On the degree of approximation of $2\pi$-periodic functions in $C^1$ with positive linear operators of the Jackson type

$C^1$ with positive linear operators of the Jackson type

Citation for published version (APA):

Schurer, F., & Steutel, F. W. (1977). On the degree of approximation of $2\pi$-periodic functions in $C^1$ with positive linear operators of the Jackson type. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 77-WSK-02). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1977

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ONDERAFDELING DER WISKUNDE DEPARTMENT OF MATHEMATICS

On the degree of approximation of

2~-periodic

functions in C1 with positive linear operators of the Jackson type

by

F. Schurer and F.W. Steutel

T.H.-Report 77-WSK-02 August 1977

functions of a real variable. The positive linear operators of the Jackson type are denoted by L _n,p (n ~ m) , where p is a fixed positive integer. The object of this paper is to study the exact degree of approximation with the operators L for functions in

c

2 1

.. The value of max

I

L (f ;x) - f (x)

n,p ~ x n,p

is estimated in terms of w(f';o), the modulus of continuity of ff t with

~

o ...

n'

Exact constants of approximation are determined for the operators L (n e;

m,

p ;::: 2) and for the Fej er operators L 1 (n e

m).

Furthermore,

n,p n,

the limiting behaviour of these constants is investigated as n + ~ and

p +~, separately and simultaneously,

1. Introduction and summary 2. Preliminary results

3. The extremal functions

4. The exact constants of approximation c for the n,p

operators L

n,p

5. The limiting behaviour of the constants c n,p

4-8

15

1. Introduction and summary

1.0. The class of real, continuous, 2~-periodic, functions of a real vari-able is denoted by C2~' Assume f ~ C2~ and let p be a positive integ~r. The positive linear operators L are then defined by the relation

n,p (1. 1) where (1.2) L (f;x)

=

n,p with A n,p such that ~

r

J

~

r

J

f(x+t)k (t)dt n,p k (t)dt .. l. n,p (n € IN)

If P

=

I we obtain the Fejer operators, while the name of Jackson is asso-ciated with Ln,Z" Approximation properties of the operators Ln,p' in partic-ular those of Ln,l and Ln,2' have been extensively studied; cf., for in-stance, Butzer and Stark [3J, Gorlich and Stark [5J, Matsuoka [7J, Schurer and Steutel [9J, where the survey paper [5J deserves special mention because of its wealth of references.

In view of the kind of problems we shall be concerned with in this report, we mention here the following result of Wang Hsing-hua [12J, where the exact constant of approximation of functions in C2~ by the Jackson operators is determined. He obtains (1.3) sup -~ _;;:.. .n+h;JN maxl~ 2(fi x) -f(x)

I

x , 3

--- = -

₂ w{f' _{, n+l} 2...) *) (N = [~J + 1)

assuming f to be nonconstant. As usual, w(f;o) denotes the modulus of continuity of f, defined'by

w(f;o).. sup If(x) -fey)

I

Ix-YI$o

(0 > 0) .

Results analogous to (1.3) for the operators L (p • 3,4) and L (p ~ 5)

n,p n,p

may be found in [8J and [9J respectively. A similar problem for the Jackson operators in two variables was solved by Bugaets and Martynyuk [2J. ;

1.1. In [10J we investigated the local degree of approximation of continu-ously differentiable functions by Bernstein polynomials; a similar analysis was recently carried out for the well-known Meyer-Konig and Zeller operators

(cf. [11]).

This report is concerned with related problems. Here the setting is the I

class C2~ of real, continuously differentiable, 2~-periodic, functions of a real variable. The degree of approximation is measured in terms of the modulus of continuity of f', denoted by wI' In particular, we shall deal with the problem of determining the exact constants of approximation of

functions in

e

2 1

by the operators L • Furthermore, the limiting behaviour

~ n,p

of these constants will be investigated.

More specifically, for f nonconstant and assuming n eo IN and p IS IN fixed, the exact constant of approximation for the Jackson type operator

L (p ~ 2) is defined by n,p (1.4) c _n,p :- _{ ILn (f;x) -f(x)1 nip" sup ~~,p---wI (f ;~) ~ X eo

whereas for the Fejer operators the definition reads

(1.5) c _n, 1 : ...

The norming is prescribed by asymptotic properties; the Fejer operators L 1

n, differ in this respect from the operators L (p ~ 2) • In order to keep the

n,p

constants c bounded. definition (1 .4) in case n

...

1 is replaced by

1 , P

(1.6) (p = 3,4, ... ) •

Assuming p e IN fixed the exact constant of approximation for the sequence of operators {L } (n IS IN) may then be defined as (d. (1.3»

n,p (I.7) c (p) : ... sup

neJN

Furthermore, the exact constant of approximation for the whole class of operators L (n €: lN, p ;:: 2) is defined by

n,p

(1.8) c := sup c(p) •

p;::2

1.2. We now give a brief sketch of the contents of the various sections. Section 2 contains all those preliminary results which will be needed fre-quently in the sequel. Relevant material on the kernels of the operators Ln,p is gathered together from [9J, whereas also some inequalities are given that are useful when estimating integrals over these kernels. In section 3 the so-called e~tremaL functions are introduced; just as in the investigation of the Bernstein polynomials and the Meyer-Konig and Zeller operators, they playa crucial role in determining the constants c • The

n,p pattern of deducing the extremal functions is in part similar to the pro-cedure given in [10J; a serious complication however is caused by the con-straint of periodicity. The material of sections 2 and 3 is then used in

. 4 • 2 8 7 7 *) . .

sect10n to establ1sh that c •

;,r

=

1.12 3 91. The proof 1S qU1te

intricate, mainly because a number of different cases of nand p have to be examined separately, each of them needing a different approach. In partic-ular, cases p = 2 and p

=

3 have to be investigated in considerable detail.

The numerical values of the constants c for the first few values of n n,p

and p ;:: 2 are given. The exact constant of approximation in the case of the Fejer operators is determined in the last part of section 4; we show that

(1) 'IT

C

=

'4

(cf. (1.7)).

en

=

1(1)10; 15(5)50;

Moreover, a table with the values of c 1

n,

75; 100) is given. In section 5 the limiting behav-iour of c is considered as n + ~ and p + ~, separately and

simultaneous-n,p

ly (in both orders). For instance, it is proved that lim lim c

=

lim lim c

=

~

=

0.97720502 •

n~ p~ n,p p~ n-+oo n,p 11"

A separate discussion is devoted to the asymptotic degree of approximation by Fejer operators. It turns out (viz. definitions (1.4) and (1.5)) that

their performance is essentially worse than the operators L (p ;:: 2). n,p

The value of lim c 1 is determined. Section 5 also contains a table with n~ n,

the numerical values of c : = lim c (p '" 2 (l ) 10; 20 (l 0) 50) •

p n~ n,p

2. Preliminary results

2.0. Approximation properties of the operators L p were investigated in _{. n,}

[9J. From that paper we need the following four lemmas.

Lemma 2.1. If

v

and ~ are positive integers then

(2.1) (2.2)

S(v,~)

:-

J

o

( . _Sl.nt ) 2v _dt

₌

t2~ '" (-vI )v+~'lT V~l • 2 2 1 ~ - L. (-1)J ( .v)(2v-2j) ~-2 (2~-1)! j-O J (v ;:: ~) , (v > ~) •

Lemma 2.2. For any positive integer p and for k '" O,I, ••• ,2p-2 the following asymptotic equivalence holds

'IT

J

t k

(:t~ l~t)2p

dt ... 2k+ln2p-k-l

S(p,p-~k)

(n -+ "") •

°

Lemma 2.3. If k and p satisfy the conditions of lemma 2.2 then

Lemma 2.4. The coefficients \.l(n,p) in k the expansion (s~n ~nt Zp 1..1 (n,p) np-p 1..1 (n,p) (2.3)

_h)

...

+ 2

I

cos kt s~n

a

k=l k are given by (2.4) lJ. (n,p) ... k

with the usual convention that (:) •

a

if a < b.

2.1. We proceed by proving a few inequalities that will be used for estimat-ing integrals over the kernel (2.3). These integrals will be encountered in section 4.

Lemma 2.5. For n € IN one has

where (2.5) sin t 2 ...;;..;;;.;;;..~ ~ exp(-~at ) . t n s~n -n 8 'IT a -

-Z

10g(2) ,. 0.366039 • 'IT Proof. As (2.6) n sin ~ ~ (n+l)sin __ t_ n n+l

it is sufficient to show that

Put

sin t ~ exp (-fat~ t sin t 2 f(t):- t - exp(-~at ) • 'IT (O < t ~ '2) , (a < t

Expansion in Taylor series shows that f is positive on the interval (0,3;J.

3 'IT 'IT • 'IT

Furthermore, f is decreasing on

[8' 2J,

wh~le f ('2) • O. This proves the

In order to investigate the behaviour of the constants c

3 ,p as defined in

(1.4) we need a slight improvement of lemma 2.5 if n

=

3. This is given by Lemma 2.6.

sin t ~ exp(-!bt2)

3 sJ.n .

'3

t where

Proof. Putting v := - we have to show that t

3

(0 < t s;; ~) ,

A simple computation shows that £ is positive and increasing on (O,~J.

1 1T

The interval

"3 '6

J is taken care of by noting that fl decreases, i.e. that f . loS concave t ere, an h d f(_1T)

=.

0

6

Lemma 2.7. For n € IN one has

(2.7) sin t n sin .t n 1 1 2 s;; exp(-

6

(1

-z)t )

n

Proof. Inequality (2.7) can be rewritten in the form

. t

sJ.n

-_ -_ n_ exp (

1

(1;;) 2)

t 6 n

n

From this it follows that it is sufficient to show that f(t)

-:-;si~

t

ex~_(~t2ris

decre_asing on (O,1T). One has

(0 < t < 1T) •

(n € IN) •

1 -2 1 2 2

f f (t) == - t exp (- t ) {-3 sin t + 3 t co s t + t sin t} ,

3 6

and the expression between brackets is easily seen to be negative on (0,1T).

•

2.Z. The next result will be needed to obtain upper and lower bounds for the

11'

integral t(cos t)ZPdt, wfiich occurs in section 4.1.4.

J

z

o

Lemma Z.8.

(Z.8) _Sl.n . t +

6

1 . 3 Sl.n t + 40 Sl.n t s t s nn t 3 . 5 • +

6

1 . 3 Sl.n t + 40 un t + 3 , 5 (11'

2

-'"i2'Q

149). Sl.n t 7

Proof. It is well known that

1 3 3 5 arcs in x - x + '6 x + 40 x + .•• =

~

j=O a. xZj+1 2J +-1 with a

ZJ+l ' > 0 for all j. Writing t :- arcsin x it follows that

co t =

_L

_{aZj +1} (sin t)Zj+l j=O and also co (Z.9) 11'

₂

...

L

_{a Zj +1}

.

j-O

(1 xl s

1) , (It

I

s !) ₂ As a

Zj+1 > 0 the left-hand side of inequality (2.8) obviously holds; the right-hand side follows from (2.9) by observing that

co t == sin t +

6

1 . 3 Sl.n t + 40 Sl.n t + 3 , 5

L

_aZj+l ( . Sl.n t )2j+l s j=3 co s sin t + 1 . 3 3 . 5 . 7

L

6 Sl.n t +45' Sl.n t + Sl.n t a 2j +1

.

j=3

Finally we note that (C£. [6J, p. 97)

1! 1!

r

2 (sin t) 2p dt ==

r

2 (cos t)2Pdt == 11'(2E)! (2.10)

₎

J

22p+1 (p!)2 ' 0 0

a result that will be frequently needed in section 4.

3. The extremal functions 3.0. As in (1.1) let

-'IT 'IT

J

f(x+t)k _n,p (t)dt,

where k is given by- (1.2). Assuming n E :IN and p E IN fixed, we shall

n,p determine d defined by n,p (3. 1) d ... sup{IA (f;x)1 x E

m,

f E F } n,p n,p n where /:, (f ;x) := L (f·x)

-

f(x) n,p n,p , and _{F := F is deffned by ~ ~_~} n

(3.2) F

=

{f:

[ -'IT,

'IT J -;.

m

; f E _C21 _'IT' w (f· -) 'IT s; I}

.

1 ' n 3. 1. Lemma 3. 1 • d ... sup IA f

I

n,p fEFo n,p

-where, defining f by f(t)

=

f(-t),

(3.3) Fo

=

(f E F ;

! ...

f~ f(O) ... O,'f'(t) ~ 0 for t E [O,'lTJ} ,

and A f is defined, for f €

F

O' by n,p

'IT

(3.4) _An,pf

₌

_An,p(f;O)

₌

J

_{f(t)kn,p(t)dt.}

-'IT

Proof. As for x €

m

and f € F also· f EO: F, where f is defined by

x x

f (t)

=

f (t+x), we have L (f ;x) ... L (f ;0). Hence, it~' is no restriction

x n,p n,p x

to take x ... O. As L is linear and f - f (0) EO: F i f f € F, it is no

restric-n,p

tion to take f(O) ... O. Furthermore, as

k

...

k we have

f+f n,p n,p

an pf

_,

_-

= An pf

_,

= An p

_,

-2-; hence it is no restriction to take f such that f ... f. Finally, it is no restriction to assume that A f ~ 0, as for f €

F

we have -f €

F.

It follows that for even f €

F

with f(O) - 0 we have

f

€ FO

and

i

~ f if we define

f

by f(O) - 0 and

if

(t) - max(O.f'(t» for t € [O,nJ,

...

and by symmetry on [-n,O]. As f(t) ~ f(t) for all t it follows from E3.4)

...

that 6 f ~ 6 f. This proves lemma 3.1. •

n,p n,p

3.2. We now have to maximize

7r

6 f... Jr f(t)k

(t)dt-n,p n,p

-n n

2

for f € FO' We first prove two general lemmas.

Lemma 3.2. Let K be a finite, nondecreasing function on [-!,~J and for fixed n € Jl.Il let G] := G] ,n be defined by

G

1 = {g:

[-itD

-+]R ;S=g, g(O) -0, gf continuous,

wl(g;~)

S I} • Then

(1

J

g(t)dK(t)

-!

r!

=

J

gl(t)dK(t)

-!

where

8

₁ :-

8

1,n is defined by

8

1(0) - 0 and

(i

< t <

.i!.!. ,

j = 0,

!

I,

!

2 , ••• ) • n n

Proof. The proof of this lemma involves exactly the same steps as the proof of Theorem 3.1 in [IOJ. This is apparent if we write the Bernstein polynomial as (cf. [10], formula (1.1»

I

r

BnCf;x)'" J f(t)dK(x,t).

o

Lemma 3.3. Let K be a finite, nondecreasing function on

[-!,!]

and for fixed n € Jl.Il let G

2 := G2,n be defined by

Then

r!

g(t)dK(t)...

J

82(t)dK(t) ,

-!

where g2 := gZ,n is defined by

gz(!) ...

0 and

r;(t)

III -j (2j-1 2j+l. - 0, +1 :: 2 , ••• ) 4m < t < 4m ,J

_-

,

(3.5)

-,

. . + I I. g Z( t) ... -(j+D (-L.. _2m+l < t <....l..:L _{2m+ 1 ,J} • ... 0,

:!:

1 ,

:!:

z , .•. )

Proof. For g E G

Z

we have, using integration by parts,

~ ~

(3.6) Dg:-

f

g(t)dK(t)

=

f

g' (t)

-~

dK(u)dt •

-!

t

We first state and prove three propositions.

if n == 2m ,

if n - 2m+l

Proposition (i). It is no restriction to take g concave, i.e. to take g' nonincreasing.

Proof. For g E G

Z

we define an even function

S

by

S(!)

=

0,

and

VI

g (t)

=

sup g' (s)

(-!

~ t ~ 0)

t:>:s:>:O

It is easily verified that

g'

is nonincreasing and that

g

E G_Z' As

.

VI , , v,

f!

g (-t) - g' (-t) == g (t) - g (t) ~ 0 on [0,

D

and t dK(u) is nonincreasing, it

V

follows from (3.6) that Dg ~ Dg. Proposition (ii) • * G 2 :'" {g: Then sup Dg

₌

gEG Z Let

[-!,

D

sup* gEG 2 -+ lR ; Dg

..

g

=

g; gO) == 0, gl nonincreasing, gf continuous except for fInitely many

jumps,Wl(g;~)~

I}.

Proof. By proposition (i)gt_~y be taken __ ~o be noni?creasing. Furthermore any

gf with g ~

G;

is the pointwise limit of functions g~ with gn €

G

Z and having the same wI value. By (3.6) this proves proposition (ii).

*

Proposition (iii). It is no restriction to assume that g e G

2 satisfies (3.7) g'(t) ... g'(t-Ii) - 1 1

(-i

+

Ii

1 S t S

D .

Proof. If for g € G* condition (3.7) is violated anywhere for t

=

to with l I Z

to €

[-i

+

Ii '

Zn)' then g can be replaced by go e G; as indicated in figure

3.1 below, where the graphs of g' and go are shown. Here go is obtained from gf for t < 0 as follows:

1

(t S to -

Ii

and g I (t) < g f (t 0) + 1)

(otherwise) , and by symmetry for t > O.

gf go •••• -- •• l\ !. - - -1 t - -

_o

n

o

Figure 3.1. '-t

+.!

, 0 n

Clearly, Dg

O ~ Dg (cf. the proof of proposition (i», and go e

G;

if g e

G;.

Hence we may restrict our attention to functions g satisfying (3.7), with

the possible exception of the point t ==

2~'

which does not affect the value

of Dg. This proves the proposition.

Now let g €

G;

satisfy

0.7)

and let gl(t) == !<g'(t-O)+g'(t+O» be defined for t in [-i,!]; redefining gl in this sense at discontinuity points does not affect (3.7). Then, as we have g 1(0) ... 0 and g ,

(-in)

== -g'

(in)

==

L

it follows in view of (3.7) that for j == O,l, •.• ,n we have (3.8) g'(-i + _n

i)

==

~

₂ - j _•

We now replace g by

g

€

G;

obtaine~ by join~ng the straight lines tangent to the graph of g at the points

(-!

+J., g(-! +1», i.e. with tangents given by

n n

(3.8). Als g is concave we have g ~ g and hence Dg ~ Dg. Finally we show that g E G;. Writing Yj == g(-i

+~)

for g € G; satisfying (3.8), we have

y. 1

J+ 2Y. J + Y. J-1

== _

1

n

with YO == Y_n == O. It follows that all fun~ti?ns i~

G;

satisfying (3.8) have

graphs that pass through the points

(-! +.1, J.

2(1

_:1»

for j == O,l, .•• ,n.

n n

Thus the graph of

g

passes through these points, and hence g is identical ,...

with the function g2 as defined in (3.5). Clearly, from the previous propo-'"

sitions and the construction of g2 it follows that Dg

2 ~ Dg for g € G2• This proves lemma 3.3.

A sketch of the functions gl and

8

2 with their derivatives is given in figure 3.2 below for n == 4.

_

..

-

-

--,

Figure 3.2.

3.3. We are now in a position to prove the main result of this section.

Theorem 3.1. _{Let d n,p be defined as in} (3. I). Then

11'

J

,.., (3.9) d III f (t)k (t)dt n,p n n,p -11' ...

-where f is defined by f (0) = 0, f is even, and

n n n

(E

(j + 1)11' 2m < t < 2m , j ' " 0, I , ••• ,m-I) (m III 1,2, ••• ) (! ₂ < (2j-l).1I' _4m < t < (2j+l)1I' _4m < 11'.

_{, =}

J' _{m,m+ , •.• , m ,} 1 2 )

(2..

< t < (j + 1 ) 11' • 0 1 ) 2m+ 1 2m+ 1 ' J

= , , ••• ,

m -' {j+! (3.11) f 2m+ICt)

=

(m = 0, I , ••• )

(

~< t < (j+1)1I' . = 2m-j+~ _{2m+l 2m+ 1 ,J} m+ I ,m+ 2, ••• ,2m) • Proof. The....,function ... fn' functions 8 1 ,n and g2 ,n

except for a linear transformation, consists of the put together. To be precise we have

- t

(0 s t S ~)

=

{'!I'

SI,nC;)

,.., 11' - t l fn ( -2) + 11' g _2,n C - -_11' 1)

for t € [O,1I'J, and by symmetry elsewhere. One easily verifies that (by good

11' .... 1 ...., 11'

luck) the jumps at (or close to) t :II

'2

of fn are such that WI (f

n ;

ii)

= I,

-'

and hence that fn is the pointwise limit of derivatives of functions in F_O

(cf. (3.3». Finally we have 11'

)

f(t)k (t)dt s n,p

J

11'

:2

11' k (t)dt}:+ n,p

... 'IT

o

'IT f 2(t)k n,p

(t)dt-J

2

81

,n (;)k n,p (t)dt 'IT 'IT 2 'IT J

r

k' (t)dt + n,p .

J

,...,

t + 'IT g2 n(; - l)k (t)dt· , n,p 'IT

r

J

f

n (t)k n,p (t)dt. 1! 2

This proves the theorem.

Corollary. The (extremal) functions

f

are given by

n .

•

(m'" 1 t 2, ••• ) '"" (3.13) _{f 2m+ 1} (t) (m = 0, 1 , 2, ••• ) t where a+ :- max(O,a).

-A sketch of f together with its derivative is given in figure 3.3 below

n ,...,

for n ... 4 and n - 5 respectively. (Because of symmetry the graph of f is n only shown on [O,'lTJ.)

11" 4 11" 2 5'/T

8'

1r 0 figure 3.3. 11" 5 211"

T

211"

5'"

4. Thaexact constants of approximation cn,p for the operators Ln,p

411" 11"

5"

4.1. Case p ~ 2. Assuming n E

m

and p ~ 2 fixed, the exact constant of

approximation cn,p was defined in the introductory section by (1.4). Both in (1.4) and in the definition of d in (3.1) it is not an essential

restric-n,p

tion to take wI in fact equal to one. It now follows from theorem 3.1 that

11"

(4. 1) c

₌

nip d = nip

_r

,.... f (t)k (t)dt

n,p n,p _J n n,p (n,p" 2,3, ••• )

-11"

and, according to definition (I.6)

(p .. 3,4 •••• ) .

By means of (3.13) and (1.2) it immediately follows that

11"

The main object of this section is to determine the exact constant of

approximation for the whole class of operators L (n e IN,p ~ 2), i.e. to n,p

determine the value of c :'" sup p~2 sup neJN c n,p

Solving this problem turns out to be a quite cumbersome task; a good many particular cases have to be considered. The analysis is split up in several subsections.

4.1.0. Defining

(4.3)

f

(t) '"

!Itl

+ h

(t) ,

n n

we conclude from (4.1) that

(4.4) _{c '" Sl(n,p) + S2(n,p) ,} n,p where 'IT S 1 (n,p) ... nIP

_r

~Itlk p(t)dt, J n, -'IT 'IT S2(n,p) -nIP

J

h (t)k n n,p (t)dt • -'IT

As will become apparent Sl(n,p) is by far the main contributor to the con-stant c t whereas S2(n,p) rapidly becomes very small when p increases

n,p

(cf. table 4.1 of section 4.1.11). Because of this phenomenon partition (4.4) is made. We proceed by investigating Sl(n,p) and S2(n,p) separately. Using (1.2) we easily find that

(4.5)

f

nO?t [

sin

~

1

2P

.dt

o

n sin

~I

n;;J T 1 (n,p) SI (n,p) .. - - - .. ---.;:;.;. ... - - - - == n'IT ,... N(n, p) •

T

~

IP

J

[nSi~

\J2

P

dt

_J

_[Sin

t

,]2P~t

o

un

Ii

0 n sin

...!..

nIP-,[ sin t

J

2p t . t dt 'n Sl.ll --n

Taking into account lemma 2.4 we note that the denominator in the right-hand side of (4.5) is equal to (4.6)

lflJo

(n, p)

/P

N(n,p) ... ----...---2n2p- 1

where lJ(n,p) is given by (2.4). Formula (4.6) will be used in the sequel for

o

small values of p. We also need a lower bound for N(n,p).

4.1. 1. Letmn.a 4.1.

(4.7) N(n,p) ~

l

Vi

.fii1 erf(~vpa) 11' \ , - .

where (cf. (2.5»

a ... ~ log(~) ... 0.366039 •

11'2

Proof. In view of (4.5) we have

(n e 1N, P E: IN) ,

nlfyp

t 2 N(n,p) ,.. 2

[Si~

IP

t ] p dt

~

I

n SLn

---o

nIP)

lI'IP

2

r

Sl.n -

.

_yp

t

j

2p _dt

of

In sin

~

• I.. nip

An application of letmn.a 2.5 gives

!!j.

( sin..!:..

r

p

lfYP

2

J

IE

dt ~

r

exp(-at2)dt

₌

. t 0 _I.. n SLn-_nIP) 0

This lower bound for N(n,p) is increasing with p. Numerical values can be obtained from Abramowitz-Stegun ([lJ, p. 311).

4.1.2. The next result furnishes an upper bound for the expression Tt(n,p) occurring in (4.5). Lemma 4.2. (4.8) _T1(n,p)s 3 _{1 {I-exp(-3(l-Z)'lfP)}+p_}1 1 2

--L

2 -(2p+l) 1 TT 2 2(1 - 2) n n _(n,p

₌

_{2,3, ••• ) •}

Proof. Assuming n ~ 2, P ~ 2 and taking into

TTvP [ sin..!.

12

P

r

Tt(n,p) .,.

J

t _{n sin}

liJ

_--=

t dt + 0 nip

By means of the inequality

(4.9) sin x ~ if 2 x (0 s X s 'If

'2) ,

account (4.5) one has

~

r

sin..!.

r

J

t .

IE

t

TTIp _l n Sl.n-_nIP)

a crude estimate for R can be deduced as follows.

nTT nTT

2

dt

=

(4.10) 2 2

J

r

sin t ) p t n sin

tJ .

dt s TT l n

f

t-2p+1 dt < t- 2p+1 dt _...

-L

_TT 2 2-(2p+l) _• p-l

Moreover, using lemma 2.7 one has

. t Sl.n

--rp

1 1 t2

--.----t-

S exp(-

'6

(1 -2)

'"P')

(0 < t n Sl.n - - n

nip'

and hence TT

1T/P

(l sin

~

J

2p 11";;

I .

t .

vP

_t dt:s; J t exp(-

~

(1 --I)t2)dt

-o

n S1n - - 0 n nip. 3 1 1 2 • --~ {I - exp (-

'3

(I -

2')

11" p)} • 2(1 -~) n n

This, together with (4.10), establishes lemma 4.2.

4.1.3. We proceed with an estimate for S2(n,p), the second contributor to the constant c (cf. (4.4». Taking into account that h (t)

=

0 if

n,p n

I

t i s ~, and using (1.2) we have

n1l"

p J1I"n h (t) [si.n ntt) 2p dt p

JT

n h (2t)

r

sin t

J

2p dt

- 11" n S1n 11" n n . t n Sl.n Ii (4.11) S2 (n, p) .. - - - 1 1 " - - - -n

=

- - - - n - 1 I " - - - -2

=

rn

p J [Sin !nt 1 2p r 2 Lemma 4.3. (4.12) sin

!tJ

dt

;p

J

o

0 T 2(n,p) = ~.--...-N(n,p) • ( . 4 1 9(1 - 2) n 2 16 ) + -':::'2-"';"';'-"'1 --;::"2 311" p(1 --2) n 1 1 2 exp(- - 0 - -2)11" p) + 12 n

•

+ 11" P 2-2p+4 27 (p-2) (n2:2, p2:3) •

Proof. Let n 2: 2 be fixed. The function h is approximated by a polynomial q n

of low degree such that h (t) S q(t) for t 2: O. In order to be able to

per-n

form the resulting integration easily when h is replaced by q in the

ex-n

pression for T2(n,p),~.~<:~o~se a polynomial of odd degree. One simply

.. ·4n2\'. 3 -, .. . '

verifies that q(t)

=

~ t will do, i.e. that 2711"

Taking into account (4.11) and using (4.13) one has mr 'IT pt3[

Si~

ttJ2Pdt

(

pt3[ si: ttJ2Pdt} (4.14) T2(n,p) < 32

{J

+

₌

- 27'IT2 _{n Sl.n - n Sl.n}

_ii

'IT n 'IT 2 sin

~

r

n'IT

.rp [

T

_{3 [ sin t fP} 32

{I

J

t 3 n

J

dt} • - 27'IT2

P

_{Sl.n -}. p t dt + pt _{n Sl.n -}. t

!rIP

₂ nIP- 'IT n

An estimate similar to (4.10) takes care of the second integral in the

right-2

h d _{an Sl. e o . ; t l.S gl.ves rl.se to the contr]. utl.on 27(p-2)} 'd f (4 14) h ' " ' b ' 'IT

P

2-2p+4 . _l.n

(4.12). The other integral in (4.14) may be handled by lemma 2.7 as follows.

'lTvP

r

sin..i.

J

2p

J

t 3· R P t dt ~ / n s i n -~IP ~

nip.

=!

f

2 'IT P

.!6.

4 t exp(-

~(1

- -!)t)dt <

~

n

)

~ 4 1 1 2 3(1 - -Z)t )dt

=

n 1 1 t exp(- 3(1 - -Z)t)dt • n

Performing the resulting integration one arrives at (4.12).

Lemmas 4.1, 4.2 and 4.3 will be used in the sequel to obtain upper bounds for the constants c if nand p are not too small, say ~ 5. The estimates

n,p

are. rather poor for small values of nand p (for instance when p

=

2 or p

=

3). In that case a different approach will be needed.

4.1.4. Case n

=

2, p ~ 2. We next consider the behaviour of the sequence {c

Z ,p }. According to'formulae. (4.1), (3.12), (4.3) and (2.10) we have

'IT (4.15) cZ,p -

21P

f

f

_Z(t)k₂,p(t)dt

=

-'11" where ( 0 (0 s t s !) 4 (4.16) hZ(Zt) - 'IT 'IT S 3'1f) t - 7; (- S t 4 8 , . 1l' _(3'IT 'IT -t + - - s t < -) _• 2 8 - 2 00

Letmna 4.4. The sequence {c

2,p}Z is increasing and (4. 17)

_{= -}

2

₌

1.12837917 •

Proof. In view of (4.15) the first assertion of the lennna amounts to proving that (4.18) 1T (2P

+l)lP{

r

o

'IT 3 2

- J

r

< 2 (p+ 1) 21. J

o

'IT . 2p t(cos t) dt +

oI2

h 2(2t) (cos t)2Pdt} < 2p+2 tCcos t) dt + 'If

I2

h 2(2t) (cos t)2P+2dt} •

o

Using letmna 2.8 one obtains

C4.19)

o

1! 2

J

tCcos t) 2Pdt S l! 2

J

{sin t +

~sin\

+

Zo

sin5t +

(~- \i~)sin7.t}

(cos t)2Pdt =

o

_ --l-

+

l{--l- - --l-}

+

.1..{--l- _

~ + 1 } +

2p+l 6 2p+l 2p+3 40 2p+l 2p+3 2p+5

'IT 149 1 3 3 1

'!T

-. . 1 . ( 2 p + 2 .

J

2

Agal.n uSl.ng emma 2.8 the l.ntegral t cos t) dt can be treated l.n the

o

same fashion to obtain a lower bound. Furthermore, applying lemma 2.7 if n

=

2 we get in view of (4.16) (4.20)

_J

2

o

'!T 2; 11 h2 (2t) (cos t) 2p dt = _r 2 ₂ h2 (2t) (cos t) P dt :S ) '!T (2

J

! 4 !Xl (t -

~)exp(-Pt2)dt:S

J

(t -

~)exp(-Pt2)dt

=

'IT

4

r .

2

J

exp(-u )du 1 2

= -

_2p exp(-

.E!...)

₁₆

- -

'!T

41;

'!Tip' 4

According to [1], p:-298 one has

x +

Vx2

+

2

2 < exp(x )

_f

x

2

exp (u )du :S

-x +

Vx2

+

~

1T

(x ;::: 0) •

Using the left-hand side of this inequality in (4.20) one easily verifies that ! (4.21)

J

r2 h2 (2t)(cos t) p dt 2

o

4 2 <

-:r:T

exp(- P;6) • 'IT P 1T . ( 4 )

J2

2p+2 .

Estl.mate .19 and a lower bound for t(cos t) dt, together wl.th

o

(2p+l)1P

{2P~1

+

~(2P~1

-

2P~3)

+ 4 3 0(2Pl+1 - 2P:3 +

2P~S)

+ ( 'IT 149)1 1 3 3 1 ) + 4

(e:)}

< +

2 -

120 \:2p+J - 2p+3 + 2p+5 - 2p+7

22

exp - J6 'IT P

An elementary, but tedious, computation shows that this is indeed the case

for p ~ 8; we omit the details. The constants c

2,p (p

=

2,3, ••• ,8) can be evaluated explicitly (cf. table 4.1 of section 4.1.11). Taking these data into account it follows that the monotonicity holds for the whole sequence {c

2 ,p }. This proves the first part of the lemma.

As for assertion (4.17), it is clear from (4.15) and (4.21) that the limiting behaviour of {c } is governed by

2,p (4.22) 2 2p+2( _p. ,)2y; _p 1T(2p)! We have 'IT

...

r2 p

J

J

2

a

t(cos t)2p dt (p -+- (0) • 'lT1P co

r

·2 u 2p

J

J u(cos --) du ~ u exp(-u )du 2

= -

1

2

a

a

IP

a

_{(p -+- co) •}

This, together with an application of Stirling's formula in (4.22) proves (4.17).

The remaining part of this section will be devoted to showing that

2

sup sup c

=

--p~2 nElN n,p ; ;

4.1.5. Case n ~ 4, p ~ 5. Here we shall be concerned with estimating the exact constants of approximation c if n ~ 4 and p ~ 5. In view of (4.7)

n,p and using [IJ, p. 311 we find that

(4.23) N(n,p)

~

!

~ erf(~

VSa)

> 1.4579

en

E IN, P ~ 5) •

An application of lemma 4.2 with n 4, p

=

5 yields (4. 24) 8 -25TI

2

2 -13 8 2 -13

T 1 ( 4, 5) :;:; 5 (I - exp ( ₁₆ ) ) + 5TI 2 < 5 + 5n 2 < I. 6061 •

The last estimate, taking into account formula (4,-8), holds for all n ~ 4

and p ~ 5.

Finally, if we use lemma 4.3 ~n case n n ~ 4 and p ~ 5 one has

4, p

=

5, it follows that for all

Using formulae (4.4), (4.5), (4.11) we obtain from (4.23), (4,24), (4.25)

that

2

c < I • 1169 <

n,p

liT

(n ~ 4, p ~ 5) ,

4. 1.6. Case n = 3, p ~ 5. When we try to show that c₃ <~with the help

,p ; ;

of formulae (4.7), (4.8) and (4. 12), it turns out that these are not quite

adequate in this case. A small modification is needed: we slightly sharpen the estimate (4.7) for N(n,p) in case n

=

3. This can be accomplished as

follows. Using lemma 2.6 and (4.5) we have

N(3,p) 3

rp (

.

t

J

2p

l

s~n-r 2

rn

= )

p dt > 0 3 sin t_ 3/p

~

( sin ...E._

j'

2P r 2 ; ;

J

.

3 sin _t_ p dt ~ 0 ~

3!p.

2 exp(-bt )dt

₌

~

_/[

erf(~

/Ph) ,

0 8 3 where b

=

-r

log(z) TI 0.328658 •

Consequently, for all p ~ 5 one has (cf. [1], p. 311) (4.2A) N(3,p) > 1.5386 .

Lemmas 4.2 and 4.3 yield

(p ~ 5) ,

(p ~ 5) •

In view of (4.26) one obtains c 3 ,p < I. 1177 2 <

-.fIT

(p ~ 5) •

4.].7. Case n ~ 3, P

=

4. Next we shall deal with the constants c 4' Also n,

in this case it turns out that a mere application of lemmas 4.1, 4.2 and 4.3 is not accurate enough to ensure that c

n,4 < ~. However, inequality (4.7) of lemma 4.] can easily be replaced by a sharper bound as follows. Taking into account (4.6) and (2.4) we have

(4.27') N(n,4)

=

~(151n7

+ 70n 5

+ 49n3 + 45n) > 1.5059 315n7

en

e IN) •

Lemmas 4.2 and 4.3 supply the following bounds for T

1(n,4) and T2(n,4). T 1(n,4) 25 -32~2 Tr22-7 < 1.5882 s 16(1 - exp( 25

»

+ ₃ (n ~ 5) , (25 + 2 2 T 2(n,4) s 54 _432~2 625 ) exp (8Tr ) ₂₅ + ...:!!... 216 < 0.0716 (n ~ 5) • As a consequence of the above results one finds

c 4 < 1.1022 < ~

n, ; ; (n ~ 5) •

The constants c

3

,

4 and c4

,

4 will be taken care of by computing these numbers explicitly (cf. table 4.1 of section 4.1.1]). Of course, N(3,4) and N(4,4)

c~n.be evaluated by using (4.27), and in these cases the bound given in

(4.27) can be improved. This, together with lemma 4.2 and a small modifica-tion of lemma 4.3, also shows that both constants c

3 4 and c4 4 are smaller

than ~ • ' ,

4.1.8. Up to now we have been working with the estimates of lemmas 4.1, 4.2 and 4.3. In case both nand p are not too small, they apparently give satis-factory results. It also turned out that when nor/and p is small (f~r

instance, n

=

3 or p - 4), then some modifications were needed. However, in case p - 2 or p - 3 the aforementioned lemmas do not work; the bounds

2 2

they supply are not accurate enough to show that c < - - c <

-n,2 liT' n,3 liT

(n € ~). Consequently, a different approach will be needed with which we

shall now be concerned. We first note that it is easy to obtain good bounds on N(n,2) and N(n,3). In view of (4.6) and (2.4) we have for n E ~

(4.28) N(n,2) - 1T12(2n 3 + n)

~

.! 1T

Ii

> 1.4809 , 6n3 3 (4.29) r.:- 5 3 N(n,3) ::: 1Tv3( lIn + 5n + 4n) >

.!..!.1T

13

> 1.4963 • 40n5 40 m1'

We now consider the integral

oJ;'

t[ns~:nt~}2Pdt,

which, apart from a

factor p, is identical with T1(n,p) (cf. (4.5». Using (2.6) we deduce that

(4.30) I n+l,p (n+ 1 )1T

r

2

r

sin t

J

2p :- 0

J

tl(n+l)sin

-E..

dt s; n+l + (n+ 1) 1l' 2 2 . p

J

t

r

I,(n+l)sin Sl.n t

....L1

]

dt

s; ~ n+ 2 (n+ 1) n n1T ":2p ,~ + 2 <{(n+l)sin(2(n+l»} I n,p nn 2 + 2 -2p

=

I + 1T (2p): (n+l)-2p+1 ( . ( n1T

»

2 2 2 Sl.n 2(n+l) n,p 2 p+ (p!) where we have used (2.10).

Repeated application of (4.30) for a fixed nO € IN gives (4.31) 2 nO+8 I ~ I + 1T (2, p) !

l

nO+s,p nO'p 2p+2 2 . 1 2 (p!) Jana+ (s € IN) .-2p+l J •

A similar procedure will be used to obtain an estimate for the integral

(4.32)

I

T

nh

(~)[

sin t J2P n n n sin S

~ n

dt , 2

which, apart from a factor p, is identical with T

2(n,p) (cf. (4.11». In order to do this we need the following lemma.

Lemma 4.5. Let the function h

n be defined by (4.3). Then for n € IN one has

(4.33)

2

~ 1 04 n ~ . 2 t

. T u n n

Proof. As h (2t) is an even function that is identically zero on

_[a,E2],

it n n

is sufficient to p::ove the inequality for the range

¥

~ t ~

T.

Owing to the definition of f as given by the formulae (3.12) and (3.13) the cases

n

n

=

2m and n

=

2m+l must be considered separately. We first assume that n a 2m. It is then easy to verify (cf. figure 4.1) that on the interval

'Jl' m~

[2

t

T

J one has

2m h2 ( :!:) < ~ t 2 ~ m 2 'Jl' • 2 t

m m 1T Hn

2iii'

1T m~ m~

This establishes inequality (4.33) for the range

2

~ t ~

T'

I f

T

~ t ~ m~

the function 2m h2 (:!:) is again approximated by a quadratic function. mm

Actually, one has

Now we determine A ~ J such that

m'lf <

(T -

t S m'lf) •

m2

Dividing through by -- , we can write this relation as

'If

8(

~ _ ~)2 .... , 2 . 2 t

2 2m '" A'If Sln 2m'

If we put a

inequality

:= ; -

im,

then 0 S /l S ~ and A must be chosen such that the

*

holds. In order to obtain a value of A as small as possible we take A

=

A with

2 2

'If - 8/l

The expression

2

2

has a unique maximum on [0'4J, which is attained at 'If

'If cos /l

/l = 0.5957. Consequently A* = 1.0397 < 1.04.

This proves the asserted inequality (4.33) if n

=

2m. When n

=

2m+l, one can

d . 1 . f ' 2 2 d

procee In exact y the same way, l.e. the same unctlons i t an

2 2 2 2t

m 1T - n(m1T - t) can be used to approximate nh (-) on the intervals

['If

_2'

(2m+ 1 )

₄

1T _{J and} [ (2m+ I )

₄

'If _{,m'lf] respectlvely. Fl.nally, the lnterval} • n. n .

[m'lf,(m+

~)'If]

is taken care of by noting that if n is odd nh (l!) is constant

2 " n n

there (cf. figure 4.1

j,

where~s

1.04

¥

sin2

~

is still increasing on this

interval. This completely proves the lemma. •

We note that the factor 1.04 in (4.33) is certainly not best possible; however, it cannot be replaced by 1.

In order to estimate the integral (4.32) it will be convenient to have the graphs of the function nh (2t) available for the first few values of n. For

n n

this we refer to figure 4.1 below, in the construction of which formulae (3.12), (3.13) have been used.

n=7

o

1):

_12:.

_1T 51T 31T 71T

21T

-

91T

-

51T

₄

I I1T 311"

- -

131T 71T

2 4 _{4 2 4 4 2 4} 2-_

fisure 4. 1 •

Using lemma 4.5 and taking into account (2.6), (2.10) and (4.9) we have for

n ;;:: 5 n1T 31T n1T J2

[. (P

J2

f2

(4.34) J := nh (1!) Sln t dt _{= + S} n,p n n . t 1T n Sln

ii

1T 31T

"2

2

T

31T n11" S

J

2 5h (2t) [ sin t ]2 P dt + 0.2611"

r

(sin t) 2P dt _< 5 5 5 . t (n sin 1;) 2p-2 1! SlnS

-

31T n 2 2 t

n~1

.

-2p+2 < J S _,p + 0.261T _j=3

I

(n sin

fn)

(j+l)1T 2

f

..1.!

2 0.261T2(2E)! n-l . -2p+2 = J +

I

_{(n sin}

1:!)

S,p _{22p+l (p!) 2 j=3} 0.261T2(2E)! n-l ;5; J S +

I

.-2p+2 < J + ,p _22p+J_{(p!)2 j=3} J s,p (sin t)2p dt ... ;5; 2 00 0.261T (2E):

I

.-2p+2 22p+1 (p!) 2 _j=3 J

Inequalities (4.29), (4.31) and (4.34) will be used to dispose of the case p ... 3 and n sufficiently large. This is the subject of the next section.

4.1.9. Case n ~ 3, P = 3. In section 4.1.11 we shall show, by way of working an example in detail, how the constants c _n,p may be computed explicitly. This can be accomplished by hand when both the values of nand p are small. Proceeding in that fashion it can be shown that we have

where 8 1 (5,3) ... 21T/3 -4 175l1r ]013 (1686 + 1246 + 666 + 246 + 9 25 49

~

81 +

....L) ...

121 ... 0.955187 J 8 (5 3) _{2 ' ... 1T} 2

13

_

_17511T 20/3 = 0.017208 • 2 1T 1T 1246 246 56 6 (2 cos - +cos -- 1)(1686- - - - + - + - ) == 5 5 9 49 81 121 Accordingly c 5,3

=

0.972395.

Using these data and taking into account (4.29) and the definitions of the

integrals I and J respectively, we obtain

n,p n,p

1_{5,3 ... 0.485382} J

Applying (4.31) and (4.34) in the case p

=

3 yields for n > 5 511"2 -6 I _n,3 _< I _{5,3 +}

_'""'6'4

( . 511") _Sl.n

₁₂

co

I

j=6 j-5 < 0.485635 > 2 co J 1.311" \' .-4 J_nt3 S 5,3 + 32 L J < 0.016693 'c j=3

From these results and (4.29),it follows that

(n > 5)

and hence

c 3 < 1.0072 <

JL

n,

lIT

(n > 5) •

The constants c3 3' c4 3 will be computed explicitly; their values are

,

,

contained in table 4.1 of section 4.1.11.

4.1.10. Case n ~ 3, p

=

2. The case p

=

2 is treated in a similar way as case p

=

3. However, because inequalities (4.31) and (4.34) give a better performance for p "" 3,than for p "" 2, the value of nO in (4.31) must be taken greater than 5, and inequality (4.34) must be slightly modified. In quite an analogous way as (4.34) was derived, we deduce that for a fixed nO EO: IN there holds

(4.35)

In order to obtain satisfactory results from (4.31) and this formula in case p

=

2 (Le., to show that from a certain index on we have c 2 <

JL-).

the

n, ; ;

number nO in (4.31) and (4.35) cannot be chosen too small. Numerical compu-tations show that the choice

no

=

19 will do. As a consequence the constants ., c 2 (n

=

2,3, ••• ,19) must be computed explicitly (cf. section 4.1.11). An

n,

application of (4.31) and (4.35) in case p = 2 and nO = 19 implies that.

1

19

,

2 and JI9

,

2 must be known. In view of their definitions this means that 81 (19,2) and 82(19,2) (cf. (4.4» must be available. Th~.s ~s_accomplished

by computing c

8

1(19,2) - 0.939283 , 82(19,2) - 0.114679 , c 19 ,2 - 1.053963 •

In view of (4.28) one has

1]9,2

=

0.696484 t J

19,2

=

0.085035 •

Applying (4.31) and (4.35) if nO

=

19, P

=

2 gives for n > 19

Using 00 2 -4 I nt2 S 1 19 ,2 +

~

(sin lZ0rr)

r

j-3 < 0.697716 ~ j=20 0.39rr2 00 J S _J 19 ,2 +

r

j-2 < 0.135636 ., n,2 8 j=10

these results and taking into account (4.28) one has S2(n,2) < 0.1832 and hence c 2 < 1.1255 n, 2 <

-;;

(n > 19) (n > 19) •

4.1.11. What remains to be done is the explicit computation of the values of a few particular exact constants of approximation, because these could not be estimated adequately using the various results of the preceding sections. In particular, we have to compute c

2 ,p (p

=

2,3, •••• 8) (cf. section 4.1.4),

c3 4 and c4 4 (cf. section 4.1.7), c3 3 and

"

,

c

n,2 (n = 2,3, ••• ,19) (cf. section 4.1.10).

c4 3 (cf. section 4.1.9) and

,

In what follows we shall first work a specific example in detail and, using the exhibited pattern of

compu-tation, sketch how the numbers c may be evaluated in general. n,p

Let n

=

3 and p

=

2, i.e. we shall compute c3 2' In view of (4.4) the compu-

_,

tat ion is split up in two parts. An application of lemma 2.4 gives

(4.36) [Sin

~tJ4

• 19 + 32 cos t + 20 cos 2t + 8 cos 3t + 2 cos 4t •

Sl.n

it

(4.37) As (4.38)

o

11'

r

J

t ( . 3)4 1T

l

Sl.n

itJ

3/2 r

!

I

t

I .

I d t ,.. 381T

J

Sl.n

zt

0 k cos kt dt

=

(-1) -1 k2 (k € IN) , [ sin

:2tJ

4 t 2 dt • 1 Sl.n

zt

it easily follows from (4.36), (4.37) and (4.38) that we have

312

{2

1184}

(4.39) SI(3,2)'" 41T 11' -

-r7:I ,..

0.994499 •

By (3.13) and (4.3) the function h3 is defined as follows.

0 (0 :;; t :;; -) 11' 3 h 3(t) ,.. t-2! 3 (!! 3 :;; t s; 23 1T ) 11' _{(211' < t} s; 11') 3 3 -Consequently, 11'

,.. 312

J

S2(3,2) A3 2 , -11'

(. 3] 4

l

Sl.n - t h3 (t) .

f

dt

=

Sl.n

zt

~ . 3 4 11'

r

3/2{ r3 11'

r

un 'it1

= -

J

(t--)l 1911' 3 sin It 11' 2 ) dt +

J

3 211' 3

Performing the integrations in such a way that the upper limit of integra-tion is taken to be 1T, and observing that

I

t cos kt dt ,..

~

t sin kt +

;Z

cos kt + C _{(k " 0)} t

(4.40) 11

=~{

1911

f

[

. 3] 4

11 Sl.n

'2t

(t--) dt+ 3 . I un

it

11

f

211 [ . 3 Sl.n - t (-t + 211) 2 3 • 1 un

it

11 3

"3

3/2 { 20 8 2

+ 1911 32 cos t +4 cos 2t

+'9

cos 3t +

16

cos

==

~

11

12 -

2~;:Z

== 0.073318 By (4.39) and (4.40)

5 r::' 56812

c₃,2 == 8₁ (3,2) + 8_{2 (3,2)} ==

4

11 v2 - 5711 ... 1.067817 •

From the details given in this example it will be clear how the numbers c

n,pc~c~

can be computed in general. Basic to the computation is lemma 2.4, together Cc with formulae (3.12) and (3.13). To simplify the integrations necessary to find 8

2(n,p), we take the upper limit of integration always equal to 11. Using (4.38) and (4.4) one has

(4.41) where

~k

:=

ll~n,p)

{ np'-p } t llO + 2

r

llk cos kt dt

=

k=1 [np-p+l ] 2

r

k=1 ll2k-l } 2 ' II (2k-l)

o

n p SI(n,p) S2(n,p) c n,p ~ 2 2 1.02101971 0.02906470 1.05008441 2 3 1.05446061 0.01232470 1.06678530 2 4 1.07207314 0.00527063 1.07734377 2 5 1.08292455 0.00229302 1.08521757 2 6 1.09027557 ·0.00101363 1.09128920 2 7 1.09558275 0.00045408 1.09603682 2 8 1.09959366 0.00020564 1.09979930 3 2 0.99449879 0.07331833 1.06781712 3 3 0.98901886 0.01968274 1.00870161 3 4 0.99741667 0.00630664 1.00372332 4 2 0.97402783 0.09169919 - - ' - -1.06572702 -. 4 3 0 •. 26556373 0.01810182 0.98366555 4 4. 0.97416993 0.00511832 O~3r7928825 5 2 0.96414008 0.10591303 1.07005311 6 2 0.95706753 0.10846097 1.06552849 7 2 0.95275288 0.11140501 1.06415789 8 2 0.94948067 0.11267841 1.06215908 9 2 0.94722655 0.11429915 1.06152571 10 2 0.94543526 0.11442277 1.05985803 11 _{2 0.94410674 O. 1 14861 17 1.05896790} 12 2 0.94301421 0.11491012 1.05792433 13 2 0.94216282 0.11517397 1.05733678 14 2 0.94144449 0.11503006 1.05647455 15 2 0.94086449 0.11504856 1.05591305 16 2 0.94036537 0.11492429 1.05528966 17 2 0.93995148 0.11491780 1.05486928 18 2 0.93958969 0.11475319 1.05434288 19 2 0.93928338 0.11467940 1.05396278 Table 4.1

In the case n ,.. 2m+l a computation analogous to that performed in (4.40) shows that we have

(m+j)1f (4.42) _{S2(n,p) ,..}

_!1fm;P

2 _+.=.!!.!.2. 2n

rn

m

I

r

_L\

rnp-p llk cos kt)l

I

--~-

J

n

1fllO j=l k=1 k 2 . i !

n

By use of (2.4) both expressions (4.41) and (4.42) can be evaluated. When n ,.. 2m a similar expression for S2(n,p) can be derived, which we re-frain from giving here.

By means of these formulae the exact constants of approximation c may be n,p

computed for any values of nand p, although the amount of computational work involved grows quite rapidly. For small values of both nand p it can be done by hand, cf. the example of c

3,2 in this section and the value of c5,3 as given in section 4.1.9.

We incJ'Y-sl~_.here a table containing the numerical values of the constants

c that were mentioned in the beginning of this section. These data were n,p

computed on the Burroughs 7700 of the Computing Centre of the Eindhoven University of Technology.

4.1.12. Taking into account (4.17) and the estimates for the .constants c _{. n,p} in sections 4.1.5, 4.1.6, 4.1.7, 4.1.9, 4.1.10, together with the contents of table 4.1 and formula (4.2), we have the following theorem and corollary.

Theorem 4.1. Let c _n,p be the exact constant of approximation for the operator L as defined in (1. 4). Then

n,p

lim 2 1.12837917

c :- sup sup c ,.. c _{== -}

₌

.

pe:2 nE:lN n,p p-+- 2,p

liT

1

Corollary. Let f e: C

21f and let w1(f;o) := w(f';o) be the modulus of con-tinuity of ff, then for n e: IN and p ,.. 2,3, ••• one has

(4.43) max x

where the value

~

is best possible in (4.43).

4.2. Case p

=

1. Taking into account definition (1.5) and the results on the extremal functions of section 3, we have (cf. formulae (3.1), (3.2) and theorem 3.1) c

=

n, I -'IT 'IT

J

f

n (t)k l(t)dt n, where according to (1.2) and lemma 2.4

We write (cf. (4.3) and (4.4» where 'IT (n €: IN) , 8) (n, 1) 1

J

~Itl(s~n

Int)2 dt = -2'ITfI, _{Sl.n 2t '} -'IT 'IT 8 2 (n, 1) 1

J

h (t) (s~n !nt)2 dt

= -

_{2'ITn n Sl.n}

It

-'IT

By lemma 2.4 it is easily shown that SI(n,1) decreases monotonically to zero when n ~ ~. In contrast to the case p ~ 2 here the term S2(n,l) is the main contributor to c 1 when n becomes large (cf. table 4.2). In order to

(1) n,

determine c

=

sup

n€:JN account that h (t)

=

0

n

Cn,l we need an upper bound for S2(n,I). Taking into

i f

I

t

I

:s; :!! one has for n €: IN n 1 = -'IT tl'lt:

J

h (t)(s~n ~nt)2 _dt 2

r

n Sl.n

It

= 2 _'lTn 'rr _.... 'IT

-n 2 ... ~~ .... ---~- -n'IT

~ o~~

I2

sin2t dt < O. 13 'IT

=

0.4084 ,

1! 2 by an application of lemma 4.5. " _ . ' , , -h n

(¥-)

[s~n ~J

2 dt Sl.n -n

Furthermore, one easily verifies that (4.45) _{c I ,]} = 5](1,1) = 11"

i ...

0.78539816 •

The constants c 1 can be computed by use of formulae that, apart from a n,

factor nIP, are identical with 34.-41) and _(4.~2!. Again, if n is even a slight modification of (4.42) is needed. Table 4.2 contains the numerical values of the constants c I (n - 1(1)10; 15(5)50; 75; 100), together with

n,

the corresponding SI(n,l) and S2(n,I). In particular we have

(4.46) 5₁(3,1)

=

4 -

11" J; 4

=

0.3610.

As SI(n,l) is decreasing formulae (4.44) and (4.46), together with (4.45), imply the following theorem.

Theorem 4.2. Let c 1 be the exact constant of approximation for the

n,

operator L i a s defined in (1.5). Then n, c (1) := sup n€lN 11" cn,l - c I ,1 ...

4 ...

0.78539816 • ] Corollary. Let f € C

211" and let oo}(f;o) := oo(f';o) be the modulus of con-tinuityof

f',

then for n E IN one has for the Fejer operators

I

I

11" 11"

(4.47) max L_n,l(f;x) - f(x) $

4

oo

1(f;;) ,

x

1i'

n SI(n,l) S2(n,l) c n,1 1 0.78539816 0.00000000 0.78539816 2 0.46708828 0.03225063 0.49933891 3 0.36098498 0.09918559 0.46017058 4 0.29024945 0.14267826 0.43292772 5 0.24780813 0.18108477· 0.42889291 6 0.21526979 0.19874041 0.41401020 7 0.19202812 0.21748199 0.40951010 8 0.17297283 0.23021746 0.40319029 9 0.15815205 0.24358775 0.40173980 10 0.14550948 0.25152964 0.397039]2 15 0.]0567648 0.28328256 0.38895904 20 0.08380591 0.30040337 0.38420928 25 0.06989794 0.31195429 0.38185222 30 0.06017516 0.31973275 0.37990791 35 0.05298480 0.32578270 0.37876749 40 0.04742132 0.33032428 0.37774560 45 0.04298732 0.33410194 0.37708926 50 0.03935787 0.33708842 0.37644628 75 0.02795996 0.34677163 0.37473159 100 0.02188545 0.35195525 0.37384069 Table 4.2

5. The limiting behaviour of the constants c n,p

5.0. In this section we investigate the limiting behaviour as n + ~ or/and

p + = of the exact constants of approximation c • It turns out thaa there

n,p

are four cases to be considered, viz. n + ~, p ~ 2; n ~ 2, p +

=;

n +

=,

p + = and n +

=,

p ... 1, the last case corresponding to the Fejer operators which have a degree of approximation different from those of the Jackson type.

5.]. Case n +

=,

p ~ 2. Let d be given by (3.9), i.e. let

n,p

1T

d ...

n,p

I

f

n (t)k n,p (t)dt (p ~ 2) •

-1T

As a guide to norming we regard k as the probability density of a random n,p

variable (r.v.) T • For the expectation ET and variance var T we

n,p n,p n,p have ET

...

0, var T n,p n,p By lemmas 2.2 and 2.3 it is var T n,p -} -2 6p n 1T

=

ET2 =

I

t2k (t)dt n,p n,p -1T

easily verified that

Denoting the probability density of a r.v. X by Ix we generally have for "

a > 0 (cf. [4J, p. 45)

and therefore, letting n + =,

(5. 1) g (t)

=

~

g

~ T n T

2 n,p n,p

=

~

k {2t).-t (

I

It follows by dominated convergence

(f

(t) s altl + bnt2; cf. (5.6» that n

for n + ~ one has

n7T _co 71'

r

n

I

f

(t)k (t)dt =

£

(1!)k (2t)dt,

J

*

-

f (t)g (t)dt 2 n n,p n n n,p n p -71' n7T -00

-T

where (cf. formulae (3.12) and (3.13» co

*

.

n"'" 2t

I I

\'

f (t) := 11m

2

f (--)

=

!

t + l

n+co n n j=I

Summing up we have the following theorem.

Theorem 5.1. For p ~ 2

71'

c : = lim c = lim nip

p n+co n,p ~ -71'

J

£

(t)k (t)dt

=

n n,p ~

=

Ip

I

Itlg (t)dt + 4/P"

~

f

(t -

f)8

_pCt)dt p j=l .J where g is defined as in (5.1). p 1!.J.. 2

Remark. On account of lemma 2.1, theorem 5.1 may also be written in the form

(5.2) c =a +b , p p p where

=

IpS(p,p-D ap S(p,p) (5.3) b =

27

r

J

p S(p,p) j=l 0 (t

-l!)

(sin t)2P d 2 + t t •

The computation of a causes no difficulties. For small values of p the

p

integrals S(p,p) and S(p,p-~) can be evaluated by (2.1) and (2.2) respec-tively, whereas for large values of p it is preferable to use numerical integration. Expression (5.3) for b is suitable for numerical integration

when p is not too small, p ~ 4, say. If p • 2 or p

=

3 however, the right-hand side of (5.3) converges too slowly. To speed up the convergence we proceed as follows. where co b

=

21P

J

f(t)(si~

t) 2p dt , p S(p,p) f(t)

=

l

j=l

o

(t -

.i!)

2 + (t ~ 0) •

It is easily verified that

where g is a ~-periodic function and

Hence (5.4) b

=

p S(p,p)

21P

{ f

o

21P

{f

=~-'ITS (p,p) o _ <.-.. 2 ' . 2p

let -

!) (S1~t)- dt -'IT . 4 t _ . .

o

'IT . 2 2 . t 2p <XI

12

2 [(Sin(t+..l.!2'IT)] p } (t -

*)

(S1~

) dt -

.l

(t -

*)

''IT d t . J =0

_o

t + . .J..!. ₂

In view of lemma 2.1 the first integral in the right-hand side of (5.4) is easy to evaluate. The speed of convergence of the infinite series in (5.4) is, even for p

=

2, quite satisfactory, and the range of integration is now finite. Formula (5.4) has been used to compute b • Table 5.1 contains the

p

numerical values of a ,b and c for p

=

2(1)10; 20(10)50.

p a _p b c p P 2 0.93607757 0.10939991 1.04547748 3 0.93748742 0.01299377 0.95048119 4 0.94674058 0.00373073 0.95047131 5 0.95278406 0.00126211 0.95404617 6 0.95684686 0.00044531 0.95729217 7 0.95975273 0.00016018 0.95991291 8 0.96193220 0.00005838 0.96199158 --9 0.96362957 0.00002150 0.96365107 10 0.96498689 0.00000799 0.96499488 20 0.97109617 0.00000000 0.97109617 30 0.97313267 0.00000000 0.97313267 40 0.97415086 0.00000000 0.97415086 50 0.97476175 0.00000000 0.97476175 Table 5.1

5.2. Case n ~ 2, p + ~. We now investigate the limiting behaviour of c

n,p for n ~ 2 fixed and p + ~. We begin by considering cn/p T ,where c will

n,p

be given a convenient value. Leaving out the details of the computation, one has 1 t gcnin_{p T} (t)

= -::--

k ( - ) + g(n) (t) n,p cnl'P n,p cn/p (p +~) , where ~

(f (

2 2 \ \-1

=

exp\- t (n -1) dt \ 12n2c 2 ) ) -~

exp(

-2 2 2 . 11 2 '

I f we take (n -1)/(6n c )

=

1 it follows that V6'P (n -1) Tn,p is

asymptotic~lly standard normal. ~y dominated convergence, putting

~-:--~' --~. -. .~-

'IT a

J

f

(t)k (t)dt· n,p n n,p

..

-'IT -a 'IT n,p

We have now proved the following theorem.

Theorem 5.2. For n ~ 2

'IT

lim c • lim nIP

p-+<o n, p p-+<o

J

f

n (t)k n,p (t)dt

=

,f;[

V-;

I~ n -'IT Vn"-l 1 =

-/21r

2

We note that if n" 2 we have lim c

2 .. -- (cf. (4.17)). Furthermore, it p-+<o ,p ; ;

{ C }""

follows that lim is a decreasing sequence.

n,p 2

p-+<o

5.3. Case n + "", p + 00. From theorems 5.1 and 5.2 we obtain

. (cf. table 5.1)

'IT

lim c - lim nIP

n-+<o n , p n-+<o

J

f

n (t)k n,p (t)dt· \

V-;

II

= 0.97720502 , p-+<o p-+<o

where the limits may be taken in both orders.

Proof. It is an immediate consequence of theorem 5.2 that

lim lim c

=

V1 .

On the other hand we obtain from theorem 5.1, using n-+<o p-+<o n , p

dominated convergence for integrals and sum, lim c .. p-+<o P

· ( J

o

" 00 [ sin

;'J

2p u P du + 2 u

rp

l

j=l co

f

u exp(-

~)dU

=

v1 '

o