Journal of Mathematical Analysis and Applications 236, 125᎐147 Ž1999. Article ID jmaa.1999.6442, available online at http:rrwww.idealibrary.com on
Mean and Almost Everywhere Convergence of Fourier᎐Neumann Series ´ Oscar CiaurriU and Jose ´ J. Guadalupe† Departamento de Matematicas y Computacion, ´ ´ Uni¨ ersidad de La Rioja, Edificio J. L. Vi¨ es, Calle Luis de Ulloa srn, 26004 Logrono, ˜ Spain Email:
[email protected],
[email protected]
and Mario Perez ´ † Departamento de Matematicas, Uni¨ ersidad de Zaragoza, Edificio de Matematicas, ´ ´ Ciudad Uni¨ ersitaria srn, 50009 Zaragoza, Spain Email:
[email protected]
and Juan L. Varona‡ Departamento de Matematicas y Computacion, ´ ´ Uni¨ ersidad de La Rioja, Edificio J. L. Vi¨ es, Calle Luis de Ulloa, srn, 26004 Logrono, ˜ Spain Email:
[email protected] Submitted by Daniel Waterman Received November 18, 1998 Let J denote the Bessel function of order . The functions xy␣ r2y  r2y1r2 J␣q  q2 nq1Ž x 1r 2 ., n s 0, 1, 2, . . . , form an orthogonal system in L2 ŽŽ0, ⬁., x ␣q  dx . when ␣ q  ) y1. In this paper we analyze the range of p, ␣ , and  for which the Fourier series with respect to this system converges in the L p ŽŽ0, ⬁., x ␣ dx .norm. Also, we describe the space in which the span of the system is dense and we show some of its properties. Finally, we study the almost everywhere convergence of the Fourier series for functions in such spaces. 䊚 1999 Academic Press
Key Words: Bessel functions; Fourier series; Neumann series; mean convergence; almost everywhere convergence; Hankel transform. * Research of the first author supported by Grant ATUR97r013 of the UR. † Research supported by Grants PB960120C0302 of the DGES and API98rB12 of the UR. ‡ URL: http:rrwww.unirioja.esrdptosrdmcrjvaronarwelcome.html 125 0022247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
126
CIAURRI ET AL.
1. INTRODUCTION AND NOTATION Let J be the Bessel function of order . For ␣ ) y1, the formula ⬁
H0 J
␣q2 nq1
Ž x . J␣q2 mq1 Ž x .
dx x
s
␦n m 2 Ž 2 n q ␣ q 1.
,
n, m s 0, 1, 2, . . . Žsee w15, Chap. XIII, 13.41 Ž7., p. 404; and 15, Chap. XIII, 13.42 Ž1., p. 405x., provides an orthonormal system jn␣ 4⬁ns0 in L2 ŽŽ0, ⬁., x ␣ dx . w L2 Ž x ␣ ., from now onx, given by jn␣ Ž x . s '␣ q 2 n q 1 J␣q2 nq1 Ž 'x . xy ␣ r2y1r2 ,
n s 0, 1, 2, . . . .
For each suitable function f, let Sn f be the nth partial sum of its Fourier series with respect to the system jn␣ 4⬁ns0 . Series of this kind are a particular case of series Ý nG 0 a n J␣qn , which are usually called Neumann series, so that we refer to Sn f as a Fourier᎐Neumann series. In w14x, one of the authors studied the mean convergence in L p Ž x ␣ . of these Fourier series. In this context, some operators and spaces were introduced. In this paper we extend these results and also study the almost everywhere convergence. For ␣ ) y1, let us define the integral operator H␣ by H␣ Ž f , x . s
xy␣ r2 2
⬁
H0
f Ž t . J␣ Ž 'xt . t ␣ r2 dt,
x ) 0,
for suitable functions f. This is a modified Hankel transform: the Žnonmodified. Hankel transform is the integral operator with kernel J␣ Ž xt .Ž xt .1r2 and unweighted Lebesgue measure. See w3, 12, 8x for some modified and nonmodified Hankel transforms. In the case ␣ G y 12 , the Hankel transform satisfies 5 H␣ f 5 L⬁Ž x ␣ . F C 5 f 5 L1 Ž x ␣ . ,
f g L1 Ž x ␣ . ,
with some constant C independent of f. Moreover, H␣ can be defined in L2 Ž x ␣ . satisfying H0⬁ Ž H␣ f . gx ␣ dx s H0⬁ Ž H␣ g . fx ␣ dx, H␣2 s Id, and 5 H␣ f 5 L2 Ž x ␣ . s 5 f 5 L2 Ž x ␣ . . From these results and interpolation we obtain 5 H␣ f 5 Lq Ž x ␣ . F C 5 f 5 L p Ž x ␣ . ,
f g L pŽ x␣ . ,
for 1 F p F 2, where q denotes, here and in the rest of the paper, the conjugate of p, that is, 1rp q 1rq s 1.
FOURIER ᎐ NEUMANN SERIES
127
The Hankel transform of the function jn␣ is H␣ Ž jn␣ , x . s '␣ q 2 n q 1 PnŽ ␣ , 0. Ž 1 y 2 x . w0, 1x Ž x . , where PnŽ ␣ ,  . Ž x . is the nth Jacobi polynomial of order Ž ␣ ,  .; see, for instance, w5, Chap. 8.11, Ž5., p. 47x Ža thorough description of Jacobi polynomials can be found in w4, Chap. X; 13x.. Remark. There is a delicacy with this formula. Actually, H␣ was defined, as a first step, as a Lebesgue integral for suitably integrable functions. Then, H␣ is extended to L p spaces where the integral representation is no longer valid for some functions. Now, the integrals from w5, Chap. 8.11, Ž5., p. 47x are improper Riemann integrals. Hence, the proper understanding of those integrals should be lim Nª⬁
xy␣ r2 2
H0
N ␣
jn Ž t . J␣ Ž 'xt . t ␣ r2 dt
s '␣ q 2 n q 1 PnŽ ␣ , 0. Ž 1 y 2 x . w0, 1x Ž x . . Since jn␣ w0, N x is an integrable function, the integral form of H␣ is valid here and we can conclude that lim H␣ Ž jn␣ w0, N x , x . s '␣ q 2 n q 1 PnŽ ␣ , 0. Ž 1 y 2 x . w0, 1x Ž x . ,
Nª⬁
where the limit holds in the almost everywhere sense. Finally, the L p boundedness of the operator H␣ for 4Ž ␣ q 1.rŽ2 ␣ q 3.  p  2 and the fact that lim N ª⬁ jn␣ w0, N x s jn␣ in L p yields H␣ Ž jn␣ , x . s '␣ q 2 n q 1 PnŽ ␣ , 0. Ž 1 y 2 x . w0, 1x Ž x . in L p. Similar comments apply to Lemma 3 below. Since the Hankel transform of jn␣ is supported on w0, 1x, not every function f g L p Ž x ␣ ., 1  p F 2, can be approximated in norm by its Fourier series Sn f. As a first approach, any such function should, at least, have its Hankel transform supported on w0, 1x. But we also deal with spaces L p Ž x ␣ ., p ) 2 where H␣ is not defined and so, we need to describe the functions that we want to approximate in a different, but, in some sense, similar way. The main tool here is M␣ , the multiplier for the Hankel transform. For ␣ G y 12 and y 14  Ž ␣ q 1.Ž 12 y 1p .  41 , M␣ is a bounded operator from L p Ž x ␣ . into itself Žthis is known as Herz’s theorem, see w7x.. Also, H␣ Ž M␣ f . s H␣ Ž f . w0, 1x ,
128
CIAURRI ET AL.
for f g L p Ž x ␣ . l L2 Ž x ␣ ., M␣2 f s M␣ f for f g L p Ž x ␣ . and ⬁
H0
f Ž x . M␣ Ž g , x . x ␣ dx s
⬁
H0 g Ž x . M
␣
Ž f , x . x ␣ dx
Ž 1.
for f g L p Ž x ␣ . and g g Lq Ž x ␣ .. DEFINITION 1. For each ␣ and p with ␣ G y y 1p .  14 , let us define the L p Ž x ␣ . subspace
1 2
and y 14  Ž ␣ q 1.Ž 12
Ep , ␣ s f g L p Ž x ␣ . : M␣ f s f 4 s M␣ Ž L p Ž x ␣ . . . It is clear that, for f g Ep, ␣ l L2 Ž x ␣ ., the Hankel transform of f is supported on w0, 1x and so these spaces are suitable for our purposes. The spaces Ep, ␣ have some interesting properties: For s  r, Es, ␣ ; Er, ␣ and the inclusion is continuous and dense. Besides, the dual space is Ž Ep, ␣ .X s Eq, ␣ . Let us also consider, for each ␣ G y1 and each suitable p Žwe go into the details later., the L p Ž x ␣ . subspace ⬁
Bp , ␣ s span jn␣ Ž x . 4 ns0
Ž closure in L p Ž x ␣ . . .
In w14x, one of us showed that S n f ª f in the L p Ž x ␣ .norm for any f g Bp, ␣ , if ␣ G y 12 and max
½
4 4 Ž ␣ q 1. , 3 2␣ q 3
5
½
 p  min 4,
4 Ž ␣ q 1.
5
2␣ q 1
;
moreover, for this range of p, we showed that Bp, ␣ s Ep, ␣ . Therefore, jn␣ 4⬁ns0 is a basis for the space Ep, ␣ . By the way, notice that for ␣ G y 12 , 4 Ž ␣ q 1. 2␣ q 3 p
p
4 Ž ␣ q 1. 2␣ q 1
m y m
1 4
 Ž ␣ q 1.
Ž ␣ q 1.
ž
1 2
y
ž
1 p
1 2
/
y 
1 p 1 4
/
,
.
Our purpose in this paper is to improve and extend these convergence results, and show additional properties of the Ep, ␣ spaces. In particular, we find some conditions on ␣ ,  , and p under which the functions jn␣q  Ž n s 0, 1, 2, . . . . are a basis for Ep, ␣ . For instance,  can be taken so that ␣ q  is half an integer, which makes the functions jn␣q  better known. The almost everywhere convergence of S n f is studied, as well.
FOURIER ᎐ NEUMANN SERIES
129
Also, we can interpret the convergence in the following way: changing the parameters, we take jn4⬁ns0 , which is orthogonal in L2 Ž x ., and we study the convergence in L p Ž x .. This is a typical solution in the study of mean convergence of Fourier series. For instance, in the case of Jacobi polynomials PnŽ ␣ ,  . Ž x .4⬁ns0 , Pollard w10x studied the convergence in the natural space L p ŽŽy1, 1., Ž1 y x . ␣ Ž1 q x .  . and, later, Muckenhoupt w9x described the behavior in L p ŽŽy1, 1., Ž1 y x . a Ž1 q x . b .. Similar situations occur with other orthogonal systems ŽLaguerre, Hermite, Freud weights, Bessel and Dini.. We are interested in the approximation of functions in L p Ž x ␣ . by Fourier series in the system jn␣q  4⬁ns0 . So, our first target is to determine the range of p, ␣ , and  for which jn␣q  g L p Ž x ␣ . for all n g ⺞. We do this in Section 2. In Section 3 we state some of the main results of this paper: the uniform boundedness and convergence of the partial sum operator of Fourier᎐Neumann series. The proofs are given in Sections 6 and 7. The mean convergence can only hold for functions in the closure of the linear combinations of the functions jn␣q  . In Section 4 this space is shown to coincide with Ep, ␣ under some conditions on p, ␣ , and  . Some applications are given in Section 5. Throughout this paper, unless otherwise stated, we use C Žor C1 . to denote a positive constant independent of n Žand all other variables., which can assume different values in different occurrences. Also, in what follows, a n ; bn , for a n , bn ) 0, means C F a nrbn F C1. 2. THE SPACES Bp, ␣ ,  We use here the wellknown estimates Žsee w4; 15, Chap. III, 3.1 Ž8., p. 40; 15, Chap. VII, 7.21 Ž1., p. 199x.: J Ž x . s
x 2 ⌫ Ž q 1.
q O Ž x q2 . ,
xª0q ,
Ž 2.
and
J Ž x . s
(
2
x
cos x y
ž
2
y
4
/
q O Ž xy1 . ,
x ª ⬁.
Ž 3.
LEMMA 1. Let a ) y1, 1  p  ⬁. Then, jna g L p Ž x b . for all n s 0, 1, 2, . . . if and only if b ) y1 and y 14  Ž b q 1.Ž 12 y 1p . q a y2 b . Fur
130
CIAURRI ET AL.
thermore, in this case,
¡n ;~n ¢n
yŽ aq1.q2Ž bq1.r p
5
jna 5 L p Ž x b .
yŽ2 aybq1.r2
if p  4,
,
Ž log n .
1r4
yŽ5 r6qa.qŽ6 bq4.rŽ3 p.
if p s 4,
,
if p ) 4,
,
Proof. Inequalities b ) y1 and y 14  Ž b q 1.Ž 12 y 1p . q a y2 b follow from Ž2. and Ž3.. Then, estimates such as Ž12. below Žsee w1, 2x. show that 5 jna 5 L p Ž x b . is bounded above by a constant times the righthand side. The lower bound follows from more precise estimates for the Bessel functions, as shown in w1, 2x. For a similar expression, see w11x. As a consequence, the following definition makes sense. DEFINITION 2. For each ␣ ,  , and p with ␣ ) y1, ␣ q  ) y1, 1  p  ⬁, and y
1 4
 Ž ␣ q 1.
1
ž
2
y
1 p
/
q
 2
,
let us define ⬁
Bp , ␣ ,  s span jn␣q  Ž x . 4 ns0
Ž closure in L p Ž x ␣ . . .
Note that we assume ␣ ) y1 in the definition of Bp, ␣ ,  ; however, we require ␣ G y 12 for Ep, ␣ . Actually, the boundedness of M␣ can be studied also for ␣ ) y1, so that the definition of Ep, ␣ can be extended to the whole range ␣ ) y1. But in the case ␣  y 12 , the H␣ transform does not have as good properties as in the case ␣ G y 12 . As a consequence, the spaces Ep, ␣ do not behave for ␣  y 12 like for ␣ G y 12 . Thus, some of the results in this paper are established for ␣ ) y1, but we require ␣ G y 12 when Ep, ␣ appears. The following lemma proves that Bp, ␣ ; Bp, ␣ ,  , under some conditions on ␣ ,  , and p. LEMMA 2.
Let ␣ ) y1, ␣ q  ) y1, and 1  p  4 such that y1 4
 Ž ␣ q 1.
ž
1 2
y
1 p
/
y
<< 2
.
Then, jn␣ s
⬁
Ý an , k jk␣q  , ksn
Ž 4.
FOURIER ᎐ NEUMANN SERIES
131
pointwise and in L p Ž x ␣ ., where an , k s
2 '␣ q 2 n q 1 ␣ q  q 2 k q 1 ⌫ Ž 1 y  . ⌫ Ž ␣ q  q k q n q 1 .
'
⌫ Ž 1 q k y n . ⌫ Ž 1 y  y k q n . ⌫ Ž ␣ q k q n q 2.
.
Ž 5. Remark. If  g ⺞, then ⌫ Ž1 y  .r⌫ Ž1 y  y k q n. should be replaced by y Žy y 1.Žy y 2. ⭈⭈⭈ Ž1 y  y k q n. in formula Ž5.. Proof. The pointwise convergence and Ž5. follow from w15, Chap. V, 5.21 Ž1., p. 139x Žconditions ␣ ) y1 and ␣ q  ) y1 are required.. Strictly speaking, condition  f ⺞ should also be assumed, following w15x. But this is only a formal requirement to get a n, k in the form of Ž5.. For the L p convergence, we need only prove that the series converges: that the sum is precisely jn␣ then follows from the fact that this holds in the almost everywhere sense. If  is an integer, then there are only finitely many a n, k / 0 and the series in Ž4. is a finite sum. If  is not an integer, Stirling’s formula for the gamma function gives, for each fixed n, < a n , k < ; k 2 y3r2 , Also, from Lemma 1, p  4 and
y1 4
k ª ⬁.
 Ž ␣ q 1.Ž 12 y 1p . q 2 , we have
5 jk␣q  5 L p Ž x ␣ . ; kyŽ ␣q q1.q2Ž ␣q1.r p . These estimates and
y1 4
 Ž ␣ q 1.Ž 12 y 1p . y ⬁
Ý < an , k < 5 jk␣q  5 L
p
Žx␣.
 2
prove that
 ⬁.
ksn
3. UNIFORM BOUNDEDNESS AND CONVERGENCE OF FOURIER᎐NEUMANN SERIES Let us consider the partial sums of the Fourier series with respect to the system jn␣q  4⬁ns0 : n
Sn Ž f , x . s
Ý c k Ž f . jk␣q  Ž x . , ks0
ck Ž f . s
⬁
H0
f Ž t . jk␣q  Ž t . t ␣q  dt.
We are interested in the study of the uniform boundedness of the partial sum operators Sn : L p Ž x ␣ . ª L p Ž x ␣ . .
132
CIAURRI ET AL.
Our result is THEOREM 1. Let ␣ ) y1, ␣ q  ) y1, and 1  p  ⬁. There exists a constant C ) 0 such that 5 Sn f 5 L p Ž x ␣ . F C 5 f 5 L p Ž x ␣ . ,
f g L p Ž x ␣ . , n g ⺞,
if and only if 43  p  4 and y
␣qq1 2 y
1 4
 Ž ␣ q 1.  Ž ␣ q 1.
ž ž
1 2 1 2
y y
1 p 1 p
/ /
q q
 2
 2
, 
1 4
Ž 6. .
Proof. See Section 6. COROLLARY 2. y
Let ␣ ) y1, ␣ q  ) y1, 43  p  4, and
␣qq1 2 y
1 4
 Ž ␣ q 1.  Ž ␣ q 1.
ž ž
1 2 1 2
y y
1 p 1 p
/ /
q q
 2
 2
, 
1 4
.
Then, Sn f ª f in L p Ž x ␣ . for all f g Bp, ␣ ,  . Proof. Bp, ␣ ,  is the closure in L p Ž x ␣ . of the orthogonal system, so this is just a standard consequence of Theorem 1. Regarding the almost everywhere convergence of Fourier᎐Neumann series, we have THEOREM 3. y
Let ␣ ) y1, ␣ q  ) y1, 43  p  4, and
␣qq1 2 y
1 4
 Ž ␣ q 1.  Ž ␣ q 1.
ž ž
1 2 1 2
y y
1 p 1 p
/ /
q q
Then, Sn f ª f almost e¨ erywhere for any f g Bp, ␣ ,  . Proof. See Section 7.
 2
 2
, 
1 4
.
FOURIER ᎐ NEUMANN SERIES
133
4. THE HANKEL TRANSFORM OF ORDER ␣ FOR jn␣q  AND THE SPACES Bp, ␣ ,  AND Ep, ␣ Theorem 1 and Corollary 2 are more interesting if we can describe the space Bp, ␣ ,  . In this section, we find some conditions under which Bp, ␣ ,  and Ep, ␣ coincide. As we pointed out, the Hankel transform of order ␣ for jn␣ can be written in terms of the nth Jacobi polynomial of order Ž ␣ , 0.. It is not difficult to obtain H␣ Ž jn␣q  . from known results about integrals of products of Bessel functions that can be expressed in terms of hypergeometric 2 F1 functions. But the relation between H␣ Ž jn␣q  . and the Jacobi polynomials of order Ž ␣ ,  . is not easily found in the literature. For instance, it does not appear in the standard references w4, 15, 5x. For the sake of completeness, in this section we obtain H␣ Ž jn␣q  . explicitly in terms of PnŽ ␣ ,  .. For ␣ ,  ) y1 with ␣ q  ) y1,
LEMMA 3.
H␣ Ž jn␣q  , x . s 2y 
'␣ q  q 2 n q 1 ⌫ Ž n q 1. ⌫ Ž  q n q 1.
Ž1 y x.

= PnŽ ␣ ,  . Ž 1 y 2 x . w0, 1x Ž x . . In particular, suppŽ H␣ Ž jn␣q  .. : w0, 1x. Proof. We use the formula ⬁
y
H0 t
J Ž at . J Ž bt . dt b a y y1 ⌫
s
ž
2 ⌫ Ž q 1. ⌫ =2 F1
ž
qyq1
ž
/
2 qyq1 2
/
qyq1 yyq1 b2 , ; q 1; 2 , 2 2 a
/
Ž 7.
valid when 0  b  a, q y ) y1, and ) y1; here, 2 F1 denotes the hypergeometric function Žsee w5, Chap. 8.11, Ž9., p. 48; 15, Chap. XIII, 13.4 Ž2., p. 401x..
134
CIAURRI ET AL.
Taking a s 1 and x s b 2 in Ž7., and making the corresponding changes of variable and parameters Ž s ␣ , s ␣ q  q 2 n q 1, s  . we get H␣ Ž jn␣q  , x . s
'2 n q ␣ q  q 1 ⌫ Ž ␣ q n q 1. 2  ⌫ Ž  q n q 1. ⌫ Ž ␣ q 1. =2 F1 Ž ␣ q n q 1, yn y  ; ␣ q 1; x . ,
which is valid for ␣ ) y1 and  ) y1 in the interval 0  x  1. Now, we have 2 F1
Ž ␣ q n q 1, yn y  ; ␣ q 1; x . s Ž1 y x.
 2 F1
Ž yn, ␣ q  q n q 1; ␣ q 1; x . ,
where ␣ ,  ) y1, n s 0, 1, 2, . . . , and PnŽ ␣ ,  . Ž x . s
⌫ Ž n q ␣ q 1. ⌫ Ž ␣ q 1. ⌫ Ž n q 1.
2 F1
ž
yn, ␣ q  q n q 1; ␣ q 1;
1yx 2
/
␣ ,  ) y1. Therefore, H␣ Ž jn␣q  , x . s
'␣ q  q 2 n q 1 ⌫ Ž n q 1. 2  ⌫ Ž  q n q 1.
 Ž 1 y x . PnŽ ␣ ,  . Ž 1 y 2 x . ,
x g Ž 0, 1 . . Now, let us calculate H␣ Ž jn␣q  , x . for x ) 1. To do that, let us take x s a2 , b s 1, s ␣ q  q 2 n q 1, s ␣ , and s  in Ž7.. In this way, 12 Ž q y q 1. s 0, y1, y2, . . . , so the coefficient 1r⌫ Ž 21 Ž q y q 1.. vanishes and we get H␣ Ž jn␣q  , x . s 0. THEOREM 4.
Let ␣ G y 12 ,  ) y 12 , 43  p, with y
1 4
 Ž ␣ q 1.
ž
1 2
y
1 p
/

1 4
.
If p  2, assume further y Then Bp, ␣ ,  s Ep, ␣ .
1 4
 Ž ␣ q 1.
ž
1 2
y
1 p
/
y
<< 2
.
,
FOURIER ᎐ NEUMANN SERIES
135
Proof. Case p s 2. The spaces B2, ␣ ,  and E2, ␣ are well defined. Also M␣ Ž jn␣q  . s jn␣q  . In other words, B2, ␣ ,  ; E2, ␣ . If they were not equal, by the Hahn᎐Banach theorem there should exist some T g Ž E2, ␣ .X , T / 0, such that T Ž jn␣q  . s 0 ᭙ n. But Ž E2, ␣ .X s E2, ␣ , so there exists g E2, ␣ , / 0, such that H0⬁ jn␣q  x ␣ dx s 0 for every n. Then 0s
⬁
H0 j
s kn
␣q  ␣ x n
1
dx s
Ž␣ ,  . n
H0 Ž H . P ␣
⬁
H0
Ž H␣ . Ž H␣ jn␣q  . x ␣ dx
 Ž 1 y 2 x . Ž 1 y x . x ␣ dx
for every nonnegative integer n. Now, the Jacobi polynomials PnŽ ␣ ,  . Ž x . are a complete orthogonal system with respect to the measure Ž1 y x . ␣ Ž1 q x .  dx on Žy1, 1.. A change of variable proves that the polynomials PnŽ ␣ ,  . Ž1 y 2 x . are a complete orthogonal system with respect to the measure Ž1 y x .  x ␣ dx on Ž0, 1.. Thus, H␣ s 0 on Ž0, 1.. Since g E2, ␣ , we also have H␣ s 0 on Ž1, ⬁.. Therefore, H␣ s 0 and we arrive at the contradiction s 0. Case p ) 2. Note that ␣ ,  , and p meet the requirements of Definition 2. Also, by the preceding case, we have jn␣q  g E2, ␣ ; Ep, ␣ . Thus, Bp, ␣ ,  ; Ep, ␣ . Now, let f g Ep, ␣ . Given ) 0, there exists a function g g L2 Ž x ␣ . l pŽ ␣ . L x such that 5 f y g 5 L p Ž x ␣ .  . let h s M␣ g; then h g L2 Ž x ␣ . l pŽ ␣ . L x and M␣ h s h, so that h g E2, ␣ l Ep, ␣ s B2, ␣ ,  l Ep, ␣ . Since M␣ is continuous, 5 f y h 5 L p Ž x ␣ . s 5 M␣ f y M␣ g 5 L p Ž x ␣ .  C . As h g B2, ␣ ,  , there exists hX g span jn␣q  4⬁ns0 such that 5 h y hX 5 L2 Ž x ␣ .  . The inclusion E2, ␣ ; Ep, ␣ gives 5 h y hX 5 L p Ž x ␣ .  C1 , so that, by the triangle inequality, 5 f y hX 5 L p Ž x ␣ .  C2 . This gives the inclusion Ep, ␣ ; Bp, ␣ ,  . Case p  2. By Lemmas 1 and 3, jn␣q  g L2 Ž x ␣ . and H␣ Ž jn␣q  . is supported on w0, 1x, so that M␣ jn␣q  s jn␣q  . Since jn␣q  g L p Ž x ␣ . by Lemma 1, it follows that jn␣q  g Ep, ␣ . Therefore, Bp, ␣ ,  ; Ep, ␣ . The equality follows if we prove that the only operator T g Ž Ep, ␣ .X such that T Ž f . s 0 for all f g Bp, ␣ ,  is T s 0. For such an operator, we have T Ž jn␣ . s 0 for every n G 0, since jn␣ g Bp, ␣ ,  by Lemma 2. On the other hand, by the duality Ž Ep, ␣ .X s Eq, ␣ , where 1rp q 1rq s 1, there exists some g Eq, ␣ such that TŽ f . s
⬁
H0 fx
␣
dx,
f g Ep , ␣ .
136
CIAURRI ET AL.
In particular, ⬁
H0 j
␣ ␣ n x
dx s 0,
n G 0.
Ž 8.
Under the present conditions on p and ␣ , the preceding case gives Bq, ␣ , 0 s Eq, ␣ , so that g Bq, ␣ , 0 . This, together with Ž8. and Corollary 2, gives s 0.
5. APPLICATIONS Some properties of the spaces Ep, ␣ can be obtained from Theorem 4. Two examples are given here, after this preliminary result. Let ␣ G y 12 ,  s 0, and 43  p  4 ¨ erifying
COROLLARY 5.
y
1 4
 Ž ␣ q 1.
ž
1 2
y
1 p
/

1 4
.
Then, Sn f ª M␣ f in L p Ž x ␣ . and almost e¨ erywhere for all f g L p Ž x ␣ .. Proof. Let f g L p Ž x ␣ ., and so M␣ f g Ep, ␣ . Then, by Theorems 4 and 3, S nŽ M␣ f . ª M␣ f in L p Ž x ␣ . and almost everywhere. So, we only need to show that SnŽ M␣ f . s S nŽ f ., and this is clear because, by Ž1., ⬁
H0
⬁
⬁
Ž M␣ f . jn␣q  x ␣ dx s H f Ž M␣ jn␣q  . x ␣ dx s H fjn␣q  x ␣ dx. 0
COROLLARY 6. y 12 ,
Let ␣ G y 12 , y 12    1,
y
½
max y
1 4
0
,y
1 4 1 4
 Ž ␣ q  q 1. y
 2
,y
␣q1 2
ž
1 2
y
1 p
4 3
/
 p  4 with ␣ q  G
1 2
and y
4
,
y   Ž ␣ q 1.
5
 min If p  2, assume further  
1

1 4
½
ž
1 2
y
1 p
/
 1 1 , y . 4 4 2
q 2  Ž ␣ q 1.Ž 21 y 1p ..
5
FOURIER ᎐ NEUMANN SERIES
137
Then, Ep, ␣ l L p Ž x ␣q  . ; Ep, ␣q  . Proof. Let f g Ep, ␣ l L p Ž x ␣q  .. By Theorems 4 and 3, S n f ª f almost everywhere. Since f g L p Ž x ␣q  ., Corollary 5 Žwith ␣ q  instead of ␣ . gives Sn f ª M␣q  f almost everywhere. Then, f s M␣q  f almost everywhere, that is, f g Ep, ␣q  . Let ␣ G y 12 ,  g Žy 12 , 12 .,
COROLLARY 7. y 21 ,
½
max y
1 4
,y
1
y
4
 2
,y
␣q1 2
4 3
y   Ž ␣ q 1.
5
 min
½
max y
1 4
,y
1 4
q
 2
,y
␣q1 2
 p  4 with ␣ q  G
q
 2
5
½
ž
1 2
y
½
/
p
1 1  , y , 4 4 2
5
 Ž ␣ q  q 1.  min
1
1
ž
2
1
y
p
/
1 1  , q . 4 4 2
5
If p  2, assume further y 14 q 2  Ž ␣ q 1.Ž 21 y 1p . and y  q 1.Ž 12 y 1p .. Then, Ep, ␣ l L p Ž x ␣q  . s Ep, ␣q  l L p Ž x ␣ ..
1 4
y 2  Ž ␣ q
Proof. The inclusion ‘‘: ’’ is clear by Corollary 6. The inclusion ‘‘= ’’ follows also by Corollary 6 with ␣ q  instead of ␣ , and  instead of y . Theorem 4 gives different bases for Ep, ␣ for different values of  . It seems interesting to obtain the expressions for the change of basis between jn␣ 4⬁ns0 and jn␣q  4⬁ns0 in Ep, ␣ . COROLLARY 8.
½
max y
1 4
Let ␣ G y 12 , y 12    1, 43  p  4 with
,y
1 4
y
 2
,y
␣q1 2
y   Ž ␣ q 1.
5
 min
½
ž
1 2
y
1 p
/
1 1  , y . 4 4 2
5
138
CIAURRI ET AL.
If p  2, assume further y 14 q 2  Ž ␣ q 1.Ž 12 y 1p .. Then, both jn␣ 4⬁ns0 and jn␣q  4⬁ns0 are bases of the space Ep, ␣ and the change of basis is gi¨ en by jn␣ s
⬁
Ý an , k jk␣q  ,
jn␣q  s
ksn
⬁
Ý bn , k jk␣ , ksn
where an , k s bn , k s
2 '␣ q 2 n q 1 ␣ q  q 2 k q 1 ⌫ Ž 1 y  . ⌫ Ž ␣ q  q k q n q 1 .
'
⌫ Ž 1 q k y n . ⌫ Ž 1 y  y k q n . ⌫ Ž ␣ q k q n q 2. 2y
'␣ q  q 2 n q 1 '␣ q 2 k q 1 ⌫ Ž 1 q  . ⌫ Ž ␣ q k q n q 1. ⌫ Ž 1 q k y n . ⌫ Ž 1 q  y k q n . ⌫ Ž ␣ q  q k q n q 2.
,
.
Ž 9. Proof. By Theorem 4, Bp, ␣ , 0 s Ep, ␣ , so that jn␣ g Ep, ␣ . Also, by Theorem 4, Ep, ␣ s Bp, ␣ ,  . Then, Corollary 2 gives SN jn␣ ª jn␣ in L p Ž x ␣ . as N ª ⬁, that is, ⬁
jn␣ s
Ý an , k jk␣q  , ks0
in L p Ž x ␣ ., where an , k s
⬁
H0
jn␣ jk␣q  x ␣q  dx.
In a similar way, jn␣q  g Bp, ␣ ,  s Bp, ␣ , 0 and jn␣q  s Ý⬁ks0 bn, k jk␣ in L p Ž x ␣ ., where bn , k s
⬁
H0
jn␣q  jn␣ x ␣ dx.
Finally, w15, Chap. XIII, 13.41 Ž2., p. 403x gives a n, k s bn, k s 0 for k  n and Ž9. for k G n. Similar expressionsX for the change of basis between different bases jn␣q  4⬁ns0 and jn␣q  4⬁ns0 in Ep, ␣ can be obtained. Details are left to the reader.
6. PROOF OF THEOREM 1 6.1. Necessary conditions Let us begin by showing that conditions Ž6. are necessary for the uniform boundedness in Theorem 1. Assume Sn is uniformly bounded.
FOURIER ᎐ NEUMANN SERIES
139
Then the operators given by Tn Ž f , x . s Sn Ž f , x . y Sny1 Ž f , x . s jn␣q  Ž x .
⬁
H0
f Ž t . jn␣q  Ž t . t ␣q  dt
are uniformly bounded as well, i.e., 5 Tn f 5 L p Ž x ␣ . s
⬁
H0
f Ž t . jn␣q  Ž t . t ␣q  dt 5 jn␣q  5 L p Ž x ␣ . F C 5 f 5 L p Ž x ␣ . ,
with a constant C independent of n and f. By duality, this means 5 t  jn␣q  5 Lq Ž t ␣ . 5 jn␣q  5 L p Ž x ␣ . F C, where 1rp q 1rq s 1. Taking n fixed Žit suffices n s 0. and applying the first part of Lemma 1 gives Ž6.. Now, provided Ž6. holds, the norm estimates of Lemma 1 give
¡n ;~n ¢n
yŽ ␣ q  q1.q2Ž ␣ q1.r p
5 jn␣q  5 L p Ž x ␣ .
yŽ ␣ q2  q1.r2
if p  4,
, 1r4
,
if p s 4,
yŽ5 r6q ␣ q  .qŽ6 ␣ q4.rŽ3 p.
,
if p ) 4,
Ž log n .
and
¡n
yŽ ␣ y  q1.q2Ž ␣ q1.r q
5
x  jn␣q 
~
5 Lq Ž x ␣ . ; n
yŽ ␣ y2  q1.r2
,
if q  4 Ž i.e., p )
4 3 4 3
1r4
,
if q s 4 Ž i.e.,
yŽ5 r6q ␣ y  .qŽ6 ␣ q4.rŽ3 q.
,
if q ) 4 Ž i.e.,
¢n
Ž log n .
., p s ., p  43 . .
This implies 34  p  4. 6.2. Sufficient conditions Let us now assume 43  p  4 and Ž6., and prove the uniform boundedness of the partial sum operators Sn . They can be written as Sn Ž f , x . s
⬁
H0
f Ž t . K n Ž x, t . t ␣q  dt,
where n
K n Ž x, t . s
Ý ks0
jk␣q  Ž x . jk␣q  Ž t . .
140
CIAURRI ET AL.
The next lemma gives a suitable decomposition of the kernel K nŽ x, t . associated to Sn . For a similar formula, with a different proof, see w14x. Let n g ⺞ and ) y1. Then
LEMMA 4. n
Ý 2 Ž q 2 k q 1. Jq2 kq1Ž x . Jq2 kq1Ž t . ks0
s
xt
xJq1 Ž x . J Ž t . y tJ Ž x . Jq1 Ž t .
x y t2 2
qxJXq2 nq2 Ž x . Jq2 nq2 Ž t . y tJq2 nq2 Ž x . JXq2 nq2 Ž t . . Proof. Using the equality Jy1Ž z . q Jq1Ž z . s 2z J Ž z . Žsee w15, Chap. III, 3.2, p. 45x. to express Jy1 and J q2 in terms of J and J q1 yields the formula xt
xJ Ž x . J y1 Ž t . y tJ y1 Ž x . J Ž t .
x y t2 2
yxJq2 Ž x . J q1 Ž t . q tJ q1 Ž x . J q2 Ž t . s 2 J Ž x . J Ž t . . This now gives n
Ý 2 Ž q 2 k q 1. Jq2 kq1Ž x . Jq2 kq1Ž t . ks0
s
xt
xJq1 Ž x . J Ž t . y tJ Ž x . Jq1 Ž t .
x y t2 2
yxJq2 nq3 Ž x . Jq2 nq2 Ž t . q tJq2 nq2 Ž x . Jq2 nq3 Ž t . . Finally, use the formula zJq1Ž z . s J Ž z . y zJX Ž z . Žsee w15, Chap. III, 3.2, p. 45x. to take out Jq2 nq3 . From the definition, we have Sn Ž f , x . s xy␣ r2y  r2y1r2 ⬁
n
H0 Ý Ž ␣ q  q 2 k q 1. J
=
ž
␣q  q2 kq1
ks0
= t ␣ r2q  r2y1r2 f Ž t . dt
Ž 'x . J␣q q2 kq1 Ž 't .
/
FOURIER ᎐ NEUMANN SERIES
141
so that Lemma 4 with s ␣ q  leads to Sn f s W1 f y W2 f q W3, n f y W4, n f , where W1 Ž f , x . s 12 xy␣ r2y  r2q1r2 J␣q q1 Ž x 1r2 . H Ž t ␣ r2q  r2 J␣q  Ž t 1r2 . f Ž t . , x . , W2 Ž f , x . s 12 xy␣ r2y  r2 J␣q  Ž x 1r2 . H Ž t ␣ r2q  r2q1r2 J␣q q1 Ž t 1r2 . f Ž t . , x . , W3, n Ž f , x . s 12 xy ␣ r2y  r2q1r2 JX Ž x 1r2 . H Ž t ␣ r2q  r2 J Ž t 1r2 . f Ž t . , x . , W4, n Ž f , x . s 12 xy ␣ r2y  r2 J Ž x 1r2 . H Ž t ␣ r2q  r2q1r2 JX Ž t 1r2 . f Ž t . , x . , and s q 2 n q 2 s ␣ q  q 2 n q 2. Here, H denotes the Hilbert transform on Ž0, ⬁., which is defined by HŽ g, x. s
⬁
H0
gŽ t. xyt
dt
Žthe integral must be considered as a principal value.. Thus, we can conclude that the partial sum operators S n are uniformly bounded if we can prove that the operators W1 , W2 are bounded and the operators W3, n , W4, n are uniformly bounded for n G 0. We use good estimates for the Bessel functions and the A p theory of weights to prove the boundedness of the Hilbert transform. Let us start with the bounds for the Bessel functions and their derivatives. From the estimates Ž2. and Ž3. it follows that, for ) y1, J Ž x . F C x ,
x g Ž 0, 1 x ,
Ž 10 .
J Ž x . F C xy1 r2 ,
x g w 1, ⬁ . ,
Ž 11 .
with a C constant depending on . Moreover, we need bounds for the Bessel functions J␣q q2 nq2 Žand their derivatives. with constants independent of n. So, we make use of the bounds J Ž x . F Cxy1r4 Ž < x y < q 1r3 .
y1 r4
JX Ž x . F Cxy3r4 Ž < x y < q 1r3 .
1r4
,
,
x g Ž 0, ⬁ . ,
Ž 12 .
x g Ž 0, ⬁ . ,
Ž 13 .
with some constant C independent of . They follow from those of w1, 2x, for instance, and were already used in w14x.
142
CIAURRI ET AL.
6.3. Some results on Hilbert transforms and A p theory To analyze the boundedness of the Hilbert transform, some notation and previous results are necessary. As usual, for 1  p  ⬁ we write q s prŽ p y 1., i.e., 1rp q 1rq s 1. A weight is a nonnegative Lebesguemeasurable function on Ž0, ⬁.. The class A p Ž0, ⬁. w A p , for shortx consists of those weights w such that, for every subinterval I : Ž0, ⬁.,
ž
1
HIw
/ž
1
prq yq r p
HIw
/
F C,
where C is a positive constant independent of I, and < I < denotes the length of I. The A p constant of w is the least constant C verifying this inequality and is denoted by A p Ž w .. We refer the reader to w6x for further details on A p classes. Fix 1  p  ⬁; then the Hilbert transform H is a bounded linear operator on L p Ž w ., for any weight w g A p . The norm of H: L p Ž w . ª L p Ž w . and the A p constant of w depend only on each other, in the sense that given some constant C which verifies the A p condition for w, another constant C1 depending only on C can be chosen so that 5 H 5 F C1 , and vice versa. Therefore, for a sequence wn4ng ⺞ uniformly in A p , i.e., with some constant C verifying the A p condition for every wn , the Hilbert transform is uniformly bounded on L p Ž wn ., n g ⺞. Let us see some auxiliary results related with A p weights: LEMMA 5. Then
Let u, ¨ , w be weights on Ž0, ⬁. and ␥ be a positi¨ e constant.
Ža. w Ž x . g A p if and only if w Ž␥ x . g A p ; both weights ha¨ e the same A p constant. Žb. w g A p if and only if ␥ w g A p ; both weights ha¨ e also the same A p constant. Žc. If u, ¨ g A p , then u q ¨ g A p and A p Ž u q ¨ . F A p Ž u. q A p Ž ¨ .. Žd. If u, ¨ g A p and 1rw s 1ru q 1r¨ , then w g A p and A p Ž w . F C w A p Ž u. q A p Ž ¨ .x. Proof. Parts Ža. and Žb. are trivial. Part Žc. follows easily from the inequality
ž
1
prq
HŽ u q ¨ . I
yq rp
/
F min
½ž
1
prq
uyq r p
HI
/ ž ,
1
prq
¨ yq r p
HI
/ 5
.
Part Žd. is a consequence of Žc. and the fact that u g A p m uyq r p g A q , with A q Ž uyq r p . s w A p Ž u.x q r p.
FOURIER ᎐ NEUMANN SERIES
143
The proof of the next lemma is not difficult, but cumbersome, so we omit it. For the weight in Žc., observe that x r < x 1r2 y 1 < 2 ; x r near 0, x r < x 1r2 y 1 < s ; < x y 1 < s near 1 and x r < x 1r2 y 1 < s ; x rqsr2 near ⬁, whence the three conditions follow. LEMMA 6.
Let r, s g ⺢. We ha¨ e
Ža. x g A p m y1  r  p y 1. Žb. Set ⌽ Ž x . s x r if x g Ž0, 1. and ⌽ Ž x . s x s if x g Ž1, ⬁.. Then, ⌽ g A p if and only if y1  r  p y 1 and y1  s  p y 1. Žc. x r < x 1r2 y 1 < s g A p m y1  r  p y 1, y1  s  p y 1, and y1  r q sr2  p y 1. r
To simplify the notation, in the rest of this section we write s ␣ q  and s ␣ q  q 2 n q 2. 6.4. Boundedness of the operators W1 and W2 From the definition, it follows that 5 W1 f 5 L p Ž x ␣ . F C 5 f 5 L p Ž x ␣ . , if and only if 5 Hg 5 L p Ž x ␣y p r2q p r2 < Jq1 Ž x 1r2 .< p . F C 5 g 5 L p Ž x ␣y p r2 < J Ž x 1r2 .
Cx ␣y p r2qp r2 Jq1 Ž x 1r2 .
F ⌽ Ž x . F C1 x ␣y p r2 J Ž x 1r2 .
yp
are enough. According to the bounds Ž10. and Ž11., we have x ␣y p r2qp r2 Jq1 Ž x 1r2 . x ␣y p r2 J Ž x 1r2 .
p
yp
F G
½ ½
Cx ␣qp , Cx
␣ y p r2qp r4
if x g Ž 0, 1 . , ,
Cx ␣y p , Cx
␣ y p r2qp r4
if x g Ž 1, ⬁ . , if x g Ž 0, 1 . ,
,
if x g Ž 1, ⬁ . .
Let us try ⌽Ž x. s
½
if x g Ž 0, 1 . ,
x r, x
␣y p r2qp r4
,
if x g Ž 1, ⬁ . .
Ž 14 .
144
CIAURRI ET AL.
By Žb. in Lemma 6, conditions Ž14. and ⌽ g A p hold if
␣ y p F r F ␣ q p, y1  r  p y 1,
Ž 15 .
y1  ␣ y pr2 q pr4  p y 1. The third line is equivalent to y1 4
 Ž ␣ q 1.
ž
1 2
y
1 p
/
q
 2

3 4
,
which follows from Ž6.. For the inequalities in Ž15. involving r, it suffices to show that max y1, ␣ y p 4  min p y 1, ␣ q p 4 . This follows from ␣ ) y1, ␣ q  ) y1, 1  p  ⬁, and Ž6., as well. The case of W2 is entirely similar. 6.5. Uniform boundedness of the operators W3, n Here, 5 W3, n f 5 L p Ž x ␣ . F C 5 f 5 L p Ž x ␣ . , if and only if 5 Hg 5 L p Ž x ␣y p r2q p r2 < JX Ž x 1r2 .< p . F C 5 g 5 L p Ž x ␣y p r2 < J Ž x 1r2 .
F Cx ␣y p r2qp r8 w < x 1r2 y < q 1r3 x
pr4
yp
G Cx ␣y p r2qp r8 w < x 1r2 y < q 1r3 x
pr4
x ␣y p r2qp r2 JX Ž x 1r2 . x ␣y p r2 J Ž x 1r2 .
, .
It suffices to prove that g A p uniformly in n Žrecall that s ␣ q  q 2 n q 2., with
Ž x . s x ␣y p r2qp r8 w < x 1r2 y < q 1r3 x
pr4
.
From Lemma 5, we have
Ž x . g A p unif.
m
Ž 2 x . g A p unif.
m
x ␣y p r2qp r8 w < x 1r2 y 1 < q y2r3 x
m
x ␣y p r2qp r8 < x 1r2 y 1 < p r4
pr4
qyp r6 x ␣y p r2qp r8 g A p unif.,
g A p unif.
FOURIER ᎐ NEUMANN SERIES
145
where the last equivalence follows from
w < x 1r2 y 1 < q y2r3 x
pr4
; < x 1r2 y 1 < p r4 q yp r6 .
Now, Lemmas 5 and 6 prove that those weights belong to A p uniformly. 6.6. Uniform boundedness of the operators W4, n Finally, 5 W4, n f 5 L p Ž x ␣ . F C 5 f 5 L p Ž x ␣ . , if and only if 5 Hg 5 L p Ž x ␣y p r2 < J Ž x 1r2 .< p . F C 5 g 5 L p Ž x ␣y p r2y p r2 < JX Ž x 1r2 .
F Cx ␣y p r2yp r8 w < x 1r2 y < q 1r3 x
ypr4
yp
G Cx ␣y p r2yp r8 w < x 1r2 y < q 1r3 x
ypr4
x ␣y p r2 J Ž x 1r2 . x ␣y p r2yp r2 JX Ž x 1r2 .
, ,
so let us put
Ž x . s x ␣y p r2yp r8 w < x 1r2 y < q 1r3 x
yp r4
,
and show that g A p uniformly in n. Indeed,
Ž x . g A p unif.
m
Ž 2 x . g A p unif.
m
x ␣y p r2yp r8 w < x 1r2 y 1 < q y2r3 x
yp r4
g A p unif.,
and
žx
␣ y p r2yp r8
w < x 1r2 y 1 < q y2r3 x
ypr4 y1
/
; xy ␣q p r2qp r8 < x 1r2 y 1 < p r4 q yp r6 s x ␣y p r2yp r8 < x 1r2 y 1
y1
q w p r6 x ␣y p r2yp r8 x
y1
,
so that Lemmas 5 and 6 lead to the desired conclusion. The proof of Theorem 1 is now complete.
146
CIAURRI ET AL.
7. PROOF OF THEOREM 3 We only need to prove that SnŽ f, x . converges to some g Ž x . almost everywhere. This, together with Corollary 2, gives g s f almost everywhere. Now, recall that Sn f s Ý nks0 c k Ž f . jk␣q  , where ck Ž f . s
⬁
H0
f Ž t . jk␣q  Ž t . t ␣q  dt.
It follows from Lemma 1 that x  jn␣q  g Lq Ž x ␣ .; moreover, 5 x  jn␣q  5 Lq Ž x ␣ . F Cn ␦ for some constant ␦ s ␦ Ž p, ␣ ,  .. Thus, c n Ž f . F 5 f 5 L p Ž x ␣ . 5 x  jn␣q  5 Lq Ž x ␣ . F C 5 f 5 L p Ž x ␣ . n ␦ . Now, according to w15, Chap. III, 3.31 Ž1., p. 49x we have J Ž x . F
2y x ⌫ Ž q 1.
1 )y . 2
,
Therefore, jn␣y  Ž x . s ␣ q  q 2 n q 1 J␣q q2 nq1 Ž 'x . xyŽ ␣q q1.r2 F
' '␣ q  q 2 n q 1 2yŽ ␣qq2 nq1. x n ⌫ Ž ␣ q  q 2 n q 2.
,
so that cn Ž f .
jn␣q 
Ž x . F C 5 f 5 L pŽ x ␣ .
n ␦q1r2 Ž xr4.
n
⌫ Ž ␣ q  q 2 n q 2.
and the series Ý⬁ns 0 c nŽ f . jn␣q  Ž x . converges. ACKNOWLEDGMENTS The authors thank the referees for their valuable comments, which helped us to improve this paper in its final version.
REFERENCES 1. J. A. Barcelo Bandlimited functions: L pconvergence, Trans. Amer. ´ and A. Cordoba, ´ Math. Soc. 313 Ž1989., 655᎐669. 2. A. Cordoba, The disc multiplier, Duke Math. J. 58 Ž1989., 21᎐27. ´ 3. A. J. Duran, ´ On Hankel transform, Proc. Amer. Math. Soc. 110 Ž1990., 417᎐424.
FOURIER ᎐ NEUMANN SERIES
147
4. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, ‘‘Higher Transcendental ´ Functions,’’ Vol. 2, McGrawHill, New York, 1953. 5. A. Erdelyi, ´ W. Magnus, F. Oberhettinger, and F. Tricomi, ‘‘Tables of Integral Transforms,’’ Vol. 2, McGrawHill, New York, 1954. 6. J. GarcıaCuerva and J. L. Rubio de Francia, ‘‘Weighted Norm Inequalities and Related ´ Topics,’’ Mathematical Studies, Vol. 116, NorthHolland, Amsterdam, 1985. 7. C. S. Herz, On the mean inversion of Fourier and Hankel transforms, Proc. Nat. Acad. Sci. U.S. A. 40 Ž1954., 996᎐999. 8. J. M. Mendez and M. M. Socas, A pair of generalized Hankel᎐Clifford transformations ´ and their applications, J. Math. Anal. Appl. 154 Ž1991., 543᎐557. 9. B. Muckenhoupt, Mean convergence of Jacobi series, Proc. Amer. Math. Soc. 23 Ž1969., 306᎐310. 10. H. Pollard, The mean convergence of orthogonal series. III, Duke Math. J. 16 Ž1949., 189᎐191. 11. K. Stempak, A weighted uniform L p estimate of Bessel functions: A note on a paper of K. Guo: ‘‘A uniform L p estimate of Bessel functions and distributions supported on S ny 1 ,’’ Proc. Amer. Math. Soc., to appear. 12. K. Stempak and W. Trebels, Hankel multipliers and transplanting operators, Studia Math. 126 Ž1997., 51᎐66. 13. G. Szego, ¨ ‘‘Orthogonal Polynomials,’’ 4th ed., American Mathematical Society Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, RI, 1975. 14. J. L. Varona, Fourier series of functions whose Hankel transform is supported on w0, 1x, Constr. Approx. 10 Ž1994., 65᎐75. 15. G. N. Watson, ‘‘A Treatise on the Theory of Bessel Functions,’’ 2nd ed., Cambridge Univ. Press, Cambridge, U.K., 1944.