On convergence and Riesz means of a series of Bessel functions

RO C ZN IK I POLSK IEG O TOWARZYSTWA MATEM ATYCZNEGO Séria I: PRACE MATEMATYCZNE XXII (1980)

S. R. Agrawal and С. M. Patel (Baroda, India)

On convergence and Riesz means of a series of Bessel functions

1. Introduction. Let Zf be the class of all Lebesgue integrable functions with p-th power (1 ^ p < oo) over [ a ,b ] , and let C be the class of all functions continuous over [ a ,b ] , 0 < a < b.

Let J v(t) and Y^it) be the Bessel functions of the first and the second kind respectively, for v ^ —1/ 2, and let

Cy{&, P) = -Ma) Yv(fi) — J v(fi) Yv(a).

Denote by j x < j 2 < j$ < ... the successive positive zeros of J v(t) and by 7i < Ï2 < Уз < • •• those of cv(at,bt),0 < a < b.

Define

(1.1) <pv(0 = (in t)ll2J v (t)

<pv(0) = lim <pv(t);

r-»0 +

for t > 0,

and

(1.2) C%4t) = y / t c v(tym,bym), a ^ t < b Consider the series

(1-3) £ am(pv{xjm), 0 x ^ 1,

m= 1 and

(1.4) £ d „ C ÿ (x ), a U x H b ;

m= 1 and the typical Riesz means:

(1.5) R*n(x) = ^ (*/«), 0 < x ^ l , and

(1.6) « W = î 1 - ^ i C W , a ü x ç b ,

m= 1 \ *>„ /

with a > 0, where j n < An < j n + l and y„ < Bn < y„ + 1. If the series (1.3) and (1.4) are Fourier-Bessel series corresponding to a function / (cf. [2]), then the Riesz-means (1.5) and (1.6) will be denoted by R f,(x ,f) and Q l ( x , f ) respectively, and we shall have

(1.7) K ( x , f ) = [f{t)<Pl{t,x)dt, O ^ x ^ l ,

where

and

(¹.⁸)

where K ( t , x )

= 1 V A n / J v + l\ J m )

Qn(x, f ) = J / ( i ) ^ ( f , x)dt, a ^ x ^ b,

I 1

7m \ 71 Jv ( a y j Cv (xym, bym) cv (tym, bym)

B; 2 {v2(aym) - v 2(bym)}

Titchmarsh [6] has studied the convergence of Fourier-Bessel series (1.4) corresponding to a function of bounded variation in the neighbourhood of a point and of class L1. Gupta and Khoti [4] have considered the Riesz summability of (1.4). The authors of the present paper also have established certain properties regarding convergence of (1.4) in [1] and have studied certain Riesz means of (1.4) in [2].

Taberski [5] studied the-Riesz-means (1.5) of the series (1.3). The purpose of this paper is to study the typical Riesz-means (1.6) of the series (1.4).

2. Fundamental lemmas. Properties of 9*n(t,x). The first three lemmas follow from Lemmas 3.1 and 3.2 in [1] (K m and K m(oi) respectively denote suitable positive constants depending upon v and v, a, throughout the paper):

Lemma 2.1. For large values of n, Bn ~ n.

Lemma 2.2. On the rectangle R whose vertices are at ± B i, Bn± B i in the w-plane, where В will be made to tend to infinity and w = u + iv,

= О

= 0 {y/bjx e ~ ^ {b~x))

wcv(xw, aw)cv(tw, bw) ( ° x)

cv(aw, bw) for t > x and n sufficiently large.

Lemma 2.3.' On the rectangle R, cv (xw, aw) cv (aw, bw) for a < $ < b, and w = u + iv.

Lemma 2.4. Let v ^ —1/2. Then (i) for a > 0,

(2.1) \9a„(t, x)| ^ n K 1 (a), t, x e [ a , b ], n ^ 1;

(ii) for 0 < a ^ 1,

(2.2) \œH{t, x)| ^ » Г , х б [ а , Ч , о 1

, ri \t —

arcd (iii) /o r a > 1,

(2.3) lflj(*,x)| ^ . . ^ 3 ^ ,2 » t , x e [ a , b ] , x , n ^ 1.

n |t — x|

P ro o f. Let

4 wa \ nwcv(xw, aw)cv(tw ,bw)

G(w) = y /x t 1— — --- :---r -t---,

\ Bn J cv (aw, bw)

where a ^ x < t ^ b.

The residue of G(w) at w = ym is given by (cf. [2], Lemma 4) xt I j _ \ y i J hay„)c,(xym,bym)c.(tym,bym)

K ) 2{Jv2( a y J - J v2(i-Tj}

Taking R as the contour of integration, we shall have

j B r, + aoi

(2.4) f i ( t , x ) = - — J G(w)dw +

2 Я i g — cei

+- 1 2я l* — ooi because on account of Lemma 2.2

(u±Bi)a

J y/Xt wa + 1- cv(xw, aw)cv(tw ,bw)

в: cv (aw, bw) dw;

|G(w)| ^ K4 1 — в:

B(t - дс)

, w = u ± B i, 0 ^ и ^ Bn,

SO that

в„ ± Bi

lim J G(w)dw = 0.

B-* oD _{± B i} Also, for 0 < a < 1,

(B„ + vi)a (2.5)

and for a > 1,

m < K 5(oc)-— , - o o < v < o o ,

D„

(2.6) 1- (Bn + v if

Bl

к s (a)

< b„ ’

к , ^v

N ^ Bn, M > B„.

6 K ’ If 0 < a < 1, by (2.4) and (2.5), we have

0° a oo a

|0“(t,x)| ^ 2K1K M \ - W e-'>«-x4 v + 2K b \ - - e - ' ’(' - x4 v

0 0 D n

< K 2 ( a )

na (t — x)a +1 ’ by using Lemma 2.1.

Similarly, by an interchange of t and x, i f x > £ , 0 < a < l , K 2 (a)

n“( x - t) a This proves (2.2).

Again, if a > 1, by (2.4) and (2.6), we obtain by a similar reasoning, K ( t , x ) \ < *з(«)

n (x — t)^{2 ’}

where n is so chosen that Bn > K/(t — x), for K , a constant.

This proves (2.3).

The case, a ^ x ^ t ^ b, follows as in [2], Lemma 4.

Lemma 2.5. For v > —1/2, v Ф 0, and any a > 0, f b

‘j |0 “(t,x)M r K 8 (a), x e [ f l,b ] , n > 1.

a

P ro o f. We have,

i K b . x W t = { Т ,П+ T " + / } 10Ж X)M( « K , ( a),

a a x — l /п x + 1 /n

by using Lemma 2.4.

Lemma 2.6. Let f x{t) = tv + 112 for te\_a,b~\. Then (i) for 0 < a ^ 2 , a < x < b,

(2.7) Ш * , / , ) - / , (x)| ^ K ,(«) + » 1,

and (ii) for a > 2 , a < x < b,

(2.8) Ш * . Л ) - Л М < « , . ( - ) - > ! •

P ro o f. 1Ву (1.8), for a < x < b,

n - y / x " / f m \ y2mJ 2(aym)cv(xym,bym) е л х . л ) - I ( ‘ “ И j H a y j - J H b y j x

x J tv + l cv(tym, b y j d t - / V ( l Ут \ ^v(fly«)Cv(xyw,b y J

L 1 к ) J h a y m) - J t ( b y m) X

X (fev Jv (aym) - a v J v {bym)}

(cf. [2], Lemma 5).

Considering

H(w) = 2 1- wa \ bvcv(xw,aw) — avcv(xw,bw) wcv(aw, few)

and Я as the contour of integration, we obtain (cf. [2], Lemma 5)

j B„ + ooi . oci

Qan( x , f 1) - x v+i/2 = -1— - j H(w)dw— -—г (P) J H(w)dw,

B „ - o o i 2 n i

since, by Lemma 2.3,

Bn ± B i

J H(w)dw-> 0 , as £-> oo

± B i

Hence,

(2.9) |Π( x ,/ i) - /i( * ) l

11

n 1 - (Д .+ 1ИГ fov + l/ 2 e ~ |t)|(b —x) _|_ flv + 1/2 M ( x - a )

V * î +*>2

dv + 2K n * r’ " 1

£ £*

+ ---— J ——- (fev +1/2 e~v(b~x) + av + 1/2 e~v(x~a)} dv,

= I ! + / 2, say.

Now, if 0 < a ^ 1, by (2.5)

2 К 1г K 5 (a) °? va bv + ll2e - v{b~x) + av + ll2e~v(x~a)

(2 .1 0 ) - ^ S r dv

c К и («) j 1 , 1

(2.11)

B* [{b — x f (x — a f Similarly, if a > 1, then by (2.6)

* 1 3 ( « )

h ^ Bt

1 1

(b —x )2 + -(x — a)2 Moreover, for any a > 0

(2.12)

2К п Г(а) f bv + l12 _ av + 1/2

12 = л ~ ; ~ + -

nBt l ( b - x f ( x - a f

^ K 14(«) j 1 t 1

b: (b — x f (x — a f

The lemma now follows from (2.9) to (2.12) and Lemma 2.1, by choosing n so large that

' 1 1

B„ > max

b — x x —a J

Lemma 2.7. The following estimates are valid for v ^ —1/2, v Ф 0:

(i) I f 0 < a < i, a < x < b, n ^ i,

(2.13) U f f ( t , x H f - l | X 15( « ) - j ^ r + . 1 1 хи* n’ f x - a f

(ii) if ol = 1, a < x < b, n > 1, ь

(2.14) + +

(iii) if 1 < a < 2 , a < x < b, n > 1, ь

(2.15) \$æn( t , x ) d t - l \

, 4 . log n i 1 1

^ --- 1---- :--- — H— r - :---~ + '

nx n(x — a)2 na (x — a f ri* (b — x f ( ’ and (iv) if a > 2 , a < x < b, n > 1,

(2.16) I jæ n( t , x ) d t - l \

r, / ч . log n 1 1 1

^ *18 (a) --- 1---- ;---ttH --- — + '

nx n(x — a)2 n2(x — a)2 n2(b — x)2 P ro o f. Let a < d < x < b, and let

f 2(t) — tv + 1/2 for a ^ t < d,

= dv + 112 for d < t ^ b . Then (cf. [2], Lemma 6),

(2.17) Ш х , / 2) - / 2( х ) К S K i t . x t Iv + 1,2<fc +

a

b

+ dv+1/2 J 11 - ( t/xy +ll2\\e*n(t , x)| d t+

a

+ (d/xy +1/2 m x J J - M x ) . Let, now, 0 < a < 1; then by (2.2),

K 2( a) (2.18) j W , x ) | C ^ d t «

a n a [ X I /

Also,

(2.19) J |l - ( t / x ) v + I'2| № ((,x)|dr

x — 1 /n x x + 1/n b

= { I + S + J + I Ï

d x —1 /n x x + l / n

= ^1 + ^2 + ^з + ^4? say.

Д + 1 an* (.x — d f

v + 1 / 2

|0S (L x)| dt

By (2.2),

(2.20)

Similarly,

(².²¹)

, „ x 2(«) 'V '" 1-(« A ),+1' 2 ^ *!<>(«) 'l < . . J --- :--- T T T ^ d t <

i (x-()* +1

K „ ( « )

xn

xn

(2.22)

Again, by (2.1), in a similar manner,

K 2 0 (oi)

and 7, ^ K 20 (a)

n x nx

By (2.17) to (2.22) and using Lemma 2.6, we obtain

(2.23) \Ql(x, f 2)—f2(x)\ < <f + 1/2* 2i(a) ^{1 1} na(x —d f xri*

1 1

H— r:--- ^r + -

n* (x — à f t f (b — x f Also, by (1.8) and definition of / 2,

ь

(2.24) \&xn( t , x ) d t - \

a

= S 0 U t,x )d t + d - ’' - l >2 [{Q’„ ( x ,f2) - f2(x) } - f t' + 4 26 i(t,x)dt].

a a

Using (2.23), (2.24) and (2.18), and taking limit as d-+a, we have for a < x < b,

№ ( t , x ) d t - 1| < K „ (x )

a

1 1 1

xrf + n*(x — a f + n*(b —x f J for n > 1. This proves (2.13).

If a = 1, everything in (2.18) and (2.22) goes well by substituting a = 1.

In case of (2.20) and (2.21), (2.25) _{/ 1 ^} K 19( a ) -lo g n

and JL ^ K 19(a)log n

nx nx

Using (2.17) to (2.19), (2.22), (2.24) and (2.25), and Lemma 2.6, it follows by taking limit as d->a, that (2.14) is proved.

In case 1 < a ^ 2, by (2.3) of Lemma 2.4,

(2.26) $ m , x ) \ t v+i/2dt *22 (* K^{v+ 1/2} nx Also, by a similar analysis that is used in (2.19),

(2.27) f |1 — (0c)v + 1/2l \6^(t,x)\dt 2* i 9 (a) log n + 2K20 (a) nx

The proof of (2.15) is, now, completed by (2.17), (2.24), (2.26), (2.27) and (2.7).

Similarly, for a > 2, the proof of (2.16) follows from (2.17), (2.24), (2.26), (2.27) and (2.8).

3. Classes of functions and typical means. Let for / б С [ о ,Ь ] ,

0)(ô, / ) = sup \ f ( t ) - f ( x ) \ , t, x g [a, b ] .

I t - x \ i â

We say that / е Л я [а ,Ь ], if co(S,f) = 0 (S a) as <5->0 + , 0 < а < 1. The integral modulus of continuity of f e l? will be denoted by

('OpiàJ) = sup { J \ f ( x ± h ) - f ( x ) \ pdx}llp.

O^n^o a

Theorem 3.1. Let (1.4) be the Fourier-Bessel series of f e C , f ( a) = f(b)

— 0. I f v ^ —1/2, v ^ 0, then (i) for 0 < а < 1,

(3.1) max \Q*n(x, f ) - f ( x ) \ ^ K 23 (a) co(l/nf / ) , n ^ 1;

a^x^b

(ii) for a = 1,

( l o g n

(3.2) max \Q*„(x, f ) - f (x)\ < X 24m 1 , / , n > 1;

a^x^b \ ft

and (iii) /o r а > 1,

( logn

(3.3) max \Q*n(x, f ) - f (x)\ ^ X 25(a)col--- , / , n > 1.

a^x^b \ П

P ro o f. We write

(3.4) Qan(x, f ) - f ( x ) = $ { f{ t)-f(x )} 0 * n(t,x)d t+ f(x)[$ 0 * n( t , x ) d t - \ ]

= U„(x)+V„(x), say.

For 0 < а < 1,

\ f ( t ) - f ( x ) \ ^ со (1/n*, /) { n a|t - x | + l}, so that by Lemmas 2.4 and 2.5, we obtain,

(3.5) ш ( 1 Д Л /) [ К 8(а) + К 2(а )< ' ('" + J ’ -7Т~пг +

a x + l / n 11

x + l / n

+ nl+r Kiitx) J \t — x\dt~\

x — l In

^ К 26(а)со(1/пя, / ) for all х е [ а , Ь ] . Also, by Lemma 2.7, for x e [a + 1/n, b — 1/n], (3.6) |F„(*)I

^ / 4 f 1 + ^ _ a )na , 1 + (x —а)ия 1 +(b —x)n*

^ co(l/n , / ) X 15 (a) <--- ;--- + — --- — +-

xn~ n*(x —a)* n*(b — x )*

= K 21 (a) co (1/n*,/).

For a ^ x < a + l/n, b —1/n < x < b, by Lemma 2.5,

(3.7) |F„U) ^ {1 + К 8(а)}со(1/пя, / ) { 1/п1_я + 1} = К 28(а)со(1/пя, / ) By (3.4) to (3.7), we get (3.1) proved.

If a = 1, we take,

\ f ( t ) - f ( x ) \ ^ œ / ) | ÏQg n \*~*1 + 1 j . so that, by a similar treatment as in the previous case,

(3.8) ll/.M I K „ ( o t ) Л

Similarly, by (2.14), for a+ 1/n ^ x ^ b — 1/n, and by Lemma 2.5 for a ^ x < a + l/n , b — l/n < x ^ b,

(3.9) IК M l « / ) •

Estimation (3.2) is now proved by (3.4), (3.8) and (3.9). The treatment in case a > 1, is similar.

The following corollaries follow immediately from Theorem 3.1, Lemma 2.4 and Lemma 2.7:

Corollary 3.1.1. I f / е Л а [а ,Ь ], 0 < a < 1, /(a ) = f(b) = 0, then

Ш * , / ) - / М 1 = 0 ( " - 2), « > i .

Corollary 3.1.2. Let f e C [ a ,b ] and /ef f (a) = f (b) = 0. TTien /or v ^ —1/2, v # 0 and a > 0, tlte typical Riesz-means Q„{x, f ) converges uniformly to f (x) in [a, b].

Corollary 3.1.3. Let f e l l , 1 ^ p < oo. Then for v ^ —1/2, v # 0, and a > 0 ,

\ i m ] m x , f ) - f ( x ) \ pdx = 0.

л - o o a

Theorem 3.2. I f the series (1.4) is the Fourier-Bessel series of f e L p, 1 ^ p < oo, and if v ^ - 1/2 , v ^ 0 , 0 < a < l , f/ien

(3.10) { S m x , f ) - f ( x r d x } " '

a

a K 31(a) {cop(l/n " ,/)+ W * (l/n ,f)}, n > 1, where

w ; ( & , f) = { J i/(x )|M x } 1" + « ^ f

/ м

Ь- ô

a + <5J

/ ( * ) ^{p 1 1 /p} d x f +

b - 0 +<*1 1

a + <5

b — ô (x — a f

+ { J i / M I ' d * } 1" -

p ^ i / p

dx> + <5a<( J

J 1 e + à

f i x ) ( b - x f

p ) 1/p

dx> +

b

b — S1

P roof. As in (3.4), we have

(3.11) {J I «: [J !J|/(l)-/(x)| |(>;(r,x)|dr|'V x]1"’ +

a a a

+ [ J l / ( * ) l p | J 0£(f. x) d t - l \pdx]llp = / + /', say.

a a

b x — 1 fn x x + \ /n b

(3.12) / = [ J {( S +

s

a a x — 1 /n x x + 1 /n

^ /1 + /2 + / 3 + / 4 , say.

By Lemma 2.4,

K 2{ a) (3.13) /1 <

n

К 2 ( a )

n*

Jl ! лгх

J { J | / ( x + a ) - / ( x ) | pd x }1/p

l / n u + a

1 /P

du Л + 1

< K 2(«) * у €0р(ц, / ) ,а+ 1

l / n

< /C2(a)cop(l/na, / ) . fa na u + l

,a + 1 du

l / n

Similarly, (3.14)

Again, by Lemma 2.4,

= i K 31(a)cop(l/na, / ) .

/4 ^ i X 31(a)û>p( l K , / ) .

I 2 < n K ^ U { J I f ( x - u ) - f ( x ) \ d u } pd x f lp ^ \ K 3l (a)ftip(l/n“, / ) .

a 0

Similarly, 73 can be evaluated and therefore, (3-15) I 2 + U < i X 31(a)cup(l/na, / ) .

Further,

a + l / n b — l / n b b

(3.16) '' = [ ( 1 + 1 + 1 ) I / (x)lp 11 0ü(t. x)dt —l\pdx] llp

a a + l / n b — l / n a

— + f 2 + f 3 » 8аУ•

a + l / n

By Lemma 2.5,

(3.17) /', + / , « K „ (a)[{ 1 |/(* )|'< fr} , " 4 { j I f i x r d x } 1"’].

Also, by (2.13),

b b - 1 /n1

(3.18) Г2 ^ * 3 i ( « ) b - l / n a + 1/n

f i x )

( b - 1 In

+ I (a + t/n^

X

f i x ) i x - a f

p ) t/p dx> +

» ~)1/P ( b - t / n

d x \ + <^ f

J U+1/и

f i x ) p ) t/p dx >

( b - x f The theorem is now proved by (3.11) to (3.18).

Theorem 3.3. A necessary and sufficient condition for the series (1.4) to be the Fourier-Bessel series of a function f e C , vanishing at a and b, is that

(3.19) lim Q*„ix) — f {x),

GO

uniformly in [ a ,b ].

P ro o f. The necessity of condition (3.19) follows from Corollary 3.1.2.

For the sufficiency part, we have, by using (1.6) and the orthogonality of sequence {C^(x)} (cf. Titchmarsh [6], p. xiv),

(3.20) ( :1 - dm = bm 1 Ql (X) CS> (x) d x , where

, = 7Г2 Ут Jv jaVm) 2 {J?(aym) - J v2(bym)} '

In view of (3.19), by taking limit as n->^{g o} in (3.20), we obtain,

(3.21) dm = К

b

$ f i x ) C {ZHx)dx, m — 1 , 2 , 3 , . . .

(3.21) exhibit the coefficients dm in (1.4) as the Fourier-Bessel coefficients of / , hence the theorem is proved.

Theorem 3.4. Series (1.4) is the Fourier-Bessel series of a function f e l } , or f e U, 1 < p < oo, if and only if

(3.22) lim J \Qan( x ) - fi x ) \d x = 0,

n ~*cc a

or

b

(3.23) sup J \Ql(x)\pdx < oo

n > 1 a

respectively.

P ro o f. Let us consider series (1.4) to be the Fourier-Bessel series of / e L p, 1 < p < oo. Then

(3.24) { ï m x j r d x } 1»

a

« [ J {(’ Г + ’ T + î w m e r u t . x n d t y d x y »

a a x — 1/n x + 1/n

^ / l + / 2 + ^ 3 » S a Y -

If 0 < a < 1, then as in the proof of (3.13),

/1 ^ *Щ- J 1 If ( x - u ) \ pdx}llpdu ^ K 32( a ) ||/ ||p.

fl 1 /n W u + a

If a > 1, only a has to be replaced by 1, so that for all a > 0, we have,

(3.25) « K 32( « ) ||/||„ and h < К г2(и)\\/\\р.

In a similar way,

(3.26) I 2 ^ K 33(a) II/ ||p.

The necessity of condition (3.23), now, follows from (3.24) to (3.26).

The sufficiency part may be proved as in Bary ([3], Vol. I, Theorem 3, p. 165-167).

In case / e L1, the necessity of condition (3.22) follows from Corollary 3.1.3. The sufficiency part may be proved as in the case of Lp.

4. Convergence of Fourier-Bessel series. Throughout this section we shall assume that (1.4) is the Fourier-Bessel series of the function / .

Theorem 4.1. Let / e L 2, 0 < a < 1, v ^ —1/2, v Ф 0, and let

00 J

(4.1) £ - щ - {m2( l/n * ,/) + H ? ( l/n ,/) } < 00.

n = l n

Then the Fourier-Bessel series (1.4) of f (x) converges absolutely and uniformly in [a, b~\.

P ro o f. By the theorem of best approximation and the Parseval’s relation,

00 b

(4.2) £ \dm\2 ^ J \f(x)-Q *n( x , f ) \ 2dx.

m = n - h i a

00

Set r„ = £ \dm\2. Then by (4.1), (4.2), Theorem 3.2 and Bary ([3],

m = n + 1

Vol. II, p. 157),

I Id j K 34 E

n = 1 n = 1

(4.3)

n ^{< 0 0}.

Since,

C{m M = 0 ( 1/yJ, as m->co (cf. [1], Lemma 2), the theorem follows from (4.3).

Theorem 4.2. Let f e C be a function of bounded variation over [a ,b ] such that f (a) = f (b) = 0. I f moreover,

0° 1

(4.4) £ — {co(l/na, f ) } l/2 < со, 0 < a < 1,

n= 1 П

then the Fourier-Bessel series (1.4) of f converges absolutely and uniformly in [a, b~\.

P ro o f. By (3.1) and (4.2), for 0 < a < 1,

(4-5) r„ ^ K 23( o t) c o ( l /n \f ) \\f ( x ) - Q : ( x , f) \ \1.

Since, / is of bounded variation, it is bounded, so that (4.6) ffli ( 5 ,/) = 0(0) and W?(<5 ,/) = 0(0).

Therefore, by Theorem 3.2, (4.5) and (4.6), we have as in (4.3),

(4.7) £ I4J « K 35 £

m = 1 n= 1 П

The theorem, now, follows from (4.4) and (4.7).

Following is a theorem of Lorentz type (cf. [3], Vol. I, p. 215-216):

Theorem 4.3. Let / е Л а [ а , Ь ] , 1/p — \ < a < 1, 1 ^ p ^ 2, f (a) — f (b)

= 0. Then

( 4 -8 ) { m = n + 1

Î

P ro o f. By (4.2) and Corollary 3.1.1,

00

(4.9) I I4.I2 = 0 ( n - 2-).

m = n + 1

Now, by Holder’s inequality (cf. [3], Vol. I, p. 216) and (4.9), (4.10)

к = 0, 1 , 2 , 3 , . . .

(4.8), now, follows from (4.10) and

00 00 2 k + i n

1 14.1'= I I 14.1'.

m = n + l k = 0 m = 2 * n + l

By putting p = 1 and p = 2, respectively, the following theorems of Bernstein type and Zygmund type (cf. [3], Vol. II, p. 154, 161) are established:

I К

= -L 1

\p ^

К_{3 6} p(a + l/²) - l

П {2 -p(a+l/2)+l

Г,

Theorem 4.4. I f 1/2 < a < 1 in the hypothesis of Theorem 4.3, then (1.4) converges absolutely and uniformly in [a ,b ].

Theorem 4.5. I f 0 < a < 1 in the hypothesis of Theorem 4.3, then the Fourier-Bessel coefficients of (1.4) have the order given by

d„ = 0 (n ~ a), as n-+cc.

In particular, (1.4) converges absolutely and uniformly in [a ,b ].

5. Appendix. We consider the special case v = 1/2. By the identity (cf.

[7], § 3.4), (5-1)

we can write

Ci/2(a, P) sin (p- a ) ,

(5.2) / ( * ) ~ _{m — 1}

z

Also, if v = ± | , we may choose ([1], Lemma 1),

(5.3) Ут mn

b — a for large n.

Using (5.1) to (5.3), we obtain,

v m u x , m ux

(5.4) f ( x ) ~ X 1 am cos --- + br, sin

m = i I b - a b - a Г a ^ x ^ b, where

(5.5) an

2 (b — a)dm sin я 2т ч/Ь

mub b — a

and bm =

2 (b — a) dm cos - и2т^/Ь

mub b — a

Hence, (5.4) represents, a Fourier series corresponding to / in [ a ,b ] , which is absolutely and uniformly convergent under the conditions of Theorem 4.5.

The following theorem is, thus, easily established:

Theorem 5.1. Let f e Aa [a, b ], 0 < a < 1, / ( a ) = / ( b ) = 0, and let its Fourier series be represented by (5.4). Then its Fourier coefficients (5.5) have the order given by

In particular, the Fourier series (5.4) of f converges absolutely and uniformly in [a, b ] .

References

[1] S. R. A g ra w a l and С. M. P a te l, On convergence of Fourier-Bessel series, Bull. Acad.

Polon. Sci., Sér. Sci. Math., Astronom. et Phys. 23 (1975), p. 1255-1263.

[2] —, — On the convergence and Riesz summability o f a Fourier-Bessel series, Jnânabha, Sect. A, 5 (1975), p. 145-160.

[3] N. K. Bary, A treatise on trigonometric series, Vols. I and II, Pergamon Press, 1964.

[4] D. P. G u p ta and B. P. K h o ti, On Riesz summability of a series o f Bessel functions, Publ. Math., Debrecen 17 (1970), p. 273-282.

[5] R. T a b e r sk i, Some properties o f Fourier-Bessel series, V, Bull. Acad. Polon. Sci., Sér.

Sci. Math., Astronom. et Phys. 15 (1967), p. 253-259.

[6] E. C. T itc h m a r sh , A class o f expansions in series o f Bessel functions, Proc. London Math. Soc. (2) 22 (1924), p. xiii-xvi.

[7] G. N. W a tso n , A treatise on the theory of Bessel functions, Cambridge, University Press, 1952.

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