ABSOLUTE CONVERGENCE OF THE DOUBLE SERIES OF FOURIER HAAR COEFFICIENTS

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1 Acta Mathematica Academiae Paedagogicae Nyíregyháziesis 6, SSN ABSOLUTE CONVERGENCE OF THE DOUBLE SERES OF FOURER HAAR COEFFCENTS ALEANDER APLAKOV Abstract. this aer we study the absolute covergece of the double series of Fourier-Haar coefficiets of the class PBV.. troductio The roblems related to the behaviour of sigle series of Fourier-Haar are well studied [9]. Namely, P. Uliaov [4] ad B. Golubov [8] received the results related to the roblems of absolute covergece of the series of Fourier Haar coefficiets. Some geeralizatio of these results related were received by Z. Chaturia [3], T. Akhobadze [], U. Gogiava [7] ad by the author []. the term of modulus of smoothess the roblem of absolute covergece of the series of Fourier-Haar coefficiets was studied by V. Krotov []. Multidimesioal aalogies corresodig to the results of V. Krotov were formulated i the works of V. Tsagareishvili [3] ad G. Tabatadze []. The estimates of Fourier coefficiets of fuctios of bouded fluctuatio with resect to Walsh system were studied i [] ad with resect to Vileki system were studied by G. Gát ad R. Toledo [4]. We cosider the double Haar system {χ x χ m y :, m =,,,...} o the uit square = [, ] [, ]. As usual, L deotes the set of all measurable fuctios defied o, for which f = f x, y dxdy < ad C is the sace of cotiuous fuctios o equied with maximum orm f c = max f x, y. x,y f f L, the C,m f = is the, mth Fourier-Haar coefficiet of f. f x, y χ x χ m y dxdy Mathematics Subject Classificatio. 4C. Key words ad hrases. Fourier-Haar coefficiets, bouded variatio, absolute covergece. 33

2 34 ALEANDER APLAKOV We say that f Li α o [, ], if f + h, + η f, c = O h + η α We have the followig theorem., α, ]. Theorem A []. a Let f Li α o [, ], α, ]. f β > ad γ + < β α+, the m γ C,m f β <. = m= b Let γ + = β α+, for some α,. The there exists a fuctio f α Li α for which m γ C,m f α β =. = m= The case for γ = was cosidered earlier by V. Tsagareishvili [3]. Let f L. The artial itegrated modulus of cotiuity are defied by } ω δ, f = su { f x + u, y f x, y : u δ, } ω δ, f = su { f x, y + v f x, y : v δ. We also use the otio of the mixed itegrated modulus of cotiuity. defied as follows ω, δ, δ, f = su { f x + u, y + v f x + u, y f x, y + v + f x, y : t is ot difficult to show that ω, δ, δ, f ω δ, f ω δ, f. u δ, v δ }, f L. t is We study the roblem of absolute covergece of the series of Fourier-Haar coefficiets for the classes of fuctios with bouded artial -variatios, which were first cosidered by U. Gogiava see [5] for = ad [6] for >. Defiitio. A fuctio f : R is said to be of bouded artial -variatio f PBV if there exists a costat K such that for ay artitio we have : x < x < x <... < x, : y < y < y <... < y m, V f = su y V f = su x f x i, y f x i+, y K, su i= f x, y j f x, y j+ K. m su j= Give a fuctio f x, y, eriodic i both variables with eriod. Deote by h f x, y = f x + h, y f x, y, h f x, y = f x, y + h f x, y, h,h f x, y = h h f x, y = h h f x, y = f x, y f x + h, y f x, y + h + f x + h, y + h.

3 ABSOLUTE CONVERGENCE OF SERES Mai Results The mai results of this aer are reseted i the followig roositios. Theorem. Let f PBV, ad β > +. The = m= C,m f β <. Theorem. Let f PBV, ad α <. The = m= [ + m + ] α C,m f <. Theorem 3. Let f PBV, ad β >, α + < β +. The = m= [ + m + ] α C,m f β <. Sice Li PBV i case > the sharess of Theorems -3 follows from the works [, 3]. 3. Auxiliary results Lemma. Let f PBV,. The ω i δ, f 3 δ Vi f i =,, < δ <, where V i f is a artial -variatio of fuctio. Usig the method of [8], we ca easily obtai the validity of Lemma. Proof of Theorem. We write 4. Proof of mai results C,m f β = C, f β + C,m f β + C,m f β. = m= = m= = m=

4 36 ALEANDER APLAKOV Let = + i, m = m + j, =,,... i =,...,, m =,,..., j =,..., m. The usig Hölder iequality, from the Lemma we get C +i, f i= = Z Z i= i Z Z 6 B i= 6B 4@ i= Z ω i + + i + i + Z Z i + Z i + +, f» f x, y f x + «, y + 3 f x, y + dx C A dy 7 5 f x, y + dxc A dy C A f x, y + dxdy «dxc A dy Z i + Z i + dxdyc A 3 V f c V f Let + < β <. Usig Hölder iequality, from we get 3 i= f β, C i i= C i β f, β β c V f β c β β c [ β β ]. By 3 ad from the coditio of the Theorem we obtai C, f β = C +i, f β [ β β ] <. = = i= = Aalogously, we obtai that m= C,m f β <, for β > +.

5 ABSOLUTE CONVERGENCE OF SERES Usig Hölder iequality, by ad from Lemma we get 4 i= m j= i+ i +m +m +m j+ m j m i= i= i+ + i + f x, y χ +i x χ m +j y dxdy i+ j+ m + m + +, f x, y dxdy m + j= m j= j+ m + j m + i + +m +m ω, j m + i+ + i + dxdy j+ m + j m + +, +m ω +, f c +m +m +, m+, f +, f x, y dxdy m + m + f x, y dxdy ω m+, f c +m. Let + < β <. Usig Hölder iequality, by 4 we write 5 i= m j= C +i, m +j f β c +m β +m β = c +m[ β β + β ] = c [ β β ] m [ β β ]. By 5 ad from the coditio of the Theorem we get = m= C,m f β = c = m = = i= β [ ] β m j= m = C +i, m +j f β m[ β β ] <. The roof of Theorem is comlete.

6 38 ALEANDER APLAKOV Proof of Theorem. We write = m= [ + m + ] α C,m f = + α C, f + m= = m + α C,m f + Let β =. The from 3 we get 6 i= = m= [ + m + ] α C,m f. + i + α C +i, f c α C +i, f i= c α = c α+. By 6 ad from the coditio of the Theorem we obtai + α C, f = = c + i + α C +i, f = i= = α i= C +i, f c α+ <. = Aalogously, we obtai that m + α C,m f <, for α <. m= Let β =. The by 5 ad from the coditio of the Theorem we get [ + m + ] α C,m f = m= c = m = = m +mα α+ The roof of Theorem is comlete. i= j= m = C +i, m +j f mα+ <. Combiig the methods of Theorems - we ca rove validity of Theorem 3. Observe that the result of this aer ca be roved i the same way for dimesio more tha. Refereces [] T. Akhobadze. Geeralized BVP, φ class of bouded variatio. Bull. Georgia Acad. Sci., 633:46 48,. [] A. Alakov. O the absolute covergece of the series of Fourier-Haar coefficiets. Bull. Georgia Acad. Sci., 64:38 4,. [3] Z. A. Chaturia. O the absolute covergece of the series of Fourier-Haar coefficiets. Commet. Math. Secial ssue, :5 35, 979. [4] G. Gát ad R. Toledo. Fourier coefficiets ad absolute covergece o comact totally discoected grous. Math. Pao., :3 33, 999. [5] U. Gogiava. O the uiform summability of multile Walsh-Fourier series. Aal. Math., 63:9 6,. [6] U. Gogiava. Uiform covergece of N-dimesioal trigoometric Fourier series. Georgia Math. J., 74: ,.

7 ABSOLUTE CONVERGENCE OF SERES [7] U. Gogiava. O the absolute covergece of the series of Fourier-Haar coefficiets. Bull. Georgia Acad. Sci., 64: 3,. [8] B.. Golubov. O Fourier series of cotiuous fuctios with resect to a Haar system. zv. Akad. Nauk SSSR Ser. Mat., 8:7 96, 964. Russia. [9] B.. Golubov. Series i the Haar system. Mathematical aalysis 97, ages VNT, Moscow, 97. Russia. [] V. G. Krotov. Fourier coefficiets with resect to a certai orthoormal system that forms a basis i the sace of cotiuous fuctios. zv. VUZ. Matematika, 6:33 46, 975. Russia. [] F. Schi, W. R. Wade, ad P. Simo. Walsh series. A itroductio to dyadic harmoic aalysis. Adam Hilger Ltd., Bristol, 99. [] G. Z. Tabatadze. O absolute covergece of Fourier-Haar series. Bull. Acad. Sci. Georgia SSR, 33:54 543, 98. Russia. [3] V. Tsagareishvili. Fourier-Haar coefficiets. Bull. Acad. Sci. Georgia SSR, 8:9 3, 976. Russia. [4] P. L. Ulijaov. O Haar series. Mat. Sb. N.S., 63 5:356 39, 964. Russia. Received May 7, 5. Deartmet of Agro-Busiess Egieers, Georgia State Uiversity of Subtroical Agriculture,. Chavchavadze str., Kutaisi, 466 Georgia address: a alakov@waex.ge