Let us consider a locally compact abelian groupGwith Haar measureµ, assuming that the topology of the space G is Hausdorff

ON THE APPROXIMATION OF FUNCTIONS ON LOCALLY COMPACT ABELIAN GROUPS

D. UGULAVA

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

Let us consider a locally compact abelian groupGwith Haar measureµ, 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 onGand integrable on it with respect to the measureµto thep-th power with the usual normkfk^p={R

G|f|^pdµ}^1/pfor 1≤p <∞. L^∞(G, µ) is a space of functions, essentially bounded on G with respect to µ and having normkfk∞= 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 ofG, i.e., a complex-valued continuous function χonGwith the properties|χ(g)|= 1, ∀g∈Gandχ(g1g2) =χ(g1)·χ(g2),

∀g1, g2∈G, is called a character of the groupG. An Abelian group structure is naturally introduced into the set of all characters defined on G. The obtained group is denoted byGb and called a group dual toG. Gb is usually topologized by the following two topologies: the first one is the weakest topology containing continuous functionalsfbdefined by the formula

fb(χ) = Z

G

f(g)χ(g)dµ(g)≡ Z

G

f(g)χ(g)dg, χ∈G, fb ∈L¹(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 aidGbtransforms to a locally compact Abelian group. A functionfbof

1991Mathematics Subject Classification. 41A65, 41A35.

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

379

1072-947X/99/0700-0379$15.00/0 c1999 Plenum Publishing Corporation

form (1) defined on Gb forf ∈L¹(G) is called the Fourier transform of the functionf. Similarly to (1), we define the inverse transform ˇf off by the formula

fˇ(χ) = Z

G

f(g)χ(g)dg.

Of much importance here is the Pontryagin duality principle by which the natural mapping ofG into Gbb, which to an element g ∈Gassigns the character fg on G, is an isomorphism of topological groups. The notionsb of direct and inverse Fourier transform can be extended by the well known technique to the case of spacesL^p(G, µ) for 1< p≤2. By Plancherel’s the- orem the Fourier transform is a linear isometry ofL²(G) on L²(G) but theb inverse Fourier transform is a linear isometry fromL²(G) onb L²(G). These mappings are mutually inverse ([1], v. 2, §31). Usually, Haar measures on G and Gb are normalized so that the inversion formula f = (fb)

ˇ

holds for functions f ∈L¹(G),fb∈L¹(G). These measures are called mutually dualb 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

kfbkq≤ kfkp, (2) whereq(here and in what follows) is the conjugate number ofp(¹_p+¹_q = 1).

A great part of the approximation theory deals with questions of ap- proximation of functions on an n-dimensional Euclidean space Rⁿ and on an n-dimensional torus Tⁿ which are Abelian groups. As an approximat- ing subspace, a space of exponential type entire functions and a space of trigonometric polynomials [2] are usually taken as an approximating sub- space in the first and the second case, respectively. We know thatRcⁿ=Rⁿ andTb=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 classi- cal groups. For example, by analogy with groups Rⁿ or Tⁿ one can try to consider as an approximating set the set of functions on Gwhose Fourier transform lies in some compact K of the dual space G. But such an ap-b proach cannot simultaneously be used for all spacesL^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 defini- tion of approximating subspaces which is simultaneously applicable for all p, 1≤p≤ ∞.

By UG we shall denote a space of all symmetric compact sets from G which are the closures of neighborhoods of unity inG. Sets from UG will be called compact neighborhoods of unity. KT ={g : g =g1g2, g1 ∈ K, g2 ∈ 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 fromU^Gb we shall consider the function defined onG([3], Ch. 5,§1)

VK,T(g) = (mesT)⁻¹(1)c_T(g)·(1)c_{T K}(g), (3) which, for simplicity, will be denoted byV.

It is at once obvious thatV ∈L¹(G). Indeed, kVK,Tk ≤ 1

mesT k(1)c_Tk²· k(1)c_{T K}k²=

= 1

mesT k(1)Tk2· k(1)T Kk2=mesT K mesT

‘1/2

. (4)

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

(1)c_K(g) = Z

K

χ(g)dχ= Z

K

χ(g⁻¹)dχ= Z

K

χ(g)dχ=(1)c_K(g).

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

G

VK,T(g)dg= 1 mesT

Z

G

(1)c_T(g)·(1)c_{T K}(g)dµ=

= 1

mesT Z b

G

(1)T(χ)·(1)T K(χ)dχ= 1 mesT

Z

T

dχ= 1.

Using the functionf ∈L^p(G) and kernelV, we introduce the function PK,T(f, g) = (f∗VK,T)(g) =

Z

G

f(h)·VK,T(h⁻¹g)dh. (5)

Definition 1. LetK ∈U^Gb and p∈[1,∞]. ByW^p(K) we shall denote the set of functionsf from the spaceL^p(G, µ) for which we have the formula f(g) =PK,T(f, g) ∀g∈G and T ∈U^Gb. (6) Remark . If G = Rⁿ and K = ∆ν = {x(x1, . . . , xn) ∈ Rⁿ : |xi| ≤ νi, ν= (ν1, . . . , νn)}, then the class defined by us coincides with the known class Wνp(Rⁿ) of entire functions of exponential type ν. A functionf ∈L^p(Rⁿ) belongs to Wνp(Rⁿ) ([2], Ch. 3, §3.1) if it is analytically extendable onto

the whole n-dimensional complex Euclidean space Cⁿ, and for any ε > 0 there exists a numberAεsuch that the inequality

|g(z)| ≤Aε·exp Xn j=1

(νj+ε)· |zj| is fulfilled for allz(z1, . . . , zn)∈Cⁿ.

But forf ∈Wνp(Rⁿ) we have representation (6) ([2], Ch. 8, §8.6) with (1)c_K = 2ⁿ

Qn i=1

(xi)⁻¹sinνixi. Conversely, let (6) hold for f ∈ L^p(Rⁿ). We take ann-dimensional cube Tε={x∈Rⁿ : |xi| ≤ ε,i = 1, . . . , n}, ε >0, as a neighborhood ofT. The kernelVK,Tε will be a exponential type entire function for any vector ε with constant coordinates ε. The convolution f∗VK,Tε belongs to the same class too ([2], Ch. 3,§3.6), which by virtue of (6) means thatfbelongs toWνp(Rⁿ). ThusW^p(K) coincides withWνp(Rⁿ) forG=Rⁿ and K= ∆ν. Such an equivalence can be proved by a similar reasoning for more generalK⊂Rⁿ 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 fn ∈ W^p(K) be an arbitrary converging sequence and kfn− fk^p →0 as n→ ∞. Using the well known estimate of a convolution norm ([1], v. 2,§31) forL^p, we obtain

kPK,T(fn−f, g)kp≤ kfn−fkp· kVK,Tk1, ∀T ∈U^Gb.

ButPK,T(fn) =fn, ∀n∈N, and thusfn tends inL^p simultaneously tof and PK,T(f). Therefore f =PK,T(f) and f ∈ W^p(K). The operation of shift by the elementhwill be denoted byLh. Let f ∈W^p(K) and T ∈U^Gb. We obtain

(Lhf)(g) =f(hg) = Z

G

f(g1)VK,T(g⁻₁¹gh)dg1= Z

G

f(ξh)VK,T(ξ⁻¹h⁻¹gh)dξ=

= Z

G

(Lhf)(ξ)VK,T(ξ⁻¹g)dξ= (Lhf∗VK,T)(g)

and Lemma 1 is proved.

Definition 2. When p∈[1,2], for an arbitrary fixed compact K from the dual spaceGb we shall denote by F^p(K) the set of functions from the spaceL^p(G, µ) whose Fourier transform supports belong toK. Using Parce- val’s theorem ([1], v. 2, §31) by which R

GfΦbdg =R b

GfbΦdχ, substituting the function Φ(χ) = _mes^χ(g)_T((1)T ∗(1)T K)(χ) into it, and calculating Φ byb

the rule (χf)(b·) =fb(χ⁻¹·) ([1], v. 2,§31), for 1≤p≤2 we obtain another representation ofPK,T(f), namely:

PK,T(f, g) = (mesT)⁻¹ Z b

G

f(χ)b ·χ(g)·((1)T∗(1)T K)(χ)dχ,

where integration actually occurs on the compactT²K. IfGis a compact, then this representation ofPK,T certainly holds for all 1≤p≤ ∞.

Note the following important property of functions fromF^p(K). Iff ∈ F^p(K), 1 ≤ p ≤ 2, then f ∈ L^p1(G) for arbitrary p1 ∈ [p,∞]. Indeed, by virtue of (2), fb∈L^q(G), where q is the conjugate number of p. Since suppfb=K and q≥2, we havefb∈L²(G) and (b f)b

ˇ

=f almost everywhere onG. Thusfb∈L¹(G)b ∩L²(G) and theb L² inverse Fourier transform offb is f. But ([1], v. 2, §31) the L¹ inverse Fourier transform of fbis also f. Hence it is clear thatf is a continuous and bounded function on G. Since f ∈L^p∩L^∞, we havef ∈L^p1(G) for anyp1∈[p,∞]. Moreover, forp=∞, by (2) we obtain the inequality

kfk∞≤ kfbk1= Z

K

|fb|dχ≤

’ Z

K

dχ

“1/p

kfbkq ≤(mesK)^1/p· kfkp.

Lemma 2. If 1≤p≤2, then the setsW^p(K)andF^p(K)coincide.

Proof. Let us first assume thatf ∈W^p(K). Then for∀T ∈U^Gb we have f\∗VK,T(χ) =fb(χ)

and therefore

fb(χ)·VbK,T(χ) =fb(χ) (7) for almost allχ∈G.b

Next, by (3) we obtain 0≤VbK,T(χ) = 1

mesT (1)T ∗(1)T K(χ) = 1 mesT

Z

T

(1)KT(h⁻¹χ)dh.

Letχ /∈K. Ifh⁻¹χ /∈KT for anyh∈T, thenVbK,T(χ) = 0. If however χ /∈K and χ∈ KT, then in T there exists a set U with mesU such that χU /∈KT so that

VbK,T(χ) = 1 mesT

’ Z

U−1

+ Z

T\U−1

“

= 1

mesT Z

T\U−1

<1.

Then from (7) it follows that forχ /∈Kwe havefb(χ) = 0 almost everywhere and thereforef ∈ F^p(K).

Let nowf ∈ F^p(K) andT be an arbitrary set from U^Gb. We have PbK,T(χ) =fb(χ)·VbK,T(χ) = 1

mesT Z

T

(1)KT(h⁻¹χ)dh·fb(χ).

If χ ∈ K and h ∈T, then h⁻¹χ ∈ KT and we obtain fb(χ) on the right- hand side. If however χ /∈K, then we have the equality fb(χ) = 0 which by virtue of the fact that f ∈ F^p(K) implies P\K,T(f)(χ) = 0. Therefore P\K,T(f)(χ) =fb(χ) for almost allχ∈G. Keeping in mind that the functionsb f and PK,T(f) are continuous on G, by the uniqueness theorem on the Fourier transform of f we obtain representation (6) for any T ∈ U^Gb and g∈G.

Remark . IfG=Rⁿ, 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 functionsf ∈L^p(Rⁿ) whose generalized Fourier transform supports (in the sense of generalized functions) belong toK⊂Rⁿ.

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^p1(G) for any p1 ∈ [p,∞] holds for functions from W^p(K) for p ∈ [1,∞]. Indeed, if f ∈W^p(K), thenf =f∗VK,T for anyT ∈U^Gb. ButVK,T ∈ F¹(KT²) and, as we already know, this implies that VK,T ∈L^r(G) for all r∈[1,∞]. By the Young inequality ([1], v. 1,§20)

kfk^p1 ≤ kfk^p· kVK,Tk^r, where 1 p1

= 1 p+1

r −1. (8) If 1 ≤r ≤ _p−^p₁, then p1 takes all values from the interval [p,∞]. Let us prove that the functions from W^p(K) are continuous for any 1≤p≤ ∞. Indeed, iff ∈W^p(K), thenf ∈W^∞(K)

|f(g1)−f(g)|=

ŒŒ

ŒŒ Z

G

f(h)‚

VK,T(h⁻¹g1)−VK,T(h⁻¹g)ƒ dh

ŒŒ

ŒŒ≤

≤ kfk∞·VK,T(h⁻¹g1)−VK,T(h⁻¹g)

1,

and the continuity of functions follows from the continuity of the shift op- erator inL¹([3], Ch. 3, §5).

Let C⁰(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 ∈ C⁰(G) if for any ε > 0 there exists a compact Gε

depending on f and such that the inequality |f(g)| < ε holds everywhere

outsideGε. We shall show that for 1≤p <∞the functions fromW^p(K) belong to C⁰(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 Gε such that

ŒŒ

ŒŒf(g)− Z

Gε

f(h)V(h⁻¹g)dh

ŒŒ

ŒŒ< ε.

Next, using (4), we obtain

ŒŒ

ŒŒ Z

Gε

f(h)V(h⁻¹g)dh

ŒŒ

ŒŒ≤ kfkp·

š Z

Gε

|V(hg)|^qdh

›¹_q

≤

≤mesT K mesT

‘_2q¹

· kfk^p· sup

h∈Gε

|V(hg)|^q−q1

(forp= 1, q=∞we havekfk∞· sup

h∈Gε

|V(hg)|on the right-hand side).

It is the well known fact that the Fourier transform of a function from L¹(G) belongs tob C⁰(G) ([1], v. 2,§31). Hence by (3) it is clear thatV(hg)∈ C⁰(G), ∀h∈Gε. Since p6=∞, q >1 and the fact that V(hg) belongs to C⁰(G), together with the last two estimates, enable us to conclude that f ∈ C⁰(G). A constant function f(g) ≡ C 6= 0 on G may serve as an example showing that for p = ∞ the inclusion W^∞(K) ⊂ C⁰(G) is not valid.

If G = Rⁿ and 1 ≤ p < ∞, then the obtained inclusion Wνp(Rⁿ) ⊂ C⁰(Rⁿ) is equivalent to the well known fact that lim

|x|→∞f(x) = 0 if f ∈ Wνp(Rⁿ) which was proved by a different method in [2] (Ch. 3,§3.2).

Lemma 3. Let f ∈L^p(G),1≤p≤ ∞,K∈U^Gb,T ∈U^Gb. Then PK,T(f)⊆W^p(KT²).

Proof. From the equality

VbK,T(χ) = (mesT)⁻¹ Z

T

(1)KT(h⁻¹χ)dh

it follows thatVK,T ∈ F¹(KT²) and, by Lemma 2 we haveVK,T∈W¹(KT²).

After multiplying both sides of equality (5) by VKT²,T1(g⁻¹t), whereT1 ∈

U^Gb, and integrating them over G, we obtain Z

G

PK,T(f, g)VKT²,T1(g⁻¹t)dg= Z

G

Z

G

f(h)VK,T(h⁻¹g)dh·VKT²,T1(g⁻¹t)dg=

= Z

G

f(h)dg Z

G

VK,T(h⁻¹g)·VKT²,T1(g⁻¹t)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

Z

G

dg Z

G

ŒŒf(h)VK,T(h⁻¹g)·VKT²,T1(g⁻¹t)ŒŒdh≤

≤



 Z

G

ŒŒf(h)VK,T(h⁻¹g)ŒŒdh





p

· kVKT²,T1k_p−^p₁ <∞. Next, by the change of the variable we obtain

Z

G

PK,T(f, g)VKT²,T1(g⁻¹t)dg= Z

G

f(h)dh Z

G

VK,T(ξ)·VKT²,T1(ξ⁻¹h⁻¹t)dξ.

ButVK,T ∈W¹(KT²) and, according to the definition of the classW¹, the internal integral can be replaced byVK,T(h⁻¹t). As a result,

Z

G

PK,T(f, g)VKT²,T1(g⁻¹t)dg= Z

G

f(h)VK,T(h⁻¹t)dh=PK,T(f, t).

This implies thatPK,T(f)∈W^p(KT²).

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 K1 is an arbitrary compact from U^Gb containingK, then f ∈ W^p(K1). This can be shown by repeating the proof of Lemma 3, where VKT²,T1 is replaced by VK1,T1 for any T1 ∈ U^Gb, and taking into account that VK,T ∈ W¹(K1T1). The latter inclusion is valid because VK,T ∈ F¹(KT²). ButT can be chosen so that KT² ⊂K1T1 ([3], Ch. 3,

§1). ThenVK,T ∈ F¹(K1T1) and, by virtue of Lemma 2,VK,T ∈W¹(K1T1).

Finally, we obtainf =PK,T(f)∈W^p(K1T1).

Theorem 1. Let p∈[1,∞]. The set of functions from W^p(K) for all possible compacts fromU^Gb is dense everywhere inL^p(G, µ).

Proof. First we shall prove that iff ∈W^p(K) and χ is an arbitrary char- acter of the group G, then f ·χ ∈ W^p(K1) for some compact K1 ∈ U^Gb. Indeed, sincef ∈W^p(K), we have

χ(g)f(g) = Z

G

f(h)χ(g)VK,T(h⁻¹g)dh=

= Z

G

f(h)χ(h)χ(h⁻¹g)VK,T(h⁻¹g)dh= (f χ∗χVK,T)(g). (9)

Next we obtain

(V\K,T·χ)(χ0) = Z

G

VK,T(g)·χ(g)·χ0(g)dg=

= Z

G

VK,T(g)·(χχ0)(g)dg=VbK,T(χ⁻¹χ0).

Since VK,T ∈ F¹(KT²), this implies that χ·VK,T ∈ F¹(K1) for some compactK1∈U^Gb, i.e.,χ·VK,T ∈W¹(K1) and χ·VK,T =χVK,T ∗VK1,T1

for anyT, T1∈U^Gb. Next, by (9) we have

χ(g)f(g) = (f χ∗(χVK,T∗VK1,T1))(g) =

= ((f χ∗χVK,T)∗VK1,T1)(g) = (χf∗VK1,T1)(g), which implies thatχf∈W^p(K1).

We shall show that for allK∈U^Gbthe functions fromW^p(K) are dense everywhere inL^p(G, µ). Let us assume that the opposite is true. LetA^p be the closure of the union∪

KW^p(K) andA^p not coinciding withL^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). Sincef·χ∈ A^p for arbitrary χ∈G, thenb

Z

G

f χϕ dµ= 0, f ∈L^p(G), ϕ∈L^q(G).

The latter equality implies

fd·ϕ(χ) = 0, f·ϕ∈L¹(G)

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

Forp= 1, Theorem 1 actually establishes the density of functions from F¹(K), for all possible compacta K, in L¹(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 subspaceC⁰(G)⊂L^∞(G).

The dual space toC⁰(G) is the spaceM(G) of complex-valued measures defined on someσ-algebra containing all Borel sets inG([1], v. 1,§14). To verify this, i. e., the density ofW^∞(K)∩C⁰(G) inC⁰(G), we must repeat the proof of Theorem 1, which will lead us to the existence of a nontrivial measureµ∈M(G) such thatR

Gf χdµ= 0, ∀f ∈C⁰(G)∩W^∞(K),χ∈G.b Hence the Fourier transform of the measuref dµis equal to zero on the entire G. Thereforeb f dµ([1], v. 2,§28) and thusµ= 0, which is impossible.

Let us consider the best approximation EK(f)p,G of the function f ∈ L^p(G) by the subspaceW^p(K), i. e., the value

EK(f)p,G= inf

g∈W^p(K)kf −gkL^p(G). (10) If we obtain inf on some elementg0∈W^p(K), then it will be called the best approximation element off inW^p(K). We shall prove that such an element necessarily exists if 1≤p≤ ∞. Indeed, letf ∈L^p(G), andtn∈W^p(K) be the minimizing sequence, i.e.,

kf−tnkp≤d+εn,

where d=EK(f)p,G and ε↓ 0. Clearly, the sequencetn is bounded with respect to the norm of the space L^p, and by applying inequality (8) to p1=∞we have

ktnk∞≤C· ktnk^p≤A, whereAdoes not depend onn. Next we obtain

ktn(h⁻¹g)−tn(g)k^p=



 Z

G

tn(ξ)‚

VK,T(ξ⁻¹h⁻¹g)−VK,T(ξ⁻¹g)ƒ dξ





p

≤

≤ ktnk^p· kVK,T(h⁻¹g)−VK,T(g)k¹.

The continuity of the shift operator inL¹ ([3], Ch. 3,§5) implies equicon- tinuity of the family{tn} inL^p(G) and therefore inL^∞(G) as well. Hence by virtue of the well known theorem ([7],§5) it follows that fromtn we can extract a subsequencetnν converging uniformly on arbitrary compacts from Gto a certain functiont. Let us prove thatt∈W^p(K). The sequencetnν

will again be denoted bytn. In the first place, for any compactG1⊂Gwe have

ktkL^p(G1)≤ kt−tnkL^p(G1)+ktnkL^p(G1)≤C(G1)kt−tnkL^∞(G1)+ktnkL^p(G).

But since lim

n→∞kt−tnkL^∞(G1) = 0 and ktnkL^p(G) ≤ C, for any compact G1 ⊂G we have ktkL^p(G1) ≤C, where C is a constant not depending on G1. Thereforet∈L^p(G). Let consider an arbitrary compactG1⊂G. Since tn ∈W^p(K), we have

tn(g) = Z

G

tn(h)VK,T(h⁻¹g)dh=

= Z

G1

tn(h)VK,T(h⁻¹g)dh+ Z

G\G1

tn(h)VK,T(h⁻¹g)dh.

Assuming thatg∈G1and passing to the limit asn→ ∞, we obtain t(g) =

Z

G1

t(h)VK,T(h⁻¹g)dh+A(G\G1),

where

|A(G\G1)| ≤sup

n ktnkp· Z

G\G1

|VK,T(h⁻¹g)|dh.

Now taking into account the fact that the Haar measure is regular and mak- ing G1 tend toG, we find that t(g) =R

Gt(h)VK,T(h⁻¹g)dh and therefore t∈W^p(K). Further, for any compactG1⊂Gwe have

kf −tkL^p(G1)= lim

n→∞kf −tnkL^p(G1)≤ lim

n→∞kf −tnkL^p(G)=d and thus t is the best approximation element of f ∈ L^p(G) in W^p(K), 1≤p≤ ∞.

Forp= 2 the best approximation element off ∈L²(G) in the subspace W²(K) can be constructed explicitly in the form

fK(g) = Z

K

fb(χ)χ(g)dχ. (11)

Indeed, letS be an arbitrary element fromW²(K), i.e., fromF²(K). Then almost everywhere (S)b

ˇ

=S. Since Sb∈L¹(G) and its Fourier transforms inb the sense of the spacesL¹ andL²coincide, we have

S(g) = Z

K

S(χ)χ(g)b dχ almost everywhere onG.

Applying Parceval’s theorem we obtain kf−Sk²2=kfk²2−

Z

K

S(χ)b fb(χ)dχ− Z

K

S(χ)b fb(χ)dχ+kSk²2=

=kfk²2+ Z

K

[S(χ)b −fb(χ)]²dχ− Z

K

[fb(χ)]²dχ.

The right-hand side of this equality will is minimal if Sb=fbalmost every- where onK, i.e., by virtue of (11), if

S(g) = Z

K

S(χ)χ(g)b dχ= Z

K

fb(χ)χ(g)dχ=fK(g).

For p 6= 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 Gis the unit circumferenceT, was posed and solved by de la Vall´ee-Poussin ([10]). The space of trigonometric polynomials of order nis considered as an approximating set and it is proved that for 1≤p≤ ∞andf ∈L^p(T)

kf−σn,m(f)kp≤2 n+ 1

m+ 1En(f)p, where

(σn,mf)(x) = 1 m+ 1

n+mX

k=n

(Skf)(x) = Z2π

0

f(u)Vn,m(x−u)du,

Sk(f) is the Fourier sum of the functionf of order k, and Vn,m(x) =sin^(m+1)x₂ ·sin^(2n+m+1)x₂

2π(m+ 1) sin^2x₂ . (12) In [10], this result was obtained for the space ofC(T)-continuous functions onT. Ifm= [ⁿ₂], then the deviationkf−σn,[n/2]kp has a the best approx- imation orderEn(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≤ ∞, andK and T are the compact

symmetric neighborhoods of the unit of the dual groupG, thenb kf(g)−PK,T(f, g)kp≤

1 +mes(T K) mesT

‘1/2‘

EK(f)p,G, (13) wherePK,T(f, g)andEK(f)p,G are defined by formulas(5) and(10).

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

kf(g)−PK,T(f, g)kp≤ kf−tnkp+kPK,T(f−tn)kp≤

≤ kf−tnk^p+kf−tnk^p· kVK,Tk¹. Hence, taking into account inequality (4) and the relation lim

n→∞kf−tnk^p= EK(f)p,G, we obtain estimate (13).

Remark . To have a best order approximation in (13), we must chooseK andT fromU^Gb such that forEK the multiplier be bounded from above by a number not depending on Kand T. For example, ifT =K andPK,K is denoted byPK, then (13) gives the estimate

kf−PK(f)kp≤

1 +mesK² mesK

‘‘1/2

·EK(f)p,G,

and ^mes_mes^K_K² is bounded from above by the number not depending on K for a sufficiently wide class of groups G. For example, for Gb = Rⁿ the Haar measureµcoincides with the Lebesgue measure, and ifKis compact convex symmetric neighborhood of zero, then ^mes_mes^K_K² = 2ⁿ. IfGb=Zⁿ, and K = {x ∈ Zⁿ, −N ≤ xi ≤ N, xi ∈ Z, i = 1,2, . . . , n}, then ^mes_mes^K_K² = (^4N+1_2N+1)ⁿ < 2ⁿ. If however in the latter case K = {x∈ Zⁿ, |x| ≤ N} is an integer-valued lattice fromZⁿ contained within a ball of radiusN, then

mesK²

mesK behaves as 2ⁿ for large N . This follows from the known relation P

|k|≤N

1∼ P

|k|≤N

π^n/2[Γ(ⁿ₂ + 1)]⁻¹Nⁿ, where Γ is a Euler function of second kind.

Examples.

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

VK,T(x) = (sπ)⁻ⁿ Yn j=1

x⁻_j²·sinsxj·sin(N+s)xj.

Fors= ^N₂ this is the well known Vall´ee–Poussin kernel ([2], Ch. 8,§8.6).

IfKandT aren-dimensional balls of radiiN ands, then, after perform- ing some calculations, we find that

VK,T(x) = n Ωn

 s N+s

‘ⁿ₂

|x|⁻ⁿIⁿ2(s|x|)· Iⁿ2((N+s)|x|),

where Ωn is the surface area of then-dimensional unit sphere andIn/2 is a first kind Bessel function of ordern/2. In both cases the expression ^{mes(T K)}_mesT is bounded from above by the number (^N_N^+s)ⁿ.

2. If G = E is the unit circumference from R², then Gb = Z and the group G has a character of the form χn(t) = e^−int, t ∈ E, n ∈Z. Take K = [−n, n] = {−n,−n+ 1, . . . , n−1, n}, T = [−s, s]. Then for the kernelVK,T we obtain the above-mentioned de la Vall´ee-Poussin kernelVn,m

defined by formula (12) with m= 2s. If m is an odd number, then VK,T

does not exactly coincide withVn,m, but to obtainVn,mone should modify the definition of the kernelVK,T by means of averaging the kernels(1)c_Ki, whereT ⊂Ki⊂K.

IfGis ann-dimensional torus, then one can take, as VK,T, the product of one-dimensional kernels (12), i.e.,

VK,T(x) = (2π)⁻ⁿ Yn

j=1

(2sj+ 1)⁻ⁿ·sin⁻²xj

2 ·sin sj+1

2

‘ xj×

×sin

Nj+sj+1 2

‘

xj, tj, Nj∈Z⁺.

3. If G =Z, then Gb = E. Characters of the group G have the form χt(n) = tⁿ, 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 onE, the arc length divided intoi√

2π. We have (1)c_T(n) = 1

i√ 2π

Z

T

ξⁿdξ= 1

√2π Zt

−t

e^iθ(n+1)dθ=

=

( ^√₂

√π·(n+1) sint(n+ 1), n6=−1,

√2

√πt, n=−1, andPK,T(f, n) in this example takes the form

PK,T(f, n) =

= 1 tπ

š X∞

k=−∞

k6=n+1

f(k)sint(n−k+ 1)·sin(ϕ+t)(n−k+ 1)

(n−k+ 1)² +t(ϕ+t)

›

and ^{mes(T K)}_mesT = ^ϕ+t_t .

4. LetG =R⁺ be a multiplicative group of positive integers with the unite= 1. The groupGhas a character χ(ξ) =ξ^ix, wherex∈R,Gb =R to within an isomorphism. Take, as K and T, the intervals K= [−N, N], T = [−t, t],N, t∈R⁺. Then

(1)c_K(ξ) = ZN

−N

ξ^ixdx= 2sin(Nlnξ) lnξ ,

PK,T(f, x) = 1 tπ

Z∞

0

f(h)sin(tln^x_h)·sin((N+t) ln_h^x) hln²_h^x dh.

5. Let a = (a0, a1, . . . , an, . . .) be a given sequence, where all an are integer numbers greater than 1, and consider the Cartesian product ∆a = Pn∈N∪{0}{0,1, . . . , an−1}. If one applies the summation operation to ∆a

([1], v. 1, §10), then ∆a together with Tikhonov topology of the product becomes a compact Abelian group. This group G = ∆a is called a group of integera-adic numbers. Its dual group is the discrete group Gb=Z(a^∞) consisting of all numbers of the form

exp

2πi l

a0a1· · ·ar

‘

, l, r∈Z, r >1

([1], v. 1, §10). For the fixed natural number N we take, as a compact symmetric neighborhoodK of the unit, the set of numbers fromZ(a^∞) for whichr≤N. It is proved ([1], v. 1,§25) that the character corresponding to the numberξ= exp(2πi_a ^l

0a1·ar) is written as χξ(g) = exph 2πil

a0a1· · ·ar(g0+a0g1+· · ·+a0a1· · ·ar−1gr)i , where g0+a0g1+· · ·+a0a1· · ·ar−1gr is the sum of the first r+ 1 terms of the a-adic expansion of the element g ∈ ∆a. Assuming that the Haar measure of each point inZ(a^∞) is 1, we have

(1)c_K(g) = Z

K

χξ(g)dξ =

= XN r=0

a0a1X···ar−1

l=1−a0a1···ar

exp

” 2πil a0a1· · ·ar

’ g0+

Xr

k=1

a0· · ·ak−1gk

“•

=

= XN r=0

sin(a0a1· · ·ar−¹2)xr

sin^x₂^r , xr= 2π a0a1· · ·ar

’ g0+

Xr

k=1

a0· · ·ak−1gk

“ .

Clearly,K²=K and, takingT =K, we obtain PK(f, g1) = 1

(N+ 1)² Z

∆a

f(gg1)

”XN r=0

sin(a0a1· · ·ar−¹2)xr

sin^x₂^r

•2

dg, where integration is performed with respect to the measure dual to the taken discrete measure onZ(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 expo- nential type. (Russian)Trudy Inst. Vichisl. Matem. Akad. Nauk Gruzin.

SSR28(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, `Elements de mathematique, livre III, Topologie g´en´erale, 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 Vall´ee-Poussin, Le¸cons sur l’approximation des fonctions d’une variable r´eelle. 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