Approximation by trigonometric polynomials in the framework of variable exponent grand Lebesgue spaces

2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Nina Danelia ◽  
Vakhtang Kokilashvili
Keyword(s):  
Lebesgue Space ◽  
Variable Exponent ◽  
Trigonometric Polynomials ◽  
Lebesgue Spaces ◽  
Variable Exponent Lebesgue Space ◽  
Grand Lebesgue Spaces ◽  
Direct And Inverse Theorems ◽  
Inverse Theorems ◽  
Grand Lebesgue Space

AbstractIn this paper we establish direct and inverse theorems on approximation by trigonometric polynomials for the functions of the closure of the variable exponent Lebesgue space in the variable exponent grand Lebesgue space.

scholarly journals On Erdős–Ginzburg–Ziv inverse theorems

2007 ◽  
Vol 129 (4) ◽  
pp. 307-318 ◽  
Author(s):  
D. J. Grynkiewicz ◽  
O. Ordaz ◽  
M. T. Varela ◽  
F. Villarroel
Keyword(s):  
Inverse Theorems

Inverse theorems for some linear positive operators using Beta and Baskakov basis functions

2021 ◽  
Author(s):  
Lakshmi Narayan Mishra ◽  
A. Srivastava ◽  
T. Khan ◽  
S. A. Khan ◽  
Vishnu Narayan Mishra
Keyword(s):  
Positive Operators ◽  
Basis Functions ◽  
Linear Positive Operators ◽  
Inverse Theorems

On Global Inverse Theorems of Szász and Baskakov Operators

1979 ◽  
Vol 31 (2) ◽  
pp. 255-263 ◽  
Author(s):  
Z. Ditzian
Keyword(s):  
Rate Of Convergence ◽  
Exponential Growth ◽  
Polynomial Growth ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Baskakov Operators ◽  
Inverse Theorems ◽  
Image Position ◽  
Necessary And Sufficient

The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

scholarly journals AN INVERSE THEOREM FOR THE GOWERS $U^3(G)$ NORM

2008 ◽  
Vol 51 (1) ◽  
pp. 73-153 ◽  
Author(s):  
Ben Green ◽  
Terence Tao
Keyword(s):  
Absolute Constant ◽  
Bounded Function ◽  
Additive Group ◽  
Arithmetic Progressions ◽  
Inverse Theorem ◽  
Inner Product ◽  
Arbitrary Group ◽  
Quadratic Phase ◽  
Inverse Theorems ◽  
The Fourier Transform

AbstractThere has been much recent progress in the study of arithmetic progressions in various sets, such as dense subsets of the integers or of the primes. One key tool in these developments has been the sequence of Gowers uniformity norms $U^d(G)$, $d=1,2,3,\dots$, on a finite additive group $G$; in particular, to detect arithmetic progressions of length $k$ in $G$ it is important to know under what circumstances the $U^{k-1}(G)$ norm can be large.The $U^1(G)$ norm is trivial, and the $U^2(G)$ norm can be easily described in terms of the Fourier transform. In this paper we systematically study the $U^3(G)$ norm, defined for any function $f:G\to\mathbb{C}$ on a finite additive group $G$ by the formula\begin{multline*} \qquad\|f\|_{U^3(G)}:=|G|^{-4}\sum_{x,a,b,c\in G}(f(x)\overline{f(x+a)f(x+b)f(x+c)}f(x+a+b) \\ \times f(x+b+c)f(x+c+a)\overline{f(x+a+b+c)})^{1/8}.\qquad \end{multline*}We give an inverse theorem for the $U^3(G)$ norm on an arbitrary group $G$. In the finite-field case $G=\mathbb{F}_5^n$ we show that a bounded function $f:G\to\mathbb{C}$ has large $U^3(G)$ norm if and only if it has a large inner product with a function $e(\phi)$, where $e(x):=\mathrm{e}^{2\pi\ri x}$ and $\phi:\mathbb{F}_5^n\to\mathbb{R}/\mathbb{Z}$ is a quadratic phase function. In a general $G$ the statement is more complicated: the phase $\phi$ is quadratic only locally on a Bohr neighbourhood in $G$.As an application we extend Gowers's proof of Szemerédi's theorem for progressions of length four to arbitrary abelian $G$. More precisely, writing $r_4(G)$ for the size of the largest $A\subseteq G$ which does not contain a progression of length four, we prove that$$ r_4(G)\ll|G|(\log\log|G|)^{-c}, $$where $c$ is an absolute constant.We also discuss links between our ideas and recent results of Host, Kra and Ziegler in ergodic theory.In future papers we will apply variants of our inverse theorems to obtain an asymptotic for the number of quadruples $p_1\ltp_2\ltp_3\ltp_4\leq N$ of primes in arithmetic progression, and to obtain significantly stronger bounds for $r_4(G)$.

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