PII. S0161171203204269 http://ijmms.hindawi.com © Hindawi Publishing Corp.
MODULUS OF SMOOTHNESS AND THEOREMS CONCERNING
APPROXIMATION ON COMPACT GROUPS
H. VAEZI and S. F. RZAEV
Received 3 April 2002
We consider the generalized shift operator defined by(Shuf )(g)=
Gf (tut−1g)dt
on a compact groupG, and by using this operator, we define “spherical” modulus of smoothness. So, we prove Stechkin and Jackson-type theorems.
2000 Mathematics Subject Classification: 42C10, 43A77, 43A90.
1. Introduction. In this paper, we prove some theorems on absolutely con-vergent Fourier series in the metric spaceL2(G), whereGis a compact group.
The algebra of absolutely convergent Fourier series is a subject matter about which a good deal, although far from everything, is known (see [5, page 328]). Like many branches of harmonic analysis onT andR, the theory of absolutely convergent Fourier series is a fruitful source of questions about the corre-sponding entity for compact groups. By using some absolute convergence the-orems of the classical Fourier series, (see [1,11]), a generalized form of Stechkin [6] and Szasz theorem [1,11] of the Fourier series on compact groups is ob-tained. Thus, we solve open problems formulated in [5, page 366] (see also [3, Chapter I, page 9]).
2. Preliminaries and notation. Now, we explain some of the notation and terminologies used throughout the paper.
LetGbe a compact group with a dual space ˆG,dgdenote the Haar measure onGnormalized by the conditionGdg=1, andGf (g)dg denote the Haar integral of a functionf onG. LetUα,α∈Gˆdenotes the irreducible unitary representation ofGin the finite dimensional Hilbert spaceVα. We reserve the symboldαfor the dimension ofUα. Thus,dαis a positive integer. Also, we de-note byχαandtα
ij(i, j=1,2, . . . , dα),α∈Gˆthe character and matrix elements (coordinate functions) ofUα, respectively.
LetLp(G)be the space of all functionsf equipped with the norm
fp=
G
f (g)p dg
1/p
. (2.1)
or [10, page 99]), the spaceL2(G)can be decomposed into the sum
L2(G)=
α∈Gˆ
⊕Hα, (2.2)
where
Hα=f∈C(G):f (g)=trUα(g)C , C=HomVα, Vα . (2.3)
This theorem is one of the most important results of the harmonic analysis on compact groups. The orthogonal projectionYα:L2(G)→Hαis given by
Yαf (g)=dα
Gf (h)χα
gh−1 dh, (2.4)
where(Yαf )(g)does not depend on the choice of a basis inL2. Carrying out
this construction for every spaceHα,α∈G, we obtain an orthonormal basisˆ inL2 consisting of the functions
dαtα
ij, α∈G,ˆ 1≤i, j≤dα. Any function f∈L2(G)can be expanded into a Fourier series with respect to this basis
f (g)= α∈Gˆ
dα
i,j=1
aα
ijtαij(g), (2.5)
where the Fourier coefficientsaα
ij are defined by the following relations:
aα ij=dα
G f (g)tα
ij(g)dg, (2.6)
such thattijα(g)=tijα(g−1), whereg−1is the inverse ofg. Note that (2.5) is a
convergent series in the mean and that the Parseval’s equality
G
f (g)2dg= α∈Gˆ
1 dα
dα
i,j=1
aαij2 (2.7)
holds. The aforementioned result of harmonic analysis on a compact group can be found, for example, in [4,5,7,10].
We denote by Shuthe generalized translation operator on compact groupG defined by
Shuf (g)=
G
ftut−1g dt,
uf (g)=f (g)−Shuf (g)=E−Shu f ,
whereu, g∈GandEis the identity operator. We set
k uf= u
k−1
u f
=E−Shu kf= k
i=0
(−1)k+iCi kSh
i
uf , (2.9)
in which Sh0uf=f and Shu(Sh i−1
u f )=Sh i
uf,i=1,2, . . . , kandk∈N.
We note thatαis a complicated index. Since ˆGis a countable set, there are only countably manyα∈Gˆfor whichαα
ij≠0 for someiandj; enumerate them as{α0, α1, . . . , αn, . . .}. So,dα0< dα1< dα2<···< dαn<···. Because of that, the symbol “α < n” is interpreted as{α0, α1, . . . , αn−1} ⊂G, andˆ α≥ndenotes
the set ˆG\(α < n). Letdα, as usual, be the dimension ofUα. For typographical convenience, we writedn for the dimension of the representationUαn, n= 1,2, . . . .(See [5, page 458].)
We denote byEn(f )pthe approximation of the functionf∈Lp(G)by “Spher-ical” polynomials of degree not greater thann:
En(f )p=inf
f−Tnp:Tn∈
α<n,α∈Gˆ
⊕Hα
. (2.10)
The sequence of best approximations{En(f )p}∞n=0is a constructive
charac-teristic of the function f. In the capacity of structural characteristic of the functionfon a compact groupG, we define its Spherical modulus of smooth-ness of orderkby
ωk(f;τ)p=supE−Shu kfp:u∈Wτ, (2.11)
whereWτ is a neighborhood ofeinG. In other words,
Wτ=u:ρ(u, e) < τ, u∈G, (2.12)
whereρis a pseudometric onGandτ is any positive real number. It is easy to show the following properties ofωk(f , τ)p:
(a) limτ→0ωk(f , τ)p=0;
(b) ωk(f , τ)p is a continuous monotonically increasing function with re-spect toτ;
(c) ωk(f1+f2, τ)p≤ωk(f1, τ)p+ωk(f2, τ)p;
(d) ωk+l(f , τ)p≤2lωk(f , τ)p, l=1,2, . . . .
Lemma3.1. The following equality holds for allu, g∈G:
Shutα ij
(g)=χα(u) dα t
α
ij(g). (3.1)
Proof. Using the orthogonality relations and other formulas for matrix
elementstijα(g)(see [7, page 189]), we have
G tα
ij
tut−1g dt=
dα
p=1
dα
q=1
tα
qp(u)tijα(g)
G tα
iq(t)tαqp(t)dt
= 1 dα
dα
p=1
tα
pp(u)tijα(g)= 1 dαχα(u)t
α ij(g).
(3.2)
This proves the lemma.
The following formula is the particular event of the above lemma:
G
χαtut−1g dt=χα(u)χα(g)
dα . (3.3)
It can be called a Weyl formula.
We note that the expansion (2.5) is connected with the expansion
f (g)= α∈G
Yα(f )(g), Yα∈Hα, (3.4)
which is defined by (2.4), that is, by the equality
Yα(f )(g)= dα
i,j=1
aαijtijα(g). (3.5)
Thus, the coefficientsaα
ij are defined by (2.6). UsingLemma 3.1and the defi-nition ofYα, we obtain
YαShuf (g)= dα
i,j=1
aαij
Gt α ij
tut−1g dt
= dα
i,j=1
aαijχα(u) dα t
α ij(g)
=χα(u)dα Yα(f )(g).
(3.6)
The following are simple facts with frequent usage: iff∈Lp, then (1) Shufp≤ fp;
(2) f−Shufp→0 asu→e;
Theorem3.2. Iff∈L2andfis not constant, then
En(f )2≤
dn dn−2kωk
f;1 n
2
, n=1,2, . . . . (3.7)
Proof. Letf∈L2andSn(f , g)denote thenth partial sum of the Fourier
series (2.5), that is,
Sn(f , g)= α<n
dα
i,j=1
aαijtαij(g)= n p=0 dαp i,j=1
aαijptijαp(g). (3.8)
Using Parseval’s equality for the compact groupG, we have
E2n(f )2=f−Sn(f ) 2 2=
α≥n 1 dα dα i,j=1
aαij2. (3.9)
Using (3), it is not hard to see that
Yαkf (g)=
1−χα(u)dα
k
Yαf (g), α∈G.ˆ (3.10)
Consequently,(kf )(g)=
α∈Gˆ(1−χα(u)/dα)kaαijt α
ij. By another application of Parseval’s equality, we obtain
k uf
2 2=
α∈Gˆ
1 dα dα i,j=1
1−χα(u) dα
2kaα
ij 2 ≥ α≥n 1 dα dα i,j=1
1−χα(u) dα
2kaα
ij
2
= α≥n
1 dα dα i,j=1
1−2 Reχα(u)dα +χα(u)
2
d2α
k
aαij2.
(3.11)
Now, using Bernolly’s inequality(1+x)k≥1+kxforx≥ −1, we obtain
k uf 2 2≥ α≥n 1 dα dα i,j=1
1−2kReχα(u)
dα +
kχα(u)2 d2 α aα ij 2 . (3.12) Consequently, k uf 2 2≥ α≥n 1 dα dα
i,j=1
aαij2− α≥n
1 dα
dα
i,j=1
2kReχα(u) dα
aαij2; (3.13)
therefore,
E2
n(f )2≤kuf
2
LetΦWτbe a nonnegative integrable function vanishing outsideWτand satisfy-ing the conditionGΦWτ(g)dg=1. For example, we can takeΦWτ=ξWτ/µ(Wτ), whereµ(Wτ)is the Haar measure ofWτandξWτis the characteristic function ofWτ. Multiplying both sides of (3.14) byΦW1/n, and integrating with respect touonG, and using the equalityG|χα|2dg=1 (see [7, page 195]), we obtain
G E2
n(f )2ΦW1/n(u)du≤
G
k
uf
2
2ΦW1/ndu
+2k α≥n
1 d2α
dα
i,j=1 aαij
2
G
χα(u)ΦW
1/n(u)du
≤supk uf
2
2+
2k dn
α≥n 1 dα
dα
i,j=1 aαij2.
(3.15)
Therefore, it is not hard to see that
En2(f )2≤ω2k
f ,1 n
2
+2kdnEn2(f )2. (3.16)
Finally, we obtain
En(f )2≤
dn dn−2kωk
f ,1 n
2
, (3.17)
which proves the theorem.
This theorem is given without proof in [8] for the case wherek=1. We note that the matrix elements of unitary representationstijα(g)satisfy the relations
dα
j=1
tα
ij(g)tkjα(g)= dα
j=1
tα
ij(g)tjkα(g)=
0 ifi≠k,
1 ifi=k. (3.18)
In particular, we have
dα
j=1
tijα2=1 ⇒tijα(g)≤1 (3.19)
for allα∈Gˆandi, j=1,2, . . . , dα. Furthermore, it is obvious that|aα
ijtijα(g)| ≤ |aα
ij|; therefore, according to the sufficient condition for absolutely convergent Fourier series on the groupG, the seriesα∈Gˆ
α i,j=1|a
α
ij|is convergent. Let A(G):= {f :α∈Gˆ
α
Theorem3.3. Iff (g)∈L2(G), then
∞
n=1
ωk(f ,1/n)2
√n <+∞ ⇒f (g)∈A(G). (3.20)
This theorem is analogous to the Szasz theorem of the classical Fourier series in the case wherek=1 andG=T.
Theorem3.4. Iff (g)∈L2(G), then
∞
n=1
En(f )2
√
n <+∞ ⇒f (g)∈A(G). (3.21)
This theorem is also analogous to a theorem in trigonometric case proved by Stechkin [9].
4. Applications to compact groupSU(2). The group SU(2)consists of uni-modular unitary matrices of the second order, that is, matrices of the form
u=
α β
−β α
, |α|2+|β|2=1. (4.1)
Therefore, each elementuof SU(2)is uniquely determined by a pair of com-plex numbersαandβsuch that|α|2+|β|2=1. We have (see [5]) the relation
“(α, β)(φ, θ, ψ),” whereαβ≠0,|α|2+ |β|2=1, and the parametersφ, θ,
andψare called Euler angles defined by
|α| =cosθ
2; Argα= φ+ψ
2 ; Argβ= φ−ψ
2 . (4.2)
Letφ,θ, andψsatisfy the conditions
0≤φ <2π , 0≤θ < π , −2π≤ψ <2π . (4.3)
Also, we know that the dimension of the representationTlof SU(2)is equal to 2l+1, wherel=0,1/2,1, . . . and the matrix elements ofTlfor group SU(2) are defined by
tl
mn(u)=e−(nψ+mφ)Pmnl (cosθ)i(m−n). (4.4)
Expressingtl
mn(u)in terms ofPmnl (cosθ), we arrive at the following conclu-sion:
Any functionf (φ, θ, ψ), 0≤φ <2π, 0≤θ < π, and−2π≤ψ <2π belong-ing to the spaceL2(SU(2))such that
2π
−2π
2π
0 π
0
can be expanded into the mean-convergent series
f (φ, θ, ψ)= l
l
m=−l l
n=−l αl
mne−i(mφ+nψ)Pmnl (cosθ), (4.6)
where
αl mn=
2l+1 16π2
2π
−2π
2π
0 π
0
f (φ, θ, ψ)ei(mφ+nψ)Pl
mn(cosθ)sinθ dθdφdψ. (4.7) In addition, we obtain from Parseval’s equality that
l l
m=−l l
n=−l 1 2l+1α
l mn
2
= 1 16π2
2π
−2π
2π
0 π
0
f (φ, θ, ψ)2sinθ dθdφdψ.
(4.8) UsingTheorem 3.2, we obtain the following theorem.
Theorem4.1. Iff (φ, θ, ψ)∈L2(SU(2)), then
En(f )2≤
1+n−21ωk
f ,1 n
2
,
l≥n l
m=−l l
n=−l 1 2l+1α
l mn
2 1/2
≤
1+ 2 n−1ωk
f ,1 n
2
.
(4.9)
Using the relation between the polynomial Pn(α,β)(z)and Pmnl (z), we con-clude that
Pl
mn(z)=2−m
(l−m)!(l+m)!
(l−n)!(l+n)!
1/2
(1−z)(m−n)/2(1+z)(m+n)/2P(m−n,m+n) l−m .
(4.10) The Jacobi polynomials obtained here are characterized by the condition that αandβare integers andn+α+β∈Z+.
Now, we consider the following case.
LetL(α,β)2 [−1,1]be the Hilbert space of the functionsfdefined on the seg-ment[−1,1]with the scalar product
f1, f2 = 1
−1
f1(x)f2(x)(1−x)α(1+x)βdx; (4.11)
then, any functionfin this space is expanded into the mean-convergent series
f (x)= ∞
n=0
where the polynomials ˆPn(α,β)(x)are given by
ˆ
Pk(α,β)(x)=2−(α+β+1)/2
!
k!(k+α+β)!(α+β+2k+1) (k+α)!(k+β)!
"1/2
Pk(α,β)(x), (4.13)
αn=
1
−1
f (x)Pˆn(α,β)(x)(1−x)α(1+x)βdx. (4.14)
The Parseval’s equality
1
−1
f (x)2(1−x)α(1+x)βdx= ∞
n=0
|α|2 (4.15)
holds. The formulas (4.12), (4.14), and (4.15) are proved for integral nonnega-tive values ofαandβ. We can show that they are valid for arbitrary real values ofαandβexceeding−1. Finally, we reach the following theorem.
Theorem4.2. Iff (x)∈L2[−1,1], then the following hold for Jacobi series:
En(f )2≤
1+n−21ωk
f ,1 n
2
,
∞
l=n
αl2
1/2
≤
1+ 2 n−1ωk
f ,1 n
2.
(4.16)
Note. For the ideas similar to this paper we refer to [2] and its references.
Acknowledgments. This research was supported by Tabriz University.
We would like to thank the research office of Tabriz University for its support.
References
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H. Vaezi: Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
E-mail address:[email protected]
S. F. Rzaev: Institute of Mathematics and Mechanics, Azerbaijan Academy of Sciences, Baku, Azerbaijan
