Approximating Properties of Entire Functions of Exponential Type

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

201, 642]659 Ž1996.

0279

Approximating Properties of Entire Functions of Exponential Type Fang Gensun Department of Mathematics, Beijing Normal Uni¨ ersity, Beijing, 100875, People’s Republic of China Submitted by E. W. Cheney Received January 4, 1995

In this paper, we prove that E Ž f , Es l L p Ž R . . p Ž R . s lim E Ž f , Sp r s , m l L p Ž R . . p Ž R . ,

1 - p - `,

E Ž f , Es l L p Ž R . . p Ž R . F lim E Ž f , Sp r s , m l L p Ž R . . p Ž R . ,

p s 1, `,

mª`

mª`

E Ž f , Es X l L p Ž R . . p Ž R . G lim E Ž f , Sp r s , m l L p Ž R . . p Ž R . , mª`

X

p s 1, `; 0 - s - s , where Sp r s , m denotes the space of cardinal splines of degree m with nodes w n q 12 Ž m y 1. h x4j g Z , and Es denotes the restriction to the real line R of entire functions of exponential type s . From this connection, we solve two extremal problems of some fundamental classes of functions defined of R. Q 1996 Academic Press, Inc.

1. INTRODUCTION Both entire functions of exponential type and cardinal splines are fundamental approximating tools for the classes of functions defined on R. On the one hand, Bernstein, Krein, and Akhieser w1x and many others studied the extremal properties of entire functions of exponential type and obtained many important results. On the other hand, in the seventies, Schoenberg w2x, Karlin, Micchelli and Pinkus w3x, Marsden, Richards, and Riemenschneider w4x, and many others investigated the properties of cardinal splines, and their research gives another new and powerful tool for the approximation of classes of functions defined on R. Recently, after Supported by Natural Science Fundation of P. R. China. 642 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

APPROXIMATING PROPERTIES OF ENTIRE FUNCTIONS

643

the concept of average width was introduced by Tikhomirov w5x, some works Žsee w6]8x and the references of w6x. have demonstrated that both entire functions of exponential type and cardinal splines are optimal in the sense of average width for some fundamental classes of functions defined on R. It is the purpose of the present paper to establish some connection between approximating properties of the entire functions of exponential type and cardinal splines. The interest of this result lies in the fact that we may solve some extremal problems of approximation of some classes of functions defined on R by entire functions of exponential type. DEFINITION 1.1. Let m be a non-negative integer, and let Sh, m s  s4 denote the class of functions s, satisfying the following conditions: Ži. s g C my 1 ŽR., Žii. the restriction of s to every interval w n h q 12 Ž m y 1. h, Ž n q 1. h q 21 Ž m y 1. h. is a polynomial of degree not exceeding m. In particular, for h s 1, we write also Sm instead of S1, m . We denote by S˜1r n, m , n g N, the 2-periodic subspace of S1r n, m . Let E be a finite interval or R; we define L p Ž E ., 1 F p F `, as the classical Lebesgue space on E, 5 ? 5 pŽ E . denotes the norm of L p Ž E ., and for convenience, we write also 5 ? 5 p [ 5 ? 5 pŽR . , and finally let Sh , m , p s  s g Sh , m : s g L p Ž R . 4 ,

Sm , p [ S1, m , p .

Ž 1.1.

Denote by Lrp Ž E ., 1 F p F `, the subspace of function f in L p Ž E . for which the Ž r y 1.st derivation of f exists, is absolutely continuous on E, and 5 f Ž r . 5 pŽ E . is finite; further, Wpr Ž E . [  f g Lrp Ž E . : 5 f Ž r . 5 pŽ E . F 1 4 .

Ž 1.2.

˜ p , W˜pr be the 2-periodic subsets of L p wy1, 1x and Wpr wy1, 1x, Let L respectively, and let l p , 1 F p F `, be the Banach space of double infinite bounded sequences with the usual norm 5  yj 4 5 p [

< yj < p

žÝ /

1rp

,

1 F p - `,

jgZ

5  y j 4 5 ` s sup < y j < . jgZ

Ž 1.3.

644

FANG GENSUN

Further, Es ŽR., s ) 0, denotes the restriction on R of entire functions of exponential type s , and let Bs , p s Es Ž R . l L p Ž R . ,

1 F p F `,

Bs [ Bs , ` .

THEOREM A w2x. Ž1. For a gi¨ en sequence y s  y j 4j g Z g l p , 1 F p F `, there is a unique element Lm y g Sm, p , interpolating the gi¨ en data at the integers, i.e., Lm y Ž j . s y j ,

j g Z,

and Lm y Ž x . s

Ý yj Lm Ž x y y . ,

Ž 1.4.

where L mŽ x . g Sm, p is the fundamental spline of Sm which interpolates the data y 0 s 1, y j s 0, j / 0. Ž2. If s g Sm, p , 1 F p F `, then  smŽ j .4j g Z g l p . In w4x, the behavior of 5 Lm y 5 p , y s  y j 4j g Z g l p , p-fixed, as m tends to infinity, is investigated, and the following theorem, among others, is proved. THEOREM B w4x. lim

mª`

Let y s  y j 4j g Z g l p , 1 - p - `. Then

Ý

yj Lm Ž x y j . y

jgZ

Ý jgZ

yj

sin p Ž x y j .

p Ž x y j.

s 0,

Ž 1.5.

p

where Ý j g Z y j ŽŽsin p Ž x y j ..rp Ž x y j .. is usually called the Whittaker cardinal series. DEFINITION 1.2. Let X be a normed linear space, for a given f and a subset M of X; the quantities E Ž f , N . X s inf 5 f y g 5 X , ggN

E Ž M , N . X s sup E Ž f , N . X fgM

are called the best approximations of the element f and the subset M by subspace N of X, respectively. It is well known that the subspace of trigonometric polynomials and periodic splines are both important approximation tools for the classes of periodic functions. Velikin w9x studied the connection between approximation properties of trigonometric polynomials and periodic splines and based on this relation, he solved some extremal problems of some convolution classes of periodic functions. Among others, he proved

645

APPROXIMATING PROPERTIES OF ENTIRE FUNCTIONS

THEOREM C w9x.

˜ p , 1 F p F `, m, n g N. Then Let f g L

lim E Ž f , S˜ny1 , m . pwy1, 1x s E Ž f , Tny1 . pwy1, 1x ,

mª`

Ž 1.6.

where Tny 1 s span  1, cos p x, sin p x, . . . , cos Ž n y 1 . p x, sin Ž n y 1 . p x 4 . From Theorem C, it is a natural conjecture that lim E Ž f , Sp r s , m , p . p s E Ž f , Bs , p . p ,

mª`

1 F p F `,

Ž 1.7.

but we showed in w10x that Ž1.7. is not valid in the case of p s `; i.e., there is a function f g C ŽR. l L`ŽR., such that lim m ª` E Ž f , Sp r s , m , ` . ` ) E Ž f , Bs . ` , and we proved also that Ž1.7. is true in the case of p s 2.

2. THE MAIN RESULTS It is the main purpose of the present paper to further investigate the asymptotic connection between the approximating properties of cardinal splines and entire functions of exponential type. THEOREM 1. Ž1. Ž2. Ž3.

Let f g L p ŽR., m g N, s ) 0. Then

EŽ f, Bs , p . p s lim m ª` EŽ f, Sp r s , m, p . p , 1 - p - `; EŽ f, Bs , p . p F limm ª` EŽ f, Sp r s , m, p . p , p s 1, `; EŽ f, Bs X , p . p G limm ª` EŽ f, Sp r s , m, p . p , 0 - s X - s , p s 1, `.

THEOREM 2.

If M is a bounded subset of L p ŽR., 1 F p F `, then E Ž M , Bs , p . p F lim E Ž M , Sp r s , m , p . p . mª`

Remark 2.1. It is worth pointing out that the proof of Theorem C used some characteristic properties of finite dimensional spaces, such as the compactness of the unit ball of finite dimensional space. Since Sh, m, p and Bs , p are both infinite dimensional, the proof of Theorem 1 needs new ideas and new methods, and we want to mention also that the method of proof of Theorem 1 is different from that of w10x, where we used some particular properties of Hilbert spaces, such as Plancherel’s theorem and Riesz-Fischer’s theorem.

646

FANG GENSUN

Some extremal problems of some classes of functions defined on R could be determined by Theorems 1 and 2. Let us consider some examples. We denote by X ŽR. a normed linear space defined on R. For f g X ŽR., let f N Ž x . s f Ž x . ? x N Ž x ., and X Ž IN . [  f N : f g X Ž R . 4 , where x N Ž x ., N ) 0, is the characteristic function of the interval wyN, N x, and A is a linear subspace of X ŽR.. Set K Ž « , N, A . s min  dim F : F ; X Ž IN . , E Ž B Ž A . l X Ž IN . , F . X Ž IN . - « 4 ,

« ) 0, where B Ž A. denotes the unit ball of A in X ŽR., and dim F denotes the dimension of the finite dimensional subspace F. DEFINITION 2.1 w5x. Let s ) 0, and let A be a linear subspace of X. If lim lim

K Ž « , N, A . 2N

«ª0 Nª`

s s - q`,

then s is said to be the average dimension of A in X ŽR., and is denoted by dim A s s . DEFINITION 2.2 w6x. Given a subset M ; X ŽR., s ) 0, the quantity ds Ž M , X Ž R . . s

inf

dim AF s

E Ž M , A . X ŽR .

is said to be the average s-width in the sense of Kolmogorov. If there is a subspace AU with average dimension F s such that ds Ž M , X Ž R . . s E Ž M , AU . X Ž R . , then AU is said to be an optimal subspace for ds Ž M, X ŽR... THEOREM D w6x.

Let r, m g N, m G r, p s 1, `, s ) 0. Then

ds Ž Wpr Ž R . , L p Ž R . . s E Ž Wpr Ž R . , S1r s , m , p . p s

Kr

Ž sp .

r

,

and S1r s , m, p is optimal for ds ŽWpr ŽR., L p ŽR.., where Kr is Fa¨ ard’s constant, i.e., Kr s

4

`

p

ks0

Ý

Ž y1. k

Ž rq1 .

Ž 2 k q 1.

rq1

.

Ž 2.1.

647

APPROXIMATING PROPERTIES OF ENTIRE FUNCTIONS

THEOREM E w11x.

Let s ) 0, and 1 F p F `. Then dim Ž Bs , p , L p Ž R . . s

s p

.

Let p s 1, `, r g N, s ) 0. Then

THEOREM 3.

ds Ž Wpr Ž R . , L p Ž R . . s E Ž Wpr Ž R . , Bsp , p . p s

Kr

Ž sp .

r

.

r

.

Ž 2.2.

Proof. By Theorems 1 and D, we have E Ž Wpr Ž R . , Bsp , p . p F

Kr

Ž sp .

r

,

and from Theorems D and E, we obtain E Ž Wpr Ž R . , Bsp , p . p G ds Ž Wpr Ž R . , L p Ž R . . s

Kr

Ž sp .

Theorem 3 is proved. The exact estimate of EŽWpr ŽR., Bs , p . p , p s `, was determined in w1, p. 194x in a different way. Now we consider a new extremal problem. Let f be a local p-integrable function. The Sobolev]Wiener class of functions is defined as w7, 12x Wprq Ž R . s  f g Lrq Ž R . : 5 f Ž r . 5 p q F 1 4 , where 5 f 5 pq s

½

Ý 5 f Ž ?q j . 5 qpwy1, 1x jgZ

1rq

5

,

5 f 5 p , ` s sup 5 f Ž ?q j . 5 pwy1 , 1x ,

1 F p F `, 1 F q - `, 1 F p F `.

jgZ

It is clear that Wprp ŽR. s Wpr ŽR., 1 F p F `. THEOREM F w7x.

Let 1 - p F `, m, n g N, 1rp q 1rpX s 1. Then

ds Ž Wpr, 1 , L1 Ž R . . s E Ž Wpr Ž R . , Sny1 , m , 1 . 1 s 5 Fr , n 5 pX w0, 1x ,

648

FANG GENSUN

where Fr , n Ž x . s

`

4

Ý p Ž np . ks0

cos Ž Ž 2 k q 1 . np x y pr Ž 2 r q 2 . .

r

Ž 2 k q 1.

rq1

is the Euler spline with period 2rn. Using Theorems 1, E, and F, we have THEOREM 4.

Let 1 - p F `, r, n g N, 1rp q 1rpX s 1. Then

ds Ž Wpr, 1 Ž R . , L Ž R . . s E Ž Wpr, 1 Ž R . , Bnp , 1 . 1 s 5 Fr , n 5 pX w0, 1x , and Bnp , 1 is optimal for ds ŽWp,r 1ŽR., LŽR... Remark 2.2. Theorem 4 is an analogue of Taikov’s inequality on ˜pr , 1 - p F `, of periodic functions w13, p. 172x. Sobolev classes W We prove Theorems 1 and 2 in the case of s s p only. The general case of s ) 0 can be proved in a similar manner. In order to prove Theorem 1, we need to prove an analogue of the Markov]Bernstein inequality for cardinal splines.

3. THE MARKOV]BERNSTEIN INEQUALITY FOR CARDINAL SPLINES LEMMA 3.1 w14x.

Let s g Sm, ` , m g N, k s 1, . . . , m. Then 5 s Ž k . 5 ` F Kmyk Kmy1p k 5 s 5 ` ,

Ž 3.1.

where Km is the Fa¨ ard constant defined by Ž2.1.. The constant on the right hand side of Ž3.1. cannot be impro¨ ed, and the Euler spline is an extremal function of Ž3.1.. LEMMA 3.2 w13, p. 125x ŽStein’s inequality.. 1 F p - `. Then krm 5 f Ž k . 5 p F Cm , n 5 f 5 1y 5 f Ž m. 5 kp r m , p

Let f Ž x . g Lm ŽR., m g N, k s 1, 2, . . . , m y 1, Ž 3.2.

where Cm, k s Kmyk Kmy1 qk r m , Km is the Fa¨ ard constant. LEMMA 3.3.

Let s g Sm, p , 1 F p - `, k s 1, . . . , m. Then p

5 s Ž k . 5 p F Kmy k ? Kmy1 p p q 1

ž '

k F 2 Ž 2p . 5 s 5 p .

k

/

5 s5 p

Ž 3.3. Ž 3.4.

649

APPROXIMATING PROPERTIES OF ENTIRE FUNCTIONS

Proof. Let s g Sm, p and s Ž m. Ž x . s c j , j - x - j q 1. Then 1rp

5 s Ž m. 5 p s

ž

ÝH jgZ

jq1

< s Ž m. Ž x . < p dx

j

/ ž s

1rp

Ý < cj < p

/

jgZ

.

It is not difficult to prove that, for all j g R,

Hj

jq1

< s Ž my1. Ž x . < p dz G min bgR

Hj

jq1

s < c j < p min

< c j x y b < p dx 1

H < x y b< bgR 0

dx G < c j < p 2yp

p

1 pq1

;

therefore, we have 5s

Ž my1. 5

p

G

1 2

p

(

1 pq1

< cj <

1rp p

žÝ /

s

jgZ

1 2

p

(

1 pq1

5 s Ž m. 5 p . Ž 3.5.

By virtue of Stein’s inequality, we get m 1r m 5 s Ž my1. 5 p F Cmy1, p 5 s 5 1r 5 s Ž m. 5 1y p p m 1r m 5 s Ž m. 5 1y s K1 Kmy1 r m 5 s 5 1r . p p

Ž 3.6.

Since K1 s pr2 w13, p. 103x, from Ž3.5. and Ž3.6., we obtain m 5 s 5 p G Km Ž K1 5 s Ž my1. 5 p ? 5 s Ž m. 5y1q1r . p

G Km

2

1

p

1

m

ž ( / p

?

2

m

5 s Ž m. 5 p ;

pq1

therefore, we have p

5 s Ž m. 5 p F Kmy1 p p q 1

ž '

m

/

5 s5 p .

Ž 3.7.

By Ž3.7. and by Stein’s inequality again, we have krm 5 s Ž k . 5 p F Kmyk Kmy1 qk r m 5 s 5 1y 5 s Ž m. 5 kp r m p k

F Kmyk Kmy1 Ž p p p q 1 . 5 s 5 p .

'

Ž 3.8.

650

FANG GENSUN

Hence Ž3.3. is proved. Since w13, p. 103x 1 s K0 - K2 - ??? - K4 - ??? - K3 - K1 s K2 s

p2 8

K3 s

,

Kmy k Kmy1 F

p 2

p3 24

p 2

,

,

Ž 3.9. Ž 3.10.

- 2,

Ž 3.11.

which together with Ž3.8. complete the proof of Ž3.4.. LEMMA 3.4 w10x.

Let f g Lmp ŽR., 1 F p - `. Then 5 f Ž j . 4 5 p F 5 f 5 p q 5 f X 5 p .

Ž 3.12.

A well known theorem of Marcinkiewicz w15x establishes the equivalence of the L p-norm of trigonometric polynomials of order F m and the discrete l p2 my1-norm constituted from their values at a uniform lattice w15x. We prove an analogue of this theorem for cardinal splines which will be used for the proof of Theorem 1. THEOREM 5.

Let s g Sm, p , m g N, 1 F p - `. Then 5  s Ž j . 4 5 p F Ž 1 q 4p . 5 s 5 p .

Proof. From Lemmas 3.3, and 3.4, we have 5  s Ž j . 4 5 p F 5 s 5 p q 5 sX 5 p F 5 s 5 p q 4p 5 s 5 p s Ž 1 q 4p . 5 s 5 p .

Ž 3.13.

Theorem 5 is proved. Remark 3.1. Schoenberg w16x proved that there is a constant Mm, p which depends on m and p only, such that 5  s Ž j . 4 5 p F Mm , p 5 s 5 p ,

1 F p F `.

Denote by Q1Ž x . the characteristic function of the interval w0, 1x, and Q mŽ x . the m-fold convolution of Q1Ž x . with itself. Explicitly, we find that Qm Ž x . s

1

m

Ý Ž y1. Ž m y 1 . ! rs0

r

ž mr / Ž x y r .

my1 , q

APPROXIMATING PROPERTIES OF ENTIRE FUNCTIONS

651

where xqs maxŽ0, x ., and Q mŽ x . has compact support w0, m x and Q mŽ x . ) 0 in Ž0, m.. Finally, let Dn f Ž x . be the mth divided difference of the f Ž x ., namely, m

Ý Ž y1. my k

Dm f Ž x . s

ks0

m f x q k. . k

ž /Ž

It is well known that Dm f Ž j . s

HRQ

Ž x y j . f Ž m. Ž x . dx.

m

Ž 3.14.

Let m M Ž x . s Q mŽ x q 12 m.. Then Mm g Smy1, ` and Mm has compact support wymr2, mr2x, and let FŽ z. s

Ý

ngZ

Mmq 1 Ž n . z n ,

Fmq 1 Ž u . s F Ž e i u . ,

zgC

i g 'y 1 , u g R.

Ž 3.15.

Then by w16x, Fmq 1Ž u. is a cosine polynomial such that Fmq 1 Ž u . ) 0

for all u g R and Fmq1 Ž 0 . s 1,

Ž 3.16.

min Fmq 1 Ž u . s Fmq1 Ž p .

ugR

s2

mq1

2

ž /

`

Ý

p

ks0

Ž y1. k

Ž mq1 .

Ž 2 k q 1.

mq1

s 2 mpym Km , Ž 3.17.

where Km is the Favard constant defined by Ž2.1.. It follows from Ž3.15. that the reciprocal of F Ž z . admits a Laurent expansion, 1 FŽ z.

s

q`

Ý vn z n ,

< z < s e iu ,

Ž 3.18.

y`

which is identical with the Fourier series 1 Fmq 1 Ž u .

s

Ý

ngZ

vn e i n u ,

where vn s vy n ,

and for all n g Z, Žy1. nvn ) 0; therefore,

Ý

ngZ

< vn < s

n Ý Ž y1. vn s

n gZ

For these results, we refer to w16x.

1 Fmq 1 Ž p .

s 2 mpym Km .

Ž 3.19.

652

FANG GENSUN

Remark 3.2. Theorem 6, below, is an analogue of the Markov]Bernstein inequality which plays an important role in the proof of Theorem 1, and has also its own independent role in approximation theory. Let s g Sm, p , m g N, 1 F p - `, k s 1, . . . , m. Then

THEOREM 6.

5 s Ž k . 5 p F Ž 1 q 4p . krm Kmyk Kmy1p k 5 s 5 p

Ž 3.20.

F 2 Ž 1 q 4p . p k 5 s 5 p .

Ž 3.21.

Proof. We use the method of discretization of w14x. Assume s g Sm, p . Then by Lemma 3.3, 5 s Ž m. 5 p - q`, hence 5 s Ž m. Ž j .45 p - q`; therefore, s Ž m. Ž x . is a bounded step function with discontinuities at the points j q Ž m y 1.r2, for integer j. Let s Ž m. Ž x . s c j ,

x g j q Ž m y 1 . r2, j q 1 q Ž m y 1 . r2 . . Ž 3.22.

We may therefore write s Ž m. Ž x . s

Ý c j Q1 jgZ

ž

xyjy

1 2

Ž m y 1. .

/

Ž 3.23.

According to w16x, for m, n g N, a , b g R,

HRQ

m

Ž x y a . Qm Ž x y b . dx s Qmqn Ž a y b q m . .

Ž 3.24.

From Ž3.24., we have

HRQ

m

Ž x y i . Q1 x y j y

ž

s Q mq 1 i y j q

ž

1 2

1 2

Ž m y 1 . dx

/

Ž m q 1 . s Mmq1 Ž i y j . .

/

Ž 3.25.

Using Ž3.14., Ž3.23., and Ž3.25., we may write Dm s Ž i . s s

HRQ

Ž x y i . s Ž m. Ž x . dx

Ý c j H Q m Ž x y i . Q1 jgZ

s

m

R

Ý c j Mmq1Ž i y j . . jgZ

ž

xyjy

1 2

Ž m y 1.

/ Ž 3.26.

653

APPROXIMATING PROPERTIES OF ENTIRE FUNCTIONS

Since  c j 4 g l p , we may regard the sequence convolution Ž3.26. as a bounded linear transformation of the space l p into itself. Therefore, it follows from Ž3.16., Ž3.18. that the transformation Ž3.26. admits an inverse given by

Ý v iyj Dm s Ž j . ,

cj s

for all j

Ž 3.27.

jgZ

Žsee w14, 16x., which is the only bounded solution of Ž3.23.. Using Ž3.23., Ž3.27., Minkowski’s inequality, and Ž3.27., we have 5 s Ž m. 5 p s

ž

1rp

Ý < cj < p

/

jgZ

p 1rp

s

žÝ

Ý

/

vk D sŽ j y k . m

jgZ kgZ

F

Ý Ý < v k Dm s Ž j y k . < p kgZ

s

Ý

ž

jgZ

< vk < ?

kgZ

F

Ý

< vk <

kgZ

s

Ý kgZ

ž

< vk <

Ý Ý Ž y1.

m

Ý rs0

my r

jgZ rs0

ž

Ý Ý rs0

jgZ

m r

ž /ž

/ p 1rp

m

m

1rp

m sŽ j y k q r . r

ž /

m sŽ j y k q r . r

/

p 1rp

ž /

Ý < sŽ j y k q r . < p jgZ

/ 1rp

/

s 2 m ? 2ymp m Kmy1 ? 5  s Ž j . 4 5 p s p m ? Kmy1 5  s Ž j . 4 5 p .

Ž 3.28.

Using Ž3.28. and Theorem 5, we obtain 5 s Ž m. 5 p F Ž 1 q 4p . p m Kmy1 5 s 5 p ,

Ž 3.29.

which together with Stein’s inequality for k s 1, 2, . . . , m y 1, gives krm 5 s Ž k . 5 p F Cm , k 5 s 5 1y 5 s Ž m. 5 kp r m p

F Kmyk ? Kmy1qk r m Ž 1 q 4p . s Ž 1 q 4p .

krm

krm

p k Kmyk r m 5 s 5 p

Kmyk Kmy1p k 5 s 5 p .

Ž 3.30.

This completes the proof of Ž3.20.. Now Ž3.30. and Ž3.9. show that Ž3.21. indeed holds.

654

FANG GENSUN

4. THE PROOFS OF THEOREMS 1 AND 2 LEMMA 4.1 w1, p. 191x.

Let f g Lmp ŽR., m g N, 1 F p F `, s ) 0. Then

E Ž f , Bs . p F

Km

s

m

5 f Ž m. 5 p F

2

sm

5 f Ž m. 5 p ,

where Km is the Fa¨ ard constant defined by Ž2.1.. THEOREM 7. then

If sm g Sm, p , m s 1, 2, . . . , 1 F p F `, and 5 sm 5 p F c 0 , lim E Ž sm , Bp q d , p . p s 0,

mª`

where c 0 is independent of m. Proof. From Lemma 4.1, Lemma 3.1, Theorem 6, and 5 sm 5 p F c 0 , we have E Ž sm , Bp q d , p . p F

2

Žp q d .

m

F 4 Ž 1 q 4p .

Ž m. 5 5 sm p

p

žp d/ q

m

5 s m 5 p F 4 Ž 1 q 4p .

p

m

žp d/ q

c0 ,

Ž 4.1. which gives lim E Ž sm , Bp q d , p . p s 0.

mª`

Theorem 7 is proved. LEMMA 4.2.

Let f g L p ŽR., 1 F p - `, s ) 0, d ) 0. Then lim E Ž f , Bsq d , p . p s E Ž f , Bs , p . p .

dª0

Proof. Without loss of generality, we assume 0 - d - s . Let gs and gsq d be a best approximation of f by Bs , p and Bsq d , p in L p ŽR., respectively. It is well known that Bs , p ; Bsq d , p ; therefore, gs g Bsq d , p . From this, we have 5 gsq d 5 p F 5 f 5 p q 5 f y gs q d 5 p F 5 f 5 p q 5 f y gs 5 p F 3 5 f 5 p . Ž 4.2.

APPROXIMATING PROPERTIES OF ENTIRE FUNCTIONS

655

According to w17, p. 62x and Ž4.2. < gsq d Ž x . < F 2 Ž s q d . 1rp 5 gs q d 5 p F 6 Ž 2 s . 1rp 5 f 5 p ;

Ž 4.3.

therefore, the family  gsq d Ž x .4d ) 0 is uniformly bounded on R and hence  gsq d 4d ) 0 is a normal family. Hence, there is an analytic function g#Ž x . g Bs , such that lim gsq d Ž x . s g# Ž x .

Ž 4.4.

dª0

uniformly on every finite interval on R. Since gsq d Ž x . g Bsq d , p , by the results of w17, pp. 62, 194x, and Ž4.2., we have < gsq d Ž x . < F 6 Ž s q d . 1rp e Ž sq d .< y < 5 f 5 p ,

x, y g R, i s 'y 1 , Ž 4.5.

which together with Ž4.4. gives < g# Ž x . < F 6 e s < y < 5 f 5 p , therefore, g# g Bs . Using Fatou’s lemma, and Ž4.2., Ž4.4., we have 5 g# 5 p F 3 5 f 5 p , hence, g# g Bs , p . By Fatou’s lemma again, we obtain E Ž f , Bs , p . p F 5 f y g# 5 p F lim 5 f y gsq d 5 p dª0

F lim 5 f y gsq d 5 p F 5 f y gs 5 p s E Ž f , Bs , p . p . dª0

This completes the proof of Lemma 4.2. Remark 4.1. In the case of p s `, an analogue of Lemma 4.2 was proved in w17, p. 58x. LEMMA 4.3 w17, p. 232x ŽBernstein’s inequality.. `, s ) 0. Then

Let f g Bs , p , 1 F p F

5 fX5 p F s 5 f 5 p. LEMMA 4.4 w6x. Let f g Lmp ŽR., m g N, p s 1, `. Then there is a unique smŽ f, x . g Sm, p such that smŽ f, j . s f Ž j ., j g Z, and 5 f Ž ? . y sm Ž f , ? . 5 p F where Km is the Fa¨ ard constant.

Km

pm

5 f Ž m. 5 p ,

Ž 4.6.

656

FANG GENSUN

LEMMA 4.5.

Let f g Bs , p , p s 1, `, s - p . Then lim 5 f Ž ? . y sm Ž f , ? . 5 p s 0.

Ž 4.7.

mª`

Proof. Lemma 4.5 for the case of p s ` was proved in w3, p. 294x. In a similar way and using Lemmas 4.3 and 4.4, we obtain 5 f Ž ? . y sm Ž f , ? . 5 p F

2

p

5 f Ž m. 5 p F 2 m

s

m

žp/

5 f 5 p,

Ž 4.8.

which gives Ž4.7.. LEMMA 4.6 w17, p. 248x.

Let f g Bs , p , 1 F p F `. Then

5 f Ž j . 4 5 p F Ž 1 q p . 5 f 5 p .

Ž 4.9.

Now we give the proofs of Theorems 1 and 2. Proof of Theorem 1. Let 1 - p - `, let f Ž x . g L p ŽR., and let smŽ x ., m s 1, 2, . . . , gp Ž x ., be the best approximations of f Ž x . by Sm, p and Bp , p in L p ŽR., respectively. By Lemma 4.6, we have that 5 gp Ž j .45 p - q`; therefore, by virtue of Theorem A, there is a unique a mŽ x . g Sm, p such that a mŽ j . s gp Ž j ., j g Z, and

am Ž x . s

Ý

gp Ž j . L m Ž x y j . .

Ž 4.10.

jgZ

By the Whittaker]Shannon]Kotelnikov sampling theorem and its generalization w18x, gp Ž x . s

Ý jgZ

gp Ž j .

sin p Ž x y j .

p Ž x y j.

,

; x g R,

Ž 4.11.

which together with Ž4.10. and Theorem B gives lim 5 a m y gp 5 p s 0,

mª`

1 - p - `;

therefore, 5 f y sm 5 p F 5 f y a m 5 p F 5 f y gp 5 p q 5 gp y a m 5 p lim E Ž f , Sm , p . p F E Ž f , Bp , p . p q lim 5 a m y gp 5 p F E Ž f , Bp , p . p .

mª`

mª`

In the case p s 1, and p s `, in the same way, but using Lemma 4.5, we have lim E Ž f , Sm , p . p F E Ž f , Bs , p . p ,

mª`

0 - s - p , p s 1, `.

APPROXIMATING PROPERTIES OF ENTIRE FUNCTIONS

657

Now we assume 1 F p F `. For fixed d ) 0, let bm, d be the best approximation of sm , m s 1, 2, . . . , by Bp q d , p in L p ŽR.. Then E Ž f , Bp q d , p . p F 5 f y bm , d 5 p F 5 f y sm 5 p q 5 sm y bm , d 5 p s 5 f y sm 5 p q E Ž sm , Bp q d , p . p . Since sm is a best approximation of f Ž x . by Sm, p in L p ŽR., 5 sm 5 p F 2 5 f 5 p , therefore by Theorem 7, lim E Ž sm , Bp q d , p . p s 0;

mª`

hence, we obtain E Ž f , Bpq d , p . p F lim E Ž f , Sm , p . p .

Ž 4.12.

mª`

For this and Lemma 4.2, we have E Ž f , Bp , p . p F lim E Ž f , Sm , p . p . mª`

Theorem 1 is proved. Proof of Theorem 2. Since M is bounded in L p ŽR., EŽ M, Bp , p . p - q`. Let  « k 4k g N be a sequence tending to zero and  f k 4k g N ; M, such that E Ž M , Bpq d , p . p F E Ž f , Bp q d , p . p q « k .

Ž 4.13.

Denote by sm, k a best approximation of f Ž x . by Sm, p in L p ŽR., and denote by a m, k a best approximation of sm, k by Bp q d , p in L p ŽR.; therefore, 5 sm, k 5 p F 2 5 f k 5 p . By virtue of Theorem 7, we have lim 5 sm , k y a m , k 5 p s lim E Ž sm , k , Bp q d , p . p s 0;

mª`

mª`

Ž 4.14.

therefore, E Ž M , Bp q d , p . p F E Ž f k , Bp q d , p . p q « k F 5 f y a m , k 5 p q « k F 5 f k y sm , k 5 p q 5 sm , k y a m , k 5 p q « k F E Ž M , Sm , p . p q E Ž sm , k , Bp q d , p . p q « k , Ž 4.15. which together with Ž4.14. gives E Ž M , Bpq d , p . p F lim E Ž M , Sm , p . p . mª`

Ž 4.16.

658

FANG GENSUN

On the other hand, there is a sequence  g k 4 ; M, such that E Ž M , Bp , p . p F E Ž g k , Bp , p . p q « k ,

k s 1, 2, . . . ,

from Lemma 4.2, lim E Ž g k , Bp q d , p . s E Ž g k , Bp , p . p ;

dª0

Ž 4.17.

therefore, E Ž M , Bp , p . p F E Ž g k , Bp q d , p . p q « k s lim E Ž g k , Bp q d , p . p q « k F lim E Ž M , Bp q d , p . p q « k dª0

d ª0

F lim E Ž M , Bpq d , p . p q « k F E Ž M , Bp , p . p q « k . dª0

Letting k ª ` in Ž4.17., we have lim E Ž M , Bpq d , p . p s E Ž M , Bp , p . p .

dª0

Ž 4.18.

Combining Ž4.16., Ž4.18., we obtain E Ž M , Bp , p . p F lim E Ž M , Sm , p . p . mª`

This completes the proof of Theorem 2.

ACKNOWLEDGMENT I am grateful to the referees for their valuable comments about this paper.

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