Approximation by translates of a single function of functions in space induced by the convolution with a given function
This work addresses theoretical approximation rates for a specific function class, but the results are incremental and specialized to periodic functions and convolution-induced spaces.
The paper studies approximation of periodic functions by linear combinations of n translates of a single function, focusing on functions in a class induced by convolution with a given function. It provides upper bounds on L_p approximation convergence rates for 1 < p < ∞ and lower bounds on best approximation for p=2.
We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some methods of this approximation for functions in a class induced by the convolution with a given function, and prove upper bounds of $L_p$-the approximation convergence rate by these methods, when $n \to \infty$, for $1 < p < \infty$, and lower bounds of the quantity of best approximation of this class by arbitrary linear combinations of $n$ translates of arbitrary function, for the particular case $p=2$.