Научная статья на тему 'JACOBI TRANSFORM OF (ν,γ,P)-JACOBI-LIPSCHITZ FUNCTIONS IN THE SPACE LP(ℝ+, Δ(α,β)(T)DT)'

JACOBI TRANSFORM OF (ν,γ,P)-JACOBI-LIPSCHITZ FUNCTIONS IN THE SPACE LP(ℝ+, Δ(α,β)(T)DT) Текст научной статьи по специальности «Математика»

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JACOBI OPERATOR / JACOBI TRANSFORM / GENERALIZED TRANSLATION OPERATOR

Аннотация научной статьи по математике, автор научной работы — El Hamma Mohamed, Lafdal Hamad Sidi, Djellab Nisrine, Khalil Chaimaa

Using a generalized translation operator, we obtain an analog of Younis’ theorem [Theorem 5.2, Younis M. S. Fourier transforms of Dini-Lipschitz functions, Int. J. Math. Math. Sci., 1986] for the Jacobi transform for functions from the (ν,γ,p)-Jacobi-Lipschitz class in the space Lp(ℝ+, Δ(α,β)(t)dt).

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Текст научной работы на тему «JACOBI TRANSFORM OF (ν,γ,P)-JACOBI-LIPSCHITZ FUNCTIONS IN THE SPACE LP(ℝ+, Δ(α,β)(T)DT)»

URAL MATHEMATICAL JOURNAL, Vol. 5, No. 1, 2019, pp. 53-58

DOI: 10.15826/umj.2019.1.006

JACOBI TRANSFORM OF (v, Y,p)-JACOBILIPSCHITZ FUNCTIONS IN THE SPACE Lp(R+, A(a,ß)(t)dt)1

Mohamed El Hamma1^, Hamad Sidi Lafdal11, Nisrine Djellab1,

Chaimaa Khalil1

1 Laboratoire TAGMD, Faculté des Sciences Aïn Chock, Université Hassan II, B.P 5366 Maarif, Casablanca, Marocco

11CRMEF, Laayoune, Morocco

tm_elhamma@yahoo.fr

Abstract: Using a generalized translation operator, we obtain an analog of Younis' theorem [Theorem 5.2, Younis M.S. Fourier transforms of Dini-Lipschitz functions, Int. J. Math. Math. Sci., 1986] for the Jacobi transform for functions from the (v, 7,p)-Jacobi—Lipschitz class in the space Lp(R+,

Keywords: Jacobi operator, Jacobi transform, Generalized translation operator.

Younis [8, Theorem 5.2] characterized the set of functions in L2(R) satisfying the Dini-Lipschitz condition by means of an asymptotic estimate of the growth of the norm of their Fourier transforms.

Theorem 1. [8, Theorem 5.2] Let f € L2(R). Then the following conditions are equivalent:

where F stands for the Fourier transform of f.

The main aim of this paper is to establish an analog of Theorem 1 for the Jacobi transform in the space Lp(R+, A(a,^)(t)dt). For this purpose, we use a generalized translation operator which was defined by Flensted-Jensen and Koornwinder [5].

In order to confirm the basic and standard notation, we briefly overview the theory of Jacobi operators and related harmonic analysis. The main references are [1, 4, 6].

Let A € C, a > ft > -1/2, and a = 0. The Jacobi function of order (a,fi) is the unique even C^-solution of the differential equation

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1. Introduction and preliminaries

as h ^ 0, 0 < a < 1, ß > 0,

(2) j|A|>r |F(f)(A)|2d\ = O(r-2a(logr)-2ß) as r ^

(Da,ß + A2 + p2)u = 0, u(0) = 1, u'(0) = 0, where p = a + ß + 1, Da,ß is the Jacobi differential operator defined as

1 Dedicated to Professor Radouan Daher for his 61's birthday.

with

A(a>/s)(x) = (2 sinh x)2a+1 (2 cosh x)2fi+1,

and A(a^)(x) is the derivative of A(a ^)(x).

The Jacobi functions can be expressed in terms of Gaussian hypergeometric functions as

4>\(x) = 4>^\ 'l3\x) (^(p — i\), ^(p + 'iA), a + l, —sinh2®

where the Gaussian hypergeometric function is defined as

^ cmm\

with a,b,z € C, c / —N, a0 = 1, and am = a(a + 1) ■ ■ ■ (a + m — 1).

The function z ^ F(a, b, c, z) is the unique solution of the differential equation

z(1 — z)u"(z) + (c — (a + b + 1)z)u' (z) — abu(z) = 0,

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which is regular at 0 and equals 1 there.

From [7, Lemmas 3.1-3.3], we obtain the following statement.

Lemma 1. The following inequalities are valid for a Jacobi function <fi\(t) (A,t € R+):

(1) I0a(t)| < 1;

(2) |1 — 0A(t)|< t2(A2 + p2);

(3) there is a constant d> 0 such that

1 — 0A(t) > d for At > 1.

Let L^(R+) = Lp(R+, A(a ^)(t)dt), 1 < p < 2, be the space of p-power integrable functions on R+ endowed with the norm

\ 1/p

|f (x)|PA(«„3)(x)dxJ < ro.

Let L^(R+) = Lp(R+, dp(A)/2n), 1 < p < 2, be the space of measurable functions f on R+ such

l p

that

/ 1 rro \ 1 /p

1 fWMV) '

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where dp(A) = |c(A)|-2dA and the c-function c(A) is defined as

2p-iXr(a + 1)r(iA)

c(A) =

r(1/2 ■ (i\ + a + p + 1))r(1/2 ■ (i\ + a - p + 1)) ' Now, we define the Jacobi transform

^ c ^

/(A) = f (x)0A(x)A(«,^)(x)dx,

J 0

for all functions f on R+ and complex numbers A for which the right-hand side is well defined. The Jacobi transform reduces to the Fourier transform when a = p = -1/2. We have the following inversion formula [6].

Theorem 2. If f € L^(R+), then

1 f ~ -

/0*0 = — j /(A)</>a(-*(A).

From [3], we have the Hausdorff-Young inequality

ll/IU < C21|f ||p for all f € La,,

where 1/p + 1/q = 1 and C2 is a positive constant.

The generalized translation operator Th of a function f € La,(R+) is defined as

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Thf (x)= / f (z)K (x,h,z)A(a,) (z)dz, Jo

where K is an explicity known kernel function such that

v = 2~2pr(q + I)(cosh x cosh y cosh z)a~l3~1 _ r2^«-i/2 [■hy,«) r(I/2)r(o; + I/2)(sinh.Tsinhj/sinhz)2Q' ;

xF ^a + /3, a — /3, a + ^(1 — for \x — y\ < z < x + y,

and K(x,y,z) =0 elsewhere and

cosh2 x + cosh2 y + cosh2 2 — 1 2 cosh x cosh y cosh z

From [2], we have

(Tf )(A) = 0A(h)fr(A). 2. Main results

In this section, we give the main result of this paper. We need first to define the (v, Y,p)-Jacobi-Lipschitz class.

Definition 1. Let v,7 > 0. ^ function f € La,(R+) is said to be in the (v,Y,p)-Jacobi-Lipschitz class, denoted by Lip(v, Y,p), if

v

l|Tfe/(.T)-/(.T)||p = o((log^//j)7j as h y 0.

Theorem 3. Let f belong to Lip(v, Y,p). Then

/ |/(A)|fd^(A) = O(N-fv(logN)-fY) as N ^ Jn

Proof. Let f € Lip(v, Y,p). Then we have

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hv

! as h —>• 0.

Y

Therefore,

/ |1 - ^(h)|fl/(A)|fd^(A) < Cf||Thf(x) - f(x)|p. o

If A € [1/h, 2/h], then Ah > 1 and inequality (3) of Lemma 1 implies that

1 <¿11-^)1**.

Then

r 2/h 1 r 2/h

/ |/(A)|qcfyt(A) < — / |1 - Mh)\qk\f(\)\qdAX) J1 /h J 1/h

1 r+,x , - 1 f hqv

"Wo 11 " <^(/>)n/(A)№(A) < ^C*||Tft/(:r) - f(x)\\l = O (j '

/0 I UV ,1 J - dqk ¿H j J Wllp V(log1/h)qY

Then

/• 2N

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/ |f(A)|qdp(A)= O(N-qv(log N)-qY)) as N ^ +to. JN

Thus, there exists C4 such that

f 2N

/ |f(A)|qdp(A) < C4N-qv(log N)-qY.

N

Furthermore, we have

+,

/ |/(A)|q dp(A) =

N

/•2N /-4N /-8N / + / + /

N 2N 4N

|/(A)|q dp(A)

< C4^q"(logN+ C4(2N)"qiy(log2N)"q7 + C4(4N)-qv(log4N)-qY + ...

< C4N-qv(logN)-qY(1 + 2-qv + (2-qv)2 + (2-qv)3 + ...

< C4CkN-qv(log N)-qY),

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where Ck = (1 - 2-qv)-1 since 2-qv < 1. This proves that

/ |/(A)|qdp(A) = O(N-qv(logN)-qY)) as N ^ +to,

N

and this completes the proof. □

Definition 2. A function f € L^ ^(R+) is said to be in the (0,p)-Jacobi-Lipschitz class, denoted by Lip(0,p), if

where

(1) 0(t) is a continuous increasing function on [0, to);

(2) 0(0) = 0;

(3) 0(ts) < 0(t)0(s) for all s,t € [0, to).

Theorem 4. Let f € L^ ^(R+), 0 be a fixed function satisfying the conditions of Definition 2, and let f (x) belong to Lip(0,p). Then

r+,

/ |/(A)|qdp(A) = O(0(N-q)(logN)-qY) as r ^ +to.

N

Proof. Let f € Lip(^,p). Then we have

llT^)-/(g)llp = Q((log(l/i)7) as /w0 r+m

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/ |1 - 0A(h)|f |/(A)|fd^(A) < Cf ||Thf (x) - f (x)||p. o

If A € [1/h, 2/h], then Ah > 1 and, similarly to the proof of Theorem 3, by inequality (3) of Lemma 1, we obtain

1 <¿11-^)1**.

Then

r2/h 1 r2/h

/ |/(A)№(A) <-j: |1 - MhW'mWdAX)

J1/h J1/h

There exists a positive constant such that

/•2N ^ )

N 5 logN)<n

Thus,

/ |/(A)|q = In

r2N MN /-8N

/ + +

/N ./2N ./4N

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l/(A)|q d^(A)

~ 5 (log N)q~f 5 (log 2N)<n 5 (log 4N)<n "

" 5 (log N)<n 5 (logiV)9T + 5 (logiV)97 < gs^^l + '0(2-") + C0(2-«))2 + W(2"*))3 + ...

- (log N)fY'

where K1 = (1 - ^(2-f))-1 since (1) and (3) from Definition 2 imply that ^(2-f) < 1. This proves that

r

/ |/(A)|fd^(A) = O(^(N-f)(logN)-fY) as N JN

and this completes the proof. □

Acknowledgements

The authors would like to thank the referee for his valuable comments and suggestions.

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