ISSN 0001-4346, Mathematical Notes, 2011, Vol. 89, No. 5, pp. 689–705. © Pleiades Publishing, Ltd., 2011.
Original Russian Text © V. A. Makarichev, 2011, published in Matematicheskie Zametki, 2011, Vol. 89, No. 5, pp. 738–754.
Asymptotics of the Basis Functions
of Generalized Taylor Series for the Class H
ρ,2
V. A. Makarichev
*
NAKU “KHAI”, Kharkov, Ukraine
Received April 29, 2009; in final form, September 6, 2010
Abstract—We study the basis functions ϕ
n,k
and ψ
n,p
of generalized Taylor series for the class H
ρ,2
and obtain asymptotic expansions of the functions ϕ
(l)
n,0
and ψ
(l)
n,2·4
n
−1
. We prove the existence of
an asymptotics for the functions ϕ
n,k
and ψ
n,p
for k =0 and p =2 · 4
n
− 1. The first term of the
asymptotic expansions of these functions is obtained.
DOI: 10.1134/S0001434611050099
Keywords: generalized Taylor series, basis functions, the class of functions H
ρ,2
, functional-
differential equation.
1. PRELIMINARY RESULTS
Consider the class of functions
H
ρ
=
ϕ ∈ C
∞
[−1,1]
: ϕ
(n)
C
[−1,1]
≤ c(ϕ)ρ
n
2
n(n+1)/2
,n =0, 1, 2,...
.
For functions of this class, a series called the generalized Taylor series was proposed in [1]. Let
N
n
= {−2
n−1
, −2
n−1
+1,..., 2
n−1
} for n =0, N
0
= {−1, 0, 1},
and let
x
n,k
=
k
2
n−1
, n =0, k ∈ N
n
, x
0,k
= k, k ∈ N
0
.
It was proved in [1], [2] that if f ∈ H
ρ
for 1 ≤ ρ< 2, then
f
(l)
(x)=
∞
n=0
k∈Nn
f
(n)
(x
n,k
)ϕ
(l)
n,k
(x),
where the series on the right-hand side converges uniformly on [−1, 1] for each l =0, 1, 2,... , and the
basis functions ϕ
n,k
(x) are uniquely determined from the conditions
ϕ
n,k
∈ H
1
, ϕ
(m)
n,k
(x
m,s
)= δ
m
n
· δ
k
s
,
n =0, 1, 2,..., k ∈ N
n
, m =0, 1, 2,..., s ∈ N
m
;
here δ
m
n
is the Kronecker delta. The functions ϕ
n,k
(x) are constructed by using the compactly supported
solution of the functional-differential equation
y
(x)=2 · (y(2x + 1) − y(2x − 1)).
Note that generalized Taylor series for the class H
ρ
allow one to reconstruct functions from the values
of the derivatives at the points of sufficiently simple finite sets. Generalized Taylor series for the wider
classes of infinitely differentiable functions were constructed in [3], [4].
*
E-mail: victor.makarichev@gmail.com
689