Abstract
Greedy expansions with prescribed coefficients were introduced by V. N. Temlyakov in a general case of Banach spaces. In contrast to Fourier series expansions, in greedy expansions with prescribed coefficients, a sequence of coefficients
{
c
n
}
n
=
1
∞
{\left\{{c}_{n}\right\}}_{n=1}^{\infty }
is fixed in advance and does not depend on an expanded element. During the expansion, only expanding elements are constructed (or, more precisely, selected from a predefined set – a dictionary). For symmetric dictionaries, V. N. Temlyakov obtained conditions on a sequence of coefficients sufficient for a convergence of a greedy expansion with these coefficients to an expanded element. In case of a Hilbert space these conditions take the form
∑
n
=
1
∞
c
n
=
∞
{\sum }_{n=1}^{\infty }{c}_{n}=\infty
and
∑
n
=
1
∞
c
n
2
<
∞
{\sum }_{n=1}^{\infty }{c}_{n}^{2}\lt \infty
. In this paper, we study a possibility of relaxing the latter condition. More specifically, we show that the convergence is guaranteed for
c
n
=
o
1
n
{c}_{n}=o\left(\frac{1}{\sqrt{n}}\right)
, but can be violated if
c
n
≍
1
n
{c}_{n}\hspace{0.33em}\asymp \hspace{0.33em}\frac{1}{\sqrt{n}}
.
Maximal regularity of discrete second order Cauchy problems in Banach spaces
