Weak convergence and weak compactness in the space of almost periodic functions on the real line

1996 ◽  
Vol 106 (1) ◽  
pp. 1-11 ◽  
Author(s):  
J. Batt ◽  
M. V. Deshpande
Keyword(s):  
Weak Convergence ◽  
Real Line ◽  
Weak Compactness ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
The Real

scholarly journals On extensions of inequalities of Kolmogoroff and others and some applications to almost periodic functions

1972 ◽  
Vol 13 (1) ◽  
pp. 1-16 ◽  
Author(s):  
C. J. F. Upton
Keyword(s):  
Real Function ◽  
Real Line ◽  
Complex Function ◽  
Real Variable ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Almost Everywhere ◽  
Image Position

Let f(x) be a complex function of a real variable, defined over the whole real line, which possesses n derivatives (the nth at least almost everywhere) and is such that . Then, if k is any integer for which 0< k < n, Kolmogoroff's inequality may be written as,or, by putting ,The constant K=K (k, n) known explicitly and is the best possible, i.e., there is a (real) function for which equality holds (see Bang [1]).

On finite sums of periodic functions

2020 ◽  
Vol 27 (2) ◽  
pp. 265-269
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Analytic Function ◽  
Real Line ◽  
Real Analytic Function ◽  
Periodic Functions ◽  
Uniform Limit ◽  
The Real ◽  
Pointwise Limit ◽  
Finite Sums ◽  
Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

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