On some Properties of Conjugate Fourier-Stieltjes Series

Transcription

1 Bullein of TICMI ol. 8, No., 24, On some Properies of Conjugae Fourier-Sieljes Series Shalva Zviadadze I. Javakhishvili Tbilisi Sae Universiy, 3 Universiy S., 86, Tbilisi, Georgia (Received January 6, 24; Revised April 3, 24; Acceped June 23, 24) A heorem of Ferenc Lukács [9] saes ha he parial sums of conjugae Fourier series of a periodic Lebesgue inegrable funcion f diverge a he logarihmic rae a he poins of firs kind disconinuiy of f. The aim of his paper is o invesigae analogous problems in erms of Fourier-Sieljes series and Abel-Poisson means of he Fourier-Sieljes series. Keywords: Fourier-Sieljes series, Bounded variaion. AMS Subjec Classificaion: 42A38, 42A5.. Inroducion Le f be a 2 periodic Lebesgue inegrable funcion. The Fourier rigonomeric series of he funcion f is defined by where a i = a 2 + (a i cos ix + b i sin ix), () i= f(x) cos ixdx and b i = f(x) sin ixdx, are he Fourier coefficiens of f. The conjugae series of () is defined by (a i sin ix b i cos ix). (2) i= Le S k (f; x) be he k-h parial sum of series (2). Lukács [9] proved he following heorem. Theorem. : If f L(, ] and he finie limi lim [f(x + ) f(x )] = d x(f), + sh.zviadadze@gmail.com

2 exiss a some poin x (, ] hen ol. 8, No., lim k + S k (f; x) ln k = d x(f). B. Golubov obained a formula for he jump of a funcion of bounded p-variaion a a given poin in erms of derivaives of parial sums of is Fourier series. R. Riad [3] proved an analogous heorem of he Lukács heorem in erms of he conjugae Walsh series. G. Kvernadze, T. Hagsrom, H. Shapiro ([5]-[8]) invesigae how o deermined jumps for class of funcion generalized variaion in erms of Jacobi polynomials, also hey uilize he runcaed Fourier series as a ool for he approximaion of he poins of disconinuiies and he magniudes of jumps of a 2-periodic bounded funcion in erms of derivaive of he parial sums, hey also use inegrals. F. Móricz ([], []) generalized Lukács s heorem in erms of he Abel-Poisson means and proved esimae of he parial derivaive of he Abel-Poisson mean of an inegrable funcion a hose poins where funcions are smooh. Pinsky [2] generalized Fourier parial sums by using a family of convoluion operaors wih some classes of kernels. Q. Shi and X. Shi [4] discuss he concenraion facor mehods for deerminaion of jumps in erms of specral daa. P. Zhou and S. Zhou [9] generalize Lukács heorem in erms of he linear operaors which saisfy some cerain condiions. D. Yu, P. Zhou and S. Zhou [7] show how jumps can be deermined by he higher order parial derivaives of he of is Abel-Poisson means. The auhors ([2], [2]) examine he analogous heorems for he generalized Cesáro means, inroduced by Akhobadze ([]-[3]), as well as posiive regular linear means, and consider ([22], [23]) Lukás heorem for he funcions and series inroduced by Taberski ([5], [6]) as well as generalized Cesáro, posiive regular linear and Abel-Poisson means. 2. Formulaion of he resuls Our ineres is o sudy same asks for Fourier-Siljes series. Le he funcion f be 2 periodic and have bounded variaion on he [ ; ], he Fourier coefficiens are defined as a k = cos kxdf(x) b k = sin kxdf(x). (3) By S n (df; x) we define he n-h parial sum of Fourier-Sieljes series of he f. Also define ϕ x () = f(x + ) + f(x ) 2f(x) (f (x+) f (x )).

3 24 Bullein of TICMI Theorem 2. : For any poin x where here exis numbers f (x+), f (x ) and (ϕ x ) = o(), +, (4) we have lim n + S n (df; x) ln n = f (x+) f (x ). (5) where (ϕ x ) is a variaion of he funcion ϕ x ( ) on he se [; ]. I is naural o ask: is he analogue of he las saemen valid for a Abel-Poisson summabiliy mehod? Abel-Poisson means of he conjugae Fourier-Sieljes series is defined as d f(r, x) = k N where r k (a k sin kx b k cos kx) = Q(r, ) = k N r k sin k = Q(r, x)d(f(x + ) + f(x )), (6) r sin ( r) 2 + 4r sin 2 (/2). (7) Now we examine he analogous of he heorem 2. for Abel-Poisson mean. Theorem 2.2 : have For any poin x where here exis numbers f (x+), f (x ) we lim r d f(r, x) ln( r) = f (x+) f (x ). (8) Noe ha in Theorem 2.2 we omi condiion (4). I is naural o ask abou condiion (4). If he funcion f is absoluely coninuous hen he menioned condiion follows auomaically, bu in his case we provide he known resuls in he inroducion. Arises a quesion: is condiion (4) equivalen o absoluely coninuiy of funcion f? Proposiion 2.3: There exiss funcion f of bounded variaion which is no absoluely coninuous bu for which (4) holds. I is ineresing if here is a possibiliy o replace (4) wih a weaker condiion. Our hypohesis is ha (4) is he bes opion o guaranee (5).

4 ol. 8, No., Proofs Proof (of Theorem 2.): S n (df; x) = D k ()d(f(x + ) + f(x )) = D k ()d(f(x + ) + f(x ) 2f(x) (f (x+) f (x ))) f (x+) f (x ) D k ()d = A (n) A 2 (n). (9) Le esimae A (n). By (4) for every ε > we can choose (ε) > such ha (ϕ x ) < ε, (; ). () Le us choose n such ha /n <. We ge A (n) = /n + /n + D k ()dϕ x () = B (n) + B 2 (n) + B 3 (n). () Since D n () n for all, (see [8, Ch. II, (5.)]), by () we have B (n) n /n d (ϕ x ) < ε. (2) By esimaion D n () 2/, (; ], (see [8, Ch. II, (5.)]) and inegraion by pars wih respec o and () we have B 2 (n) 2 /n d (ϕ x ) < ε + 2 /n 2 (ϕ x )d < ε + 2ε n d = o(ln n). (3)

5 26 Bullein of TICMI B 3 (n) 2 By ()-(4) we ge I is well known ha d (ϕ x ) 2 d (ϕ x ) 2 (ϕ x ) = O(). (4) lim A (n)/ ln n =. (5) n + D n ()d ln n, herefore we ge lim n + A 2 (n) (f (x+) f (x )) ln n =. Combining (9), (5) and he las esimae we prove (5). Proof (of Theorem 2.2): d f(r, x) = Q(r, x)dϕ x () f (x+) f (x ) Q(r, x)d Le us esimae D 2 (r). Consider Therefore we have (2 ln(( r) 2 + 4r sin 2 (/2))) = = D (r) D 2 (r). (6) r sin ( r) 2 + 4r sin 2 (/2). D 2 (r) = f (x+) f (x ) (2 ln(( r) 2 + 4r sin 2 (/2))) = f (x+) f (x ) (ln( + r) ln( r)) f (x+) f (x ) ln( r). (7) By definiion of he numbers f (x+) and f (x ), for any ε > we choose = (ε) > such ha when [; ] we have f(x + ) f(x) f (x+) / < ε/2, f(x ) f(x) + f (x ) / < ε/2.

6 ol. 8, No., Therefore f(x + ) f(x) f (x+) < ε /2, f(x ) f(x) + f (x ) < ε /2. By definiion of ϕ x () we have ϕ x () = f(x + ) + f(x ) 2f(x) (f (x+) f (x ) f(x + ) f(x) f (x+) + f(x ) f(x) + f (x ) < ε. (8) Consider D (r). D (r) = Q(r, x)dϕ x () + Inegraion by pars wih respec o we have We have E = Q(r, )ϕ x() Q(r, x)dϕ x () = E + E 2. (9) Q (r, )ϕ x ()d = F + F 2. F = O(). (2) If β is a poin such ha Q (r, ) changes he sign on i, hen by (8) and inegraing by pars wih respec o we have F 2 ε β Q (r, ) d = ε Q (r, )d ε β Q (r, )d β = εβq(r, β) ε Q(r, )d εq(r, ) + εβq(r, β) + ε β Q(r, )d 2εβQ(r, β) + 2ε Q(r, )d = o() + o(ln( r)) = o(ln( r)). (2) On he oher hand, by he represenaion of Q(r, ) (see [8, Ch. III, (6.3)]) we have r sin ( r) 2 + 4r sin 2 (/2) sin 4 sin 2 (/2) = 2 co(/2).

7 28 Bullein of TICMI Then E 2 co(/2)d (ϕ x ) 2 d (ϕ x ) d (ϕ x ) = O(). (22) Finally, by (6)-(22) we prove (8). Theorem 2.2 is proved. Proof (of Proposiion 2.3): Le us inroduce he se A = {, /2, /3,...}. (23) Now choose some number γ > and define he funcion f f(x) = { n γ (n + ) γ if x = n, if x [ ; ]\A. I is easy o see ha f is no absoluely coninuous, f has bounded variaion and (4) is valid a he poin. Le as consider ϕ () because f () = when > we have ϕ () = f( + ) + f( ) 2f() (f (+) f ( )) = f(), hen (ϕ ) = 2 n / ( ) n γ (n + ) γ = 2 γ = o(), +. Proposiion 2.3 is proved. Acknowledgmen. Research suppored by Shoa Rusaveli Naional Science Foundaion gran #2/25. References [] T. Akhobadze, On generalized Cesáro summabiliy of rigonomeric Fourier series, Bull. Georgian Acad. Sci., 7 (24), [2] T. Akhobadze, On he convergence of generalized Cesáro means of rigonomeric Fourier series I, Aca Mah. Hungar., 5 (27), [3] T. Akhobadze, On he convergence of generalized Cesáro means of rigonomeric Fourier series II, Aca Mah. Hungar., 5 (27), 79- [4] B. I. Golubov, Deerminaion of he jumps of a funcion of bounded p-variaion by is Fourier series, Mah. Noes, 2 (972), [5] G. Kvernadze, Deerminaion of he jumps of a bounded funcion by is Fourier series, J. Approx. Theory, 92 (998), 67-9 [6] G. Kvernadze, Approximaion of he singulariies of a bounded funcion by he parial sums of is differeniaed Fourier series, Appl. Compu. Harmon. Anal., (2), [7] G. Kvernadze, T. Hagsrom, H. Shapiro, Locaing disconinuiies of a bounded funcion by he parial sums of is Fourier series, J. Sci. Compu., 4 (999), [8] G. Kvernadze, T. Hagsrom, H. Shapiro, Deecing he singulariies of a funcion of p class by is inegraed Fourier series, Compu. Mah. Appl., 39 (2), [9] F. Lukács, Über die besimmung des sprunges einer funkion aus ihrer Fourierreihe, J, Reine Angew. Mah., 5 (92), 7-2 [] F. Móricz, Deerminaion of jumps in erms of Abel-Poisson means, Aca Mah. Hungar., 98 (23),

8 ol. 8, No., [] F. Móricz, Ferenc Lukács ype heorems in erms of he Abel-Poisson mean of conjugae series, Proc. Amer. Mah. Soc., 3 (23), [2] M.A. Pinsky, Inverse problems in Fourier analysis and summabiliy heory, Mehods Appl. Anal., (24), [3] R. Riad, On he conjugae of he Walsh series, Ann. Univ. Sci. Budapes. Rolando Eovös, Sec. Mah., 3 (987), [4] Q. Shi, X. Shi, Deerminaion of jumps in erms of specral daa, Aca Mah. Hungar., (26), [5] R. Taberski, Convergence of some rigonomeric sums, Demonsraio Mahemaica, 5 (973), -7 [6] R. Taberski, On general Dirichle s inegrals, Anales soc. Mah Polonae, Series I: Prace mahemayczne XII (974), [7] D.S. Yu, P. Zhou, S. Zhou, On deerminaioin of jumps in erms of Abel-Poisson mean of Fourier series, J. Mah. Anal. Appl., 34 (28), 2-23 [8] A. Zigmund, Trigonomeric Series, Cambridge Universiy Press, (959) [9] P. Zhou, S.P. Zhou, More on deerminaion of jumps, Aca Mah. Hungar., 8 (28), 4-52 [2] Sh. Zviadadze, On he Saemen of F. Lukács, Bull. Georgian Acad. Sci., 7 (25), 4-42 [2] Sh. Zviadadze, On a Lukács heorem for regular linear means of conjugae rigonomeric Fourier series, Bull. Georgian Acad. Sci., 73 (26), [22] Sh. Zviadadze, On generalizaions of a heorem of Ferenc Lukás, Aca Mah. Hungar., 22 (29), 5-2 [23] Sh. Zviadadze, On some properies conjugae rigonomeric Fourier series, Georgian Mah. Journal, 2 (23),