ON THE APPROXIMATION OF FUNCTIONS ON LOCALLY COMPACT ABELIAN GROUPS
D. UGULAVA
Abstract. Questions of approximative nature are considered for a space of functionsLp(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 Lp(G, µ) or,
simply, by Lp(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 normkfkp={RG|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
b
f(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,Lpspace, character, Fourier transform, entire functions of exponential type, best approximation.
379
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
ˇ f(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 spacesLp(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 dualband pairs of such measures will be considered below. For f ∈ Lp(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(1p+1q = 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 spacesLp(G, µ), 1≤p≤ ∞,
since, generally speaking, for p > 2 the Fourier transform of a function f ∈ Lp does not exist. Therefore below we shall give a modified
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 fromUGb we shall consider the function defined onG([3], Ch. 5,§1)
VK,T(g) = (mesT)−1(1)cT(g)·(1)cT 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)cTk2· k(1)cT Kk2=
= 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
c
(1)K(g) =
Z
K
χ(g)dχ=
Z
K
χ(g−1)dχ=Z
K
χ(g)dχ=(1)cK(g).
By Parceval’s theorem ([1], v. 2,§31) we obtain
Z
G
VK,T(g)dg=
1 mesT
Z
G c
(1)T(g)·(1)cT 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 ∈Lp(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 ∈UGb and p∈[1,∞]. ByWp(K) we shall denote
the set of functionsf from the spaceLp(G, µ) for which we have the formula
f(g) =PK,T(f, g) ∀g∈G and T ∈UGb. (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 ∈Lp(Rn)
the whole n-dimensional complex Euclidean space Cn, and for any ε > 0
there exists a numberAεsuch that the inequality
|g(z)| ≤Aε·exp n X
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 c
(1)K = 2n Qn i=1
(xi)−1sinνixi. Conversely, let (6) hold for f ∈ Lp(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). ThusWp(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. Wp(K) is the shift-invariant closed subspace of the space
Lp(G).
Proof. Let fn ∈ Wp(K) be an arbitrary converging sequence and kfn−
fkp →0 as n→ ∞. Using the well known estimate of a convolution norm
([1], v. 2,§31) forLp, we obtain
kPK,T(fn−f, g)kp≤ kfn−fkp· kVK,Tk1, ∀T ∈UGb.
ButPK,T(fn) =fn, ∀n∈N, and thusfn tends inLp simultaneously tof
and PK,T(f). Therefore f =PK,T(f) and f ∈ Wp(K). The operation of
shift by the elementhwill be denoted byLh. Let f ∈Wp(K) andT ∈UGb.
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 Fp(K) the set of functions from the
spaceLp(G, µ) whose Fourier transform supports belong toK. Using
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
b
f(χ)·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 fromFp(K). Iff ∈
Fp(K), 1 ≤ p ≤ 2, then f ∈ Lp1(G) for arbitrary p
1 ∈ [p,∞]. Indeed,
by virtue of (2), fb∈Lq(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 everywhereonG. 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 ∈Lp∩L∞, we havef ∈Lp1(G) for anyp
1∈[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 setsWp(K)andFp(K)coincide.
Proof. Let us first assume thatf ∈Wp(K). Then for∀T ∈U b
G we have
\
f∗VK,T(χ) =fb(χ)
and therefore
b
f(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, thenVb
K,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
b
VK,T(χ) =
1 mesT
Z
U−1
+
Z
T\U−1
= 1 mesT
Z
T\U−1
Then from (7) it follows that forχ /∈Kwe havefb(χ) = 0 almost everywhere and thereforef ∈ Fp(K).
Let nowf ∈ Fp(K) andT be an arbitrary set from U b
G. We have
b
PK,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 ∈ Fp(K) implies P\
K,T(f)(χ) = 0. Therefore \
PK,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 ∈ UGb and g∈G.
Remark . IfG=Rn, by applying the well known Peley–Wiener theorem
([5], Ch. 6, §4) we can prove that the classes Wp(K) and Fp(K) coincide
for p > 2 too provided that by Fp(K) we shall understand the class of
functionsf ∈Lp(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 ∈ Fp(K), 1 ≤ p ≤ 2, gives that f ∈ Lp1(G) for any
p1 ∈ [p,∞] holds for functions from Wp(K) for p ∈ [1,∞]. Indeed, if
f ∈Wp(K), thenf =f∗V
K,T for anyT ∈UGb. ButVK,T ∈ F1(KT2) and,
as we already know, this implies that VK,T ∈Lr(G) for all r∈[1,∞]. By
the Young inequality ([1], v. 1,§20)
kfkp1 ≤ kfkp· kVK,Tkr, 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 Wp(K) are continuous for any 1≤p≤ ∞.
Indeed, iff ∈Wp(K), thenf ∈W∞(K)
|f(g1)−f(g)|=
Z
G
f(h)VK,T(h−1g1)−VK,T(h−1g)
dh
≤
≤ 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 G
ε
outsideGε. We shall show that for 1≤p <∞the functions fromWp(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
Next, using (4), we obtain
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∈G
ε. Since p6=∞, q > 1 and the fact thatV(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
After multiplying both sides of equality (5) by VKT2,T 1(g−
UGb, and integrating them over G, we obtain
Here changing the integration order is righful by virtue of the Fubini theorem ([1), v. 1,§13) and in view of the fact that
Next, by the change of the variable we obtain
Z
internal integral can be replaced byVK,T(h−1t). As a result, Z
Remark . While proving Lemma 3, concomittantly we have actually proved the following property of functions from Wp(K): if f ∈ Wp(K),
p ∈ [1,∞], and K1 is an arbitrary compact from UGb containingK, then
f ∈ Wp(K
1). This can be shown by repeating the proof of Lemma 3,
Proof. First we shall prove that iff ∈Wp(K) and χ is an arbitrary
char-acter of the group G, then f ·χ ∈ Wp(K
1) for some compact K1 ∈ UGb.
Indeed, sincef ∈Wp(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
\
(VK,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∈UGb, i.e.,χ·VK,T ∈W1(K1) and χ·VK,T =χVK,T ∗VK1,T1
for anyT, T1∈UGb. 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∈Wp(K 1).
We shall show that for allK∈UGbthe functions fromWp(K) are dense
everywhere inLp(G, µ). Let us assume that the opposite is true. LetAp be
the closure of the union∪
KW
p(K) andAp not coinciding withLp(G, µ).
There exists a well defined nontrivial linear functional on Lp(G) which
is equal to zero on Ap. This functional is defined by a nontrivial function
ϕ∈Lq(G) ([1], v. 1,§12). Sincef·χ∈ Ap for arbitraryχ∈G, thenb Z
G
f χϕ dµ= 0, f ∈Lp(G), ϕ∈Lq(G).
The latter equality implies
d
f·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 ∈ Wp(K), where K is an arbitrary
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 thatRGf χdµ= 0, ∀f ∈C0(G)∩W∞(K),χ∈G.b Hence the Fourier transform of the measuref dµis equal to zero on the entire
b
G. Thereforef 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 ∈
Lp(G) by the subspaceWp(K), i. e., the value
EK(f)p,G= inf
g∈Wp(K)kf −gkL
p(G). (10)
If we obtain inf on some elementg0∈Wp(K), then it will be called the best
approximation element off inWp(K). We shall prove that such an element
necessarily exists if 1≤p≤ ∞. Indeed, letf ∈Lp(G), andt
n∈Wp(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 Lp, and by applying inequality (8) to
p1=∞we have
ktnk∞≤C· ktnkp≤A,
whereAdoes not depend onn. Next we obtain
ktn(h−1g)−tn(g)kp= Z
G
tn(ξ)VK,T(ξ−1h−1g)−VK,T(ξ−1g)dξ p≤
≤ ktnkp· 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} inLp(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∈Wp(K). The sequencet nν
will again be denoted bytn. In the first place, for any compactG1⊂Gwe
have
But since lim
n→∞kt−tnkL
∞
(G1) = 0 and ktnkLp(G) ≤ C, for any compact
G1 ⊂G we have ktkLp(G1) ≤C, where C is a constant not depending on
G1. Thereforet∈Lp(G). Let consider an arbitrary compactG1⊂G. Since
tn ∈Wp(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 k
tnkp· 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) =RGt(h)VK,T(h−1g)dh and therefore
t∈Wp(K). Further, for any compactG1⊂Gwe have
kf −tkLp(G1)= lim
n→∞kf −tnkLp(G1)≤nlim→∞kf −tnkLp(G)=d
and thus t is the best approximation element of f ∈ Lp(G) in Wp(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 b
f(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 inbthe sense of the spacesL1and L2coincide, we have
S(g) =
Z
K b
Applying Parceval’s theorem we obtain
kf−Sk22=kfk22−
Z
K b
S(χ)fb(χ)dχ− Z
K b
S(χ)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 b
S(χ)χ(g)dχ=
Z
K b
f(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 ∈Lp(T)
kf−σn,m(f)kp≤2
n+ 1
m+ 1En(f)p,
where
(σn,mf)(x) = 1
m+ 1
nX+m
k=n
(Skf)(x) = 2π Z
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)2 x·sin(2n+m2+1)x 2π(m+ 1) sin2x
2
. (12)
In [10], this result was obtained for the space ofC(T)-continuous functions onT. Ifm= [n
2], 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
symmetric neighborhoods of the unit of the dual groupGb, then
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 ∈ Wp(K) be a minimizing sequence for the function f ∈
Lp(G), 1≤p≤ ∞. Then
kf(g)−PK,T(f, g)kp≤ kf−tnkp+kPK,T(f−tn)kp≤
≤ kf−tnkp+kf−tnkp· kVK,Tk1.
Hence, taking into account inequality (4) and the relation lim
n→∞kf−tnkp= EK(f)p,G, we obtain estimate (13).
Remark . To have a best order approximation in (13), we must chooseK andT fromUGb 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 +mesK
2
mesK
1/2
·EK(f)p,G,
and mesmesKK2 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 mesmesKK2 = 2n. IfGb=Zn, and
K = {x ∈ Zn, −N ≤ x
i ≤ N, xi ∈ Z, i = 1,2, . . . , n}, then mesK 2 mesK =
(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
mesPK behaves as 2n for large N . This follows from the known relation
|k|≤N
1∼ P
|k|≤N
πn/2[Γ(n
2 + 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) =eitx, 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 n Y
j=1
x−j2·sinsxj·sin(N+s)xj.
Fors= N
IfKandT aren-dimensional balls of radiiN ands, then, after perform-ing some calculations, we find that
VK,T(x) = n
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(mesT KT ) is bounded from above by the number (NN+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)cKi,
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
length divided intoi√2π. We have
c
andPK,T(f, n) in this example takes the form
and mes(mesT KT )= ϕ+tt.
4. LetG =R+ be a multiplicative group of positive integers with the
unite= 1. The groupGhas a character χ(ξ) =ξix, wherex∈R,Gb =R
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 groupG = ∆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
exp2πi 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 l
of the a-adic expansion of the element g ∈ ∆a. Assuming that the Haar
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
sinxr
2
2
dg,
where integration is performed with respect to the measure dual to the taken discrete measure onZ(a∞).
References
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2. S. M. Nikolskii, Approximation of functions of several variables and embedding theorems. (Translation from Russian) Springer, Berlin, etc.,
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Oxford, Clarendon Press,1968.
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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:
