The representation of the trilinear kernel in general orthogonal polynomials and some applications

JOURNAL

OF A~PRO~MAT~O~

THEORY

67, 93-114 (1991)

The Representation of the ~rj~i~~~r Kernel in General ~r~~~~~~~~ Pcllynomials and Some

Let {PM> (n E Z +) denote the sequence of orthodoxy ~oly~orn~~ls with respect to tbe weight I.+)( - 1 G I < 1). The representation of the kernel

is given. We use this result to construct a “‘double-humpbacked majorant” of the kernel 9*(x, y, z) to estimate the Lebesgue qua~~u~ctio~, and to compute the infinite sums

which appear in some problems of mathematical reprcscntations. 0 1991 Academic Press, Inc.

physics and in the theory of group

Let w(x) be nonnegative in [ - 1, 1] and positive almost everywhere on t-1,1], and suppose

We call MI(X) a weight function (weight). Associated with w(x) is the sequence of orthonormal polynomials ONSP( p, 3 p,( w; x) > (n E 21+ = (0, 1,2, ... >), where p,,(n) = k,x” + ” 93 0021-9045191 $3.

94

BORIS P. OSILENKER

has degree n with k, = k,(w) z=-0 and

4x1 dx= :, sl P,(x)P,(x) -1

for m=n for mfn.

The orthonormal polynomials satisfy the three-term recurrence relation [12, p. 17; 48, pp. 55-561 xP,(x)=a,p,+,(x)+u,p,(~)+a,-,p,-,(x)(p-~(x)-O,n~~+),

(1.1)

where, for n E Z + , the recurrence coefficients a, = a,(w) and U, = u,(w) satisfy k a =” n kn+l

and

u,=s-11xp;(x)

w(x) dx.

Note that

a,< 1,

k/61

(nEz+)

(1.2)

and, moreover (Rahmanov’s theorem [46,28,47]), lim a,= 4,

n-m

lim U, =O.

n-m

(l-3)

A system of orthogonal polynomials for which the recurrence coefficients satisfy (1.3) belongs to the class ME M(1, 0), introduced by Nevai [31]. Let us define

(n~~+L k=O

(1.4)

k=O

where a,, U, (n E Z + ) are the coefficients of the recursion formula (1.1). The estimate ,Ir, = o(n)

(n-4

(1.5)

holds by virtue of (1.3). By Favard’s theorem [12, p. 601 the recursion formula (1.1) completely determines the orthonormal sequence (pn} (n EZ+); therefore, many investigations are devoted to ONSP (p,} (nEZ+), defined by the recursion formula. This is of interest in scattering theory, in chain sequences,or in spectral theory of Jacobi matrices [ 1, 6, 8, 15,24, 341.

REPRESENTATION

OF THE ~IL~EAR

95

KERNEL

With appropriate conditions of the recurrence coefficients one can obtain properties of the weight NJ,asymptotics for (p,} and the zeros of ~Jx), weighted estimation, and so on [6, 8, 9, 15, 15, 19, 24, 31-34, 49, XI]. We also remark that the polynomials are generalized e~genfu~ctio~sin k2 for the Sturm-Liouville singular difference operator (see 16, Chap. V Sect. I])

where w= (~,),Aw,=w,+~-w,,Vw,,=w,-w,_, a = (a,},

qn = a,- 1 + a, + u,

(nE~+h

A fundamental role for the treatment of expansions of functions in orthogonal polynomials is played by the ~hr~sto~el-~arboux s~mrnati~~ formula

We consider the trilinear kernel

fQEZ+; -1dx,y,z
(1.6)

k=O

w~~~~possessesthe reproduction property: for every polynomial %zfx)= i: CkPk(X)

(nEzi;XE[-ll,

I])

k=O

the relation

Is dz) Y, zlw(z) dz -1 %n(x, (nEZ+;x,yEC-L 11) CkPk(X) Pk(Y)

holds. The following problem often arises in mathematical physics and in t

96

BORISP.OSILENKER

theory of group representations [lo, 45, 511: determine the sum of the series i

4%Pk(X) PkG(Y) i?%(z)= liT(x, Y, -G21,

(1.7)

i.:L,=-$

(1.8)

k=O

where a=

i

a,ER1,n,#oo,nEz+

I

is any given sequence. For example, in [ 10,451 for the orthogonal ultraspherical polynomials (C;(X)) (s >O) it is shown that

= 2-“%[F(x)] x sin

i

p4 [sin Msin /3sin ~1’ --2x

C(+PfYsinB+Y-ccsinCI-P+Ysina+P-Y r-1 2

2

2

2

I

’

where O< ~1,p, y < 71,and a triangle can be drawn with sides cl, 6, y, assuming that the sum of any two of the sides is less than or equal to rc; otherwise the infinite sum is 0. The main purpose of this paper is to state a representation of the kernel $BJx, y, z). We apply this result to the computation of the infinite trilinear sums (1.7), (1.8) and to the construction of a “donble-humpbacked majorant” of the trilinear kernel Bn(x, y, z). Some of our results were discussed in [35-371. Other appii~tions of the representation of the trilinear kernel sB~and of the estimation of a “doublehumbacked majorant” will be given in forthcoming articles. 2. A FORMUG FOR IVHEKERNEL: COMPUTATION OF THE TRILINEAR SUM Let us define the function 7’,(x) by the formulae T,(x) = cos(n arccos x) (12= 1,2, ...).

T*(x)=1

(-l<“X
Then T,(x) is a polynomial of degree n with a positive leading coefficient. The system

RE~~SENTATI~N

OF THE TRI~~~EAR

KERNEL

97

is the system of orthonormal polynomials with weight wocxk-$+

(-1
1)

(~hebyshev po~ynom~aIsof the first kind). We have xp;O’(x) = sp;“~pz,(x)+ $p;“A~(x) and, consequently, the system (pfp)(~>) (n GZ,) ~~-~~ass.gutting

@>?I belongs to Nevai’s

x = cos 01,y = cos p, r+ = cos(cl- p), <- = cos(a -i-/3), we obtain

and by the ~hristo~el-Darboux su~ation facula the sums on the righthand side becomes a sum of two fractions with denominators z - [- an z - i, . So, if the recurrence relation (1.1) belongs to Nevai’s M-class, one expects that the trilinear kernel has two peaks near z = i ._ and z = [ + . Note that

so

and

wherex=cosa, y=cosfl, z=cosy (O
98

BORIS P. OSILENKER

LEMMA 2.1. For a general ONSP {p,} tation is valid:

(Z-i-xi+

(n E Z +) the following

represen-

-Z)%(X,Y,Z)

=A,(x,y,z)+B,(x,y,z)+E,(x,y,z)(nE~+;

-l
(2.4)

where

+ P?z(X)Pn+*(Y) P,(Z) + P,(X) P,(Y) Pn+Z(Z)l;

(2.5)

n-2 B,(x,

Y, z) =

C k=O

C(ak

-

+)@a:

- 2ak

-

1)

+(ak-l-~)(2a:-,-2ak-l-1)1pk(x)pk(~)pk(z) n-2 +2

akak+l c k=O

cak+l-

~)[Pk+2(X)Pk(Y)Pk+2(Z)

+Pk+2(X)Pk+2(Y)Pk(Z)+Pk(X)Pk+2(~)Pk+2(Z)1 n-l +2

1 k=O

akak+l(ak-

i)bk+2(X)Pk(Y)Pk(Z)

(2.6)

+Pk(X)Pk+2(y)Pk(Z)+Pk(X)Pk(Y)Pk+2(Z)1;

and J%(x,Y,

Z)=

i ek(&.hz) k=O

(nE~+;x,y,zEC-L

with

+ [akPk+1(X)+ak--pk--1(X)lPk(y) x [akPk+l(Z)+UkPk(Z)+ak-lPk~l(Z)l -;"kPk(x)Pk(.dPk(z)

11)

(2.7)

iz-il-xc+ -z)Pk(x)Pk(Y)Pk(z) = (1-f 2xyz - x2 - y2 - z2)pk(x)pk(y)~k(z) =Pk(X)Pk(Y)Pk(Z)+2ui:Pk+l(X)Pk+I(y)Bk+l(Z)

t2a,-,a:p,,,(x)Pk--1(Y)Pk+l(Zf

-t2ak-,a2,Pk+l(~)Pk+l(Y)Pk-l(Zb 4”‘24-~

akPk,l(X)Pk-l(u)ak-r(Zf

+2ak-,a~~k-,(x)~k+,(y)Pk~I(Z)

$2aka:-lPk-l(x)Pk-lIjY)Pk+l(Z) C2a:-,akPk-l(x)Pk+l(Y)Pk-1(Z) +2a:-,pk-L(X)Pk-1(Y)Pk-1(Z)+ekdx,y,z) -akak+lPk+2(X)pk(y)Pk(Z)-‘(a~_1

~u~)~k~~~~k~~)~k~z~

~ak-.2ak-1~k-ZfX~~k(~~~k(Z~-~k~k+1~k~~~~k-i-2f~~klkf~~ -(a~-I+a:)pk~x)pk(y)pk(z)-rak-2ak-ipk(x)Pk--2(YbPk(Z) -“k

ak

+ 1 PktX)

-ak-2ak~lPk(X)Pk(YfPk-2(z),

Pk(Y>

Pk

+ dZ)

-

(ai

_ 1 +

a:>

!?k@)

PI;(y)

Pk@)

100

BORIS P. OSILENKER

where e,(x, y, z) is defined by (2.8). We regroup similar terms of the relation

For the proof of formulae (2.4)-(2.8) we consider the following “basic” terms (the others are treated in a similar manner):

xl=kk,(2a~Pk+l(X)Pk+l(V)Pk+~(z) + Cl -3GL

+ ~~mkmdYh(4

+2a~-,p,-,(x)p,-,(y)p,-,(z)},

Using peI(x)=O have

and 2ai-3u:+i=(ak-

$)(2az-2ak-1)

(k~z+),

we

Next, note that P-~(X) = p-r(x) = 8; then n-l c

IL=2

aka~+1Pk+rZ(X)Pk(Y)Pk-t-2(2)

k=O n-2 akak+1Pki-2(X)Pk(Y)Pk+-2(2) -k?,

=2a,-,a~p,+,(x)p,-,(y)p,+,(z) n-2 +2

c k=O

akQk+

1 (a kel

-

~)Pk+2(X)Pk(YbPk+2(Z).

In a similar way

Formulae (2.4)-(2.8) are a consequence of the last three formulae. Remurks. (1) The above result holds in particular for D.P. in Nevai’s class. (2) For an even weight w(x) on I-1, I] formulae (2.4)-(2.g) were a~~on~ced in [35]. (3) A simiIar result can be obtained for the tri~~nearform

where (qn) (n E Z ,) is the system of the functions or the polynomials of the second kind [45,51]. (4) Many papers are devoted to the investigations and the apphcations of the kernel 9Jx, y, z) for the classical polynomials and its generalizations (cf. [4, 7, 10, 11, 13, 14, 17, 18, 20-23, 25-27, 38-44, 51]). (5) The following curious result can be inferred from Lemma 2.2.

102

BORIS P. OSILENKER

COROLLARY 2.2. For an ONSP (p,} (n E Z,) with respect to an even weight function w(x) on [ - 1, l]? we have the representation for all xE[-1, l] andnEZ+:

2(x - l)* (x + +, i

p;(x)

k=O

n-2 +6

f)P/c(x)PZ+z(x)

tak+l-

akak+l c k=O n-2

+

c k=O

[(ak-

+)(2a:-2ak-1)

n-l +6

c k=O

akak+l(ak-~)P~(X)Pk+2(X)~

In fact, since w(x) is even u, = 0 (n E Z + ) and for x = y = z (-ldX
(z-1:-)(~+-z)=(x-l)*(2X+l)

Throughout this paper we consider ONSP (pn) (n E Z,) with respect to the weight w(x) on [ - 1, 11, for which the following hypothesis is valid: there exists a positive LL-integrable function q(x) such that (nEZ+; -1
IPn(X)l G v(x)

(2.9)

The following statement can be inferred from Lemma 2.1 LEMMA

2.3. If (1.3) and (2.9) hold, then

I(z-2;-)(l+ d Mx,

-Z)%n(X,Y,Z)l y, z) + ux,

y, z) + E,(x, y, z)I

G CJlmcp(x)P(Y) cp(z)

(nEB+; -l
(2.10)

We indicate some examples of ONSP {p,} (n E Z, ), satisfying the condition (2.9). Throughout C, C,, C,,, ... denote positive constants, independent of n E Z + and x, y, z E (- 1, 1). The same symbol does not necessarily denote the same constant from line to line. 1. Let w,,~(x) = (1 -x)”

(1 + x)~ be a Jacobi weight on [ - 1, l] with

parameters a and P(a, fl> - 1). For classical ortbonorma~ (with ~ei~~~ w,,~, Jacobi polynomials: {pFB’(x)} (n E Z ,.) jp’“‘“‘(x)l < C,,( 1 - x)n bEZ.ti

(4+

l/4) (1 + x)-iP/2i-

l/4)

-l-$)

and

are valid. Zn this case for the recurrence 6oef~~~e~ts ,$ = [,wq2

n

=

n

4n(n+a)(n~Bf(n+ct+P-t2) (2nfcl+B-1)(2n-1-afB+2)2(2ni-c!+p+r)

I 1-2(a2+p”)*o Z---j4 16n2

the condition

is fulfilled. 2. Consider orthonormal system (p,) recurrence relation ( 1.1), where

(n E Z,)

generated by the

(n = 1, 2, “..).

n (k~1)(11-4a:.,jf2I~~()dClog(n+1) k=O

In [33] (cf. also [SO]) P. Nevai proved that there exist positive constants C, 9C, independent of x 6 [ - I, 1] and n E Z ~ such that lp,(x)i
(n&Z+; -1
I).

If we suppose

k=O

Ip,,(x)l -GC(1 -x2)-1’4

[w(x)]-“‘2

(?ZGZ+; -l
104

BORIS P. OSILENKER

and

hold [15, 31, 501. In particular, Bernstein-&ego polynomials [48, pp. 44,451, AskeyWilson q-polynomials (p,( x; a, b, c, dlq)} max(lql, Id lbl, I4 I4 < 1) [2, 3, 51, and the orthogonal polynomial system {0t)(x; q)} (a > 0, )qj < 1) introduced by M. E. H. Ismail and F. S. Mulla [19] satisfy (2.11). 3. In [32], P. Nevai studied ONSP {p,} (n E Z + ), defined by the three-term recurrence relation (1.1) with

a =i+(-w ----+o “2

1) 0n

n

u, =

w)“D+o n

i 0n

2

’

where E, D are absolute constants. In this case there exist three positive numbers a, b, and c such that pgv) < c( 1 - x’) --b 1x1--a

(nEZ+; -l
and w(x)>c-’

Ixl”(1-X2)b.

Note that for such parameters X” < c log@ + 2)

(nEZ+).

If w(x) = lXla (1 -x2)b, then one has a generalized Jacobi polynomial. Let us define the auxiliary functions

where p(x) is the majorant of (p,> (FEZ+) (c$. (2.~~~and

k=O

are the first partial sums of the orthogonal series (9.7).

and (2.15)

exists for every x, y, z E ( - 1, 1) and

4sin cc+BcYsinCt$.B-YsinP+Y-asina-~4Y 2 2 2 = f

2

K(cos G!,cos p, co5 y; 2)

d(&)~A”k(COS(x,cos p, cos y)

+ &(cos

a, cos

p, cos

y) + E&OS

OE,cos

p, CBS y)],

@.ilis)

where the kernel 9$(x, y, z) is defined by (1.6). fn 6o~sc~~e~cc of the formula (2.4) and the definition (2.12)

106

BORISP. OSILENKER

(z- L xi+ -z) Rx, Y,z;1) = n-m lim M~,(x, Y, z) + k(x, Y, z) + EL5 Y, z)l n-1 - Em C 4~k)C~k(x, Y, z) -f-Bk(x, Y, z) + JJk(x,Y, z)l. n--tmk=O In view of the relations (1.3), (2.9), (2.10), (2.15), the first term of the last sum vanishes for all X, y, z E ( - 1, 1). The second term can be estimated with

and from (2.14) we obtain that the series f k=O

&k)[Ak(&

y, z)+Bk(x,y,

z)+~k(x,.bz)]

converge absolutely for x, y, z E ( - 1, 1). Putting x = cos IX, y = cos B, z = cos y and by (2.3), we have the formula (2.16), in accordance with our statement.

3. A CONSTRUCTIONOF THE “DOUBLE-HUMPBACKED MAJORANT": ESTIMATION OF THE LEBESGUEQUASIFUNCTIONS

The estimations of Lebesgue functions of the kernel play an important role in the treatment of expansions of functions in orthogonal polynomials. We begin with the construction of the majorant. It is well known [29; 30, p. 2621 that the nonnegative function F35, r)

(nE~,;5,YlE(a,b)c(-l,l))

is called a “humpbacked majorant” for the sequenceF,(& q) in the variable n at the point 5 if the following conditions are satisfied: (1) for all n E Z + and i;, q E (a, b) IFrAt, ur)l G&X5, ~1; (2) for fixed n E Z + , l E (a, b), the function F,*(L$ye)is nondecreasing on (a, 5) and nonincreasing on (4;,b). We say that the function ~Jx, y, z) (- 1 < x, y, z < 1; n E Z +) (cf. (1.6), (2.12)) has on (- 1, 1) a “double-humpbacked majorant” 3:(x, y, z) in the

variable z at the points c + (cf. (2.1)) if on each of the ~~ter~a~sf - I, XJJ) and (xy, 1) it possessesa ‘“humpbacked majorant” at the points i_. , i + ) respectively; i.e., for fixed x: and y, 9:(x, y, z) is ~o~~e~r~asi~g on ( - I,( _ )>~o~ncr~asing on ([ __, xy), nond~creas~n.~on (.xy95 + )$and xaonincreasing on ([ + , 1). The following assertions play a significant sole in our estimations.

(neZ+;

-B
43.1)

Proof: We show, that there exist ‘“humpbacked majorants” e function .&CC,y, z) on (- 1, xy) and (ny, 1) at the points g- a respectively. We construct the “humpbaek~d majorant’” for C&(x, y, z) (xy, 1) at the point 5 + ; on the interval ( - 1, ny) the ~~~str~~t~on can be deduced in a similar way. At first, consider the case

ri=xyfJ(1-x2)(1--2)Zf,

6,=~&1-x2)(1-y2),

i.e., x # y.

$=--$$(I-5+:.

(3.2)

By the relation (1.5) S,-+O, &--+O (n -+ co). It follows from (1.31, (1~5)~ (2.9), (2.10) that the following estimations are valid:

108

BORIS P. OSILENKER

where the constants C > 0 do not depend on 12E Z + and x, y, z E ( - 1,l). Thus we define the “humpbacked majorant” by

/

1 CJCl

0

e-(x, Y, z) =

1

if xydz
J(1-x2)(1-y2)r+-z’ 1 + l) (1 -x”)(l - y")'

if

1 (l-x2)(1-y2)1-i+’ 1 1 Wi J(1-x2)(1-y2)z--+’

C(n+l)

1

l+

-6,

-6,Gz<5+

if i+
+4

if [+ +&
where the constants C > 0 are independent of n E Z + and X, y, z E ( - 1, 1). In fact, when xy
lb-i-)(1:+ -z)l2JU -x2)(1 -Y”) b-i,

I.

So, for all x, y E ( - 1, 1) and y1E Z + the estimate

I%(x, Y,z)ldmx,y,2)

(nE.z+;xy
holds. Next, by the defining relation

a%(& Y,z)>O(xyQz
-&J,

a%+,Y, 4 8Z


and

n+l %3x,y,r+ - &J= c (1 -x2)(1-y’)’ %(x, y, i, + 42) = C(n + 1)

1

1

(l-x2)(1-y2)l-i+’

Consequently, the function 3:(x, y, z) is nondecreasing on (xy, c+ ) and nonincreasing on ([ + , 1). So $,*(x, y, z) is a “humpbacked majorant” for BJx, y, z) on (xy, 1) at the point i+ . Now consider w,

Y) = j’ %3x, y, z) dz XY

(nEZ+;

-1
This integral can be estimated in the following way; in virtue of the majorant 9,*(x, y, z)

Hence, using (3.2), we have

1 +c(~+1)(1-x’)(l-y2)6. +C(a-tl) + CN$z

1 1 J(l-x2)(1-y*pi+ 1 Inl-i+

f?T*

J(1 -x2)(1 -” y2) 4 1 ?2+1 JtL In 0 is independent of n E Z + and x, y E ( - 1, 1), in accordance with out statement. We still must consider the remaining case 5, = 1,

i.e.,x=y,[-=2x2-l.

In this case we define the ‘“humpbacked majorant” by

I

CNn--LL

s&x,

y, 2) =

C(fi+1)

f-X21-Z’

1 (1-X2f2’

if

x2~z<

if

1-6,gz
l-6,,

where C> 0 are the absolute constants and 6,= (NJ(n + l))(H -x’).

110

BORIS P. OSILENKER

Obviously, the function ~Jx, x, z) is nondecreasing on (x2, 1 - 6,) and nonincreasing on (1 - a,,). Furthermore, as above, we have by straightforward calculation

1 l--s, dz Wx,x,+z
2

’

j-l

dz

1-x2 1 1 In dcA$---1-x2 6, + C(n+ 1) (1 -x2)2&z. By the defining relation

where the constant C > 0 is independent of IZE Z+ and x f ( - 1, I), It coincides with (3.1) as x = y. We have completed the proof of our assertion. By virtue, of “symmetry” of the function gz(x, y, z) we can construct “double-humpbacked majorants” in the variables x and y. COROLLARY 3.2. Assume that ONSP {pn} (neZ+) with the weight w(x) satisfy (1.3) and (2.9). Then for Lebesgue’s quasifunctions 1, the following estimations hold:

The constants C > 0 in the relations (3.5) are independent of II E Z, and x, YE (-1, 1). In fact, the first estimation (3.5) can be deduced from (3.1), and, consequently,

This shows the validity of Corollary 3.2.

REPRESENTATION

OF THE TRILINEAR

KERNEL

111

We consider the pointwise estimation for the partway sums of. the following Fourier expansions

and

This problem arises, for example, in the Fourier method for partial difference equations [723,pp. 121-124-J. GoRoLLmY 3.3. Let QNSP (p,) (n E 72+) satisfy (1.3) and (2.9). Then the following (1)

statements ure valid:

at every x, y F ( - 1, 1) the following

~§tirn~t~o~

is valid~ where the constant C > 0 is ~~de~~ndent of A n E Z +.) and X,YE(--l, 1); (2) at every x E ( - 1, 1) the estimation

holds; the constant CT> 0 here is independent of the fun&cm v~r~ab~e~nEZ+, xE(-1,i). It can be Seen without difficulty

that

f and the

112

BORIS P. OSILENKER

and

k$‘of,~,(x)= s’, ~)-(Y, z)%)n(x, Y>z)W(Y)w(z)4 dz, from which by (3.5) the results follow. Remark. The methods of Section 3 give us an opportunity to investigate ONSP {p,} (n E Z, ) for which, instead of (2.9), the estimation IP,(X)l d MA+)

(nEz+;

-l
holds, but the right-hand side of (3.5) becomes more complicated.

ACKNOWLEDGMENTS The author thanks Professor P. Nevai for his attention and encouragement. I am very indebted to Professor A. Magnus for his valuable advice which improved the presentation of the paper. Thanks are also due to the refereesfor their suggestions.

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