This work was intended as an attempt to extend the results on localization of Fourier-Laplace series to the spectral expansions of distributions on the unit sphere. It is shown that the spectral expansions of the distribution on the unit sphere can be represented in terms of decompostions of Laplace-Beltrami operator. It was of interest to establish sufficient conditions for localization of the spectral expansions of distribution to clarify the latter some relevant counter examples are indicated

Ahmedov, Anvarjon

Nurullah, Ahmad Fadly

Rakhimov, Abdumalik

This work was intended as an attempt to extend the results on localization of

Fourier-Laplace series to the spectral expansions of distributions on the unit sphere.

It is shown that the spectral expansions of the distribution on the unit sphere can be

represented in terms of decompostions of Laplace-Beltrami operator. It was of

interest to establish sufficient conditions for localization of the spectral expansions

of distribution to clarify the latter some relevant counter examples are indicated

Akhmedov, Anvarjon

Rasedee, Ahmad Fadly Nurullah

Rakhimov, Abdumalik A.

The International Islamic University Malaysia Repository

Localization of Fourier-Laplace series of distributions

Universiti Putra Malaysia Institutional Repository

Malaysian Journal of Mathematical Sciences 7(2): 315-326 (2013)      Localization of Fourier-Laplace Series of Distributions     1,2Anvarjon Ahmedov, 2*Ahmad Fadly Nurullah  and 3Abdumalik Rakhimov   1Department of Process and Food Engineering, Faculty of Engineering, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia  2Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia  3Department of Science in Engineering, International Islamic University Malaysia, 50728 Kuala Lumpur, Malaysia  E-mail: anvar@eng.upm.edu.my and ahmadfadlynurullah@yahoo.com  *Corresponding author   ABSTRACT This work was intended as an attempt to extend the results on localization of Fourier-Laplace series to the spectral expansions of distributions on the unit sphere. It is shown that the spectral expansions of the distribution on the unit sphere can be represented in terms of decompostions of Laplace-Beltrami operator. It was of interest to establish sufficient conditions for localization of the spectral expansions of distribution to clarify the latter some relevant counter examples are indicated.    Keywords: Distributions, Fourier-Laplace series, localization, Riesz means, Laplace-Baltrami operator.    1.  INTRODUCTION Let NS  be unit sphere in 1.NR +    1 2 2 2 21 2 1= { :| | = ... = 1}.N N NS x R x x x x+ +∈ + + +   The sphere NS is naturally equipped with a positive measure ( )d xσ  and with an elliptic second order differential operator - s∆ , namely the Laplace-MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES  Journal homepage: http://einspem.upm.edu.my/journal Anvarjon Ahmedov, Ahmad Fadly Nurullah & Abdumalik Rakhimov  316 Malaysian Journal of Mathematical Sciences  Beltrami operator on the sphere. This operator is symmetric and nonnegative, and it can be extended to a nonnegative self-adjoint operator on the space ( )2 NL S . Where ( )2 NL S  denoted the 2L -space associated with the measure ( )d xσ  on the sphere. For the self-adjoint extension of the Laplace-Beltrami operator we use again the same symbol s∆ and by { }, = 0,1,2,....k kλ  we denote the set of the eigenvalues of the Laplace-Beltrami operator ,s∆ which is increasing sequence of nonnegative eigenvalues = ( 1), = 0,1,2,...k k k N kλ + − with finite multiplicities 10 1( )! ( 1)!=1, = , = , 2! ! !( 2)!NkN k N ka a N a k kN k N k−+ + −− ≈ ≥− (and written as such) tending to infinity. We denote by ( )kjY x  the eigenfunctions of the Laplace-Beltrami operator corresponding to kλ :   ( ) ( )( ) = , = 1,2,..., ; = 0,1,2,... .k ks j k j kY x Y j a kλ∆   The system of eigenfunctions of the Laplace-Beltrami operator form an orthonormal basis in  ( )2 NL S . To any measurable function f  we assign its spectral expansion:                                     =0( ) = ( , ),kkf x Y f x∞∑                                        where  ( ) ( )=0( , ) = ( ) ( ) ( ) ( ), = 0,1,2,...akk kk j jj NSY f x Y x f y Y y d y kσ∑ ∫  The operator  ( ) = ( ) ( , , ) ( )NSE f x f y x y d yλ λ σΘ∫  is called spectral expansions of f , where  ( ) ( )< =1( , , ) = ( ) ( ).akk kj jjkx y Y x Y yλ λλΘ ∑∑    In this work we investigate spectral expansions related to distributions. We denote by ( ) ( )=N ND S C S∞  the space of infinite times differentiable functions with the topology of uniformly convergence with Localization of Fourier-Laplace Series of Distributions    Malaysian Journal of Mathematical Sciences 317  respect to derivatives of all order. Let ( )' ND S  denotes the conjugate space of ( )ND S  i.e. the space of the linear functionals defined on ( )ND S  which are called distributions on sphere. From ellipticity of Laplace-Beltrami operator we obtain that ( )NE C Sλϕ ∞∈  for all ( )NC Sϕ ∞∈ . With the help of the latter remark we define:  Definition. The spectral expansions of the distribution ( )' Nf D S∈  is distribution ( )' NE f D Sλ ∈  defined as follows:  < , >=< , >,E f f Eλ λϕ ϕ  for any ( )ND Sϕ ∈ .  The correctness of this definition can be established using the fact that the operator λE  is linear and bounded from ( )ND S  to ( )' ND S . Furthermore we have   Theorem 1.1  Let ( )ND Sϕ ∈ , then the partial sums of the Fourier-Laplace series of the ϕ  converges in the topology of the space ( )ND S    ( ) ( ).nE x xϕ ϕ→   From here we obtain the following   Theorem 1.2  Let ( )' Nf D S∈ , then the partial sums of the Fourier-Laplace series of the f  converges in the topology of the space ( )' ND S    ( )< ( ), ( ) > < ( ), ( ) >, .NnE f x g x f x g x g D S→ ∀ ∈     Our aim is to prove convergence of  E fλ  in classical means, but it can be proved only in domains where f  coincides with local integrable functions.  Anvarjon Ahmedov, Ahmad Fadly Nurullah & Abdumalik Rakhimov  318 Malaysian Journal of Mathematical Sciences  For any distribution ( )' Nf D S∈  we assign its Riesz means of the spectral expansions of f  as follows  =0 =1( ) = < , ( , , ) > = 1 ( ) < , ( ) >,an kk kkn j jk jnE f x f x y n Y x f Y yαα α λλ+ Θ −  ∑ ∑   where < , ( ) >kjf Y y  is the value of the functional f  on the eigenfunction ( )kjY y  of the Laplace-Beltrami operator.   Ashurov and Anvarjon (2011) considered the spectral expansions of functions from Nikol’skii classes ( )a NpH R  related to selfadjoint extensions of elliptic differential operators A(D) of order m  in .NR  They then constructed a continuous function from Nikol’skii class with >p a N⋅  such that the Riesz means of the spectral expansion diverge at the origin. This result demonstrates the sharpness of the >p a N⋅  condition by Alimov (1976) for uniform convergence of spectral expansions, related to elliptic differential operators.   In 1983 Meaney (1984) presented a localization theorem for expansions in eigenfunctions of the Laplace–Beltrami operator on a compact two-point homogeneous space. When 1=X S  the theorem proved is just the usual localization principle for pseudo functions. The proof of his theorem mimics that of the one-dimensional case. In his paper, he manages to avoid the issue of localization of summability methods, as treated in Bonami and Clerc (1973).  Results in the work of Zhizhiashvili and Topuriya (1979) discuss the theory of Fourier-Laplace series on a N-dimensional sphere where 3.N ≥ The theory of the Fourier-Laplace series has been developed intensively in various directions. From the entire variety of problems investigated in this field, the woks in Zhizhiashvili and Topuriya (1979)  restrict the presentation of the theory of these series on a N-dimensional sphere 3.N ≥   Anvarjon, Norashikin in Anvarjon and Norashikin (2012) proved a localization theorem for nonspherical partial sums, that is, for Fourier series under summation over domains bounded by level surfaces of elliptic polynomials. The result provides a natural and intrinsic characterization of the approximation of the functions by Fourier–Laplace series.  Localization of Fourier-Laplace Series of Distributions    Malaysian Journal of Mathematical Sciences 319  For the latest investigations related to the convergence problems of the spectral expansions of the elliptic differential operators we refer the readers to Ashurov and Anvarjon (2010), Ashurov et al. (2010), Anvarjon (2010), Alimov (2006) and what we personally believe that provides a more complete insight is Alimov et al. (1992).  We define the Sobolev space ( )s NpH S as follows: for 1 < <p ∞ , > 0s , ( )s NpH S  is the collection of all ( )Npf L S∈  such that  ( ) 2= ssf I h−− ∆  for some ( )Nph L S∈  with the norm   ( ) ( ) .s N Np pH S L Sf h=   If < 0s ; ( )s NpH S  is the collection of all ( )' Nf D S∈  of the form ( )= ksf I h− ∆  with ( )2k s Nph H S+∈ , where k  is a natural number such that 2 > 0k s+  and  2( ) ( ) .s N k s Np pH S H Sf h +=    If = 0s , ( ) ( )0 =N Np pH S L S .  If = 2p , we have  ( ) .)(>)(,<1=2=120=2∞⋅+ ∑∑NSLkjkjkajlkkNSsHxYYff λ    It is important to remark here that the Nikolskii classes ( )2l NH S  are Hilbert spaces and Schwartz space ( )ND S  is dense in ( )2 , > 0l NH S l . The equivalent definition of the norm in Nikolskii spaces ( )2 , > 0l NH S l− may be given by the following   1222( )0 ( ) ( ),supNl Nl NH Su H S H Sf ufu−≠ ∈< >=                                  (2)  Anvarjon Ahmedov, Ahmad Fadly Nurullah & Abdumalik Rakhimov  320 Malaysian Journal of Mathematical Sciences  2. MAIN RESULTS The main results of the work are the following:   Theorem 2.1  Let 2 , > 0lf H l−∈  and f  vanishes in some domain .NV S⊂  If 1,2Nlα −≥ + then the Riesz means ( )nE f xα  uniformly converges to  0  on any compact .K V⊂     In the case when distribution coincides with normal function we have,   Theorem 2.2  Let a distribution 2 , > 0lf H l−∈  coincides with the continuous function ( )g x  in some domain .NV S⊂   If 12Nlα −≥ +  then  ( ) = ( )lim nnE f x g xα→∞ in any compact .K V⊂   Proof of Main Results The proof of the main facts are based on the following Lemmas.   Lemma 2.3  Let NV S⊂  be a domain and K V⊂  be any compact set. Then uniformly on x K∈  one has   ( )12\2( , , ) .NNL S Vx y n Cnαα−−Θ ≤   Proof. The proof is straightfoward. From ( )\ = ØNS V K∩  we conclude that there exist 0 > 0r  such that, for any \Ny S V∈  and x K∈  one has 0( , ) .x y rγ ≥  Based on the later we divide the integration region \NS V  into two parts as follows:  122\( , , ) ( )NS Vx y n d yα σ  Θ ≤  ∫   1 12 22 21 21 10( , , ) ( ) ( , , ) ( ) = .rn nx y n d y x y n d y I Iα αγ pi pi γ piσ σ≤ ≤ − − ≤ ≤      ≤ Θ + Θ +         ∫ ∫  Localization of Fourier-Laplace Series of Distributions    Malaysian Journal of Mathematical Sciences 321  To estimate the 1I  and 2I  we use the following:   Lemma 2.4  Let ( , , ),x y nαΘ  be the kernel of Riesz means of the spectral expansions  1. If  ,2 1 2nnpi piγ− <+ ,n → ∞  then one has  ( ) ( )( ) ( ) ( )( )( 1)/2 ( 3)/21 1( 1)/2 ( 1)/2( , , ) = (1)sin sin /2 sin sin /2N NN Nn nx y n Oαα αγ γ γ γ− −+ +− +Θ +  ( )( )11 ;sin /2 Nnγ−++  2. if 0 γ pi≤ ≤ , > 1n  then one has   ( , , ) = (1) ,Nx y n O nαΘ  3. if 00 γ γ pi≤ ≤ ≤ , > 1n  then one has   1( , , ) = (1) ,Nx y n O nα α− −Θ   for the proof see Anvarjon (2011).  Using this Lemma we first estimate 1I  as follows   ( ) ( )1 31 12 21 1 21 10 0sin ( ) sin ( )N NN Nr rn nI c n d y c n d yα αγ pi γ piγ σ γ σ− −− −− + − −≤ ≤ − ≤ ≤ −≤ +∫ ∫  11 23 110( ) .Nrnc n d y c nαγ piσ−−−≤ ≤ −+ ≤∫    Anvarjon Ahmedov, Ahmad Fadly Nurullah & Abdumalik Rakhimov  322 Malaysian Journal of Mathematical Sciences  For the estimation of 2I  we use (3) of Lemma 2.4. We have   11 1 2 22 41( ) = .N NN NnI c n d y Cn n Cnαα αpi γ piσ− − −− − − −− ≤ ≤≤ ≤∫   Finally the norm of the Riesz means of the spectral function of the Laplace-Beltrami operator can be estimated as follows:  ( )121 2\2( , , ) .NNL S Vx y n I I Cnαα−−Θ ≤ + ≤  Lemma 2.3 is proved.  We denote                               ( ) ( ) ( )22, , = 1 , ,ll sx y n x y nα αΘ − ∆ Θ                       Using the summation by parts we have   ( )2=0 =12( , , ) = 1 1 ( ) ( )an l kk kkl k j jk jnx y n Y x Y yαα λ λλ Θ − +  ∑ ∑   ( ) ( ) ( ) ( ) ( )1 2 2 21=0= 1 1 , , 1 , , .n l l lk k nkx y k x y nα αλ λ λ−+ + − + Θ + + Θ  ∑ The norm of ( )2, ,l x y nαΘ  can be estimated as follows  2212 2102( , , ) (1 ) (1 ) ( , , )l lnl k k LkLx y n x y kα αλ λ−+=Θ ≤ + − + Θ∑   22(1 ) ( , , ) .ln Lx y nαλ+ + Θ                                  (4)  For the second summand we have   212 2( \ )(1 ) ( , , ) (1 ) .l Nl N ln L S Vx y n C nααλ−− ++ Θ ≤ +   Localization of Fourier-Laplace Series of Distributions    Malaysian Journal of Mathematical Sciences 323  Here we have used Lemma 2.4. For any > 0l  the following estimation holds  ( ) ( ) 12 211 1 (1 ) , = 1, 2, ...,l llk k C k k Kλ λ −++ − + ≤ +   Applying the latter to the first summand in (4) gives  211 112 2 21 1 2( \ )0 1(1 ) (1 ) (1 ) (1 ) .l Nl l Nn nlk k k L S Vk kc k c kααλ λ−− −−−+= = + − + Θ ≤ + +  ∑ ∑     Finally we obtain   2122 ( \ )( , , ) .l NN llL S Vx y n Cnαα−− +Θ ≤   Lemma 2.5  Let ( )2 , > 0l Nf H S l−∈  and let V  be some domain in .NS   If = 0f  in ,V  then one has   12( ) ,N ln lE f x Cn fαα−− +−≤   uniformly in any compact .K V⊂   Proof. From that f  vanishes in the domain ,NV S⊂  and from (2) we obtain for any ( )Nu C S∞∈    ( )\2< , > .l Nl H S Vf u f u−≤                           (5)  Note that the operators ( ) 21 , > 0ls l−− ∆  are continuous from 2L  to 2 , .lH i e  for any 2u L∈  we have,  2 222(1 ) .llsH LLu u u−= − ∆ ≤                                (6)     Anvarjon Ahmedov, Ahmad Fadly Nurullah & Abdumalik Rakhimov  324 Malaysian Journal of Mathematical Sciences  Taking into account (5) and (6) one has   2 ( \ )( ) , ( , , ) ( , , ) l Nn l H S VE f x f x y n f x y nα α α−= < Θ > ≤ Θ   21 122 ( \ )( , , ) (1 ) ,NNll lL S Vf x y n C n fαα−− +− −Θ ≤ +   where 1 .2N lα −≥ +   Proof of Theorem 2.2.  Let ( )2 , > 0l Nf H S l−∈  and f  coincides with g  in ,NV S⊂  where ( )g x  is continuous function on ( ).NS  For all functions ( )Ng C S∈  Riesz means ( )nE f xα  uniformly converges to g  on any compact .NK S⊂   Let consider a new distribution ( ) = ( ) ( ),F x f x g x x V− ∈  and ( ) = ( ), \ .NF x f x x S V∈  It is obvious that ( )( ) l NF x H S−∈  and vanishes on .V  Then for any compact K V⊂  we have   1 12( ) (1 ) .Nn lE F x C n Fαα−− +−≤ +   By choosing ,lα , such that 1>2N lα − +  we can make ( ) 0nE F xα → , as n → ∞ .  The representation ( ) = ( ) ( )n n nE f x E F x E g xα α α+   and the fact that ( )nE g xα  uniformly converges to )(xg  on any compact K V⊂  completes the proof of Theorem 2.2.    Localization of Fourier-Laplace Series of Distributions    Malaysian Journal of Mathematical Sciences 325  ACKNOWLEDGEMENT  This paper has been supported by Ministry of Higher Education (Mohe) for its MyPhd sponshorship and Universiti Putra Malaysia under Research University Grant (RUGS), project number 05-01-09-0674RU.   REFERENCE Alimov, Sh.A. 2006. Dependence of the convergence domain of spectral expansions on the geometry of the set of discontinuity of the function being expanded. Mathematical Notes. 79(1-2): 165-177.  Alimov, Sh. A. and Ashurov, R. R. 1992. Multiple Fourier Series and Fourier Integrals: Commutative Harmonic Analysis –IV. Springer-Verlag,  1-97.   Anvarjon, A. and Ashurov, R. R. 2011. On the divergence of spectral expansions of elliptic differential operators. Malaysia Journal of Mathematical Sciences. 5(2): 185-196.  Anvarjon, A. and Norashikin Abdul Aziz. 2012. Approximation of functions from LlogL (loglogL) ( )NS  by Fourier-Laplace series. Journal of Computational and Applied Mathematics. 236(15): 3827-3835.  Anvarjon, A. 2011. The estimations for maximal operators. Applied Mathematics Letter. 24(5): 615-619.   Anvarjon, A. 2010. The Generalized Localization for the N-fold Fourier Integrals. Malaysian Journal of Mathematical Sciences. 4(2): 209-227.  Ashurov R. R. and Anvarjon, A. 2010. On summability of N-fold Fourier integrals corresponding to pseudodifferential operators. Journal of Pseudo-Differential Operators and Applications. 1(4): 417-432   Ashurov R. R., Anvarjon, A. and Ahmad Rodzi. 2010.  The generalized localization for multiple Fourier integrals. Journal of Mathematical Analysis and Applications. 371 (2): 832-841.   Anvarjon Ahmedov, Ahmad Fadly Nurullah & Abdumalik Rakhimov  326 Malaysian Journal of Mathematical Sciences  Bonami, A. and Clerc, J. L. 1973. Sommes de Cesa’ro et multiplicateues de de’veloppements et harmonique sphe’riques. Trans. Amer. Math. Soc. 183:183-223.   Li, L. and Yu, C. 2007. Convergence rate of Cesaro means of Fourier-Laplace series. Journal of Mathematical Analysis and Applications. 325(2): 808-818.  Meaney, C. 1984. Localization of spherical harmonics expansion. Monatsh Math, 65-74.   Rakhimov, A. A. 1983. On the almost everywhere localization of Fourier-Laplace series. Manuscript of Diploma Thesis, Preprint Tashkent State University, 1-14.   Rakhimov, A. A. 2003. On the localization conditions of the Riesz means of distribution expansions of the Fourier series by eigenfunctions of the Laplace operator on sphere. J. Izvestiya Vuzov, Phys. Math. Sci. (3-4): 47-50.   Zhizhiashvili, L.V. and Topuriya S.B. 1979. Fourier-Laplace series on a sphere. Journal of Soviet Mathematics. 12(6): 682-714.    

Localization of fourier-laplace series of distributions

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Localization of fourier-laplace series of distributions

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