Some estimates for the generalized Fourier transform associated with the Cherednik-Opdam operator on r Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2018, Volume 20, Issue 3, P. 78 86

y/IK 517.98

DOI 10.23671 /VNC.2018.3.18031

SOME ESTIMATES FOR THE GENERALIZED FOURIER TRANSFORM ASSOCIATED WITH THE CHEREDNIK-OPDAM OPERATOR ON R

S. El Ouadih1, R. Daher1, H. S. Lafdal1

1 Department of Mathematics, Faculty of Sciences Ai'n Chock, University Hassan II,

Route d'ElJadida, Km 8, B.P. 5366 Maarif 20100 Casablanca, Morocco E-mail: [email protected], [email protected], [email protected]

Abstract. In the classical theory of approximation of functions on R+, the modulus of smoothness are basically built by means of the translation operators f ^ f (x + y). As the notion of translation operators was extended to various contexts (see [2] and [3]), many generalized modulus of smoothness have been discovered. Such generalized modulus of smoothness are often more convenient than the usual ones for the study of the connection between the smoothness properties of a function and the best approximations of this function in weight functional spaces (see [4] and [5]). In [1], Abilov et al. proved two useful estimates for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus, using a translation operator. In this paper, we also discuss this subject. More specifically, we prove some estimates (similar to those proved in [1]) in certain classes of functions characterized by a generalized continuity modulus and connected with the generalized Fourier transform associated with the differential-difference operator Tin L^ (R)- F°r this purpose, we use a generalized translation operator.

Key words: Cherednik-Opdam operator, generalized Fourier transform, generalized translation. Mathematical Subject Classification (2010): 34K99, 42A63.

1. Introduction

In fl], Abilov et al. proved two useful estimates for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus, using a translation operator.

In this paper, we prove some estimates in certain classes of functions characterized by a generalized continuity modulus and connected with the generalized Fourier transform associated to Tin L2a ^(R) analogs of the statements proved in fl, 2-4]. For this purpose, we use a generalized translation operator.

In section 2, we give some definitions and preliminaries concerning the generalized Fourier transform. Some estimates are proved in section 3.

2. Preliminaries

In this section, we develop some results from harmonic analysis related to the differential-difference operator T(a>@\ Further details can be found in [5] and [6]. In the following we fix parameters a, /3 subject to the constraints a ^ /3 ^ — \ and a > —

© 2018 El Ouadih S., Daher R., Lafdal H. S.

Let p = a + P + 1 and A € C. The Opdam hvpergeometric functions G^«'f) on R are eigenfunctions T(a'f)Gl«'f) (x) = iAG(x) of the differential-difference operator

T^)f(x) = f{x) + [(2a + 1) coth x + (2fj + 1) tanh x] ^ ~ P/(-s)

that are normalized such that Gi«'f) (0) = 1. In the notation of Cherednik one would write

T(a'f) as

T(h + fc2)/(aO = /'(«) + {^bi + 13^} (/(®) " /(-*)) " ^ +

with o; = fci + k,2 — \ and /3 = k.2 — Here k\ is the multiplicity of a simple positive root and k.2 the (possibly vanishing) multiplicity of a multiple of this root. By [5] or [6], the eigenfunction G™ is given bv

G^\x) = ~ ^¿tf'V) = + ^ sinh^r1'^),

where =2 F\ -y-; a + 1; — sinh2 x^j is the classical Jacobi function.

Lemma 2.1 [7]. The following inequalities are valid for Jacobi functions ^>«'f(x)

(i) K'f(x)l < 1;

(ii) 1 - (x) < x2(A2 + p2).

Denote L« f (R), the space of measurable functions /on R such that 11/II2'«'f = i J f (x)|2A«'f (x) dx^ <

R '

where

Aaf (x) = (sinh |x|)2a+1(cosh |x|)2f+1.

The generalized Fourier transform of / € Cc(R) (the space of continuous functions on R with compact support) is defined by

H/(A) = J /(x)G{«'f)(-x)Aa,f (x) dx fa all A € C.

R

The inverse transform is given as

R

horo

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

Ca,l3{ AJ =

r(|(p + ¿A))rQ(a -/3 + 1 + ¿A))' The corresponding Plancherel formula was established in [5], to the effect that

J |/(x)|2A«'f (x) dx = |(|H/(A)|2 + |H/(A)|2) da(A),

R 0

where f(x) := f (-x^d da is the measure given bv

dA

da(A) =

According to [6] there exists a family of signed measures such that the product

formula

G(a.0 (^fl (y) = | ^(z) d^ (Z)

holds for all x,y € R and A € C, where

/

(x,y,z)Aaji3 (z) dz, xy = 0;

d^Of (z) =

d^ (z),

d^y(z),

y = 0;

x = 0

and

(x,y,z) = Ma,p| sinh x x sinhy x sinhz| 2a ^g(x,y,z,x)+

1 — ax + ax + ax +

-1- ^ X.7/.Z 1 1 w z.y.:X 1

cothx x cothy x coth z(sin x)2

(sin x)2^ dx,

if x, y, z € R\{0} satisfy the triangular inequality ||x — y|| < |z| < |x| + |y|, and Kajj(x, y, z) = 0 otherwise. Here

a

x =

x,y,z

cosh a;+cosh y—cosh z cos x / n.

sinh x sinh y ' XV T1 U'

0,

xy = 0

(Vx,y,z e R, x e [0,1])

and

g(x, y, z, x) = 1 — cosh2 x — cosh2 y x cosh2 z + 2 cosh x x cosh y x cosh z x cos x-

x, y € R

(i) Ka,fi(x,y,z) = Xa,fi(y,x,z);

(ii) (x,y,z) = Xa,fi(—x,z,y);

(iii) Kaji (x,y,z) = Kaji ( — z, y, —x).

The product formula is used to obtain explicit estimates for the generalized translation operators

f (y) = i f (z) d^ (z).

It is known from [6] that

H rX^/(A) = G(a'^)(x)H / (A),

(1)

for / e Cc

n

X

For a > — we introduce the Bessel normalized function of the first kind ja defined by

j 2 n

n\T(n + ck + 1)' In the terms of j«(x), we have (see [8])

VhxJa{hx) = 0{ 1), hx ^ 0, (2)

°Q /1 \ra/x Y

where J«(x) is Bessel function of the first kind, which is related to ja(x) by the formula

. , 2ar(a + 1) . .

w) =-^-J<*(XJ- (3)

Lemma 2.3 [9]. Let a ^ /3 ^ a / Then for \v\ ^ p, there exists a positive constant c0 such that

I1 - (x)l ^ co11 - ja(Ax)|.

For / € La f (R), we define the finite differences of first and higher order as follows: Ah/ = Ah/ = (ria' f) + T-h f) - 2/)/, Ah/ = Ah (Ah-1/) = (Tha'f) + T-ahf) - 2/) '/, k = 2,3,...,

where / is the unit operator in the space La f (R).

The generalized modulus of continuity of a function / € La f (R) is defined by

w(/,^2 'a ' f = sup |Ah/1|2 a f, 5> 0.

0<h^<S 2'a 'f

3. Main Result

The goal of this work is to prove some estimates for the integral

(/) = | (|H/(A)|2 + |H/(A)|2) da(A),

N

in certain classes of functions in La f Lemma 3.1. If/ € Cc(R), then

H ft ,ß)/ (A) = (-x)H/(A).

(4)

< For / € Cc(R), we have

Hfa' ß)/(A) = |, ß)/(-y)Ga ' ß)(-y)A« ,ß(y) dy = J, ß)/(y)Gia ' ß)(y)A« ,ß(y) dy

/(z)K , ß(x, y, z)Aa , ß (z) dz

Gia'ß)(y)A«;ß(y) dy

= / (z)

, y, z)Aa>ß (y) dy

Aa,ß(z) dz.

Since Ka,ß(x,y,z) = Käß(—x,z,y), it follows from the product formula that

(A) = Ga,ß)(-*)| / (z)Gia'ß)(z)A«;ß(z) dz

R

= G(a,ß)(-x) / /(_z)Ga'ß)(-z)Aa;ß(z) dz = G^(-x)H/ (A). >

Lemma 3.2. For / € L;,ß(R) then

II A/ Il2,a,ß = 22 f K'ß(h) - l|2fc (|H/(A)|2 + |Hf(A)|2) da(A).

< From formulas (1) and (4), we have

H(A/)(A) = (G^ß)(h)+ G^(-h) - 2)(/)(A)

A

and

H(Af)(A) = (g^(—h) + Go^(h) — 2) H(f)(A).

Since

and is even, then

H (Af)(A) = (h) — l) k H (f )(A)

H(Ahf)(A) = 2fc(y^(h) — l)'H(f)(A). > Now by Plancherel Theorem, we have the result.

Theorem 3.1. Given k and f € L2a ^(R). Then there exist a constant c > 0 such that, for all N > 0

jn (f ) = °(W(f,cN-

and

< Firstly, we have

J2(f) ^ J |ja(Ah)| d^(A) + J |1 — ja(Ah)| d^(A), (5)

NN

with d^(A) = (|Hf (A)|2 + |Hf (A)|2)da(A). The parameter h > 0 will be chosen in an instant. In view of formulas (2) and (3), there exist a constant ci > 0 such that

Then

J \ja(Xh)\dii(X) <Cl(hN)-a-hj%(f).

N

Choose a constant c2 such that the number C3 = 1 — C\C2 2 is positive. Setting h = c2/N in the inequality (5), we have

J(/) < J |1 - ja(Ah)| d^(A). (6)

N

Bv Holder inequality and Lemma 2.3 the second term in (6) satisfies

1 - ja (Ah) | d^(A) = J |1 - ja (Ah) | X 1 d^(A)

NN

\ 1/2fc / ^ 1—1/2fc J |1 - ja(Ah)|2k d^A)) ( J d^(A)

NN

\ 1/2fc

I |1 - ja(Ah)|2k d^(A H (Jn(/))2—1/k

N

1 / +r \1/2k

<¿1 y |l-^)|2fcdMA) {JN{f)?-l/k.

\ AI '

N

From Lemma 3.2, we conclude that

+^0

y |1 - (h)i2k d^(A) < ¡Ah/112,a,ß'

N

Therefore

y ii-i^A^i^^^iiAh/ii^^a))2-1^.

N

For h = c2/N, we obtain

Consequently by raising both sides to the power k and simplifying by (JN(f))2k we finally obtain

for all N > 0. The theorem is proved with c = c2. >

Theorem 3.3. Let f € L2a fi(R). Then, for all N > 0,

N \ 1/2N

"(/> N—1Vß = O | N—(l + 1)4k—1 Jl2(/)

1=0

< From Lemma 3.2, we have

l|Ah/||2'a'f = 22k J K'f(h) - 1|2k (|H/(A)|2 + |H/(A)|2) da(A). 0

This integral is divided into two

N

1=1+1= '' +

0 0 N

where N = [h-1]. We estimate them separately. From (i) of Lemma 2.1, we have the estimate

/2 < */ (|H/(A)|2 + |H/(A)|2) da(A) = C4JN(/).

N

/1

N

/1 < h4k |(A + p)4k (|H/(A)|2 + |H/ (A)|2) da(A) 0

N — 1 ¿+1

= h4k ^ / (A + p)4k (|H/(A)|2 + |H/ (A)|2) da(A) !=0 f

N -1

< h4^ (l + p +1)4k (J2(/) - J2+1(/)) . 1=0

From the inequality l + p + 1 ^ (p + 1)(1 + 1) we conclude

N -1

/1 < (p + 1)4kh4fc£ a, (J,2(/) - Ji2+1(/)) 1=0

with a, = (l + 1)4fc.

For all integers m ^ 1, the Abel transformation shows

m m

(J,2(/) - J,2+1(/)) = a0 JQ (/) + - a,-1) J,2(/) - am^+1 (/) 1=0 1=1

m

< aoJo2(/) + £> - a,-1) J,2(/), ,=1

because amJm+1(/) ^ 0. Hence

/1 < (p + 1)4kN-4k J2(/) + £ ((l + 1)4k - l4') J,2(/)) '

since N ^ 1/h Moreover by the finite increments theorem, we have (l + 1)4k — 14k ^ 4k(1 + 1)4k-1. Then

h < (p+i)4kn-4k J/) + 4k ¿; (i + i)4k-1 Ji2(/^.

Combining the estimates for ^ and 12 gives

K/II2,^ = ° (N-4k 2 (i + i)4k-1 J2(/^,

which implies

"(/, N-1) W = O ^N-2k (l + 1)4k-1Jl2(/)) j , and this ends the proof. >

References

1. Abilov V. A., Abilova F. V. and Kerimov M. K. Some Remarks Concerning the Fourier Transform in the Space L2(R), Comput. Math. Math. Phys., 2008, vol. 48, pp. 885-891.

2. flamma M. E., Daher R. and Khadari A. On Estimates for the Dunkl Transform in the Space L2(Rd,wk(x) dx), Ser. Math. Inform., 2013, vol. 28, no. 3, pp. 285-296.

3. Daher R., Ouadih S. E. Certain Problems on the Approximation of Functions by Fourier-Jacobi Sums in the Space L^T'^K, Alabama J. Math., 2016, vol. 40, pp. 1-4.

4. Ouadih S. E., Daher R. New Estimates for the Generalized Fourier-Bessel Transform in the Space L0,,ni South. Asian Bull. Math., 2017, vol. 41, pp. 209-218.

5. Opdam E. M. Harmonic Analysis for Certain Representations of Graded Hecke Algebras, Acta Math,., 1995, vol. 175, no. 1, pp. 75-121.

6. Anker J. P., Ayadi F., and Si£ M. Opdams Hypergeometric Functions: Product Formula and Convolution Structure in Dimension 1, Adv. Pure Appl. Math,., 2012, vol. 3, no. 1, pp. 11-44.

7. Platonov S. S. Approximation of Functions in L2-Metric on Noncompact Rank 1 Symmetric Space, St. Petersburg Mathematical J., 2000, vol. 11, no. 1, pp. 183-201.

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9. Bray W. O. and Pinsky M. A. Growth Properties of Fourier Transforms via Module of Continuity, J. Fund. Anal., 2008, vol. 255, pp. 2256-2285.

Received February 24, 2016 Final version January 19, 2018

Владикавказский математический журнал 2018, Том 20, Выпуск 3, С. 78^86

НЕКОТОРЫЕ ОЦЕНКИ ДЛЯ ОБОБЩЕННОГО ПРЕОБРАЗОВАНИЯ ФУРЬЕ, АССОЦИИРОВАННОГО С ОПЕРАТОРОМ ПЕРЕДНИКА - ОПДАМА

Эл Оуади С.1, Дагер Р.1, Лафдаль X. С.1

1 Департамент математики, Факультет наук, Университет Хасана II, Марокко, Маариф 20100 Касабланка E-mail: salahwadih@gmail. com, [email protected], [email protected]

Аннотация. В классической теории приближения функций на К+, модуль гладкости в основном строится посредством операторов сдвига f (•) ^ f (• + y). Поскольку понятие оператора сдвига было расширено в различных направлениях (см. [2] и [3]), были обнаружено много других обобщенных модулей гладкости. Часто при изучения взаимосвязи свойств гладкости функции и наилучшего приближения этой функции в весовых функциональных пространствах такие обобщенные модули гладкости оказываются более удобными, чем обычные (см. [4] и [5]). В работе [1] Абилов и др. для преобразования Фурье в пространстве квадратично интегрируемых функций доказали с использованием оператора сдвига две полезные оценки на некоторых классах функций, характеризуемых обобщенным модулем непрерывности. В данной статье мы также обсуждаем этот вопрос. Более конкретно, мы доказываем некоторые оценки (аналогичные доказанным в [1]) в классах функций, характеризуемых обобщенным модулем непрерывности и связанных с обобщенным преобразование Фурье, ассоциированное с дифференциально-разностным оператором T(а'в) в пространстве L^e (R)-Для этой цели мы используем обобщенный оператор сдвига.

Ключевые слова: оператор Чередника — Опдама, обобщенное преобразование Фурье обобщенный сдвиг.

Mathematical Subject Classification (2000): 34К99, 42А63.