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ON SOME PROBLEMS OF APPROXIMATION ON NONCOMPACT SYMMETRIC SPACES
S.S.PLATONOV Petrozavodsk State university 185018 Petrozavodsk, Karelia,Russia
1 Introduction
The modern technique of harmonic analysis on symmetric spaces makes possible to extend a lot of problems of classical harmonic analysis to symmetric spaces. In particular we can study the problems of function approximation on symmetric spaces.
Let X be a Riemannian symmetric space. Let LP(X), 1 < p < oo, be the set of all measurable functions / defined on X, for which the norm
-a
l/(*)lp dxSj
i Ip
is finite, dx being the Riemannian measure on X. Let us denote by Loo(X) the set of all continuous bounded functions on X, endowed with the norm
WfWoo ■= sup |/(x)|.
x£X
In classical case X is the circle S1 or E, the apparatus of approximation being the set of trigonometric polynomials or the set of entire functions of exponential type respectively (see [1], [2]). Let Vv — Vv{X) be the set of all trigonometric polynomials of degree < v for X = S1 or the set of all entire functions of exponential type < v for X = E. The best approximation of a function f(x) E LP(X) in Lp-metric is obtained by
E.U)P = inf ||/-*||,.
One of the main problems of approximation theory is to determine relations between the degree of decrease El/(f)p as v —► oo and the intrinsic properties of / (its smoothness, its modulus of continuity, etc.) By means of direct Jackson-type theorems upper bounds of E„(f) by the continuity modulus of / have been found. Another result of the approximation theory is the description of certain function classes in terms of the best approximations and converse approximation theorems of which important elements are the Bernstein-type inequalities.
The problems of approximation on a compact symmetric space X of rank 1 by the spherical harmonic polynomials have been active studied in the recent years, especially the case when X is the n-dimensional sphere Sn. For compact symmetric space of rank 1 X the spherical harmonic polynomials are defined as linear combinations of eigenfunctions of the Laplace — Beltrami operator V, in the case X = Sn the spherical harmonic polynomials are the classical spherical polynomials. It is not my aim here to discuss these investigations, so I refer to the papers of Nikolskii and Lizorkin [3], Ragozin [4], Rustamov [5], Platonov [6,7,8], Luoqing [9], Kamzolov [10], Ivanov [11]. The reader can find there a bibliography of recent researches on this subject.
We consider the case when X is a Riemannian noncompact symmetric space of rank 1. For this case there are only a few results. A general approach to the approximation on an arbitrary Riemannian manifold X was developed by Lizorkin in [12], but the entire vectors of finite degree, which were used by him as apparatus of approximation, are not functions on X. Another possible apparatus of approximation on Riemannian manifold is the set of functions from L2(X) with bounded spectrum. The problems of the approximation theory on the n-dimensional Lobachevsky space by the functions with bounded spectrum was considered by Lizorkin and Petrova in [13], [14].
Further we shall consider certain problems of approximation theory in Z,2-metric on arbitrary Rieman-nian symmetric space of rank 1 by the functions with bounded spectrum (see [15]). The main subjects are: direct Jackson-type theorems, the description of Nikolskii — Besov spaces in terms of the best approximations, Bernstein-type inequalities. The special case of noncompact symmetric spaces of rank
1 is the n-dimensional Lobachevsky space and a lot of of these results are generalization of the results from [13], [14].
2 Functions with bounded spectrum and Bernstein-type inequalities
Any Riemannian symmetric space X can be realized as the quotient space G/K, where G is a real semisimple connected Lie group with finite center, К is a maximal compact subgroup of G. Let G = NAK be the Ivasava decomposition of G, M be the centralizer of A in К, В = K/M. By dx we denote the element of G-invariant measure on X, by db and dk the elements of Л'-invariant normalized measures on В and К respectively. Let g, t, a and n be the Lie algebras of the Lie groups G, K, A and N respectively, o* meaning the real dual space of a. Let £ be the set of all restricted roots (£ С a*), be the set of all positive restricted roots,
p==jE“-
a€S+
By <, > we denote the Killing form on 0. This form is positive defined on a. For each A £ o' let H\ £ a be determined by А(Я) =< H\, H > for H £ a. For A,/iGa* we define
< Х,ц>:=< Н\,Нц > .
a* and a can be identified by the correspondence A —► H\. Let
a; = {A6a*: <a,A>0 Va £ £+}
be the positive Weyl chamber.
For g £ G by A(g) £ a we denote the unique element such that
g = n • exp A(g) ■ u,
where ueK,neN. For x = gK £ X = G/K and b = kM £ В = K/M we define
A(x,b) = A(k~1g).
By Cc(X) we denote the set of all continuous complex-valued functions on X with compact support.
The Fourier transform of an arbitrary function / £ Cc(X) is the function on a* x В defined by the formula
/(Л, 6) = J f(x)e^ix+p^x'bUx, A £ a*, be В x
( see [16], [17]). Then the Plancherel formula
J \f(x)\2dx= J \f(X,b)\2\c(X)\-4Xdb (1)
x а*+хв
holds, where c(A) is the Harish-Chandra’s c-function, d\ is the Lebesque measure on a* properly normalized.
The Fourier transform f{x) —* f(X,b) uniquely extends from Cc(X) to an isomorphism of the Hilbert space L2(X) to the Hilbert space L2(a*+ x В, |c(A)|-2 dX db) with the Plancherel formula still remaining in force.
A function / £ L2{X) is called function with bounded spectrum of type и > 0 if /(A, b) = 0 for |A| > is, where |A| =< А, А >^2). We denote by Jv the set of all this functions. It can be proved that Jv С C°°(X).
¿From here on let X be a noncompact Riemannian symmetric space of rank 1. In this case dim a’ = 1. We choose the basis vector Hq £ a* such that Hq £ and \Hq\ = 1. We will identify a" with Ж. by
means of the correspondence t *-> tHo, t £ M. By this identification the set a+ correspond to the set of positive real numbers. By d(x,y) we denote the distance from x to y, where x,y £ X. Let
cr(x]t) = {y £ X : d(y,x) = t}
be the sphere in X of radius t and center x. By dfix(y) we understand the surface element of a(x]t) and let |cr(i)| be volume of a(x,t).
For / £ Cc(X) we introduce the function Stf
(Stf)(x) = j^jj- J f(y)dpx{y), t> 0.
<r(ar;t)
The operator St is called averaging operator. By continuity St can be extended to L2(X) and moreover
I|S|/II> < № VOO, v/eisW,
where || • ||2 is the norm in L2{X).
We define the finite differences A*/, k = 0,1, 2,..of a function / 6 L2(X) with a step t > 0 by the rules
A?/(*) := /(*), Alf(x) = Atf(x) := f(x) - S,f(x),
Affix) := A,(Akr'f(x).
We can also write
A */(*) = (/ —S,)‘/(*) where I is the identity operator. Let V be the Laplace — Beltrami operator on X.
Lemma 1 Let $ £ Jv, v>\, <.0, k = 1,2,.... Then there exist positive constants ci,c2 (c\ = ci(X,k),
c2 = c2(X,k)) such that
||©‘*||2 < C2«'”||*||2; (2)
||A‘®||2 < c3(1/i)2*||i>||2, (3)
The proof see in [15]. The inequalities (2), (3) and their analogs are called the Bernstein type inequalities. They play an important role for the converse approximayion theorems.
3 Direct Jackson-type theorems
For f £L2{X) let
w*(/,6)2:= sup ||Af/||2> <5>0.
0 <t<6
#
2 is called the spherical continuity modulus of order k. By
EAfh :=#infj|/-*||2.
we define the best approximation of a function / £ L2(X) by functions with bounded spectrum of type v in L2.
The following theorems are called direct Jackson-type theorems.
Theorem 1 There exists positive constant C4 = c4(X, k), k = 1,2,..., such that for f £ L2(X), u > 1
Ev(f) < C4Wjfc(/,l/i/)2, (4)
Let s £ N = 1, 2,..., V be the Laplace — Beltrami operator on X. In general / £ L2(X) the element Vf is a distribution on X.
Theorem 2 Let £ L2(X). Then there exists positive constant C5 = Cs(X,A:,s) such that
for v > 1
The main elements of the proof of Theorem 1 are the following Lemmas 2 and 3. Using the Harish-Chandra’s formula for spherical functions (see [17]) for A £ a* we denote by tp\(x) the spherical function
<px{x) = Jdk, x = gl< eX = G/K.
K
The spherical function <p\(x) depends only on the distance between the points x end o — el\ £ X = G/K, e being the unit element of G, and we can write
(p\(x) = <px(t), t = d(x,o).
Lemma 2 For every f £ L2(X)
S7f(\,b) = <px(t)f(\,b)
where f i—► / is the Fourier transform on X.
Lemma 3 For all A £ a*, t > 0 the following inequalities hold:
1) kM*)l < i;
2) 1 - px(t) < t2(< A, A > + < p,p >);
3) 1 — <p\(t) > c if \t > 1 where c = c(X) > 0 is a constant.
Proof of Theorem 1.
It follows from the Plancherel formula that
OO
El(fh = J J\f(X,b)\2dn{\)db, (5)
v B
where d/i(A) = |c(A)|-2dA. From Lemma 2 it follows that
1 - tp\(l/v) > c
for A > v. Then from (3), Lemma 3 and the Plancherel formula we obtain
00 OJ. -
E2Mh < c-2t ff( 1 - (MW) |/(A, 6)1 dfi(X) db <
V B
00 Oi- ~
< c-2t ff( 1 - MU'')) l/(A, 6)1 dn(X) db =
= c~2k\\(l - Si/U)kf(x)\\2 < C-2kul(f} l/v)2,
Hence
Eu{f) < c4ujk(f, \/v)2
with C4 = C .
The proof of Lemmas 2-3 and Theorem 2 see in [15].
4 Nikolskii — Besov spaces
Let r > 0, k and s be any nonnegative integers such that 2k > r — 2s > 0. By definition a function /(*) 6 Hi = H'2(X) if/,©/.........V’f 6 L2(X) and
uk(V’f,bh<AJ6'-2• V6> 0
for a positive constant Aj. For / £ H£ by hr2(f) we denote the seminorm
Uk(V’f,6) 2
¿>0
H2 is a Banach space with respect to the norm
||/lk = ||/||2 + AStf). (6)
The next theorem gives the description of the soaces H2 in terms of the best approximations.
1 r / r\ * \ *
h2(/) := sup-----------—2, .
Theorem 3 If f £ H2 then for u > 1
EAfh<C6~--
Conversely, if f £ L2{X) and for u > 1 there exists a constant A = A(f) such that
EAfh <
l/r
then f £ H2 and
\\I\\h; < C7GI/H2 + A).
Here C6,C7 are constants, independent of f but which can depend on X,k,r,s.
Corollary 1 The space H2 does not depend on the k,s £ TL+ — {0,1,2,...} such that 2k > r — 2s > 0.
These results can be transferred to Nikolskii — Besov spaces B2q = B2q{X), r > 0, 1 < q < 00. Let k, s £ be any such that 2k > r — 2s > 0. By definition a function / £ B2q if /, Vf,..., Vs f £ L2(X) and the seminorm b2q(f) is finite, where
bUf) =
(/T.^ withl<g<oo,
sup —^1 with q = 00.
¿>0 <5r 5
B2q is a Banach space with respect to the norm
ll/Ik, = l№ + »5, (/)• (7)
Note that .ßjoo = ^2-
Theorem 4 Lei a > 1 be arbitrary real number (for example we can take a = 2). A function f £ L2{X) belongs to the space Bif and only if br2q(f) < 00, where
anr? Eqan(f)2 with q < 00
n=0
sup anr Ean(f)2 Mn/A 9 = 00.
«=0,1,2,...
6r29(/) =
Moreover the norms (7) and
\\f\\2 + br2q(f)
are equivalent.
The proofs of Theorems 3 and 4 follow the scheme of corresponding theorems in [2] where the case X = En is considered. The main elements of proof are Theorem 2 and Bernstein type inequalities (2) — (3). See [15] for details.
I would like to thank K.V.Matyushichev for help with the language.
The work supported by the Russian Foundation of Fundamental Research grant 95-01-01391
References
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[3] Nikolskii, S.M., Lizorkin, P.I. (1989) Approximation on the sphere — A survey, Banach Center Publications, Vol. 22, pp. 281-292.
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14] Petrova, T.V. (1992) Approximation on the hyperboloid in L2-metric, Trudy MIRAN, Vol. 194, pp. 215-228 (in Russian).
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