An estimate for the Lebesgue (0, 1 (-functions of some polynomial-like systems

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X I (1967)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1967)

H. Ratajski (Poznań)

An estimate for the Lebesgue (0, 1 (-functions of some polynom ial-like systems

1. Introduction. Let {<-pn{x)} be an orthonormal system with respect to a positive Lebesgue-integrable weight-function ^q(x) in < —1 , 1>, i.e.

I (0 when n Ф Tc,

\ (pn{cc)(pk{x)Q{x)dx = \

_ ■i [ 1 when n = к .

The system {cpn(x)} is called polynomial-like in <—1, 1) if the kernel

K v{t, x) = <Pj{t)cpj(x)

7 —0 is of the form

Г p

(!) K ( t , x) = ^ F k(t,x ) £ 7i%(Pv+i{t)(pv+i{x),

i l,7 = - p

where p and r are positive integers independent of v, the coefficients y[jl are uniformly bounded for all i, j, Tc and all v, and F k(t,x) are measu­

rable functions of two variables satisfying the condition

(2) F k(t, x) = 0 -

\t—x\

uniformly in t, x e ( —l , 1>, t Ф x\ if г+г < 0, we set 9ov+i(x) = 0. The definition of the polynomial-like orthonormal system was given by G. Alexits (see [1], p. 158).

In this note we deduce a certain estimate for the Lebesgue (O,1)- functions

L4?(x) = ¹

n-\-1 П у р С А , ®)

r = 0

Q(t)dt

of a given polynomial-like system. This estimate is important for (0,1)- summability problems of Fourier series with respect to the orthonormal system {(pn{x)}. In our considerations we restrict ourselves to the interval

1 4 2 H. R a t a j s k i

<—1,1>; the case of an interval <u, 6) can easily be reduced to the pre­

vious by the substitution и = —1+2(x —a)/{b—a).

2. We shall present an extension of Theorem 3.4.3 of [1], p. 186.

Theorem. Let {(pn(x)} be a polynomial-like orthonormal system with weight function q{x) in the interval <— 1, 1)>. Suppose that there exist con­

stants Сг > 0, 0 + C2 < 1, C3 > 0, C4 > 0 such that for all x e ( —l , 1) П

(3) q(x) + x Ca , ^ < p l ( x ) < C s?in (П = 0 , 1 , 2 , . . . ) ,

where l n > 0, K+il^n < 04 and Xk -> oo us ft -> oo.

Then

П

(4) < ОД^°2)/4^(С2-3)/4( ^ + ) (1+С2)/4 v = 0

for every x e ( —l, 1) and n + 1, where C is a positive constant depending neither on x nor on n.

Proof. Let en be positive numbers tending to zero (en < |).

A specific definition of en will be given later. The characteristic functions

П П

of sets of points {t, x) for which £ A„(t, x) + 0 or £ K v(t, x) < 0 will

v=0 v = 0

be denoted by P n(t, x) and N n{t, x), respectively.

Then

n 1

1 $ (oo) = -——-

V f

v = 0 -1 1

1 n 1

■ . f -^n{f x)K v(tj x) g{t)dt = A n-\-Bn.

n-\-1 J

v = 0 -1

1° We shall consider in detail the case 0 < ж + 1 — 2en. Write i

f P n(t, x ) K v(t, x)g(t)dt - i

x ~ en 1 X+ En

= ( / + / ) p n(t, oo)Kr(t, x)g{t)dt+ j P n{t, x )K v(t, x)g(t)dt = I vl-\-Iv2

— 1 x + en X~ En

and set

\P n( t,x )F k{t,x) for te<— 1, x —eny w <a?+en, 1>, gk(t, x) =

I 0 for te ( x —en, x-\-en).

Estimate for the Lebesgue (0, 1)-functions 143

By Schwarz inequality, (I) and (3), we get

n r p n n 1

Ż !y[rS]V+*M у [/?*«, *)?>,+<«)e(o««rr

j>=0 k = \ i , j = —p v =0 v=0 —1

r p n 1

= 0 (1)w 2 Ś {2d / » ( * ’ “O^W eW ***]2}1'2.

k = l i = —P v =0 —1

The condition \t—x\ ^ en implies |</&(/, x)\ ^ P n(t, x)\Fk(t, x)\ = 0 ( l / e n).

Therefore, if n is fixed, the function gk(t, x) is bounded. Consequently, the integrals under the sign of summation are Fourier coefficients of the square-integrable function gk(t, x). Applying Bessel inequality together with (2) and (3), we obtain

n 1 1

J

9C- 6 /yj 1

- » , „ ( / + / ) dt Further,

/

X + e ,

Analogously,

Hence,

1— e„ 1 0(11

(/ + J)

— 1 XĄ-er

dt

( t - x ) 2( l —t2f 2

X+En ^~Bn (it - x ) 2( l - t f 2 О (1) £-(i+o2)

/ « Г dt

(t—x)2{ l - t 2r 2 = 0( l ) ^ 1+a‘K

A I i,A = 0 (1) «2%<1+CW2.

Similarly, the Schwarz inequality, and (3) yield

Х -\ -ег

l l 2 < Pn{t, x) g(t)dt J" Kl(t, x) g(t)dt

X— X— £fi

X + e n 1 X Ą -B n

<

J

j

0(1) J c2

x+en ^

= 0(1)1, Г ^ - = 0(1)A,4“°2.

J £„2

^ <Pk{x) k= 0

1 4 4 H . R a t a j s k i

Hence,

П

v = 0

Now let

П

v = О

n

0(1)пЧ*е«-с^ ( 2 к ) т -

v=0

-4~Г

v=0 '

for n large enough. Then

Af ^< 1 +°2),2 = о (1) и 1'2 ^eL 1-c^>'2 (^ Л„)1,2.

v = 0

Combining the results Ave get inequality (4) for A n. The same estimate also holds for B n.

i

2° If 1 — 2sn < x < 1, we write j P nK vgdt as the sum of the in- - i

tegrals

1—Зе^ 1

ivl = f Pn(t, x)K,{t, x)Q(t)dt, Iv2

J Pn(t, x)Kv(t, x)q{t)dt.

—1 1—Zsn

Eeasoning as in 1°, it can easily be observed that П

y i t i l =

v = 0

and П П

y ; i i « i = о ( 1 ) «1'г41- с^ ( у л ) 1,а.

v = 0 v=0

Thus, estimate (4) is established.

In case of — 1 < x < 0 the proof runs analogously. We may notice that conditions (3) are satisfied by the normalized Jacobi polynomials (a > —1, > —1) and q(x) — (1— х)а{1-\-х)^ with Xn — n 2 a +2, where a = max(a, /9) ^ (see [2], p. 418).

R eferences

[1] Gr. A le x it s , Konvergenzprobleme der Orthogonalreihen, Berlin 1960.

[2] И. П. Н а т а н с о н , Конструктивная теория функций, Москва-Ленинград

1 9 4 9 .