. Chernykh's results about the estimate of the best $L_2$-approximation of periodic function $f$ by trigonometric polynomials by its $L_2$-modulus of continuity, in the case of functions with values in Hilbert space.

scholarly journals A global theory of flexes of periodic functions

2004 ◽  
Vol 173 ◽  
pp. 85-138 ◽  
Author(s):  
Gudlaugur Thorbergsson ◽  
Masaaki Umehara
Keyword(s):  
Periodic Function ◽  
Trigonometric Polynomial ◽  
Morse Theory ◽  
Smooth Function ◽  
Plane Curves ◽  
Periodic Functions ◽  
Global Theory ◽  
Convex Plane ◽  
Vertex Theorem ◽  
Smooth Periodic Function

AbstractFor a real valued periodic smooth function u on R, n ≥ 0, one defines the osculating polynomial φs (of order 2n + 1) at a point s ∈ R to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex (resp. clean minimal flex) of the function u on S1 if and only if φs ≥ u (resp. φs ≤ u) and the preimage (φ - u)-1(0) is connected. We prove that any smooth periodic function u has at least n + 1 clean maximal flexes of order 2n + 1 and at least n + 1 clean minimal flexes of order 2n + 1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves.

Quantum circuits for simple periodic functions

2014 ◽  
Vol 14 (9&10) ◽  
pp. 763-776
Author(s):  
Omar Gamel ◽  
Daniel F.V. James
Keyword(s):  
Quantum Computing ◽  
Periodic Function ◽  
Quantum Circuits ◽  
Special Importance ◽  
Periodic Functions ◽  
Single Period ◽  
Shor's Algorithm ◽  
Toffoli Gates ◽  
One To One

Periodic functions are of special importance in quantum computing, particularly in applications of Shor's algorithm. We explore methods of creating circuits for periodic functions to better understand their properties. We introduce a method for constructing the circuit for a simple monoperiodic function, that is one-to-one within a single period, of a given period $p$. We conjecture that to create a simple periodic function of period $p$, where $p$ is an $n$-bit number, one needs at most $n$ Toffoli gates.

APPROXIMATIVE PROPERTIES OF ABEL–POISSON-TYPE OPERATORS ON THE GENERALIZED HÖLDER CLASSES

2021 ◽  
Vol 1 ◽  
pp. 76-83
Author(s):  
Yuri I. Kharkevich ◽  
◽  
Alexander G. Khanin ◽  
Keyword(s):  
Modulus Of Continuity ◽  
Applied Mathematics ◽  
Continuous Functions ◽  
Elliptic Type ◽  
Periodic Functions ◽  
French Mathematician ◽  
First Order ◽  
Function Classes ◽  
In The Beginning ◽  
Poisson Type

The paper deals with topical issues of the modern applied mathematics, in particular, an investigation of approximative properties of Abel–Poisson-type operators on the so-called generalized Hölder’s function classes. It is known, that by the generalized Hölder’s function classes we mean the classes of continuous -periodic functions determined by a first-order modulus of continuity. The notion of the modulus of continuity, in turn, was formulated in the papers of famous French mathematician Lebesgue in the beginning of the last century, and since then it belongs to the most important characteristics of smoothness for continuous functions, which can describe all natural processes in mathematical modeling. At the same time, the Abel-Poisson-type operators themselves are the solutions of elliptic-type partial differential equations. That is why the results obtained in this paper are significant for subsequent research in the field of applied mathematics. The theorem proved in this paper characterizes the upper bound of deviation of continuous -periodic functions determined by a first-order modulus of continuity from their Abel–Poisson-type operators. Hence, the classical Kolmogorov–Nikol’skii problem in A.I. Stepanets sense is solved on the approximation of functions from the classes by their Abel–Poisson-type operators. We know, that the Abel–Poisson-type operators, in partial cases, turn to the well-known in applied mathematics Poisson and Jacobi–Weierstrass operators. Therefore, from the obtained theorem follow the asymptotic equalities for the upper bounds of deviation of functions from the Hölder’s classes of order from their Poisson and Jacobi–Weierstrass operators, respectively. The obtained equalities generalize the known in this direction results from the field of applied mathematics.

scholarly journals Mean-periodic functions

1980 ◽  
Vol 3 (2) ◽  
pp. 199-235 ◽  
Author(s):  
Carlos A. Berenstein ◽  
B. A. Taylor
Keyword(s):  
Partial Differential Equations ◽  
Periodic Function ◽  
Entire Functions ◽  
Convolution Equation ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Exponential Polynomial ◽  
Periodic Functions ◽  
Open Questions ◽  
Constant Coefficients

We show that any mean-periodic functionfcan be represented in terms of exponential-polynomial solutions of the same convolution equationfsatisfies, i.e.,u∗f=0(μ∈E′(ℝn)). This extends ton-variables the work ofL. Schwartz on mean-periodicity and also extendsL. Ehrenpreis' work on partial differential equations with constant coefficients to arbitrary convolutors. We also answer a number of open questions about mean-periodic functions of one variable. The basic ingredient is our work on interpolation by entire functions in one and several complex variables.

scholarly journals Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials

2021 ◽  
Vol 19 (1) ◽  
pp. 1047-1055
Author(s):  
Zhihua Zhang
Keyword(s):  
Fourier Series ◽  
Trigonometric Polynomial ◽  
Algebraic Polynomial ◽  
Fourier Coefficients ◽  
Random Processes ◽  
Qualitative Theory ◽  
Trigonometric Polynomials ◽  
Algebraic Polynomials ◽  
Fourier Approximation ◽  
Random Differential Equations

Abstract Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an m m -order differentiable function f f on [ 0 , 1 0,1 ], we will construct an m m -degree algebraic polynomial P m {P}_{m} depending on values of f f and its derivatives at ends of [ 0 , 1 0,1 ] such that the Fourier coefficients of R m = f − P m {R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series R m {R}_{m} is a trigonometric polynomial, we can reconstruct the function f f well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.

scholarly journals On multidimensional Jackson's theorem

2021 ◽  
pp. 30
Author(s):  
S.A. Pichugov
Keyword(s):  
Modulus Of Continuity ◽  
Periodic Functions ◽  
Linear Polynomial ◽  
Multiple Variables ◽  
Polynomial Methods ◽  
Methods Of Approximation

We have found the best linear polynomial methods of approximation of continuous periodic functions of multiple variables in uniform metric with concave modulus of continuity.

scholarly journals The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Jifeng Chu ◽  
Ting Xia
Keyword(s):  
Periodic Function ◽  
Lyapunov Stability ◽  
Stability Criterion ◽  
Normal Forms ◽  
Sufficient Condition ◽  
Periodic Functions ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Time Periodic ◽  
Damped Oscillator

Leta(t),b(t)be continuousT-periodic functions with∫0Tb(t)dt=0. We establish one stability criterion for the linear damped oscillatorx′′+b(t)x′+a(t)x=0. Moreover, based on the computation of the corresponding Birkhoff normal forms, we present a sufficient condition for the stability of the equilibrium of the nonlinear damped oscillatorx′′+b(t)x′+a(t)x+c(t)x2n-1+e(t,x)=0, wheren≥2,c(t)is a continuousT-periodic function,e(t,x)is continuousT-periodic intand dominated by the powerx2nin a neighborhood ofx=0.

Sign in / Sign up

Export Citation Format

Share Document