© Hindawi Publishing Corp. MODULUS OF SMOOTHNESS AND THEOREMS CONCERNING APPROXIMATION ON COMPACT GROUPS

We consider the generalized shift operator defined by (Shuf)(g)=∫Gf(tut−1g)dt on a compact group G, and by using this operator, we define spherical modulus of smoothness. So, we prove Stechkin and Jackson-type theorems.


Introduction.
In this paper, we prove some theorems on absolutely convergent Fourier series in the metric space L 2 (G), where G is 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 on T and R, the theory of absolutely convergent Fourier series is a fruitful source of questions about the corresponding entity for compact groups. By using some absolute convergence theorems 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 obtained. 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.
Let G be a compact group with a dual spaceĜ, dg denote the Haar measure on G normalized by the condition G dg = 1, and G f (g)dg denote the Haar integral of a function f on G. Let U α , α ∈Ĝ denotes the irreducible unitary representation of G in the finite dimensional Hilbert space V α . We reserve the symbol d α for the dimension of U α . Thus, d α is a positive integer. Also, we denote by χ α and t α ij (i, j = 1, 2,...,d α ), α ∈Ĝ the character and matrix elements (coordinate functions) of U α , respectively.
Let L p (G) be the space of all functions f equipped with the norm We write · p instead of · Lp (G) , and L ∞ = C is the corresponding space of continuous functions, and f = max{|f (g)| : g ∈ G}. As it is known (see [4] or [10, page 99]), the space L 2 (G) can be decomposed into the sum where This theorem is one of the most important results of the harmonic analysis on compact groups. The orthogonal projection Y α : L 2 (G) → H α is given by where (Y α f )(g) does not depend on the choice of a basis in L 2 . Carrying out this construction for every space H α , α ∈Ĝ, we obtain an orthonormal basis in L 2 consisting of the functions d α t α ij , α ∈Ĝ, 1 ≤ i, j ≤ d α . Any function f ∈ L 2 (G) can be expanded into a Fourier series with respect to this basis where the Fourier coefficients a α ij are defined by the following relations: such that t α ij (g) = t α ij (g −1 ), where g −1 is the inverse of g. Note that (2.5) is a convergent series in the mean and that the Parseval's equality holds. The aforementioned result of harmonic analysis on a compact group can be found, for example, in [4,5,7,10]. We denote by Sh u the generalized translation operator on compact group G defined by (2.8) where u, g ∈ G and E is the identity operator. We set in which Sh 0 u f = f and Sh u (Sh i−1 u f ) = Sh i u f , i = 1, 2,...,k and k ∈ N. We note that α is a complicated index. SinceĜ is a countable set, there are only countably many α ∈Ĝ for which α α ij ≠ 0 for some i and j; enumerate them Because of that, the symbol "α < n" is interpreted as {α 0 ,α 1 ,...,α n−1 } ⊂Ĝ, and α ≥ n denotes the setĜ\(α < n). Let d α , as usual, be the dimension of U α . For typographical convenience, we write d n for the dimension of the representation U αn , n = 1, 2,.... (See [5, page 458].) We denote by E n (f ) p the approximation of the function f ∈ L p (G) by "Spherical" polynomials of degree not greater than n: The sequence of best approximations {E n (f ) p } ∞ n=0 is a constructive characteristic of the function f . In the capacity of structural characteristic of the function f on a compact group G, we define its Spherical modulus of smoothness of order k by where W τ is a neighborhood of e in G. In other words, where ρ is a pseudometric on G and τ is any positive real number. It is easy to show the following properties of ω k (f , τ) p : Proof. Using the orthogonality relations and other formulas for matrix elements t α ij (g) (see [7, page 189]), we have This proves the lemma.
The following formula is the particular event of the above lemma: It can be called a Weyl formula. We note that the expansion (2.5) is connected with the expansion which is defined by (2.4), that is, by the equality Thus, the coefficients a α ij are defined by (2.6). Using Lemma 3.1 and the definition of Y α , we obtain (3.6) The following are simple facts with frequent usage: if f ∈ L p , then (1) Sh u f p ≤ f p ; (2) f − Sh u f p → 0 as u → e; (3) (Y α (Sh u f ))(g) = (χ α (u)/χ α (e))(Y α f )(g) for all α ∈Ĝ. We note that χ α (e) = d α . Proof. Let f ∈ L 2 and S n (f , g) denote the nth partial sum of the Fourier series (2.5), that is, Using Parseval's equality for the compact group G, we have (3.9) Using (3), it is not hard to see that Consequently, ( k f )(g) = α∈Ĝ (1−χ α (u)/d α ) k a α ij t α ij . By another application of Parseval's equality, we obtain k u f (3.11) Now, using Bernolly's inequality (3.12) Consequently, (3.13) therefore, Re χ α (u) d α a α ij 2 . (3.14) Let Φ Wτ be a nonnegative integrable function vanishing outside W τ and satisfying the condition G Φ Wτ (g)dg = 1. For example, we can take Φ Wτ = ξ Wτ /µ(W τ ), where µ(W τ ) is the Haar measure of W τ and ξ Wτ is the characteristic function of W τ . Multiplying both sides of (3.14) by Φ W 1/n , and integrating with respect to u on G, and using the equality G |χ α | 2 dg = 1 (see [7, page 195]), we obtain (3.15) Therefore, it is not hard to see that Finally, we obtain which proves the theorem.
This theorem is given without proof in [8] for the case where k = 1. We note that the matrix elements of unitary representations t α ij (g) satisfy the relations In particular, we have for all α ∈Ĝ and i, j = 1, 2,...,d α . Furthermore, it is obvious that |a α ij t α ij (g)| ≤ |a α ij |; therefore, according to the sufficient condition for absolutely convergent Fourier series on the group G, the series α∈Ĝ α i,j=1 |a α ij | is convergent. Let A(G) := {f : α∈Ĝ α i,j=1 |a α ij | < +∞}. Using Theorem 3.2, and repeating the proof of analogous theorems (see [1,Chapter IX] or [6, Chapter II]) with some changes, we obtain the following theorems.
This theorem is analogous to the Szasz theorem of the classical Fourier series in the case where k = 1 and G = T .
This theorem is also analogous to a theorem in trigonometric case proved by Stechkin [9]. (2). The group SU(2) consists of unimodular unitary matrices of the second order, that is, matrices of the form