REALIZATION OF PEETRE K-MODULUS ON THE SPHERE

1998,18(4):427-434

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f£~tiJl1~trl REALIZATION OF PEETRE K-MODULUS ON THE SPHERE 1 )

Chen Shouyin ( F* 't~ ) Li Luoqing ( ~~7k Department: of Mathematics, Hubei Unive1'sity, Wuhan 430062, China Abstract The Peetre K-modulus and the generalized. Riesz summability operators on

the sphere are introduced. The convergence and boundedness of the Riesz operators are discussed. The strong and weak equivalence relationships of the K-moduli and the Riesz operators are presented. The Riesz operators can serve as a realization of the corresponding K-modulus. Key words Kvmodulus. Riesz operator, spherical harmonic expansion.

1

Introduction and Not.at.ions

Let 0 = {It = (x, y, z) E R 3 , IILI = 1} be the unit sphere in the Euclidean space R 3 . By C(O) and LP(O), 1 ~ p < 00, we denote the space of continuous, real-valued functions and the space of (the equivalent classes of) p-integrable functions on 0 endowed with the respective norms

IIflloo:=~~lf(~)I,

1

and

IIfllp:={Llf(~)lPd~}P, l~p
where dIL denotes the surface measure element on O. For convenience, we sometimes use L oo (0) to denote the space C(O). Let l::i. be the Laplace-Beltrami operator, and let 00

L 2(0) = $1in(O) n=O

be the decomposition of the space L 2(0) as a direct orthogonal sum of finite dimensional SO(3)invariant and SO(3)-irreducible subspaces 1i n(O). The subspaces 1t n (O) are the eigensubspaces of ~ corresponding to the eigenvalues -An = -n(n + 1):

The orthogonal projection Yn

:

L 2(O)

Ynf(IL) 1 Received

----+

1t n (O) is given by the formula

11

2n+=47r

n

.

Pn(IL· v)f(v) di/,

Aug.17,1997. The author was partially supported by NSFC Grant #19771009

(1.1)

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P n being the Legendre polynomials which are the only everywhere on the interval [-1,1] infinitely differentiable eigenfunctions of the Legendre operator Lx := (1 - x 2 ) d~2 - 2x d~r. which . in x 1 satisfy Pn (1) 1. Concerning the Legendre polynomials see [7] for more details. Apart from a multiplicative constant, the Legendre function Pn(l . ~), ~ E 0, is the only

=

=

spherical harmonic of degree n which is invariant under orthogonal transformations leaving 1 := (0,0,1) fixed. The linear space ?-In(O) of all spherical harmonics of order n is of dimension diu11i n (0 ) = 2n + 1. Consequently an L 2(0) function can be written as 00

f = LYnf, n=O

where the partial

SUl11S

of the series converge in the L 2 -t opology; moreover, 00

IIfll~ = L IIYnfll~· n=O

Let No be the set of all nonnegative intergers, and let Il.,, n E No, be the subspaces of the spherical polynomials of degree

~

n, i.e.,

TIn := ?-lo(O) EB ?-l1(0) EB ••• EB ?-In(!l). We define by E~p)(/) the best approximation of a function of degree less than or equal to n in LP(!l):

1 E LP(!l)

by spherical polynomials

As a result of the considerations in this paper we obtain (1.2)

and this inequality is best possible in the sense that a weak converse inequality E~P) (I) is valid. To achieve this we define a generalized summability operator Rn,r that will yield the strong .converse inequality of type A in the sense of [5]. The summability operator treated here is given by

n (

Rn,r(!, J.L) := ( ; 1 -

Ak ) ,\~

._ n ( (k(k + 1))r) . Yk!(J.L) = { ; 1 - (n(n + l))r Yk!(J.L),

(1.3)

which is also called the generalized Riesz means, Here Yk is given by (1.1). It is clear that

Rn,rl are in Il., and their derivates are also linear polynomial operators. The Peetre K-modulus is a very useful tool for estimating the rate of convergence of linear operators. Recently, W. Chen, Z. Ditzian and K. Ivanov [3] , and Z. Ditzian and K. Ivanov[5], and V. Totik[8] etc. considered some strong converse inequalities of approximation by linear operators, such as Bernstein, and Bernstein-Durrmeyer operators. Their results show that the order for approximation by some linear operators is completely characterized by the corresponding K-modulus, which is equivalent to the some moduli of smoothness. Very recently, Z. Ditzian (4) considered the Riesz summability operators for Fourier-Legendre expansions. He showed that the Ksmodulus

Chen & Li: REALIZATION OF PEETRE K-MODULUS ON THE SPHERE

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is equivalent to the rate of convergence of the Riesz summability operators. Furthermore he constructed a linear polynomial approximation operator with a rate comparable t.O that of the best polynomial approximation. The aim of this paper is to investigate the realization of the Peetre K-modulus on the sphere, and to consider the relationships of the Peetre K-modulus, the Riesz summability and the best spherical polynomial approximation. We refer to [1] for discussing in some respects of the Peetre K-moduli and the best approximation on the sphere. In Section 2 the main equivalence results are proved. Some applications and generalizations of the main result will be discussed in Section 3.

2

Strong and Weak Equivalence Results We first give some basic properties of the operators Rn,r which are useful in proving the

equivalence results. Lemma 2.1 Let 1 ::; p ::;

00,

and let Rn,r be defined by (1.3). Then Rn,r is of type

(p,p), i.e.,

f

E LP(O).

(2.1)

In this paper, we denote "const" an absolute positive constant which is dependent only on the parameters indicated by the index. Proof If r = 1, Z. Ditzian[4] gave a proof for (2.1). If r

2: 1,

the conclusion can be

deduced from Theorem 3.9 in [9]. We here omit the details. We refer to [2] and [6] for the summability of spherical harmonic expansions. As a corollary of Lemma 2.1, we have limn -. oo IIRn,rf - flip = 0 for all f E LP(O). That is to say {Rn,r} is an approximation process on LP(O). The following lemma presents the relationships between the Riesz summability operators

Rn,r and the differential operator ~r. Lemma 2.2 Let f E LP(O), 1 :S p

~ 00,

and let Rn,rf be defined by (1.3). Then (2.2)

Proof We note that for f E LP(O) there holds

n ( (k(k+1))r) n(n + 1) (k(k + 1)rYk(f)

1 Rn,r(R",rf - f) = - (n(n + 1)Y { ; 1 for 0 ::; k

~

n. By the definition of

s:

we have

It follows that

Combining this equation with (2.3) we get

(2.3)

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Lemma 2.2 is proved. For a given function in C 2r(O) we have the Jackson-type inequality which will be proved by following an idea of Ditzian[4]. Lemma 2.3 Let f E C 2r(o), and let Rn,r be defined by (1.3). Then for 1 ~ p ~

CO:~~p,r lI~r flip.

lIRn,rf .- flip :::; Proof For

f

00

(2.4)

E LP(O) we have from (2.2) in Lemma 2.2

(_1)r+l s r R;',rf - Rn,rf = (n(n + 1W I I Rn,rf· By direct calculations, we know that

~

and Rn,r are exchangable, (2.5)

and for all n, m,

.

f

Rm,rRn,rf = Rn,rRm,rf, Furthermore we deduce that

(1+ 1))r -

2 f D -R f - ( )r+l Rm,r - .LLm+l,r m,r - -1 (m(m

_ (_1)r+l

-

E LP(O).

1)

r ((m + 1)(m + 2)Y ~ Rm,rf

+ 2)r - m r ~r R f (m(m + l)(m + 2))r mr s » (m

and

R;"+l,rf - R m,r Rm+l,rf = (-1Y Cm(m 1+

1W - ((m + 1)~m + 2))r ) s: Rm+l,rf

-(_1)r (m+2)r- mr ~r f (m(m + 1)(m + 2))r Rm+l,r' Note that (m + 2)r - m"

f"wJ

rm r -

1

as m

It follows that

-+ 00.

IIR~,rf - R~+l,rfllp ~ IIR~,rf - Rm+ 1,r Rm,rfll p + IIR~+l,rf - R,n,rRm+l,rfll p

/

< C::?~r (lI~r Rm,rfllp + lI~r R m+

1,rfll p),

Hence Lemma 2.1 and (2.5) yield for f E C 2 r(o ) that 2

2

II R m,r f - R m+1 ,rf llp Lemma 2.1 and (2.5) also imply IIR~,rf - flip

< constp,r m 2r+1

-+

0 as n

-+ 00.

00

m=n We finally get Jackson's estimate for f E c 2r(o ) 00

m=n

r

II~ flip·

This gives

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Lemma 2.3 is proved. The following strong equivalence relation is understood the strong converse inequality in the sense of [5].

Theorem 2.4 Let f E LP(O), 1 ~ p ~ 00, and ,let Rn,r and K 2r(f, n- 2r )p be defined by (1.3) and (1.2) respectively. We have the equivalence relation: (2.6) Here and in the following A

f'V

B means there exists a positive constant "const" such that

co~st A ~ B ~ const A. Proof Choose g E C 2r(O) such that

We get

II R n,r f - flip

~ IIRn,r(f - g) - (f -

g)llp + II R n,rg -

gllp 2r ~ constp,r K 2r (! , n- )p + IIRn,rg - gllp

By making use of Lemma 2.3, we have for f E C 2r (o ) that

Combining the inequalities above we get

To prove the converse result, by making use of Lemmas 2.1 and 2.2 we have

It follows from the definition of K-modulus that

The proof of Theorem 2.4 is complete. , From Lemma 2.2 and the proof of Theorem 2.4 we deduce that

This equivalence relationship shows that the Rn,r' can serve as a realization of the Ksmodulus K 2r (! , n- 2r )p. The realization implies the natural relation for hierarchy of measures of smoothness on the K-modulus.Hence realization is important. We now present the weak equivalence results between the best polynomial approximant and the generalized Riesz summability operators.

Theorem 2.5 Let! E Yen), 1'~p ~

00,

and let Rn,r! be defined by (1.3). Then

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Val.IS

Conversely,

In particular, for 0


if, and only if,

Proof The first inequality is obvious. Concerning the second one, let Qn be a polynomial of best approximation of order less than or equal to n of f in LP(n). The following Bernstein type inequality is easy to verify:

Then for any integer m

~

0, we have by Lemma 2.1, (2.7)

Set Q -1 == 0 and write

m

Q2

m

= L{Q2i -

Q2i- 1 }

.

j=O

By Jackson- and Bernstein-type inequalities above,

Taking into account that 2k

22kr E 2k(f)p ~ 22r

j2r- 1Ej(f)p,

L j=2 k -

we obtain

1+1

2m

IIRn,rQ2 m

-

constp,r '"'""' .2r-1 ' Q2m lip ~ 2r L..J(k + 1) Ek(f)p· n

k=O

Finally we choose m such that 2m - ~ n < 2 , and we incorporate above inequality in (2.7) and get the desired 'inequality. The proof of Theorem 2.5 is complete. 1

m

By this theorem and Theorem 2.4 we have

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Theorem 2.6 Let

f E LP(O), 1 ~

p~

00,

433

and let Rn,rf be defined by (1.3). Then

conversely,

In particular, for 0 < a


if, and only if,

3

Fur-ther Results and Generalizations

In this section we deal with Rn == Rn,l and their iteration R~f := Rn(R~-l f). Recall that the definition of Rn,l we write (3.1) From Theorem 2.4 we have for

f

E

LP(O), 1 ::; p ::; 00, (3.2)

Let I be the identity operator. We may deduce the following equivalence relation (3.3) By combining and iterating the operators R n we define linear operators Ln,r as (3.4) Analogous to (2.2) we have, for k 2: 1,

This together with (3.3) implies that

It follows that Summarizing above we conclude that Ln,r may also be serve as a realization of the Peetre K-modulus K 2r (f , n- 2r )p. That 'is

.

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Theorem 3.1

For! E LP(O), 1

~ p ~ 00,

s;

Vol.18

given by (3.1) and Ln,r given by (3.4),

we have The near best polynomial approximant Qn which is a polynomial satisfying

can also serve as a realization of the K-modulus K 2r (! , n- 2r )p. That is, we have

We now define the delayed means using

Rn by

2n + 1 Vn! :== 2 3n + 1 R 2n!

n+1 - 3n + 1 Rnf·

Obviously, Vnh == h for h E II n and IIVnfilp ~ constpllfllp. These facts imply that

which makes Vnf a near best polynomial approximant. Therefore Vnf can realize the Peetre K-modulus K 2r (f , n- 2r )po References 1 Berens H, Li Luoqing. The Peetre K-moduli and best approximation on the sphere. Acta Math Sinica, 1996, 38(5): 589 -599 2 Bonami A, Clerc J L. Sommes de Cesaro et Multiplicateurs des Developpements en Harmonics Spheriques. Trans Amer Math Soc, 1973, 183: 223 - 263 3 Chen W, Ditzian Z, Ivanov K. Strong converse inequality for the Bernstein-Durrrneyer operator. J Approximation Theory, 1993, 75: 25 - 43 4 Ditzian Z. A K-functional and the rate of convergence of some linear polynomial operators. Proc Amer Math Soc, 1996, 124(6): 1773 - 1781 5 Ditzian Z, Ivanov K. Strong converse inequalities. J d'Analyse Math, 1993,61: 61 - 111 6 7 8 9

Sogge C D. Oscillatory integrals and spherical harmonics. Duke Math J, 1986, 53(1): 43 - 65 Szego G. Orthogonal Polynomials. Amer Math Soc Coli Publ, No. 23, Providence 1959 Totik V. Approximation by Bernstein polynomials. Amer J Math, 1994, 116: 995 - 1018 Trebels W. Multipliers for (C, a)- bounded Fourier expansions in Banach spaces and approximation theory. Lecture Notes in Math., Vol. 329. Berlin, New York: Springer-Verlag, 1973