Legendre transform methods and best algebraic approximation

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXI (1979)

R. L. St e n s and M. We h r e n s (Aachen)

Legendre transform methods and best algebraic approximation

1. The theory of best approximation of 2я-регiodic functions by trigono­

metric polynomials, including the fundamental theorem on the equivalence of the assertions of the theorems of Jackson, Bernstein, Steôkin, Zamansky and Butzer-Pawelke-Sunouchi for polynomials of best approximation, is a field that has reached a high level of completeness.

The corresponding theory of best approximation of functions defined on a finite interval [a, b] by algebraic polynomials, however, is far from being complete. This is due to the fact that when dealing with the corresponding classes of functions as in the trigonometric case, the approximation be­

haviour is better near the end points a and b of [a ,b ] than inside, and so is not uniformly good over the whole [a ,b ] . This fact causes basic difficulties.

The counterparts of the theorems of Jackson and Bernstein, namely, the famous results of Timan [43] and Dzjadyk [21], [22] state, for example, that a function / e C [ - l , l ] belongs to the class Lip2(/i;C) for some 0 < jS < 2 , i.e.

if and only if there exists a sequence of polynomials pn of degree ^ n such that

(*) In the following M denotes a positive constant, the value of which may be different at each occurrence, even in a given line. M is always independent of the quantities at the right margin.

(1.1) sup |/( x + b) + /(x — b) — 2 / (x)| ^ Mrjp (rj > OH1),

X ,x ± /l 6 [— 1,1]

(1-2) \ f ( x ) - p n(x)\

352 R. L. S te n s and M. W e h r e n s

In continuation of these results Gopengauz [29], [30] (2) and Teljakovskiï [42] showed that the right-hand side in (1.2) can be replaced by M ( yJ \ — x 2/ n f , i.e. (1.1) and (1.2) are equivalent to the existence of another sequence of polynomials qn of degree ^ n for which

(1.3) 1 / М ч . М а К 1^ Y ( x e [ - l , 1]; n e N ) .

This shows in particular that the approximation error for fixed n decreases like (^ 1 — x 2f towards the boundaries.

It would seem to be just as interesting to investigate necessary and sufficient conditions such that inequalities of type (1.2) or (1.3) hold for which the factor y j \ — x 2 on the right-hand side has been dropped, thus to determine precisely the class of functions / e C [ —1, 1] which satisfy (1.4) \f(x)-p„{x)\ ^ Mn~ p ( x e [ - l , 1]; n e N )

for a suitable sequence of polynomials pn or for which (1.5) En( f ; C ) = 0(п-Р) (n-> oo),

where En ( / ; C) denotes the best approximation to / by algebraic polyno­

mials of degree ^ n (cf. Section 5). This class cannot be Lip2(/?;C) but must be larger. Moreover, it is to be expected that the factor y j l —x 2 now has somehow to occur in the definition of a Lipschitz condition corresponding to (1.1). One of the results to be obtained reads that (1.4) or (1.5) is equi­

valent to

(1.6) sup ( y / l - x 2Y \ f (x + h ) + f ( x - h ) - 2 f (x)\ ^ Mt]l{

x , x ± h e [ — 1,1]

l'" ^ ( x e [ —1, 1]; q > 0).

Comparing the equivalence of (1.1) with (1.3) on one hand, and (1.4) with (1.6) on the other, one sees that the factor y / l — x 2, which improves the approximation towards the endpoints in (1.3), has been moved into the Lipschitz condition (1.6). Therefore one has two parallel theories in algebraic approximation theory. In one case the weight is attached to the order of approximation as in (1.2), (1.3), in the other to the modulus of continuity as in (1.6). As no weight factors are in the trigonometric instance, there of course exists just one theory then.

If one has the fundamental theorems on best approximation by algebraic polynomials of the second form in mind, then it is possible to solve them using an integral transform method. As is well known in the case of trigo-

(2) According to a written communication by Professor R. DeVore, Columbia, S. C , the idea of the proof of Theorem 1 of [30] is principally wrong. But the particular results cited are correct (see [19], Section 7).

nometric approximation, the finite Fourier transform plays a basic role. In the case of non-periodic functions defined on the finite interval [ —1, 1], say, it is near at hand to choose a transform defined via an orthogonal system of algebraic polynomials, e.g. the Jacobi polynomials Although the theory of Jacobi series and transforms have made much progress in the recent years (see e.g. [l]-[5 ], [26], [27], [37]) many basic problems remain unsolved.

But for the special case a = /? = —1/2, namely the Chebyshev poly­

nomials, it was possible to reach almost the same level of completeness as in the trigonometric case (see [11]—[15], [40]). Now we wish to investigate another special case, that of the Legendre polynomials; these are the Jacobi polynomials for a = = 0. This set is the most “natural” orthogonal system of Jacobi polynomials, because it is orthogonal relative to the weight function 1.

Using the Legendre transform enables one to present a systematic treatment and reduces many of the various approaches in the algebraic case to a standard procedure, leading to unification and greater simplicity.

One major advantage of this approach is that it covers not only the case of approximation of functions belonging to C [ —1, 1] but also approxi­

mation in the' U ( —\, l)-spaces, 1 ^ p < oo, without any weight factor occurring in the norms like y j 1 — x 2 in the Chebyshev frame. The classical results in this respect are thinly spread (cf. [20]).

Concerning the history of the subject, the underlying set of Legendre polynomials was first introduced by Legendre in 1784/5 [34]. Then in 1874 Gegenbauer [28] discovered in a more general setting the product formula (2.6) which is essential for the translation and convolution concept of the paper. In the beginning of this century Fejér (see e.g. [24]), Gronwall (see e.g. [33]), and others dealt with Legendre series and their summability.

In 1950 Tranter [45] considered the Legendre transform method for solving differential equations; this was continued by himself, Sneddon and others (see [39], Chapter 8.5, and the authors cited there). A more systematic treatment from the point of view of approximation was initiated by Butzer [6] in 1963. These are just a few names in a field that has been worked on intensively for over 100 years. Another access to the present paper may be found by specializing the above mentioned papers about Jacobi series and transforms.

The paper is organized as follows. In Sections 2-4 the Legendre trans­

form theory is presented in a systematic form, the treatment being more or less self-contained. It would be possible to apply this transform to the solution of certain partial differential equations (cf. e.g. [45], [39], Chapter 8.5) in a rigorous form as was carried out by means of Fourier transform methods in [8], Chapter 7, and Chebyshev transform methods in [11]. This is the classical field of applications of transform methods. However, in this

3 5 4 R. L. S te n s and M. W e h r e n s

paper the focus is placed upon problems of best approximation. Particular approximation processes on C [ —1, 1], £?(—1, 1), 1 ^ p < go, defined by singular integrals in the Legendre frame, will be considered in [17].

Specifically, Section 2 is concerned with elementary properties of the Legendre transform such as (Legendre-) translation and convolution. Whereas Section 3 deals with the (Legendre-) derivative and integral in the strong sense, in Section 4 pointwise interpretations of the foregoing concepts are given. In Section 5 the Legendre transform is applied in proving the funda­

mental theorem on best algebraic approximation. Here the class of functions / satisfying (1.41 is described by means of the (Legendre-) derivative and modulus of continuity. In Section 6 condition (1.4) is characterized by pointwise moduli of continuity defined via the classical differences.

The authors would like to thank Professor P. L. Butzer for a critical reading of the manuscript as well as for valuable suggestions. The contribu­

tion of M. Wehrens was supported by DFG research grant Bu 166/27 which is gratefully acknowledged.

2. Basic properties of the Legendre transform. In the following let X be one of the spaces Zf(— 1, 1), 1 ^ p < oo, or C [ —1, 1] endowed with the norms

11/11,:= { i J I / ( u ) l 'du}1" (1 « P < oo), - 1

II/lie := sup |/( x ) |.

As usual we denote the Jacobi polynomial of degree n e P := { 0 ,1 ,2 ,...} , order (a,)?), a ,/i > — 1, by P{*,p), i.e.

a - x f (1 + * / P*-»(x) = (~ 7* -jL - [ ( 1 - х Г “( 1 +х Г+г] ( x e ( —1,1)) and in the special case of the Legendre polynomials one writes P„(x) instead of P<°’0)(x). For these polynomials one has

(2.1) |P„(*)I ^ P„(l) = 1 ( x e [ — 1, 1] ; neP) ,

(2.2) (1 — x2)P"(x) — 2xP'n(x)-I-n(n + 1)Pn(x) = 0 (neP; x e R : = real line),

(2.3) P'n( l ) = n(n+l)/2 (neP).

For a proof of these properties as well as for the other properties in this section see e.g. [38], [41].

The basic role in this paper is played by the Legendre transform, defined for f e X by

(2-4) ^ [ / ] ( & ) = / (k) := j f f ( u ) P k(u)du (k e P ).

- 1

The operator associates to each / e X a sequence of real (complex) numbers { / (/c)}£°=0, the (Fourier-) Legendre coefficients.

Le m m a 1. Assuming f , g e X and ce R, we have

(») № U m \ ^ \ \ f \ \ x (k e P);

(ii)

& l f + g № = & 1 Л ( к ) + & Ш к ) , &l c f ] { k ) = c & u m (keP);

(iii) c p v p i a \ р / ( 2^ + 1),

(iv) i? С /] (Л) = 0 /o r all k e P iff f (x) = 0 (a.e.) (3).

The latter property is known as the uniqueness theorem for the Legendre transform.

Furthermore, we define a translation operator xh for /ie [ —1,1] by

(2 .5 ) (thf )( x) 1 i

: = — J / ( x h и y j 1 x 2 y j \ - h 2) y / l - u 2 _1 du n - i

I n

= — J / (xh + y / l — x 2 y / l —h2 cos (p)d(p ( f e X ; x e [ - l , 1]).

я 0

By substituting у = xh + ^ / l - x 2 у / 1 - h 2 cos q> the translation may be re­

written as

(**/)(*) = — f f ( y ) K ( x , h , y ) d y ( f e X ; x , h e ( - l , 1)), к _ j

K (x, h , y ) : =

Г ( l — x 2 — h 2 — y 2 + 2 x h y ) ~ 112, x h - y / l - x 2 y / l - h 2 < у < x h + ^ / l - x 2 ^ J l - h 2 ,

j 0, otherwise.

The most important property of the operator xh is that

(2 .6 ) (t„ P„)(x) = Pn{h)Pn(x) (x, h e [ - 1, 1] ; n e P ) . Using this equality, one can prove part (a) of the following

Le m m a 2. (a) The translation çperator xh is a positive linear operator from X into itself satisfying

(3) (a.e.) means that an assertion holds for all x e [ —1, 1] if X = C [ —1 , 1] , and for almost all x e [ —1 ,1 ] if X = Lp( —1, 1), 1 < p < oo.

3 5 6 R. L. S te n s and M. W e h r e n s

(i) ' ||т»||г м = 1 ( Л е [ - 1, 1]);

(H) iim | | т , / - / | | , = 0 ( Je X);

h -* 1 —

(iii) [ T */]* № )= Pk(h)f~(k) ( f e X ; h e [ —l, 1]; keP);

(iv) (t* /) W = (t,/)(A ) ( f e X ; x , h e C - 1,1]).

(b) One has for each f e X

lim Г { к ) = 0 .

fc->aо

Concerning the proof of part (b), denoting the largest root of Pk by x k, one deduces from Lemma 1 (i), (ii) and Lemma 2 (iii) for k e N that

\Г(к)\ = \ U - T Xkf T( k ) \ ^ \ \ f - T xJ \ \ x .

Since lim x k = 1 by Bruns’ inequality (cf. [41], Theorem 6.21.2), the asser- k-* oo

tion follows from Lemma 2 (ii). Part (b) is a Riemann-Lebesgue type result.

Assertion (iii) shows in particular that xh corresponds to the usual translation of a 27i-periodic function F . Indeed, denoting the finite Fourier transform by , one has (cf. [8], Proposition 4.1.1)

J\_F{- -<p)](/c) = e - ik(p. ? l F l ( k ) (k = 0 , ± 1, ± 2,...; cpeR).

This also gives the motivation for defining a convolution of two functions.

In Fourier analysis the convolution of two periodic functions F, G is given by

(F*G)(0| :=

2

Ztz —я

For functions / , g defined on [ —1, 1] the (Legendre-) convolution is given by (/* # )(* ): = 2 J {xuf){x)g{u)du = } J (zx f)(u)g(u)du,

- 1 - 1

whenever the integral exists.

Lemma 3. i f f e X , g e l } ( —1,1), then the convolution { f *g)(x) exists (a.e.) and belongs to X . Furthermore, one has

(2-7) \\f*g\\x ^ Wf hWg h ,

(2.8) и * д Г { к ) = Г ( к ) д ~ ( к ) (k e P )

If X = Zf(—1, 1), 1 ^ p < oo, the proof of (2.7) follows by l l / * 0 l l * < i } \\(j*f)(-)h\g(u)\du < ll/I U llérlli.

-1

The ca.sc — C 1) 1] can be obtained similarly) and ^2.$) is an easy consequence of Fubini’s theorem.

Equation (2.8) together with Lemma 1 (iv) yields that for all

( / i * / 2)(*) = ( /2 * /i)M (a.e.), (/i * i f г */з)) (•*) = ((/i */2) */з)(-х) (a.e.).

To each f e X let us now associate its Legendre series QO

/ w ~ Х ( 2 1 + 1 ) / '( Ц Р ,М

k = 0

with partial sums

(2.9) (Sn/)(x ) := f ( 2 k + l ) f \ k ) P k{x) ( x e [ - l , l ] ; n e P ).

k — 0

These may also be written as

(2.10) (S„/)(x) = (/./> „)(x ) ( x e [ - l , 1]; neP) , where D„ denotes the Legendre-Dirichlet kernel

(2.11) D„(x) := £ ( 2 k + l ) P t (x) ( x e [ —1 ,1 ]; n e P ), Jt= 0

which may by Christoffel’s summation formula (see [38], p. 179) be written as

( n + 1) P„(x)-P„ + l (x)

Dn(x) = v ' l - x

) ( « + 1)2,

= P'n + i(x)+P'n{x) It is well known that

x e [ l , 1), x = 1, х б [ - 1, 1].

(2-12) lim \\Snf - f \ \ p = 0 { f e U { - 1, 1)) П-* 00

provided p e (f,4 ), and that (2.12) is not valid for p outside this interval nor for the space C [ - l , l ] (cf. [35], [36]). For the Lebesgue-constants lID Ji one has (cf. [33])

lim ||DJIi n~ 112 = i J l / n .

П - * У v

3. Legendre derivative and integral. We start with the definition of a strong (or norm-) derivative.

Definition 1. If for f e X there exists g e X such that lim

A - l -

/ ( • ) - ( ? * / ) ( • )

1 - h - g ( ) = o,

x

358 R. L. S te n s and M. W e h r e n s

then g is called the strong (Legendre-) derivative of / , denoted by Dxf . For any r e N the r-th strong derivative of / is defined with D ° f = f by

Drf = Dx(Dr~xf ) ,

whenever this is meaningful. The set of all / e X for which Drf exists as an element of X is denoted by W^.

Proposition 1. I f f eW^, r e N , then

(3.1) [Dr/] '« c ) = ( * (*2+ 1) Y / W (keP).

P ro o f. If r = 1 one has on account of Lemma 1 (ii), Lemma 2 (iii) and Lemma 1 (i) that

1 ~ P k(h)

1 - h f ( k ) - [ _ D xn ( k )

<

/ ( • ) - ( * » / ) ( • )

\ —h / ( • ) - ( ! , / ) ( )

1 - h

~ ( D lf ) ( •)

( f > 7 ) ( ) (k)

(keP).

Since the right-hand side tends to zero as h->l —, it follows that 1 - P k(h)

*-»i- lim 1 —h

This yields (3.1) for r = 1 since by (2.3) i - P M

f (k) = ( D' f) (k) (kEP).

h -l —lim 1 - h ⁼ PHI) = к (k+l)/2 (kEP).

The result for r ^ 2 follows by induction. □ A simple consequence of this result is

Corollary 1. / e and Drf — 0 (a.e.) for some r EN holds if and only if f = const (a.e.).

P ro o f. Let Drf = 0 (a.e.); then / (к) = 0 for all fcsiV by Proposition 1, which in turn implies / = const (a.e.) by Lemma 1 (iv). The converse follows from the definition of Drf since (zh f ) (x) = / (x) (a.e.) if / = const (a.e.). □ In order to define an inverse operator to Dr one has to look for a function ф г e l} ( — 1, 1) whose Legendre transform is given by

kElS,

(3.2) Фг (к) = k ( k + l ) ) '

1, k = 0.

Lemma 4. (a) The functions

<Ai(*) := 2 lo g —--- 1 ( x e [ - 1, 1)), 1—x

фг := ф1 *фг- 1 (r = 2, 3,

belong to L1( —1, 1) for each r e N , and their Legendre coefficients are given by (3.2).

P ro o f. On account of Lemma 3 it suffices to consider the case r = 1.

First, an easy calculation shows that ф1е 1 } ( —1,1), and that (3.2) holds for к = 0. Now let k e N . Using the differential equation (2.2) and inte­

grating by parts, we find

f ^{2 log Pk (w) du}

_ i

— ² 1 [ — 2 log (1 — m)+ 2 log 2 —1] (-1) d

k ( k + \ ) du [(1 - u 2)P'k(u)-]du 1

k(k+ 1) J (1 + u)Pk(u)du 2

k(k+ 1) ’ where we have also used the fact that Pk( 1) = 1. □

For r e N the (Legendre-) integral P can now be defined as (3.3) (Irf ) ( x ) '•= ( f *фг)(х) ( x e [ - l , l ] ; / e I ) .

Lemma 4. (b) The integral Г is for each r e N a bounded linear op­

erator from X into itself, which satisfies for f e X , r, s e N

(i) [ / 7 ] *(t) = i { к ( к + 1 ) ) f ( k ) ’ k e N ’

| Л 0), k = 0 ;

(ii) ( /r / 7 ) M = ( F I ' f H x ) = ( / ' +7 ) W (a . e.).

(iii) Moreover, for f eW£ one has

(PDrf ) ( x ) = f ( x ) - n 0) (a.e.).

The proof follows by Lemma 3, Proposition 1, and Lemma 1 (iv).

The analogue of (3.3) for r = 1 in classical analysis is (3.4) ^] f ( u ) d u = f ( x ) - f { - 1),

- 1

which holds for each continuous / with integrable derivative. If one inter-

3 6 0 R. L. S te n s and M. W e h r e n s

changes the order of integration and differentiation in (3.4) one obtains an equation which is valid for each continuous / , namely

Î f ( u)du = /(*)•

Therefore the question arises whether the equation ( D 4 rf ) ( x ) = f ( x ) - f f 0) (a.e.)

is also true for e a c h /gX . In order to prove this, we introduce an algebraic counterpart of the Steklov or integral means (cf. [8], p. 33, [13]).

If x(x;h) is the function, defined for h e ( — 1, 1) by

(3.5) x( x; h)

, (l + x ) ( l- h )

log —--- w< log

(1 —x) (1 ^{+ h )} 1 +h 0 ,

— 1 < h ^ x < 1, otherwise, these are given by

(3.6) Ahf = f * x ( - ; h ) ( h e ( - 1,1)).

Lemma 5. (a) For each / i e ( —1,1) the function x ( ’,h) of (3.5) belongs to l ! ( —1, 1), is non-negative for x e [ —1, 1] and satisfies

1 ~ P k(h) (i)

(Ü)

log - 1 1 +h

( к) = к ( к + 1 )

1|х(-;Л )||1 = 1 ( M —i,D )

k G Ï S ,

к = 0;

(b) For each h g( —1,1), the Ah are positive linear operators from X into• itself with operator norm 1 and

lim \\Ahf - f \ \ x = 0 ( f e X ) . h-* 1 —

P ro o f. Part (i) follows by partial integration as in the proof of Lemma 4. Since x(-;h) is non-negative, (ii) is nothing but the case к = 0 in (i). Concerning (b), we have by Lemma 3

\\Ahf \ \ x < 1 И -;Ь )М /1 1 * = ll/llx ( f e X) , and on the other hand

1кл i)(*)iix = imix-

This gives М л||[хх] = 1. Now one has for the Legendre polynomials by (2.8) and Lemma 1 (iii)

1А„РпУ(к) 1 ~P„(h)

n(n+l) log 1 2

1 +h Р Л ) (keP',neN).

This gives by Lemma 1 (iv) (AhPn)(x) = 1 ~ P n(h)

n(n + 1) log 1 ( x e [ - l , 1])

which tends to Pn(x) in 3f-norm for /1—> 1 —, since the term in square brackets tends to 1 by l’Hospital’s rule and (2.3). Having established (b) for / = P„ (the case n = 0 being trivial), the general case now follows by the Banach-Steinhaus theorem since the P„ form a fundamental set in X . □

One of our main theorems now reads:

Theorem 1. The following statements are equivalent for / e l , reJV :

(i) f * W rx = { f c X - , D rf e X } ,

(ii) there exists a function g ^ ^ X such that

/ k ( k + l ) \ j -до = g - до (he N),

(iii) there exists g2e X such that

f ( x ) = (Irg2)(x) + const (a.e.).

The functions g 1, g2 are uniquely determined (a.e.) apart from an additive constant, and one has

(3.7) (Drf ) ( x ) = 0i ( x ) - gî ( 0) = g2( x ) - g A 0 ) (a.e.).

P ro o f. Let f e W x . By Proposition 1 one has f k(k+ 1) V .

y — - 2 J f (k) = I D ' H (k) ( k e P ) ,

i.e. (ii) is valid with gl = Drf . On the other hand, if (ii) holds, then

/'№ ) = ( k{k + i ) ) ' в ' М = t ' W e i W = U ' g S i k ) (fceJV),

and (iii) follows by Lemma 1 (iv). Now let (iii) be satisfied with r = 1.

By calculating the Legendre transform we obtain /)(* ) = У 1 д 2) ( х) - {ч (1 1 g2))(*)

i - h 1- h

2 log

= l(Ahg2) ( x ) - g 2m 1 +h

1 - h (a.e.).

362 R. L. S t e n s and M. W e h r e n s

Since [2 log (2/(1 +/т))]/(1 — h)-> 1 for Л-*1 —, we have on account of Lemma 5 (b)

f - 4 f

1 - h - 9 2 + дЛО) x 2 log

1 +h

l — h - 1 1И* 02II*+11A 02 “ 0211* +

2 log

1- 1 +h

1 - h l02(O)i = 0 (1) ( Л - 1 - ) . This proves (ii) => (i) in case r = 1, and the general case follows by induc­

tion. Finally, the uniqueness of the functions g i f g2, as well as (3.7), can be proved by the uniqueness theorem for the Legendre transform (Lemma 1 (iv)). □

This result is very useful when deciding whether a function / e X possesses a strong derivative or not.

Corollary 2. (a) I f f e W x , r e N , and g e L l (—l , l ) , then f * g e W x and (3.8) 1У^{( /} * gf) (x) = ((1У/)*д)(х) (a.e.).

(b) The Legendre polynomials are arbitrarily often differentiable in the strong sense and

(DrP„)(x) = f — ( x e [ - l , 1]; r e N ; neP) .

(c) The Legendre integral V f of f e X belongs to Wx for each r e N and one has

( D ' V f ) ( x ) = f ( x ) - f f 0) (a.e.).

(d) The integral means Ahf o f f e X belong to Wf for each h e ( —l, 1) and (Dl Ahf ) ( x ) = $ log - 1

1 +h [ /( * ) - ( * * /) ( * ) ] (<*•<?•)•

Furthermore, if f e W f , then

(D1 Ah f ) (x) = ( Ah D ' f ) (x) (a.e.) (e) The operator Dr : Wx - * X is closed, i.e. if

Hm II f t f IIx = Hm \\Drf n- g \ \ x = 0

n -+ a o n - * a o

for a sequence {/„}*= x c= Wx and f , g e X , then f gWx and Drf = g (a.e.).

P ro o f, (a) Since / € Wx , we have / * g e X , (Drf ) * g e X and m f ) * g T ( k ) ' * ( * + l ) Y , . , „ f k ( k + l ) \ r r

~ 2 ---- j f ( k)9 (k) = ( --- 2---- \ U *9\ (k) (keP).

2 2

This means by Theorem 1 that f * g e W x , and comparing the Legendre coefficients of Dr ( / * g) and (Drf ) * g yields (3.8). The proofs of (b), (c), (d), and (e) follow along the same lines (cf. also [ 11]). □

4. The pointwise Legendre derivative and integral. Corollary 2 (b) shows that the Pn are the eigenvectors of the operator Dr corresponding to the eigenvalues [n (/i+ l)/2 ]r. On the other hand we know that the Pn satisfy the differential equation (2.2), which means that they are eigenvectors of the differential operator

d (x2 — 1 ) d

dx 2 dx

x~ — 1 dl d

dx2 dx

corresponding to the eigenvalue n(n+ \)/2. This fact leads us to a pointwise interpretation of the operator D 1.

Let us first recall the familiar class TC|2c(—1, 1) = А С 2Ж. It is the set of all functions f e C ( — 1, 1) for which the (ordinary) derivative / ' is con­

tinuous on ( —1, 1) and the second derivative f " exists for almost all x e ( —1, 1) and is integrable over each compact subinterval of ( —1,1). Let us now introduce (1 ^ p < oo)

Р - И Я - М 1:= { / e C [ - l , l ] ; / ' e C [ - l , l ] ;

/ " s C ( —1,1), Hm (x2- l ) / " ( x ) exist} (4), Р -И ^ р (_ 1Д) := { f e L P { - 1,1); there exists J e A C ^ with f ( x ) = / ( x ) a.e.,

lim (x2- l ) / '( x ) = 0 , ((x2 — \ ) f ”(x)+ 2xf'(x))eLP( — 1, 1)].

Definition 2. For f e P - W set for X = Lp( - l , l ) , 1 p < oo:

(P — Dl f )( x ) : = д (х2 —1 )/" (x ) + x /'(x ) ( / as above) for X = C [ —1, 1]:

j \ { x 2- \ )/" (x ) + x /'(x ), x e ( —1, 1), ( P - D 1f) ( x) : = ) Um_\(u2- l ) f ' ( u ) + f ' ( l ) , x = 1,

i V - l ) / " ( « ) - / '( - ! ) , x = - l .

It is easy to see that P — Dl f exists as an element of X for each / e P — Wxl . Proposition 2. I f f e P - W j , then

l P - D ln \ k ) = f ( k ) (k e P ).

( 4 : lim means lim as well as lim

W-*!- +

6 — Roczniki PTM — Prace M atematyczne XXI

3 6 4 R. L. S te n s and M. W e h r e n s

P ro o f. For X = Lp( - 1 , 1), 1 ^ p < oo, the desired result follows by partial integration, namely

Г.-+0 +

£->0 +

1 - i :

s

- 1 + E

d и2 — 1 ^ d

du 2 du

1 — t:

f

- 1 +£

' d -7 (« )

1 — и2

du 2

1 £

d Г u2- l

f ( u ) Pk (и)du

P'k (м) du

«-►0 + 1 +E du Р'Ли) du

= i f /( » ) = * №-+ 1) /~(fc) (*eP ).

- 1

In the case X = C [ —1, 1] one only has to replace f by / . □

In view of Theorem 1 we have also shown that f e P — Wf implies / e W f , and ( P - D */)(*) = (D1f ) ( x ) (a.e.). In order to prove the converse, we define a pointwise analogue of the integral J 1, given by (3.3). For f e X we call

( p - i if)( x) = ] и ы - r m d u d y

о У — 1 - l

the pointwise (Legendre-) integral of / . A few easy computations show that P — 11 is a linear operator from X into P — W f, and that

(4.1) [(/> -B ‘) ( P - / ‘) / ] ( x ) = / ( * ) - / ‘(0) (a.e.).

Now Proposition 2 yields

(4.2) /~(fc) = [ ( P - D ' ) ( P - I l) f T ( k ) = fe<fc2+1) [ ( P - J ‘)/]~№ ) (keJV).

Here we have used that [ / (0)] (к) = 0 for all fceiV. By combining (4.2) with Lemma 4 (b) (i) we find that

(4.3) (Il f ) (x) = ( P - / 1/ ) (x)+const (a.e.).

The pointwise characterization of W f and D1 now reads

Theorem 2. The following statements are equivalent to those of Theorem 1 for f e X in case r = 1 :

(i') f e P ~ W x\

(hi') there exists g3e X such that

f(x) = ( P - I 1g3)(x)+const (a.e.).

The function дъ is uniquely determined apart from an additive constant, and one has

(4.4) ( P - D ' f U x ) = (D1f ) ( x ) = 03( * ) - 0з(О) (a.e.).

P ro o f. f e P —W} implies / e W} as mentioned after Proposition 2.

Theorem 1 now yields that f (x) = (I1 g2)(x)+const (a.e.), and by (4.3) it follows that f(x) = (P — 71 gf2)(^)+const (a.e.), which is (iii') with g3 = g2.

The remark after the definition of P — 11, namely that (P — 71 g3) e P — W}, gives (iii') => (i'). The uniqueness of g3 and (4.4) follow as in Theorem 1- □ It should be also possible to consider pointwise derivatives of higher order, but the description of the associated function classes P — W f would seem to be very complicated.

5. Best approximation by algebraic polynomials. As usual, let

En( f ; X ) : = inf \ \ f - Pn\\x

be the best approximation of / e l by algebraic polynomials of degree

^ n (.'/„ = span {1, x, x 2, ..., x"}). By Weierstrass’ theorem, one knows that (5.1) lim En( f ; X ) = 0 ( / e l ) .

n-> 00

The purpose of this section is to describe the rate of convergence in (5.1) by properties of / such as smoothness and differentiability. It is convenient to introduce a modulus of continuity of f e X and Lipschitz classes of order a > 0 in the Legendre frame via

œ ï { ô ; f ; X ) = a t ( ô ; f ) : = sup \\xhf ~ f \ \ x 0 e ( - l , l ) ) , LiPi(a; X) := = 0((1 - â f ) , Ô - 1 - } . Some properties of coi are contained in

Lemma 6. (a) For f , g e X there holds

(i) œ ï ( à ; f - , X ) ^ 2 \ \ f \ \ x (<5e(— 1, 1));

(ii) lim a>i(0;f; X) — 0;

s-> i -

(iii) <aï(ô; f + g ; X) ^ w \ { 0‘ / ; X) + <o\{ô\g; X) (<5e(-1, 1));

(iv) a t ( 0 ; f ; X ) = 0 ( ( l - 8 r ) ,

<5 —»> 1 — for some a > 1 implies f = const (a.e.).

(b) I f f e W } , then

toli ( d - f - X ) ^ 6(1-<5) HD1/ U* ( ô e ( - U i ) ) . (5.2)

366 R. L. S t e n s and M. W e h r e n s

P roof. Assertions (i), (ii), (iii) of part (a) are obvious, and (iv) follows from Corollary 1. Concerning (b), it follows by Corollary 2 (d) that

(5.3) | | / -t„ / ||;

2 log

1+h In view of the inequality

( А Д 1/ ) _x < 2 lQg Т Т Г_{1 + h} \ D' f\ \ x ( h e ( - 1, 1)).

(5.4) log

1 +h <: 1 - h (0 ^ h ^ 1)

the assertion follows for non-negative h. For — 1 < h < 0 we first show that for g e X with g (0) = 0 there holds the inequality

(5.5) log

1 +h \Ahg\\x *£311011* ( - 1 < h < 0 ) . If ee(0, 1) is arbitrary, — 1 < h < 0, the integral

F / , (1+ИН1-Л ) , W 4 J

/(£ ):= J lo g —---- — - - - (тxg) (и) du h ( l - u ) ( i + h )

exists for all x e [ —1, 1] and integrating by parts yields

(2 — f)(\ —h) 1 1_£ 2 1

I (e) = log —— - J (Tx g)(y)dy+ J ---T j ( x xg)(y)dydu +

E ( l + n ) i - £ о l — U u

+ j

. 2

h 1 — и

The first term tends to zero in X-norm if £-+0+ since (2 - £ ) ( ! - /») }

E ( l + h ) j _ £

\ Ii ( e)\ \ x < log j \\(-tyg)(-)\\xdy

e(1 +h) and similarly we obtain for I2 and / 3

l-e 2 1

^ £ log ^ ^ \\g\\x = o(l) (£->0 + ) ,

Ш \ \ х ^ J -j---r j M\xdu = 21og(2 -fi) ||0 ||*, о 1 M и

(5.6) 1 ^ з ( £ ) Их = | | J - j ---Г J {'Cyg){-)dydu \\x^ 2 \ o g ( l - h ) \ \ g \ \ x

h 1 — u - 1

^ 2 log 2 \\g\\x ,

where in (5.6) the fact was used that J (ryg)(x)dy = 0 , valid because - 1

g (0) = 0. An application of Fatou’s lemma gives

||lim inf|/(e)|||* ^ lim sup \\I(e)\\x ^ 4 1 o g 2 ||/||x ^ 3 ||/ ||* .

£^►0+ £-“►<) + Now (5.5) follows by

lim inf I/ (e)| = log \(Ah g) (x)| (a.e.).

e-*0+ L+n

From (5.5) ^{and the} equation (cf. Corollary 2 ^(d))

Ik*/ - Я * = 21° g - j ^ - \\AkDlf \ \ x

inequality (5.2) follows for — 1 < <5 < 0 since [D1/ ] ^ ) = 0 by Propo­

sition 1. □

Since we wish to c’pply a general theorem on best approximation in normed linear spaces due to Butzer-Scherer [9], [101 (cf. Г71), we need the К -functional of f e X

K ( t , f ; X , = K ( t , f ) : = inf { \ \ f - g \ \ x + t \\D'g\\x ) (t > 0).

g e W x

In the case r = 1, the К -functional K ( l — S ; f ; X , W x ) and the modulus of continuity a>i(ô; f ; X) define seminorms on X for fixed <5e(—1,1); both are equivalent to another as shown in

Proposition 3. For all f e X , <5e(— 1, 1) there holds

(5.7) 1 - 5 , / ; X , Wi) ^ œï(ô-, / ; X) ^ 6 K ( l - 3 , / ; X , W}).

P ro o f. By Corollary 1 (d) one deduces Aôf e Wx for all <5e( — 1,1) and K ( l - Ô , f ) ^ \\ f — Aô f \\x + (1 — ô) WD1 Aôf \ \ x

= 111 I [ / ( - ) — (т„/ ) ( - ) ] х(м; S)du\\x + Ô

+ (1 - 0)i l o g ' I I / (• )-(* * / ) ( • )ll*

The inequality

(5.8) 1 —ô ^ 2 log -г——- ( - 1 < S < 1 ) 1 + d

now yields the left-hand side of (5.7).

3 6 8 R. L. S t e n s and M. W e h r e n s

To prove the right-hand side, we choose an arbitrary g e W x . By Lemma 6 it follows that

coLA Ô ; f ) < œL1( ô - , f - g ) + coLl (ô;g) ^ 2 \ \ f —g\\x + 6(l — 3)\\Dl g\\x . Taking the infimum over all g e W x completes the proof. □

A first application of this result is

Corollary 3. For all f e X , 3t , 32e( — 1,1) one has f tW i ; / ; X) ^ 12 max < 1,

^ 12 1 +

1 32 1 - 5 1

^ ( 3 2; / ; Х )

1 - 3 , <*iià2\ f \ X ) .

To apply the general theorem mentioned, it suffices to show the validity of associated Jackson- and Bernstein-type inequalities. These are contained in

Proposition 4. For each r e N there holds

(5.9) En( f ‘, X ) ^ M n ~ 2r ||Drf \ \ x (/eW£; nel V) , (5.10) \\D'pn\\x ^ M n 2r\\pJx (p„ePn; neP) . (5.10) \\DrPn\\x ^ M n 2r\\Pn\\x (рпе ^ п;п(=Р).

P ro o f. To f e W x we construct a sequence of polynomials p „ ( f ) e ^ n, n e N , such that

(5.11) \\f-Pn(f)\\x ^ M n ~ 2r \\Drf \ \ x {neN), where the constant M > 0 does not depend on / and n.

Indeed, consider the polynomials (see also [3], [5])

(5.12) (Knf ) (x ) = pn( f ; x ) := i J (ruf)(x)r„(u)du (neN ), - l

>•.(*):= (i } № 0, (и)]2Ли)-‘ [P » '0|(*)]2, - 1

N being the largest integer < (n/2). From [41], (7.32.5), (4.1.3), we obtain n~ 1( 1 —x)_5/2, x e (0 , 1),

(5.13) [Pj?’0)(x)]2 ^ M

1, x 6 [ — 1, 0] ,

and by [23], (16.4.20), we find that

(5.14) 1 \ r n (2 0w м 2 , N 2 + 3N + 3 N 2 i j [P?-°’(«)]2^ = --- J--- » у

Now let f e Wx . Since r„(x) ^ 0 and J rn(u)du = 2, we have - 1

II/ (-) —i f ( ïu f) ( )r„(u)du\\x

O l - N 2 1

say. By (5.13), (5.14) and (5.2) one can estimate I y and I 2 as follows

о о

ly ^ (M/N2) f ( o\ (u ; f) d u ^ ( M/N2) \\Dlf \ \ x J (1 - u ) d u ^ M n ~ 2 ||D 7 |Iy,

- l - l

1 — N~'2

I 2 ^ ( M/ N3) J co^iu; f ) ( \ - u y 5l2du

о

1 - N ~ 2

. ^ (M /N3)||D 1/llx J (\ — u)~3/2 du ^ M n ~ 2 \\Dlf \ \ x .

о

For 13 one obtains easily by (5.2)

} w‘i ( u ; f ) r n(u)du H o A ( l - N - 2; f ) ^ M n - 2 \\D'f\\x .

l - N 2

This proves (5.11) and, of course, also (5.9) for r = 1. In order to extend this result to arbitrary r e N one may use the polynomials

Pn( f', x) := X ( - 1 y + l Ç)((r„*rn* . . . * r n) * f ) ( x )

j= i --- ---

y-fold

and the same ideas as in [13] or [3]f [7] to show that these polynomials satisfy (5.11).

For the proof of (5.10) we refer to [31], [32]. □

Inequality (5.9) can be used to deduce a Jackson-type theorem for the best approximation by algebraic polynomials.

Corollary 4. I f f e Wx for some r e P ( W f = A"), then (5.15) Ел( / ; Х ) < М п - 2' т \ а - п - г ; 1 У / ; Х ) (n e N ), the constant M being independent of f and n.

P ro o f. For the operator K n defined by (5.12) we have shown that Il0-K„0ll* M n ~ 2 ||D' g ||j ( g e W ' ; n e N ) .

This yields

W f - K A O - n i x < \ W f - g ) - K „ ( I X f - g ) \ \ x + \ \ g - K ng\\x

< 2 \ \ V f - g \ \ x + M n - 2 \\Dl g\\x ( f e W ^ g e W l ' n e N ) .

3 7 0 R. L. S te n s and M. W e h ren s

By taking the infimum over all g e W x and using the left-hand side of (5.7) one obtains

(5.16) \\Dr f —К n (Drf)\\x

^ MK (n~2, Drf ) ^ Mcoli ( \ - n ~ 2;Drf ) ( f e W ' ; n e N ) , which is (5.15) in case r — 0. If r ^ 1, let Г K nDrf be the r-th integral of /C„D7 defined by (3.3). Since TK„Drf e.J/ n and [K„Dr/ ] (0) = 0 in view of the convolution theorem, one obtains by Corollary 2 (c), (5.9), and (5.16)

En( f ; X ) = E„( / — Г K nDrf ; X) ^ M n ~ 2r \ \ D J - K nDrf \ \ x

^ M n ~ 2rwli ( l - n - 2; Drf ) ( / e W f ; n e N ) . □

Now we pass to the fundamental theorem on best approximation by algebraic polynomials.

Theorem 3. Let p* = p*{f)€.J/ n be a polynomial of best approximation to J ' e X . The following assertions are eguivalent for ri , r 2e P, rl e N , 0 < a ^ 1, r2 < r-hoc < rl :

(i) (ii)

(iii) f e W p (iv)

Moreover, if 0

En( f ; X ) = 0 ( n ~ 2r- 2*) (л->оо),

\\Dr' p*\\x = 0 ( n ~ 2r~2x + 2ri) (n->oo), and \\Dr\ f - D r^p*\\x = 0 ( п ~ 2г- 2л + 2г>) K { f \ f \ X , Wp) = 0 ( t 2r+2*) (t—0 + )

< a < 1, these assertion are equivalent to

(n^-+ oo),

(v) f e W f and K (t, Drf ; X , Wf) = O (t2x) (f-> 0 + ), (vi) f e Wf and o)li( ô; Drf ; X ) = 0 ( ( l - ô f ) (<5->l-), i.e. Drf eLipi (a; X).

P roof. The equivalence of assertions (i) to (iv) is a consequence of the Butzer-Scherer Theorem. Now let 0 < a < 1 and assume (i) to be valid.

Choosing r2 = r, one finds in view of the equivalence of (i) and (iii) that E„(D'f -,X) ^ \\Drf —Drp*\\x = 0 ( n ~ 2*) (и -о о ),

and using (i)=>(iv) with Drf replacing / , the case r = 0, rx = 1 yields K (t, Drf ; X , Wx ) = 0 ( t 2*) (r->0+),

which is (v). The implication (v)=>(vi) follows immediately by Proposition 3, and finally Corollary 4 yields (vi)=>(ij.

6. Pointwise moduli of continuity. The aim of this section is to express the Lipschitz classes L ipi(a;C ) in terms of moduli of continuity defined via the usual differences

(Л1 / ) ( x ) : = f ( x + h ) - f (x), (Ahf)(x): = f (x + h ) + f ( x - h ) - 2 f (x).

Definition 3. For / e C [ - l , l ] , x g[ - 1 ,1 ] and rje(0,1), we call (6.1) Oi(fj;x; f \ C ) = Qi(rj;x; f ) : = sup {|(4/H *)I; h e l x<„ti} (/ = 1,2) the pointwise or local modulus of continuity of order i at the point x, where

Ix,n,i : = ? ] n [ - l - x , 1-^c],

Ix , n,2 ■= [ - 1/, I/] П [ - 1- X , 1- X ] n [ - 1+X, 1+ x ] .

The definition of 1Х^ Л guarantees that x + h is contained in [ —1,1] for all x g[ — 1, 1] and h e IX IIA, and similarly that x ± h is contained in [ —1,1]

for all x g[ —1, 1] and h e l xtl>2.

The difference between and the usual moduli of continuity (6.2) a>i (tj; f ; C ) = ^ ( 17; / ) : = sup {|(d[/)(x)| ; h e l x^ yi, x e [ - 1, 1]}

is that in (6.1) the supremum is taken for fixed x with respect to h, whereas in (6.2) the supremum is taken with respect to x and h. But there holds

(6.3) (0,(17; / ; C) = sup {Qi(ti;x- / ; C );x g[ - 1 , 1]} ( / = 1 , 2 ) .

We now define Lipschitz classes for i = 1,2 , a > 0 via the moduli of continuity Qi

(6.4) F -L ip , (a; C) : =

/ gC [ — 1, 1J; Qi(rj; x; f ; С) = О

l - x : I? ->0 + , XG( — 1, 1)

“P ” stands for “pointwise”, which should indicate that the behaviour of Qi(rj; x, f ;C) depends on the point x. The О-condition in (6.4) is to be understood as

0 , (17; x; f ; C ) ^ (0 < 17 < 1; x g( - 1 , 1 )),

where the constant M may depend on / but not on x and 17.

In [40] the function Л (x; rj): = min (i72/(l —x2), t]} was used in (6.4) instead of rj2/(l — x 2). We would like to show now that definition (6.4) remains unchanged if rj2/( 1—x2) is replaced by Л (x; ^7).

372 R. L. S te n s and M. W e h r e n s

Lemma 7. For |x| ^ 1, \h\ ^ \ and \x + h\ ^ 1 one has \h\ ^ 1— x 2 or

\h\ ^ 1 — (x + h)2.

P roof. Assume \h\ > 1— x 2 and show that \h\ ^ 1— (x+h)2. From this assumption it follows that \h\+x2 > 1, which means that the signs of x and h must be different since otherwise one would have

\x + h\ = |x|+ |/i| ^ x 2 + \h\ > 1.

But this is a contradiction to the hypothesis \x + h\ < 1. Therefore, \x+h\

= ||x| — |/i||, which yields the assertion since 1— (x + h)2 + 1 — \x+h\ '= 1 — |x| 4" \h\ > \h\

l -\ h \ + \ x \ ^ 2 ^ \h\

for \h\ ^ |x|, for |x| ^ \h\. □ Proposition 5. (a). I f for / e C [ —1 ,1 ], i = 1 or i = 2 ,0 < r] ^ \ and a > 0 there holds the inequality

(6.5) P i / )m i « w l - f - then

( x e [ —1, 1]; h e l x<ri<i),

(6.6) |( d '/) ( x ) | < M |Л|“ ( x e [ - l , 1]; h e l^{x^ ) .} (b) For each a > 0 one has f e P — Lipf(a; C) if and only if (6.7) Qi(rj;x; f ; C ) = 0([A(x;rj)Y) ( x e [ - 1 , 1]; rç->0+).

P ro o f, (a) If \h\ ^ 1—x2, assertion (6.6) follows from (6.5). For

\h\ > 1—x2 we consider first the case i = 1. Setting x' = x + h, h' = —h, one obtains from (6.5) since h'Glx <rlA that

(6.8) \ f ( x + h) -f( x) \

= \ f ( x ' ) - f ( x ' - h ' ) \ ^ M (h'f

l - ( x ' )

= M h2

1 — (x + h)2 Now \h\ > 1 —x2 implies \h\ ^ 1 — {x+h)2 by Lemma 7, and so (6.6) follows from (6.8).

In the case i = 2 we use the inequalities

(6.9) h2/ ( l —x 2) ^ h2/(l — (x + h)2), if xh ^ 0, (6.10) h2/( 1—x2) ^ h2/(l — (x — h)2), if xh < 0 . If (6.9) holds, one obtains (6.6) by Lemma 7 since

1 Л х + ( , ) - Л 1 - 1 . ) + 2 / м к м | - Т г | g M ( 1 - ( x + h f ) = М |Л |“ - If (6.10) is valid one uses Lemma 7 with h replaced by ( — h).

The proof of part (b) is now an easy application of (a). □

If one defines the Lipschitz classes with respect to а>{ for / = 1 , 2 , a > 0 by

LiPi (a ; C) = { / e C [ - 1 ,1 ], = 0 ( f ) , n^ 0 + } , one can easily derive from Proposition 5

Corollary 5. For each a > 0 and i = 1,2 there holds Lipf(2a; C) cz P — Lip, (a; С) c Lipf(a; C).

A first connection between P — Lipf and Lip^ is contained in Proposition 6. The following relations are valid:

(i) K (< 5 ;/;C ) ^ 3 f , C) ( /gC [ - 1,1]; - 1 < <5 < 1);

(ii) cof(<5; / ; C) ^ œ2( y / \ - ô \ / ; C) + m1( l - ^ ; / ; C)

( /gC [ - 1 , 1 ] ; - 1 < <5< 1);

(iii) P — Lipi(a; C) c= Lipf(a; C) (a > 0);

(iv) P — Lip2(a; C) c= Lipj(a; C) (0 < a < 1);

(v) / gP - Lip2 (1 ; C)=>®t (Ô; f ; C ) = <0^1-0; f ; C ) + 0 ( l - S ) (5-> 1 - ) • P roof. First observe that the translation zhf of (2.5) may be written as

( ч Л ( х ) = — S { / ( x h + U y J l - x 2 ^ l - h 2)+

л о

+ / (xh — и y / l — x 2 y / l — h2)} y / l — и2 du (x, h e [ — 1, 1]), and therefore

(6.11) \(zhf ) ( x ) - f ( x ) \

4 sup \\ { f ( x h + u ^ / T ^ h 2) + f (xh — и y / î ^ x 2 ^ f U - h 2)}—f (x)\

\ u \ < l

^ H SUD \ f ( x ' + h ' ) + f ( x ' - h ' ) - 2 f ( x ' ) \ } + \ f ( x ' ) - f ( x ) \ ,

|u|$ 1

where x' = xh and h' = и у/ l —x 2 ^ J l —h2. Since \h'\ 4 у / 2(1 — h) and

|x' — x| ^ 1 — h, (ii) follows from (6.11) and the inequality o>2 (Xrj; f ) 4 A2co2(g; / ) , A ^ 1. This in turn yields (i) since co2(rj; f ) 4 2coi(rj; / ) .

If one now assumes that / gP —Lip,(a; C) for /g{1 ,2} and some a > 0, then (6.11) may further be estimated by

(6.12) \(zh f ) ( x ) - f (x)\ 4 sup l«|o

u2( l - x 2) ( l - h 2) *'

1 — (xh)2 F Q ^ l - h ; x; f ) 4 M ^ - h f + Q ^ l - h ^ x ; f ) (x, /i g[ - 1 , 1 ] ) .

This immediately gives (iii).