E.J. Ibrahimov
On Fourier coefficients of some classes of functions and their applications in approximation theory
Abstract. In the paper, the behavior of Fourier coefficients of some classes of func- tions on an arbitrary orthogonal system is studied. The estimates of order of con- vergence to zero of Fourier-Gegenbauer coefficients are found. These estimates are precise and are of terminal character. The obtained results are used in convergence of Fourier-Gegenbauer series.
2000 Mathematics Subject Classification: 46B20, 42B25, 42B35.
Key words and phrases: Fourier-Gegenbauer series, convergence, asymptotic estima- tion, Gegenbauer transform, strong derivative and of Gegenbauer integral, general- ized shift function.
Introduction. The estimates of Fourier-Legandre functions belonging to one of the following classes: C [−1, 1], L [−1, 1] and L2[−1, 1] were given in [1]. The ob- tained inequalities were applied to the problems of convergence of Fourier-Legandre series.
In [2] these results were generalized to ultraspherical series for f ∈ Lp,µ[−1, 1] , 1 ¬ p ¬ ∞.
In the paper [12] the author obtained the estimates of Fourier-Jacobi coefficients of smooth functions of bounded variation.
Unlike the indicated papers, in this paper we study the behavior of Fourier coefficients of some classes of functions from an arbitrary orthogonal system.
Suppose µ(x) is weight function such that Z b
a
µ(x)dx = Z b
a
dµ(x) = 1.
Let ϕn(x) , n = 0, 1, ...− be the orthogonal function system with the weight
µ (x) on the segment [a, b], a
(1) f (n) =ˆ
Zb
a
f (x) ϕn(x) dµ (x)
– be the Fourier coefficients of the functions f, belonging to one of the following classes: Lp,µ[a, b] , (1 ¬ p < ∞) is a class of functions summable with the p-th degree by the weight µ (x), L01,µ[a, b] is a class of functions with integrable derivative on [a, b].
Denote by X one of the linear spaces Lp,µ, or L01,µ, by L = L (X, X), a space of linear operators acting from X to X, for which the following equality is fulfilled:
(2)
Zb
a
(Af) (x) g (x) dµ (x) = Zb
a
f (x) (Ag) (x) dµ (x) .
In the case f ∈ Lp,µ we assume g ∈ Lq,µ, where 1p +1q = 1.
We’ll say that f ∈ WXr, if ∃g ∈ X such that
(3) f (x) = (Arg) (x) + c, r = 0, 1, ..., where
A∈ L (X, X) , A0f = f, Arf = A Ar−1f , and c− is some constant.
Denote the norm f ∈ Lp, µ by
kfkLp,µ ≡ kfkp,µ=
Zb
a
|f (x)|pdµ (x)
1 p
<∞, 1 ¬ p < ∞.
kfkL01,µ≡ kfkC= sup
−1¬x¬1|f (x)| .
Further be built the operator which satisfy of condition (2) and behavior of the Fourier-Gegenbauer coefficients of the functions for which is just presentation (3) is studied.
1. Basic properties of the Gegenbauer transform. In this section, the properties of the Gegenbauer transform of some classes of functions is studied. We introduce consept of strong derivative and of Gegenbauer integral. Be established connection between there. Over this consept become clear structuring description consider of the classes of functions. The obtained results are analogues of some theorems proved in [8] for Legendre transform.
In the following let X be one of the spaces Lp,λ[−1, 1] , 1 ¬ p < ∞ or C [−1, 1]
endowed with the norms
kfkp,λ=
Z 1
−1|f (x)|pdµλ(x)
1p
, (1¬ p < ∞) ,
kfkC= sup
−1¬x¬1|f (x)| , where dµλ(x) = Γ(12Γ(λ+1))Γ(λ+12) 1 − x2
λ−12
dx.
We consider of Gegenbauer polynomials Pnλ(x) , λ > 0, n = 0, 1, ..., which form of the orthogonal system of functions on a segment [−1, 1] with weight 1 − x2λ−12, that is (see [3], p.841)
(4) Z 1
−1 1 − x2λ−12
Pkλ(x) Pnλ(x) dx =
( 0, k6= n,
Γ(12)Γ(λ+12)Γ(n+2λ)
Γ(λ)Γ(2λ)(n+λ)Γ(n+1), k = n.
Further (see [90], pp. 178 and 93)
(5) max
|x|¬1
Pnλ(x) = Pnλ(1) = Γ (n + 2λ) Γ (λ) Γ (n + 1),
(6) d
dxPnλ(x) = 2λPn−1λ+1(x) Let
Qλn(x) = Pnλ(x) Pnλ(1).
According to (5)
(7) max
|x|¬1
Qλn(x) = Qλn(1) = 1.
The Gegenbauer transform (the Fourier-Gegenbauer coefficients) defined for f ∈ X by
f (n) =ˆ Z 1
−1
f (x) Qλn(x) dµλ(x) , n∈ P := {0, 1, 2, ...} .
Lemma 1.1 Assuming f, g ∈ X and c ∈ R, we have (a) |f∧(n)| ¬ kfkX, (n ∈ P );
(b) (f + g)∧(n) = f∧(n) + g∧(n), (cf)∧(n) = cf∧(n);
(c) Qλn(x)∧(k) =
( 0, k6= n
Γ(2λ)Γ(λ+1)Γ(n+1)
Γ(λ)(n+λ)Γ(n+2λ), k = n
(d) f∧(n) = 0for all n ∈ P if f (x) = 0 (a.e.), means that an assertion holds for all x ∈ [−1, 1] if X = C [−1, 1], and for almost all x ∈ [−1, 1] if X = Lp,λ[−1, 1], 1 ¬ p < ∞.
Proof We prove (a). Letf ∈ L01,λ[−1, 1], then by (7) we have
|f∧(n)| ¬ sup
|x|¬1|f (x)|
Z 1
−1
Qλn(x) dµλ(x) ¬ kfkC,
Now let f ∈ Lp,λ. For p = 1 by (7) we have
|f∧(n)| ¬ Z 1
−1|f (x)|Qλn(x)
dµ (x) ¬ kfk1,λ, and for p > 1 by H¨older inequality
|f∧(n)| ¬ kf (x)kp,λ
Qλn q,λ¬ kfkp,λ. Thus, for all f ∈ X
|f∧(n)| ¬ kfkX.
The properties b) and d) is obviously and (c) follow by (4).
Lemma is proved.
We consider generalized is Gegenbauer shift operator (see [6]) (τtf ) (x) = Γ λ +12
Γ 12Γ (λ) Z π
0 f xt +p
1 − x2p
1 − t2cos ϕ
(sin ϕ)2λ−1dϕ.
The following important equality
(8) τtQλn(x) = Qλn(x) Qλn(t)
follows from “multiplication theorem” for Gegenbauer polynomials:
τtQλn(x) = Γ λ +12 Γ 12Γ (λ)
Z π 0 Qλn
xt +p
1 − x2p
1 − t2cos ϕ
(sin ϕ)2λ−1dϕ
=Γ (2λ) Γ (n + 1)
Γ (n + 2λ) Pnλ(x) Qλn(x) =Pnλ(x)
Pnλ(1)Qλn(t) = Qλn(x) Qλn(t) , which follows of “addition theorem” (see [3], p.1044).
Lemma 1.2 The operator τtis a positive linear operator from X into itself, satis- fying
(a) kτtk[X,X]= 1, (t ∈ [−1, 1]);
(b) lim
t→1−0kτtf − fkX= 0, (f ∈ X);
(c) (τtf )∧(n) = f∧(n) Qλn(t), (f ∈ X, t ∈ [−1, 1] , n ∈ P );
(d) (τtf ) (x) = (τxf ) (t), (f ∈ X, x, t ∈ [−1, 1]).
One has for each f ∈ X (e)
lim f∧(n) = 0.
Proof In [2] is proved, which for any f ∈ Lp,λ at p 1 the inequality is fair:
(9) kτtfkp,λ¬ kfkp,λ, (t ∈ [−1, 1]).
On the other hand
(10) k(τt1)kp,λ= Z 1
−1
Γ λ + 12 Γ 12Γ (λ)
Z π
0 (sin ϕ)2λ−1dϕ
p
dµλ(x)
!1p
= k1kp,λ.
From (9) and (10) follows, that the case X = Lp,λ, p 1 kτtk[X,X]= 1, (t ∈ [−1, 1]).
But the case X = L01,λis elementary.
By substituting y = xt +√
1 − x2√
1 − t2cos ϕ, we obtain xt−p
1 − x2p
1 − t2¬ y ¬ xt +p
1 − x2p 1 − t2. Let us assume x = cos u, t = cos v, then we have
−1 ¬ cos (u + v) ¬ y ¬ cos (u − v) ¬ 1.
Now
kτtfkC¬ sup
|y|¬1|f (y)| = kfkC.
We prove (b). Let f ∈ Lp,λ, p 1. Then by density of C in Lp,λ for arbitrary number ε > 0 exist ψ ∈ C [−1, 1], such that
(11) kf − ψkp,λ< ε
3. And by (a) of Lemma 1.2 we have
(12) kτtf− τtψkp,λ¬ kτt(f − ψ)kp,λ¬ kf − ψkp,λ<ε 3. If ψ ∈ C [−1, 1], then ∀ε > 0, ∃δ (ε) > 0,
ψ
xt +p
1 − x2p
1 − t2cos ϕ
− ψ (x) < ε 3 for all t ∈ (1 − δ, 1).
(13) |(τtψ) (x)− ψ (x)| < ε
3 from where it follow
(14) kτtψ− ψkp,λ=
Z 1
−1|(τtψ) (x)− ψ (x)|pdµλ(x)
1p
< ε 3. Now taking into account (11), (12) and (14) we obtain
kf − τtfkp,λ¬ kf − ψkp,λ+ kψ − τtψkp,λ+ kτtψ− τtfkp,λ< ε, that is equivalent
(15) lim
t→1−0kτtf− fkp,λ= 0, p 1.
And from (15) follow that
(16) lim
t→1−0kτtf − fkL01,λ = 0.
The justice of assertion (b) follows from (15) and (16).
We prove (c).
(τtf )∧(n) =Z 1
−1
Qλn(x) (τtf ) (x) dµλ(x) . By making the substitution
z = xt +p
1 − x2p
1 − t2cos ϕ in the interval integral we obtain
cos ϕ = (z − xt) 1 − x2−12
1 − t2−12
dϕ =− 1 − x2− t2− z2+ 2xzt−12
dz,
(sin ϕ)2λ−1=
1 − x2− t2− z2+ 2xzt (1 − x2) (1 − t2)
λ−12
. Then
(τtf )∧(n) = Γ (λ + 1) Γ λ +12
Γ2 12Γ (λ) Γ λ +12 1 − t212−λZ 1
−1
Qλn(x) ×
× Z xt+√
1−x2√ 1−t2 xt−√
1−x2√
1−t2 1 − x2− t2− z2+ 2xztλ−1
f (z) dzdx.
By changing the order of integration we obtain
(τtf )∧(n) = Γ (λ + 1) Γ λ +12
Γ 12Γ λ + 12Γ (λ) Γ 12 1 − t212−λZ 1
−1
f (z)×
× Z zt+√
1−z2√ 1−t2 zt−√
1−z2√
1−t2 1 − x2− t2− z2+ 2xztλ−1
Qλn(x) dxdz.
Again making the substitution in the integral putting x = zt +√
1 − z2√ 1 − t2. Then
(τtf)∧(n) = Γ λ +12
Γ 12Γ (λ) Z 1
−1
f(z)
Z π 0
Qλn
zt+p
1 − z2p
1 − t2cos ϕ
(sin ϕ)2λ−1dϕdµ(z)
= Qλn(t)
Z 1
−1
f(z) Qλn(z) dµ (z) = Qλn(t) f∧(n) .
The property (d) is evidently by definition. Remain prove (c). Let xn is greater root of Pnλ(x), one deduces from Lemma 1.1 (a), (b) and Lemma 1.2 (c) for n ∈ N that
(17) |f∧(n)| =
(f − τxnf )∧(n)
¬ kf − τxnkX. By Stiltyes inequality ([9], p.131)
ν−1
2
π
n ¬ xν¬ πν
n + 1, ν = 1, 2, ...h n 2
i,
(18) lim
ν→∞xν= 1.
The assertion (e) follows from Lemma 1.2 (b), (18) and (19). Part (e) is a Rieman- Lebesgue type result.
For functions f, g defined on [−1, 1] the (Gegenbauer-) convolution is given by
(f ∗ g) (x) =Z 1
−1
g (u) (τuf ) (x) dµ (u) , whenever the integral exists.
Lemma 1.3 If f ∈ X, g ∈ L1,λ, then f ∗g exists (a.e.) and belongs X. Furthermore, one has
(a) (f ∗ g) (x) = (g ∗ f) (x),
(b) kf ∗ gkX¬ kfkXkgk1,λ, (X = Lp,λ, 1 ¬ p < ∞,) (c) (f ∗ g)∧(n) = f∧(n) g∧(n), n ∈ P .
Proof We prove (a). By definition
(f ∗ g) (x)
= Z 1
−1
Γ λ +12 Γ 12
Γ (λ) Z π
0
f xu+p
1 − x2p
1 − u2cos ϕ
(sin ϕ)2λ−1dϕ
!
g(u)dµλ(u) . By substituting z = xu +√
1 − x2√
1 − u2cos ϕ we obtain (f ∗ g) (x)
= Γ λ +12 Γ 12
Γ (λ)
Z xu+√
1−x2√
1−u2 xu−√
1−x2√
1−u2 1 − x2− u2− z2+ 2xuzλ−1
f(z) dz
!
g(u) dµλ(u)
= 1 − x212−λ Γ λ +12 Γ 12Γ (λ)
× Z 1
−1
Z xz+√
1−x2√
1−z2 xz−√
1−x2√
1−z2 1 − x2− u2− z2+ 2xuzλ−1
g(u) du
!
f(z) dµλ(z) .
Again lay u = xz +√
1 − x2√
1 − z2cos ϕ, we’ll have (f ∗ g) (x) =
Z 1
−1
f (u) (τug) (x) dµλ(u) = (g ∗ f) (x) .
We prove (b). If X = Lp,λ[−1, 1], 1 ¬ p < ∞, then by Minkowski inequality ([4], p.179)
Z
X
Z
Y
f (x, u) dy
p
dx
1p
<
Z
Y
Z
X
fp(x, y) dx
1p
dy we obtain
kf ∗ gkp,λ= Z 1
−1
Z 1
−1
g (u) (τxf ) (u) dµλ(u)
p
dµλ(x)
!1p
¬ Z 1
−1kτufkp,λ|g (u)| dµλ(u) (19) = kτufkp,λkgk1,λ¬ kfkp,λkgk1,λ.
Let now X = L01,λ, then
kf ∗ gkC= sup
|x|¬1
Z 1
−1
g (u) (τxf ) (u) dµλ(u)
¬ sup
|x|¬1
Z 1
−1|g (u)| |(τxf ) (u)| dµλ(u) (20) ¬ kτxfkCkgk1,λ¬ kfkCkgk1,λ.
The property (b) follows from (19) and (20). Remain prove (c). Of Fubinis theorem (see [5], p.379) and Lemma 1.2 (c) we obtain
(f ∗ g)∧(n) = Z 1
−1
Z 1
−1
g (u) (τxf ) (u) dµλ(u)
Qλn(x) dµλ(x)
=Z 1
−1
g (u)
Z 1
−1
(τuf ) (x) Qλn(x) dµλ(x)
dµλ(u)
= f∧(n)Z 1
−1
g (u) Qλn(u) dµλ(u) = f∧(n) g∧(n) .
Lemma is proved.
2. Gegenbauer derivative and integral. We start with the definition of a strong (or norm-) derivative.
Definition 2.1 If for f ∈ X there exists g ∈ X such that
t→1−0lim
f (· ) − (τtf ) (· )
1 − t − g ( · )
X
= 0,
then g is called the strong Gegenbauer derivative of f, denoted by Df. For any r ∈ N the r−th strong derivative of f is defined with D0f = f by Drf = D Dr−1f, whenever this is meaningful. The set of all f ∈ X for which Drf exists as an element of X is denoted by WXr.
Lemma 2.2 If f ∈ WXr, r∈ N, then
(21) (Drf )∧(n) =
n (n + 2λ) 2λ + 1
r
f∧(n) , (n ∈ P ).
Proof Let r = 1. Using of Lemma 1.2 (c) and, Lemma 1.1 (b, a) we obtain
1 − Qλn(t)
1 − t f∧(n) − (Df)∧(n) =
f (·) − (τtf ) (·)
1 − t − (Df) (·)
∧
(n)
¬
f (·) − (τtf ) (·)
1 − t − (Df) (·) X.
Since the right-hand side tends to zero as t → 1 − 0, it follows that
t→1−0lim
1 − Qλn(t)
1 − t f∧(n) = (Df)∧(n) , (n ∈ P ).
Taking into account (5) and (6) we have
t→1−0lim
1 − Qλn(t)
1 − t = Qλn(t)0
t=1=2λPnλ+1−1(1) Pnλ(1)
(22) = Γ (n + 2λ + 1) Γ (n + 1) Γ (2λ + 1)
Γ (2λ + 2) Γ (n) Γ (n + 2λ) =n (n + 2λ) 2λ + 1 , then for r = 1
(Df)∧(n) = n (n + 2λ)
2λ + 1 f∧(n) , (n ∈ P ).
The result for r 2 follows by induction.
Lemma is proved.
A simple consequence of this result is
Corollary 2.3 f ∈ WXr and Drf = 0 (a.e.) for some r ∈ N holds if and only if f = const (a.e.).
Proof Let Drf = 0 (a.e.); then f∧(n) = 0 for all n ∈ N by Lemma 2.1, which in turn implies f = const (a.e.) by Lemma 1.1 (d). The converse follows from the definition of Drf since (τtf ) (x) = f (x) (a.e.) if f = const (a.e.). In order to define an inverse operator to Dr one has to look for a function ψr ∈ L1,λ(−1, 1) whose Gegenbauer transform is given by
(23) ψr∧(n) =
2λ + 1 n (n + 2λ)
r
, n∈ N,
Proposition 2.4 The function
ψ1(u) = (2λ + 1)Z u
−1
1 − x2−λ−12Z x
−1 1 − t2λ−12
dt
dx, u∈ (−1, 1) ,
ψr(u) = (ψ1∗ ψr−1) (u) , r = 2, 3, ...
belong to L1,λ(−1, 1) for each r ∈ N and their Gegenbauer coefficients are given by (23).
Proof First we’ll show that ψr∈ L1,λ(−1, 1) Z 1
−1 1 − u2λ−12
ψ1(u) du
=Z 1
−1 1 − u2λ−12
(
(2λ + 1)Z u
−1
"
1 − x2−λ−12Z x
−1
(1 − t)λ(1 + t)λ−12 (1 − t)12 dt
# dx
) du
¬ (2λ + 1) Z 1
−1 1 − u2λ−12
(Z u
−1
2λ(1 + x)−λ−12 (1 − x)λ+1
Z x
−1
(1 + t)λ−12dt
dx
) du
= 2λ+1 Z 1
−1 1 − u2λ−12Z u
−1
dx (1 − x)λ+1du
= 2λ+1 λ
Z 1
−1 1 − u2λ−12
(1 − u)−λ− 2−λ du
=2λ+1 λ
Z 1
−1 1 − u2λ−12(1 − u)−λdu− 2 λ
Z 1
−1 1 − u2λ−12du
= 2λ+1 λ
Z 1
−1
(1 + u)λ 1 − u2−12
du−2 λ
Z 1
0 (1 − u)λ−12u−12du
= 22λ+1 λ
Z 1
−1
√ du
1 − u2 −2 λ
Γ λ +12Γ 12 Γ (λ + 1)
= 22λ+1π
λ − 2
λ
Γ λ +12 √π Γ (λ + 1) = 2
λ
√π 4λ√
π−Γ λ + 12 Γ (λ + 1)
!
= Cλ,
from where follows that
kψ1k1,λ¬ Cλ. Now we’ll show that
ψr∈ L1,λ(−1, 1) , r = 2, 3, ....
Assume r = 2. Using Lemma 1.3 (b) we write
kψ2k1,λ= kψ1∗ ψ1k1,λ¬ kψ1k21,λ¬ Cλ2. The result for r 3 follows by induction
kψrk1,λ= kψ1∗ ψr−1k1,λ¬ kψ1k1,λkψr−1k1,λ¬ Cλr.
Now we show that for ψr(u) , r ∈ N the equality (23) holds. According of Lemma 1.3 it suffices to consider the case r = 1. Let n ∈ N. Using the differential equation (see [9], p.73)
1 − x2 d2
dx2Qλn(x) − (2λ + 1) x d
dxQλn(x) + n (n + 2λ) Qλn(x) = 0 and integrating by parts, we find
ψ1∧(n) = (2λ + 1) Z 1
−1
Z u
−1
1 − x2−λ−12Z x
−1
1 − t2λ−12dt
dx
Qλn(u) dµλ(u)
= − 2λ + 1 n(n + 2λ)
Z 1
−1
Z u
−1
1 − x2−λ−12Z x
−1
1 − t2λ−12
dt
dx
dh
1 − u2λ+12 d
duQλn(u)i
= 2λ + 1 n(n + 2λ)
Z 1
−1
d duQλn(u)
Z u
−1
dµλ(t)
du
= 2λ + 1 n(n + 2λ)
Z 1
−1
Z u
−1
dµλ(t)
dQλn(u)
= 2λ + 1
n(n + 2λ)Qλn(1) Z 1
−1
dµλ(t)
− 2λ + 1 n(n + 2λ)
Z 1
−1
Qλn(u) dµλ(u) = 2λ + 1 n(n + 2λ),
where we have also used Lemma 1.1 (c) and (7). For r = 1 the equality (23) is proved. Now assume that
ψ∧r−1(n) =
2λ + 1 n (n + 2λ)
r−1
, then by Lemma 1.3 (c) we have
ψr∧(n) = (ψ1∗ ψr−1)∧(n) = ψ∧1 (n) ψr−1∧ (n) =
2λ + 1 n (n + 2λ)
r
.
The Proposition 2.4 is proved.
For r ∈ N the Gegenbauer integral Ir can now be defined as (24) (Irf ) (x) := (f∗ ψr) (x) , (x ∈ [−1; 1] ; f ∈ X).
Proposition 2.5 The integral Iris for each r ∈ N a bounded linear operator from X into itself, which satisfies for f ∈ X and r, s ∈ N
(a) (IrIsf ) (x) = (IsIrf ) (x) = (Ir+sf ) (x), (a.e.) (b) (Irf )∧(n) = 2λ+1
n(n+2λ)
r
f∧(n), n ∈ N, (c) for f ∈ WXr one has
(25) (IrDrf ) (x) = f (x)− c, (a.e.) (d) for any f, g ∈ X the equality is fair:
Z 1
−1
(Irf ) (x) g (x) dµλ(x) =Z 1
−1
f (x) (Irg) (x) dµλ(x) .
Proof The linearing of the operator Iris evidently and boundedness follows from inequality (see Lemma 1.3 (b))
kIrfkX= kf ∗ ψrkX ¬ kfkXkψrk1,λ¬ crλkfkX. We prove (a). By definition
ψr+s= ψ1∗ ψr+s−1= ψ1∗ (ψ1∗ ψr+s−2)
= (ψ1∗ ψ1) ∗ ψr+s−2= ψ2∗ ψr+s−2= ... = ψr∗ ψs, from where we have
(IrIsf ) (x) = (Isf∗ ψr) (x) = ((f ∗ ψs) ∗ ψr) (x) (f ∗ (ψs∗ ψr)) (x) = (f ∗ ψr+s) (x) = Ir+sf(x) . We’ll prove (b). By the Proposition 2.4 and Lemma 1.3 (c) we have
(Irf )∧(n) = (f ∗ ψr)∧(n) = f∧(n) ψr∧(n) =
2λ + 1 n (n + 2λ)
r
f∧(n) , n ∈ N.
We’ll prove (c). By the Proposition 2.2 (b) and (21) we obtain (26) (IrDrf )∧(n) =
2λ + 1 n (n + 2λ)
r
(Drf )∧(n) = f∧(n) , n ∈ N
Since Qλ0(x) = 1 (see [6], p.93), then by orthoganality of polynomials Qλn(x) we have
c∧(n) = c Z 1
−1
Qλ0(x) Qλn(x) dµλ(x) =
0, n ∈ N, c, n = 0.