GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 4, 1999, 379-394

ON THE APPROXIMATION OF FUNCTIONS ON LOCALLY COMPACT ABELIAN GROUPS

D. UGULAVA

Abstract. Questions of approximative nature are considered for a space of functions L^{p}(G, μ), 1 ≤ p ≤ ∞, defined on a locally compact abelian Hausdorff group G with Haar measure μ. The approximating subspaces which are analogs of the space of exponential type entire functions are introduced.

Let us consider a locally compact abelian group G with Haar measure μ, assuming that the topology of the space G is Hausdorff. By L^{p}(G, μ) or, simply, by L^{p}(G) we denote a space of real- or complex-valued functions defined on G and integrable on it with respect to the measure μ to the p-th power with the usual norm ∥f∥_{p} = {∫_{G} |f|^{p}dμ}^{1/p} for 1 ≤ p < ∞. L^{∞}(G, μ) is a space of functions, essentially bounded on G with respect to μ and having norm ∥f∥_{∞} = vrai sup_{g∈G} |f(g)|. We shall briefly recall some definitions

and facts from the theory of commutative harmonic analysis. A unitary irreducible representation of G, i.e., a complex-valued continuous function χ on G with the properties |χ(g)| = 1, ∀g ∈ G and χ(g_{1}g_{2}) = χ(g_{1}) · χ(g_{2}), ∀g_{1}, g_{2} ∈ G, is called a character of the group G. An Abelian group structure is naturally introduced into the set of all characters defined on G. The obtained group is denoted by Ĝ and called a group dual to G. Ĝ is usually topologized by the following two topologies: the first one is the weakest topology containing continuous functionals f̂ defined by the formula

f̂(χ) = ∫_{G} f(g)\overline{χ(g)} dμ(g) ≡ ∫_{G} f(g)\overline{χ(g)} dg, χ ∈ Ĝ, f ∈ L^{1}(G, μ), (1)

and the second one is the topology of uniform convergence of characters on compact subsets of the group G. These topologies are equivalent and with their aid Ĝ transforms to a locally compact Abelian group. A function f̂ of

379 form (1) defined on Ĝ for f ∈ L^{1}(G) is called the Fourier transform of the function f. Similarly to (1), we define the inverse transform f̌ of f by the formula

1991 Mathematics Subject Classification. 41A65, 41A35.

Key words and phrases. Locally compact abelian group, L^{p} space, character, Fourier transform, entire functions of exponential type, best approximation.

1072-947X/99/0700-0379$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

f̌(χ) = ∫_{G} f(g)χ(g) dg.

Of much importance here is the Pontryagin duality principle by which

the natural mapping of G into ˆ Ĝ, which to an element g ∈ G assigns the character f_{g} on Ĝ, is an isomorphism of topological groups. The notions of direct and inverse Fourier transform can be extended by the well known technique to the case of spaces L^{p}(G, μ) for 1 < p ≤ 2. By Plancherel’s theorem the Fourier transform is a linear isometry of L^{2}(G) on L^{2}(Ĝ) but the inverse Fourier transform is a linear isometry from L^{2}(Ĝ) on L^{2}(G). These mappings are mutually inverse ([1], v. 2, §31). Usually, Haar measures on G and Ĝ are normalized so that the inversion formula f = (f̂)ˇ holds for functions f ∈ L^{1}(G), f̂ ∈ L^{1}(Ĝ). These measures are called mutually dual and pairs of such measures will be considered below. For f ∈ L^{p}(G, μ), 1 ≤ p ≤ 2, we have the Hausdorff–Young inequality by which

∥f̂∥_{q} ≤ ∥f∥_{p},  (2)

where q (here and in what follows) is the conjugate number of p (\frac{1}{p} + \frac{1}{q} = 1).

A great part of the approximation theory deals with questions of approximation of functions on an n-dimensional Euclidean space ℝ^{n} and on an n-dimensional torus 𝕋^{n} which are Abelian groups. As an approximating subspace, a space of exponential type entire functions and a space of trigonometric polynomials [2] are usually taken as an approximating subspace in the first and the second case, respectively. We know that ℝ̂^{n} = ℝ^{n} and 𝕋̂ = Z to within an isomorphism, where Z is a set of integer numbers. In both cases the Fourier transforms of elements from the approximating space have compact supports in dual spaces ([2], Ch. 3, §§3.1, 3.2). When investigating the problem of approximation of functions defined on compact or locally compact Abelian groups one should consider sets with properties similar to the properties of approximating sets of some well known classical groups. For example, by analogy with groups ℝ^{n} or 𝕋^{n} one can try to consider as an approximating set the set of functions on G whose Fourier transform lies in some compact K of the dual space Ĝ. But such an approach cannot simultaneously be used for all spaces L^{p}(G, μ), 1 ≤ p ≤ ∞, since, generally speaking, for p > 2 the Fourier transform of a function f ∈ L^{p} does not exist. Therefore below we shall give a modified definition of approximating subspaces which is simultaneously applicable for all p, 1 ≤ p ≤ ∞.

By U_{G} we shall denote a space of all symmetric compact sets from G which are the closures of neighborhoods of unity in G. Sets from U_{G} will be called compact neighborhoods of unity. KT = {g : g = g_{1}g_{2}, g_{1} ∈ K, g_{2} ∈ T } will stand for the product of the sets K and T, while (1)_{K} will denote the characteristic function of the set K. For arbitrary K and T from U_{G} we shall consider the function defined on G ([3], Ch. 5, §1)

V_{K,T} (g) = (mes T)^{−1} ˆ(1)_{T} (g) · ˆ(1)_{T K}(g),  (3)

which, for simplicity, will be denoted by V.

It is at once obvious that V ∈ L^{1}(G). Indeed,

∥V_{K,T} ∥ ≤ \frac{1}{mes T} ∥ ˆ(1)_{T} ∥_{2} · ∥ ˆ(1)_{T K}∥_{2} =

= \frac{1}{mes T} ∥(1)_{T} ∥_{2} · ∥(1)_{T K}∥_{2} = (\frac{mes T K}{mes T} )^{1/2}.  (4)

Also note that V is a real-valued function. This follows from the fact that

ˆ(1)_{K}(g) = ∫_{K} \overline{χ(g)} dχ = ∫_{K} χ(g^{−1}) dχ = ∫_{K} χ(g) dχ = \overline{ˆ(1)_{K}(g)}.

By Parceval’s theorem ([1], v. 2, §31) we obtain

∫_{G} V_{K,T} (g) dg = \frac{1}{mes T} ∫_{G} ˆ(1)_{T} (g) · ˆ(1)_{T K}(g) dμ =

= \frac{1}{mes T} ∫ ; _{G} (1)_{T} (χ) · (1)_{T K}(χ) dχ = \frac{1}{mes T} ∫_{T} dχ = 1.

Using the function f ∈ L^{p}(G) and kernel V, we introduce the function

P_{K,T} (f, g) = (f ∗ V_{K,T})(g) = ∫_{G} f(h) · V_{K,T} (h^{−1}g) dh.  (5)

Definition 1. Let K ∈ U_{G} and p ∈ [1, ∞]. By W^{p}(K) we shall denote the set of functions f from the space L^{p}(G, μ) for which we have the formula

f(g) = P_{K,T} (f, g) ∀g ∈ G and T ∈ U_{G}.  (6)

Remark. If G = ℝ^{n} and K = Δ_{ν} = {x(x_{1}, …, x_{n}) ∈ ℝ^{n} : |x_{i}| ≤ ν_{i}, ν = (ν_{1}, …, ν_{n})}, then the class defined by us coincides with the known class W_{νp}(ℝ^{n}) of entire functions of exponential type ν. A function f ∈ L^{p}(ℝ^{n}) belongs to W_{νp}(ℝ^{n}) ([2], Ch. 3, §3.1) if it is analytically extendable onto the whole n-dimensional complex Euclidean space ℂ^{n}, and for any ε > 0 there exists a number A_{ε} such that the inequality

|g(z)| ≤ A_{ε} · exp ∑_{j=1}^{n}(ν_{j} + ε) · |z_{j}|

is fulfilled for all z(z_{1}, …, z_{n}) ∈ ℂ^{n}.

But for f ∈ W_{νp}(ℝ^{n}) we have representation (6) ([2], Ch. 8, §8.6) with

ˆ(1)_{K} = 2^{n} ∏_{i=1}^{n}(x_{i})^{−1} sin ν_{i}x_{i}. Conversely, let (6) hold for f ∈ L^{p}(ℝ^{n}). We

take an n-dimensional cube T_{ε} = {x ∈ ℝ^{n} : |x_{i}| ≤ ε, i = 1, …, n}, ε > 0, as a neighborhood of T. The kernel V_{K,T_{ε}} will be a exponential type entire function for any vector ε with constant coordinates ε. The convolution f ∗ V_{K,T_{ε}} belongs to the same class too ([2], Ch. 3, §3.6), which by virtue of (6) means that f belongs to W_{νp}(ℝ^{n}). Thus W^{p}(K) coincides with W_{νp}(ℝ^{n}) for G = ℝ^{n} and K = Δ_{ν}. Such an equivalence can be proved by a similar reasoning for more general K ⊂ ℝ^{n} too if one uses the results of [4].

Lemma 1. W^{p}(K) is the shift-invariant closed subspace of the space L^{p}(G).

Proof. Let f_{n} ∈ W^{p}(K) be an arbitrary converging sequence and ∥f_{n} − f∥_{p} → 0 as n → ∞. Using the well known estimate of a convolution norm ([1], v. 2, §31) for L^{p}, we obtain

∥P_{K,T} (f_{n} − f, g)∥_{p} ≤ ∥f_{n} − f∥_{p} · ∥V_{K,T} ∥_{1}, ∀T ∈ U_{G}.

But P_{K,T} (f_{n}) = f_{n}, ∀n ∈ N, and thus f_{n} tends in L^{p} simultaneously to f and P_{K,T} (f). Therefore f = P_{K,T} (f) and f ∈ W^{p}(K). The operation of shift by the element h will be denoted by L_{h}. Let f ∈ W^{p}(K) and T ∈ U_{G}. We obtain

(L_{h}f)(g)=f(hg)=∫_{G} f(g_{1})V_{K,T} (g_{1}^{−1}gh) dg_{1} =∫_{G} f(ξh)V_{K,T} (ξ^{−1}h^{−1}gh) dξ =

= ∫_{G} (L_{h}f)(ξ)V_{K,T} (ξ^{−1}g) dξ = (L_{h}f ∗ V_{K,T})(g)

and Lemma 1 is proved.

Definition 2. When p ∈ [1, 2], for an arbitrary fixed compact K from the dual space Ĝ we shall denote by F^{p}(K) the set of functions from the space L^{p}(G, μ) whose Fourier transform supports belong to K. Using Parceval’s theorem ([1], v. 2, §31) by which ∫_{G} fΦ̂ dg = ∫_{G} f̂Φ dχ, substituting the function Φ(χ) = \frac{χ(g)}{mes T} ((1)_{T} ∗ (1)_{T K})(χ) into it, and calculating Φ̂ by the rule (χf ˆ)(·) = f̂(χ^{−1}·) ([1], v. 2, §31), for 1 ≤ p ≤ 2 we obtain another representation of P_{K,T} (f), namely:

P_{K,T} (f, g) = (mes T)^{−1} ∫ ; _{G} f̂(χ) · χ(g) · ((1)_{T} ∗ (1)_{T K})(χ) dχ,

where integration actually occurs on the compact T^{2}K. If G is a compact, then this representation of P_{K,T} certainly holds for all 1 ≤ p ≤ ∞.

Note the following important property of functions from F^{p}(K). If f ∈ F^{p}(K), 1 ≤ p ≤ 2, then f ∈ L^{p_{1}}(G) for arbitrary p_{1} ∈ [p, ∞]. Indeed, by virtue of (2), f̂ ∈ L^{q}(G), where q is the conjugate number of p. Since supp f̂ = K and q ≥ 2, we have f̂ ∈ L^{2}(Ĝ) and (f̂)ˇ = f almost everywhere on G. Thus f̂ ∈ L^{1}(Ĝ) ∩ L^{2}(Ĝ) and the L^{2} inverse Fourier transform of f̂ is f. But ([1], v. 2, §31) the L^{1} inverse Fourier transform of f̂ is also f. Hence it is clear that f is a continuous and bounded function on G. Since f ∈ L^{p} ∩L^{∞}, we have f ∈ L^{p_{1}}(G) for any p_{1} ∈ [p, ∞]. Moreover, for p = ∞, by (2) we obtain the inequality

∥f∥_{∞} ≤ ∥f̂∥_{1} = ∫_{K} |f̂| dχ ≤ ( ∫_{K} dχ)^{1/p}∥f̂∥_{q} ≤ (mes K)^{1/p} · ∥f∥_{p}.

Lemma 2. If 1 ≤ p ≤ 2, then the sets W^{p}(K) and F^{p}(K) coincide.

Proof. Let us first assume that f ∈ W^{p}(K). Then for ∀T ∈ U_{G} we have

{ \atop f ∗ V_{K,T}} (χ) = f̂(χ)

and therefore

f̂(χ) · V̂_{K,T} (χ) = f̂(χ)  (7)

for almost all χ ∈ Ĝ.

Next, by (3) we obtain

0 ≤ V̂_{K,T} (χ) = \frac{1}{mes T} (1)_{T} ∗ (1)_{T K}(χ) = \frac{1}{mes T} ∫_{T} (1)_{KT} (h^{−1}χ)dh.

Let χ ∉ K. If h^{−1}χ ∉ KT for any h ∈ T, then V̂_{K,T} (χ) = 0. If however χ ∉ K and χ ∈ KT, then in T there exists a set U with mes U such that χU ∉ KT so that

V̂_{K,T} (χ) = \frac{1}{mes T} ( ∫_{U^{−1}} + ∫_{T \U^{−1}} ) = \frac{1}{mes T} ∫_{T \U^{−1}} < 1.

Then from (7) it follows that for χ ∉ K we have f̂(χ) = 0 almost everywhere and therefore f ∈ F^{p}(K).

Let now f ∈ F^{p}(K) and T be an arbitrary set from U_{G}. We have

P̂_{K,T} (χ) = f̂(χ) · V̂_{K,T} (χ) = \frac{1}{mes T} ∫_{T} (1)_{KT} (h^{−1}χ)dh · f̂(χ).

If χ ∈ K and h ∈ T, then h^{−1}χ ∈ KT and we obtain f̂(χ) on the righthand side. If however χ ∉ K, then we have the equality f̂(χ) = 0 which by virtue of the fact that f ∈ F^{p}(K) implies { \atop P_{K,T} (f)}(χ) = 0. Therefore { \atop P_{K,T} (f)}(χ) = f̂(χ) for almost all χ ∈ Ĝ. Keeping in mind that the functions f and P_{K,T} (f) are continuous on G, by the uniqueness theorem on the Fourier transform of f we obtain representation (6) for any T ∈ U_{G} and g ∈ G.

Remark. If G = ℝ^{n}, by applying the well known Peley–Wiener theorem ([5], Ch. 6, §4) we can prove that the classes W^{p}(K) and F^{p}(K) coincide for p > 2 too provided that by F^{p}(K) we shall understand the class of functions f ∈ L^{p}(ℝ^{n}) whose generalized Fourier transform supports (in the sense of generalized functions) belong to K ⊂ ℝ^{n}.

For our further discussion it is important to note that the above-stated property that f ∈ F^{p}(K), 1 ≤ p ≤ 2, gives that f ∈ L^{p_{1}}(G) for any p_{1} ∈ [p, ∞] holds for functions from W^{p}(K) for p ∈ [1, ∞]. Indeed, if

f ∈ W^{p}(K), then f = f ∗ V_{K,T} for any T ∈ U_{G}. But V_{K,T} ∈ F^{1}(KT^{2}) and,

as we already know, this implies that V_{K,T} ∈ L^{r}(G) for all r ∈ [1, ∞]. By the Young inequality ([1], v. 1, §20)

∥f∥_{p_{1}} ≤ ∥f∥_{p} · ∥V_{K,T} ∥_{r}, where \frac{1}{p_{1}} = \frac{1}{p} + \frac{1}{r} − 1.  (8)

If 1 ≤ r ≤ \frac{p}{p−1}, then p_{1} takes all values from the interval [p, ∞]. Let us prove that the functions from W^{p}(K) are continuous for any 1 ≤ p ≤ ∞. Indeed, if f ∈ W^{p}(K), then f ∈ W^{∞}(K)

|f(g_{1}) − f(g)| = | ∫_{G} f(h)[V_{K,T} (h^{−1}g_{1}) − V_{K,T} (h^{−1}g)]dh| ≤

≤ ∥f∥_{∞} · ∥V_{K,T} (h^{−1}g_{1}) − V_{K,T} (h^{−1}g)∥_{1},

and the continuity of functions follows from the continuity of the shift operator in L^{1} ([3], Ch. 3, §5).

Let C^{0}(G) be the Banach space of continuous functions on a locally compact but not compact group G, which vanish at infinity ([1], v. 1, §11, [3], Ch. 2, §3). f ∈ C^{0}(G) if for any ε > 0 there exists a compact G_{ε} depending on f and such that the inequality |f(g)| < ε holds everywhere outside G_{ε}. We shall show that for 1 ≤ p < ∞ the functions from W^{p}(K) belong to C^{0}(G). Indeed, in view of the fact the Haar measure is regular and taking into account the integral representation (6), for any ε > 0 we can find a compact G_{ε} such that

|f(g) − ∫_{G_{ε}} f(h)V (h^{−1}g)dh| < ε.

Next, using (4), we obtain

| ∫_{G_{ε}} f(h)V (h^{−1}g)dh| ≤ ∥f∥_{p} · \{ ∫_{G_{ε}} |V (hg)|^{q}dh\}^{\frac{1}{q}} ≤

≤ (\frac{mes T K}{mes T} )^{\frac{1}{2q}} · ∥f∥_{p} · sup_{h∈G_{ε}} |V (hg)|^{\frac{q−1}{q}}

(for p = 1, q = ∞ we have ∥f∥_{∞} · sup_{h∈G_{ε}} |V (hg)| on the right-hand side).

It is the well known fact that the Fourier transform of a function from L^{1}(Ĝ) belongs to C^{0}(G) ([1], v. 2, §31). Hence by (3) it is clear that V (hg) ∈ C^{0}(G), ∀h ∈ G_{ε}. Since p ≠ ∞, q > 1 and the fact that V (hg) belongs to C^{0}(G), together with the last two estimates, enable us to conclude that f ∈ C^{0}(G). A constant function f(g) ≡ C ≠ 0 on G may serve as an example showing that for p = ∞ the inclusion W^{∞}(K) ⊂ C^{0}(G) is not valid.

If G = ℝ^{n} and 1 ≤ p < ∞, then the obtained inclusion W_{νp}(ℝ^{n}) ⊂ C^{0}(ℝ^{n}) is equivalent to the well known fact that lim_{|x|→∞} f(x) = 0 if f ∈

W_{νp}(ℝ^{n}) which was proved by a different method in [2] (Ch. 3, §3.2).

Lemma 3. Let f ∈ L^{p}(G), 1 ≤ p ≤ ∞, K ∈ U_{G}, T ∈ U_{G}. Then

P_{K,T} (f) ⊆ W^{p}(KT^{2}).

Proof. From the equality

V̂_{K,T} (χ) = (mes T)^{−1} ∫_{T} (1)_{KT} (h^{−1}χ)dh

it follows that V_{K,T} ∈ F^{1}(KT^{2}) and, by Lemma 2 we have V_{K,T} ∈W^{1}(KT^{2}). After multiplying both sides of equality (5) by V_{KT^{2},T_{1}}(g^{−1}t), where T_{1} ∈ U_{G}, and integrating them over G, we obtain

∫_{G} P_{K,T} (f, g)V_{KT^{2},T_{1}}(g^{−1}t)dg =∫_{G} ∫_{G} f(h)V_{K,T} (h^{−1}g)dh·V_{KT^{2},T_{1}}(g^{−1}t)dg =

= ∫_{G} f(h)dg ∫_{G} V_{K,T} (h^{−1}g) · V_{KT^{2},T_{1}}(g^{−1}t)dg.

Here changing the integration order is righful by virtue of the Fubini theorem ([1), v. 1, §13) and in view of the fact that

∫_{G} dg ∫_{G} |f(h)V_{K,T} (h^{−1}g) · V_{KT^{2},T_{1}}(g^{−1}t)|dh ≤

≤ ∥ ∫_{G} |f(h)V_{K,T} (h^{−1}g)|dh∥_{p} · ∥V_{KT^{2},T_{1}}∥_{\frac{p}{p−1}} < ∞.

Next, by the change of the variable we obtain

∫_{G} P_{K,T} (f, g)V_{KT^{2},T_{1}}(g^{−1}t)dg = ∫_{G} f(h)dh ∫_{G} V_{K,T} (ξ) · V_{KT^{2},T_{1}}(ξ^{−1}h^{−1}t)dξ.

But V_{K,T} ∈ W^{1}(KT^{2}) and, according to the definition of the class W^{1}, the internal integral can be replaced by V_{K,T} (h^{−1}t). As a result,

∫_{G} P_{K,T} (f, g)V_{KT^{2},T_{1}}(g^{−1}t)dg = ∫_{G} f(h)V_{K,T} (h^{−1}t)dh = P_{K,T} (f, t).

This implies that P_{K,T} (f) ∈ W^{p}(KT^{2}).

Remark. While proving Lemma 3, concomittantly we have actually proved the following property of functions from W^{p}(K): if f ∈ W^{p}(K), p ∈ [1, ∞], and K_{1} is an arbitrary compact from U_{G} containing K, then f ∈ W^{p}(K_{1}). This can be shown by repeating the proof of Lemma 3, where V_{KT^{2},T_{1}} is replaced by V_{K_{1},T_{1}} for any T_{1} ∈ U_{G}, and taking into account that V_{K,T} ∈ W^{1}(K_{1}T_{1}). The latter inclusion is valid because V_{K,T} ∈ F^{1}(KT^{2}). But T can be chosen so that KT^{2} ⊂ K_{1}T_{1} ([3], Ch. 3, §1). Then V_{K,T} ∈ F^{1}(K_{1}T_{1}) and, by virtue of Lemma 2, V_{K,T} ∈ W^{1}(K_{1}T_{1}). Finally, we obtain f = P_{K,T} (f) ∈ W^{p}(K_{1}T_{1}).

Theorem 1. Let p ∈ [1, ∞]. The set of functions from W^{p}(K) for all possible compacts from U_{G} is dense everywhere in L^{p}(G, μ).

Proof. First we shall prove that if f ∈ W^{p}(K) and χ is an arbitrary character of the group G, then f · χ ∈ W^{p}(K_{1}) for some compact K_{1} ∈ U_{G}. Indeed, since f ∈ W^{p}(K), we have

χ(g)f(g) = ∫_{G} f(h)χ(g)V_{K,T} (h^{−1}g)dh =

= ∫_{G} f(h)χ(h)χ(h^{−1}g)V_{K,T} (h^{−1}g)dh = (fχ ∗ χV_{K,T})(g).  (9)

Next we obtain

{ \atop (V_{K,T} · χ)}(χ_{0}) = ∫_{G} V_{K,T} (g) · χ(g) · \overline{χ_{0}(g)} dg =

= ∫_{G} V_{K,T} (g) · \overline{(\overline{χ}χ_{0})}(g) dg = V̂_{K,T} (χ^{−1}χ_{0}).

Since V_{K,T} ∈ F^{1}(KT^{2}), this implies that χ · V_{K,T} ∈ F^{1}(K_{1}) for some

compact K_{1} ∈ U_{G}, i.e., χ · V_{K,T} ∈ W^{1}(K_{1}) and χ · V_{K,T} = χV_{K,T} ∗ V_{K_{1},T_{1}}

for any T, T_{1} ∈ U_{G}. Next, by (9) we have

χ(g)f(g) = (fχ ∗ (χV_{K,T} ∗ V_{K_{1},T_{1}}))(g) =
= ((fχ ∗ χV_{K,T}) ∗ V_{K_{1},T_{1}})(g) = (χf ∗ V_{K_{1},T_{1}})(g),

which implies that χf ∈ W^{p}(K_{1}).

We shall show that for all K ∈ U_{G} the functions from W^{p}(K) are dense everywhere in L^{p}(G, μ). Let us assume that the opposite is true. Let A^{p} be the closure of the union ∪_{K}W^{p}(K) and A^{p} not coinciding with L^{p}(G, μ).

There exists a well defined nontrivial linear functional on L^{p}(G) which is equal to zero on A^{p}. This functional is defined by a nontrivial function φ ∈ L^{q}(G) ([1], v. 1, §12). Since f · χ ∈ A^{p} for arbitrary χ ∈ Ĝ, then

∫_{G} fχφ dμ = 0, f ∈ L^{p}(G), φ ∈ L^{q}(G).

The latter equality implies

f̂ · φ(χ) = 0, f · φ ∈ L^{1}(G)

for any χ ∈ Ĝ. Hence it follows that ([1], v. 2, §31) fφ = 0 almost everywhere on G for any function f ∈ W^{p}(K), where K is an arbitrary compact from U_{G}. By the continuity of f we readily conclude that φ = 0 almost everywhere on G, which is impossible.

For p = 1, Theorem 1 actually establishes the density of functions from F^{1}(K), for all possible compacta K, in L^{1}(G) and this density is known ([3], Ch. 5, §4, [6], Ch. 2, §8.8).

Remark. Theorem 1 does not hold for p = ∞ but remains valid for its subspace C^{0}(G) ⊂ L^{∞}(G).

The dual space to C^{0}(G) is the space M(G) of complex-valued measures defined on some σ-algebra containing all Borel sets in G ([1], v. 1, §14). To verify this, i. e., the density of W^{∞}(K) ∩ C^{0}(G) in C^{0}(G), we must repeat the proof of Theorem 1, which will lead us to the existence of a nontrivial measure μ ∈ M(G) such that ∫_{G} fχdμ = 0, ∀f ∈ C^{0}(G) ∩ W^{∞}(K), χ ∈ Ĝ. Hence the Fourier transform of the measure fdμ is equal to zero on the entire Ĝ. Therefore fdμ ([1], v. 2, §28) and thus μ = 0, which is impossible.

Let us consider the best approximation E_{K}(f)_{p,G} of the function f ∈ L^{p}(G) by the subspace W^{p}(K), i. e., the value

E_{K}(f)_{p,G} = inf_{g∈W^{p}(K)} ∥f − g∥_{L^{p}(G)}.  (10)

If we obtain inf on some element g_{0} ∈ W^{p}(K), then it will be called the best approximation element of f in W^{p}(K). We shall prove that such an element necessarily exists if 1 ≤ p ≤ ∞. Indeed, let f ∈ L^{p}(G), and t_{n} ∈ W^{p}(K) be the minimizing sequence, i.e.,

∥f − t_{n}∥_{p} ≤ d + ε_{n},

where d = E_{K}(f)_{p,G} and ε ↓ 0. Clearly, the sequence t_{n} is bounded with respect to the norm of the space L^{p}, and by applying inequality (8) to p_{1} = ∞ we have

∥t_{n}∥_{∞} ≤ C · ∥t_{n}∥_{p} ≤ A,

where A does not depend on n. Next we obtain

∥t_{n}(h^{−1}g) − t_{n}(g)∥_{p} = ∥ ∫_{G} t_{n}(ξ)[V_{K,T} (ξ^{−1}h^{−1}g) − V_{K,T} (ξ^{−1}g)]dξ∥_{p} ≤

≤ ∥t_{n}∥_{p} · ∥V_{K,T} (h^{−1}g) − V_{K,T} (g)∥_{1}.

The continuity of the shift operator in L^{1} ([3], Ch. 3, §5) implies equicontinuity of the family {t_{n}} in L^{p}(G) and therefore in L^{∞}(G) as well. Hence by virtue of the well known theorem ([7], §5) it follows that from t_{n} we can extract a subsequence t_{n_{ν}} converging uniformly on arbitrary compacts from G to a certain function t. Let us prove that t ∈ W^{p}(K). The sequence t_{n_{ν}} will again be denoted by t_{n}. In the first place, for any compact G_{1} ⊂ G we have

∥t∥_{L^{p}(G_{1})} ≤ ∥t−t_{n}∥_{L^{p}(G_{1})} +∥t_{n}∥_{L^{p}(G_{1})} ≤ C(G_{1})∥t−t_{n}∥_{L^{∞}(G_{1})} +∥t_{n}∥_{L^{p}(G)}.

But since lim_{n→∞} ∥t − t_{n}∥_{L^{∞}(G_{1})} = 0 and ∥t_{n}∥_{L^{p}(G)} ≤ C, for any compact G_{1} ⊂ G we have ∥t∥_{L^{p}(G_{1})} ≤ C, where C is a constant not depending on G_{1}. Therefore t ∈ L^{p}(G). Let consider an arbitrary compact G_{1} ⊂ G. Since t_{n} ∈ W^{p}(K), we have

t_{n}(g) = ∫_{G} t_{n}(h)V_{K,T} (h^{−1}g)dh =

= ∫_{G_{1}} t_{n}(h)V_{K,T} (h^{−1}g)dh + ∫_{G\G_{1}} t_{n}(h)V_{K,T} (h^{−1}g)dh.

Assuming that g ∈ G_{1} and passing to the limit as n → ∞, we obtain

t(g) = ∫_{G_{1}} t(h)V_{K,T} (h^{−1}g)dh + A(G \ G_{1}),

where

|A(G \ G_{1})| ≤ sup_{n} ∥t_{n}∥_{p} · ∫_{G\G_{1}} |V_{K,T} (h^{−1}g)|dh.

Now taking into account the fact that the Haar measure is regular and making G_{1} tend to G, we find that t(g) = ∫_{G} t(h)V_{K,T} (h^{−1}g)dh and therefore t ∈ W^{p}(K). Further, for any compact G_{1} ⊂ G we have

∥f − t∥_{L^{p}(G_{1})} = lim_{n→∞} ∥f − t_{n}∥_{L^{p}(G_{1})} ≤ lim_{n→∞} ∥f − t_{n}∥_{L^{p}(G)} = d

and thus t is the best approximation element of f ∈ L^{p}(G) in W^{p}(K), 1 ≤ p ≤ ∞.

For p = 2 the best approximation element of f ∈ L^{2}(G) in the subspace W^{2}(K) can be constructed explicitly in the form

f_{K}(g) = ∫_{K} f̂(χ)χ(g)dχ.  (11)

Indeed, let S be an arbitrary element from W^{2}(K), i.e., from F^{2}(K). Then almost everywhere (Ŝ)ˇ = S. Since Ŝ ∈ L^{1}(Ĝ) and its Fourier transforms in the sense of the spaces L^{1} and L^{2} coincide, we have

S(g) = ∫_{K} Ŝ(χ)χ(g) dχ almost everywhere on G.

Applying Parceval’s theorem we obtain

∥f − S∥_{2}^{2} = ∥f∥_{2}^{2} − ∫_{K} Ŝ(χ)\overline{f̂(χ)} dχ − ∫_{K} \overline{Ŝ(χ)}f̂(χ) dχ + ∥S∥_{2}^{2} =

= ∥f∥_{2}^{2} + ∫_{K} [Ŝ(χ) − f̂(χ)]^{2}dχ − ∫_{K} [f̂(χ)]^{2}dχ.

The right-hand side of this equality will is minimal if Ŝ = f̂ almost everywhere on K, i.e., by virtue of (11), if

S(g) = ∫_{K} Ŝ(χ)χ(g) dχ = ∫_{K} f̂(χ)χ(g) dχ = f_{K}(g).

For p ≠ 2, constructing a best approximation element is a difficult task even for simple cases. Moreover, we know of the cases for which it has been proved that one cannot construct a sequence of linear continuous projectors used to realize the best order approximation ([8], Ch. 9, §5.4, [9], Ch. 9, §5). In this connection, a problem is posed to construct a sequence of linear continuous operators, close in a certain sense to projectors, by means of which the best order approximation is realized. Such a problem, in which G is the unit circumference T, was posed and solved by de la Vallée-Poussin ([10]). The space of trigonometric polynomials of order n is considered as an approximating set and it is proved that for 1 ≤ p ≤ ∞ and f ∈ L^{p}(T)

∥f − σ_{n,m}(f)∥_{p} ≤ 2 \frac{n + 1}{m + 1} E_{n}(f)_{p},

where

(σ_{n,m}f)(x) = \frac{1}{m + 1} ∑_{k=n}^{n+m}(S_{k}f)(x) = ∫_{0}^{2π}f(u)V_{n,m}(x − u)du,

S_{k}(f) is the Fourier sum of the function f of order k, and

V_{n,m}(x) = \frac{sin \frac{(m+1)x}{2} · sin \frac{(2n+m+1)x}{2}}{2π(m + 1) sin^{2} \frac{x}{2}} .  (12)

In [10], this result was obtained for the space of C(T)-continuous functions on T. If m = [\frac{n}{2}], then the deviation ∥f − σ_{n,[n/2]}∥_{p} has a the best approximation order E_{n}(f)_{p}.

Theorem 2. If for a locally compact Abelian group G the function f belongs to the space L^{p}(G, μ), 1 ≤ p ≤ ∞, and K and T are the compact symmetric neighborhoods of the unit of the dual group Ĝ, then

∥f(g) − P_{K,T} (f, g)∥_{p} ≤ (1 + (\frac{mes(T K)}{mes T})^{1/2})E_{K}(f)_{p,G},  (13)

where P_{K,T} (f, g) and E_{K}(f)_{p,G} are defined by formulas (5) and (10).

Proof. Let f_{n} ∈ W^{p}(K) be a minimizing sequence for the function f ∈ L^{p}(G), 1 ≤ p ≤ ∞. Then

∥f(g) − P_{K,T} (f, g)∥_{p} ≤ ∥f − t_{n}∥_{p} + ∥P_{K,T} (f − t_{n})∥_{p} ≤
≤ ∥f − t_{n}∥_{p} + ∥f − t_{n}∥_{p} · ∥V_{K,T} ∥_{1}.

Hence, taking into account inequality (4) and the relation lim_{n→∞} ∥f − t_{n}∥_{p} = E_{K}(f)_{p,G}, we obtain estimate (13).

Remark. To have a best order approximation in (13), we must choose K and T from U_{G} such that for E_{K} the multiplier be bounded from above by a number not depending on K and T. For example, if T = K and P_{K,K} is denoted by P_{K}, then (13) gives the estimate

∥f − P_{K}(f)∥_{p} ≤ (1 + (\frac{mes K^{2}}{mes K} ))^{1/2} · E_{K}(f)_{p,G},

and \frac{mes K^{2}}{mes K} is bounded from above by the number not depending on K for a sufficiently wide class of groups G. For example, for Ĝ = ℝ^{n} the Haar measure μ coincides with the Lebesgue measure, and if K is compact convex symmetric neighborhood of zero, then \frac{mes K^{2}}{mes K} = 2^{n}. If Ĝ = Z^{n}, and K = \{x ∈ Z^{n}, −N ≤ x_{i} ≤ N, x_{i} ∈ Z, i = 1, 2, …, n\}, then \frac{mes K^{2}}{mes K} = (\frac{4N+1}{2N+1})^{n} < 2^{n}. If however in the latter case K = \{x ∈ Z^{n}, |x| ≤ N\} is an integer-valued lattice from Z^{n} contained within a ball of radius N, then \frac{mes K^{2}}{mes K} behaves as 2^{n} for large N . This follows from the known relation ∑_{|k|≤N} 1 ∼ ∑_{|k|≤N} π^{n/2}[Γ(\frac{n}{2} + 1)]^{−1}N^{n}, where Γ is a Euler function of second

kind.

Examples.

1. Let G = ℝ^{n}. Then Ĝ = ℝ^{n} and the group G has a character χ(x) = e^{itx}, t ∈ ℝ^{n}. We respectively take, as sets K and T, n-dimensional parallelepipeds with ribs of length 2N and 2s: K = {−N ≤ x_{i} ≤ N}, T = {−s ≤ x_{i} ≤ s}, i = 1, 2, …, n. Since the measure is normalized, we readily obtain

V_{K,T} (x) = (sπ)^{−n} ∏_{j=1}^{n} x_{j}^{−2} · sin sx_{j} · sin(N + s)x_{j}.

For s = \frac{N}{2} this is the well known Vallée–Poussin kernel ([2], Ch. 8, §8.6).

If K and T are n-dimensional balls of radii N and s, then, after performing some calculations, we find that

V_{K,T} (x) = \frac{n}{Ω_{n}} (\frac{s}{N + s})^{\frac{n}{2}} |x|^{−n}I\frac{n}{2} (s|x|) · I\frac{n}{2} ((N + s)|x|),

where Ω_{n} is the surface area of the n-dimensional unit sphere and I_{n/2} is a first kind Bessel function of order n/2. In both cases the expression \frac{mes(T K)}{mes T} is bounded from above by the number (\frac{N+s}{N})^{n}.

2. If G = E is the unit circumference from ℝ^{2}, then Ĝ = Z and the group G has a character of the form χ_{n}(t) = e^{−int}, t ∈ E, n ∈ Z. Take K = [−n, n] = {−n, −n + 1, …, n − 1, n}, T = [−s, s]. Then for the kernel V_{K,T} we obtain the above-mentioned de la Vallée-Poussin kernel V_{n,m} defined by formula (12) with m = 2s. If m is an odd number, then V_{K,T} does not exactly coincide with V_{n,m}, but to obtain V_{n,m} one should modify the definition of the kernel V_{K,T} by means of averaging the kernels ˆ(1)_{K_{i}}, where T ⊂ K_{i} ⊂ K.

If G is an n-dimensional torus, then one can take, as V_{K,T}, the product of one-dimensional kernels (12), i.e.,

V_{K,T} (x) = (2π)^{−n} ∏_{j=1}^{n}(2s_{j} + 1)^{−n} · sin^{−2} \frac{x_{j}}{2} · sin (s_{j} + \frac{1}{2})x_{j} ×

× sin (N_{j} + s_{j} + \frac{1}{2})x_{j}, t_{j}, N_{j} ∈ Z^{+}.

3. If G = Z, then Ĝ = E. Characters of the group G have the form χ_{t}(n) = t^{n}, n ∈ Z, t ∈ E. Let K = {e^{iθ} : −φ ≤ θ ≤ φ, 0 < φ ≤ π}, T = {e^{iθ} : −t ≤ θ ≤ t, 0 < t ≤ π}. Take, as a dual measure on E, the arc length divided into i\sqrt{2π}. We have

ˆ(1)_{T} (n) = \frac{1}{i\sqrt{2π}} ∫_{T} ξ^{n}dξ = \frac{1}{\sqrt{2π}} ∫_{−t}^{t} e^{iθ(n+1)}dθ =

= \begin{cases}\frac{\sqrt{2}}{\sqrt{π}·(n+1)} sin t(n + 1), & n ≠ −1, \\ \begin{matrix}√ & \overline{2} \\ √ & \overline{π} \end{matrix} t, & n = −1, \end{cases}

and P_{K,T} (f, n) in this example takes the form

P_{K,T} (f, n) =

= \frac{1}{tπ} \{ ∑_{\begin{matrix}k=−∞ \\ k≠n+1 \end{matrix}}^{∞} f(k) \frac{sin t(n − k + 1) · sin(φ + t)(n − k + 1)}{(n − k + 1)^{2}} + t(φ + t)\}

and \frac{mes(T K)}{mes T} = \frac{φ+t}{t}.

4. Let G = ℝ^{+} be a multiplicative group of positive integers with the unit e = 1. The group G has a character χ(ξ) = ξ^{ix}, where x ∈ ℝ, Ĝ = ℝ to within an isomorphism. Take, as K and T, the intervals K = [−N, N], T = [−t, t], N, t ∈ ℝ^{+}. Then

ˆ(1)_{K}(ξ) = ∫_{−N}^{N} ξ^{ix}dx = 2 \frac{sin(N ln ξ)}{ln ξ},

P_{K,T} (f, x) = \frac{1}{tπ} ∫_{0}^{∞}f(h) \frac{sin(t ln \frac{x}{h}) · sin((N + t) ln \frac{x}{h})}{h ln^{2} \frac{x}{h}} dh.

5. Let a = (a_{0}, a_{1}, …, a_{n}, …) be a given sequence, where all a_{n} are integer numbers greater than 1, and consider the Cartesian product Δ_{a} = P_{n∈ℕ∪{0}}{0, 1, …, a_{n} − 1}. If one applies the summation operation to Δ_{a} ([1], v. 1, §10), then Δ_{a} together with Tikhonov topology of the product becomes a compact Abelian group. This group G = Δ_{a} is called a group of integer a-adic numbers. Its dual group is the discrete group Ĝ = Z(a^{∞}) consisting of all numbers of the form

exp (2πi\frac{l}{a_{0}a_{1} ⋯ a_{r}} ), l, r ∈ Z, r > 1

([1], v. 1, §10). For the fixed natural number N we take, as a compact symmetric neighborhood K of the unit, the set of numbers from Z(a^{∞}) for which r ≤ N. It is proved ([1], v. 1, §25) that the character corresponding to the number ξ = exp(2πi\frac{l}{a_{0}a_{1}·a_{r}}) is written as

χ_{ξ}(g) = exp [\frac{2πil}{a_{0}a_{1} ⋯ a_{r}} (g_{0} + a_{0}g_{1} + ⋯ + a_{0}a_{1} ⋯ a_{r−1}g_{r})],

where g_{0} + a_{0}g_{1} + ⋯ + a_{0}a_{1} ⋯ a_{r−1}g_{r} is the sum of the first r + 1 terms of the a-adic expansion of the element g ∈ Δ_{a}. Assuming that the Haar measure of each point in Z(a^{∞}) is 1, we have

ˆ(1)_{K}(g) = ∫_{K} χ_{ξ}(g) dξ =

= ∑_{r=0}^{N} ∑_{l=1−a_{0}a_{1}···a_{r}}^{a_{0}a_{1}···a_{r}−1} exp [\frac{2πil}{a_{0}a_{1} ⋯ a_{r}} (g_{0} + ∑_{k=1}^{r} a_{0} ⋯ a_{k−1}g_{k})] =

= ∑_{r=0}^{N} \frac{sin(a_{0}a_{1} ⋯ a_{r} − \frac{1}{2})x_{r}}{sin \frac{x_{r}}{2}} , x_{r} = \frac{2π}{a_{0}a_{1} ⋯ a_{r}} (g_{0} + ∑_{k=1}^{r} a_{0} ⋯ a_{k−1}g_{k}).

Clearly, K^{2} = K and, taking T = K, we obtain

P_{K}(f, g_{1}) = \frac{1}{(N + 1)^{2}} ∫_{Δ_{a}} f(gg_{1})[ ∑_{r=0}^{N} \frac{sin(a_{0}a_{1} ⋯ a_{r} − \frac{1}{2})x_{r}}{sin \frac{x_{r}}{2}} ]^{2}dg,

where integration is performed with respect to the measure dual to the taken discrete measure on Z(a^{∞}).

References

1. A. Hewitt and K. Ross, Abstract harmonic analysis. Springer, Berlin, etc., v. 1, 1963, v. 2, 1970.

2. S. M. Nikolskii, Approximation of functions of several variables and embedding theorems. (Translation from Russian) Springer, Berlin, etc., 1975; Russian original: Nauka, Moscow, 1969.

3. H. Reiter, Classical harmonic analysis and locally compact groups. Oxford, Clarendon Press, 1968.

4. D. G. Ugulava, On the approximation by entire functions of exponential type. (Russian) Trudy Inst. Vichisl. Matem. Akad. Nauk Gruzin. SSR 28(1988), No. 1, 192–202.

5. K. Iosida, Functional analysis. Springer, Berlin, etc., 1965.

6. V. P. Gurarii, Groups methods in commutative harmonic analysis. Commutative harmonic analysis, v. II. (Encyclopaedia of mathematical sciences, v. 25) (Translation form Russian) Springer, Berlin, etc., 1998; Russian original: Itogi Nauki i Tekhniki, VINITI, Moscow, 1988.

7. N. Burbaki, Èlements de mathematique, livre III, Topologie générale, Chap. 10, Hermann, Paris, 1961.

8. P. Laurent, Approximation et optimisation. Collection enseignement des science, 13, Hermann, Paris, 1972.

9. R. A. De Vore and G. G. Lorentz, Constructive approximation. Grundlehren der Math. Wiss. 303, Springer, Berlin, etc., 1993.

10. J. Ch. de la Vallée-Poussin, Leçons sur l’approximation des fonctions d’une variable réelle. Paris, 1919.

(Received 2.09.1997)

Author’s address:

N. Muskhelishvili Institute

of Computational Mathematics

Georgian Academy of Sciences

8, Akuri St., Tbilisi 380093

Georgia