On the uniform convergence and $L$convergence of double Fourier series with respect to the Walsh-Kaczmarz system

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1 From the SelectedWorks of Ushangi Goginava 23 On the uniform convergence and $L$convergence of double Fourier series with respect to the Walsh-Kaczmarz system Ushangi Goginava Available at:

2 On the Uniform Convergence and L-Convergence of Double Fourier Series with Respect to the Walsh-Kaczmarz System Ushangi Goginava Abstract In this paper we study approximation of rectangular partial sums of double Fourier series with respect to the Walsh-Kaczmarz system in the spaces C and L. From this results we obtain different criterion of the uniform convergence and L-convergence of double Fourier- Kaczmarz series. Introduction L. Zhizhiashvili [3]. part. 2, Chap. 3 has established certian approximation properties of rectangular partial sums of the double trigonometric Fourier series in spaces C and L. The analogous question for double Fourier series with respect to the Walsh-Paley system were treated in the works [2,8]. We will study approximation by rectangular partial sums of double Fourier series with respect to Walsh-Kaczmarz system to function f L p I 2 in the 2 Mathematics Subject Classification: Primary 4A5; Secondary 42C. Key words and phrases: Walsh-Kaczmarz system, convergence in L- norm, bounded variation in the sense of Hardy, uniform convergence.

3 norm of L p for p = or p = see Theorems, 2 and 3. From this theorems one can obtain different criteria for the oniform convergence and L -convergence of double Fourier series with respect to the Walsh-Kaczmarz system, in particular, we established two-dimensional version of the Dini- Lipschitz condition see Corollary and Corollary 2. Results of somewhat different type can be obtained by using the variation of a function. Jordan [7] introduced a class of functions of bounded variation and, applying it to the theory of the Fourier series, he proved that if a continuous function has bounded variation, then its Fourier series converges uniformly. In 96 Hardy [6] generalized the Jordan criterion to the double Fourier series and introduced for the function of two variables the notion of bounded variation. He proved that if the continuous function of two variables has bounded variation in the sense of Hardy, then its Fourier series converges uniformly in the sense of Pringsheim. The analogous result for double Walsh-Fourier series is verifed by Moricz [8]. The author [3] has proved that in Hardy s theorem there is no need to requere the boundedness of V,2 f, in particular, it is proved that if f is continuous function and has bounded partial variation f P BV then its double trigonometric Fourier series converges uniformly on [, 2π] 2 in the sense of Pringsheim. The analogous result for double Walsh-Fourier series is established in [4]. In this paper we study analogical question in case of the Walsh-Kaczmarz system see Theorem 4. A double series is said to converge in the sense of Prinsgeim if its partial rectangular sums converge. 2

4 2 Definitions and Notations We shall denote the sets of all non-negative integer by N, the set of real numbers by R and the set of dyadic rational numbers in the unit interval [,] by Q. In particular, each element of Q has the form p 2 n for some p, n N, p 2 n. Let r x be a function defined on [, by, if x [, /2 r x =, if x [/2,, r x = r x. The Rademachere system is defined by r n x = r 2 n x, n and x [,. Each n N it is possible write uniquely as m n = n k 2 k, k= where n k = or. The numbers n k called the binary coefficients of n. Given x [,, the expansion x = x k 2 k, 2 k= where each x k = or, will be called a dyadic expansion of x. If x [, ] \Q, then 2 is uniquely determined. By the dyadic expansion x Q we choose the one for which lim k x k =. The dyadic sum of x, y [, in terms of the dyadic expansion of x and y is defined by x y = x k y k 2 k. k= 3

5 by The Walsh system in the Paley enumeration {w n x : n N} is defined m w n x = r j x n j. j= Now we recall the definition of the Walsh-Kaczmarz system {ψ n x : n N}. Set ψ x =, while for n, which is given by expression with n m =, set m ψ n x = r m x r m j x n j. Let us consider the kernel of Dirichlet for the Walsh-Paley system: D n x = and for the Walsh-Kaczmarz system K n x = j= n j= n j= ω j x ψ j x. We will need the well-known equality for the Dirichlet kernel of the Walsh- Paley system [see 5, p. 27] 2 n, if x [, /2 n ; D 2 n x =, if x [/2 n,. The transformation τ n for x [, defined by n τ n x = x n k 2 k x j 2 j. k= j=n We will apply the transformation τ n also for integers p, given by τ n p = n j= x n j 2 j, 3 4

6 where p = n j= Given n and p < 2 n,we set x j 2 j. I n p = [p2 n, p 2 n It is evident that the transformation τ n x maps the segment I n p on the segment I n τ n p. and It is known [] that K n x = D 2 k x r k x D m τ k x for n = 2 k m, m < 2 k 4 D m τ k x 2k τ k p for x I k p, p =, 2,..., 2 k. 5 We consider the double system {ψ n x ψ m y : n, m N} on the unit square I 2 = [, [,. If f L I 2, then ˆf n, m = is the n, m-th Fourier coefficient of f. f x, y ψ n xψ m ydxdy The rectangular partial sums of double Fourier series with respect to the Walsh-Kaczmarz system are defined by S M,N f; x, y = M N m= n= ˆf m, n ψ m xψ n y. As usual, denote by L I 2, p the set of all measurable functions defined on I 2, for which f = f x, y dxdy 5

7 is finite. Furthermore, let C I 2 be the set of all functions f : I 2 R that are uniformly continuous from the dyadic topology of I 2 to the usual topology of R with the norm see [9], pp. 9- f C = sup x,y I 2 f x, y f C I 2. The total modulus of continuity, and the total integrated modulus of continuity are defined by ω f; δ, δ 2 C = sup { f x u, y v f x, y C : u < δ, v < δ 2 }, ω f; δ, δ 2 = sup { f x u, y v f x, y : u < δ, v < δ 2 }, while the partial moduli of continuity, and the partial integrated moduli of continuity are defined by ω f; δ C = ω f; δ, C and ω 2 f; δ 2 C = ω f;, δ 2 C f C I 2, ω f; δ = ω f; δ, and ω 2 f; δ 2 = ω f;, δ 2 f L I 2. We also use the notion of the mixed modulus of continuity, and the mixed integrated modulus of continuity are defined as follows ω,2 f; δ, δ 2 C = sup { f x u, y v f x u, y f x, y v f x, y C : u < δ, v < δ 2 }, f C I 2, ω,2 f; δ, δ 2 = sup { f x u, y v f x u, y f x, y v f x, y : u < δ, v < δ 2 }, f L I 2. A function f : I 2 R is said to be of bounded variation in the sense of Hardyf HBV I 2 if there exists a constant K such that for any partition : x < x < x 2 < < x n, 6

8 we have 2 : y < y < y 2 < < y m, V,2 f = V f = sup y V 2 f = sup x sup 2 n i= m j= f x i, y j f x i, y j f x i, y j f x i, y j K, n sup i= sup 2 n i= m j= f x i, y f x i, y K, f x, y j f x, y j K. Definition [4] We say that the function f : I 2 R is bounded partial variation f P BV I 2 if V f and V 2 f are finite. Given a function f x, y, periodic in both variables with period, for j < 2 m and i < 2 n and integers m, n we set m j f x, y = f x 2j2 m, y f x 2j 2 m, y, n i f x, y 2 = f x, y 2i2 n f x, y 2i 2 n, mn ji f x, y = n i m j f x, y = 2 m j n i f x, y 2 = f x 2j2 m, y 2i2 n f x 2j 2 m, y 2i2 n f x 2j2 m, y 2i 2 n f x 2j 2 m, y 2i 2 n. Furthemore, set λ m = and λ m j = τ m j for j < 2 m and 2 W m m f; x, y = j= λ m j m j f x, y, 2 W n 2 n f; x, y = λ n i n i f x, y 2, W mn f; x, y = 2 m j= i= 2 n i= λ m j λ n i mn ji f x, y. 7

9 2. Main Results Theorem Let M, N be positive integers such that M = 2 m j, j < 2 m and N = 2 n i, i < 2 n, for some integers m, n. If f C I 2, then ω f; 2, m 2 n C 2 W S M,N f f C m f C 2 W 2 n f C 4 W mn f C. Theorem 2 Let M, N be positive integers such that M = 2 m j, j < 2 m and N = 2 n i, i < 2 n, for some integers m, n. If f C I 2, then c { ω f; Corollary Let f C I 2 and S M,N f f C m ω 2 m 2 f; n ω C 2 n,2 f; C ω f; m as m, 2 m C ω 2 f; n as n, 2 n C 2, } mn. m 2 n C ω,2 f; 2, mn as n, m, m 2 n C then the double Fourier series with respect to Walsh-Kaczmarz system converges uniformly on I 2. Theorem 3 Let M, N be positive integers such that M = 2 m j, j < 2 m and N = 2 n i, i < 2 n, for some integers m, n. If f L I 2, then c {ω f; S M,N f f m ω 2 m 2 f; n ω 2 n,2 f; 2, } mn. m 2 n In this paper the constant c is absolute constant and may different in different contexts. 8

10 Corollary 2 Let f L I 2 and ω f; m as m, 2 m ω 2 f; n as n, 2 n ω,2 f; 2, mn as n, m, m 2 n then the double Fourier series with respect to Walsh-Kaczmarz system converges to f in L-norm. Theorem 4 Let f C I 2 P BV I 2. Then the double Fourier series with respect to Walsh-Kaczmarz system converges uniformly on I Proof of Main Results Proof of Theorem. From 4, we write = = S M,N f; x, y f x, y [f x u, y v f x, y] K M u K N v dudv [f x u, y v f x, y] D 2 m u D 2 n v dudv [f x u, y v f x, y] D 2 m u r n v D i τ n v dudv [f x u, y v f x, y] D 2 n v r m u D j τ m u dudv [f x u, y v f x, y] r n v r m u D j τ m u D i τ n v dudv 9

11 We obtain by 3 that I C = 2 nm It is well-known that /2 m = I II III IV. 6 /2 n [f x u, y v f x, y] dudv ω f; 2,. 7 m 2 n C a I m j = I m 2j I m 2j ;, if u I m 2j b w 2 m u =, if u I m 2j ; c t = u 2 m is a one-to-one mapping of I m 2jonto I m 2j. Thus, by 3, and a-d II = 2 m 2 n = 2 m 2 n = 2 m I m r= I m I nr r= I m I n 2r [f x u, y v f x, y] r n v D i τ n v dudv [f x u, y v f x, y] r n v D i τ n v dv du I n 2r C [f x u, y v f x, y] D i τ n v dv [f x u, y v f x, y] D i τ n v dv du 2 n = 2 m r= I m I n 2r [ f x u, y v f x u, y v 2 n ] D i τ n v dudv

12 = 2 m I m I n [ f x u, y v f x u, y v 2 n ] D i τ n v dudv 2 n 2 m r= D i τ n r I m I n [ f x u, y v 2r2 n f x u, y v 2r 2 n ] dudv. From 5, we have II 2 m i I m I n n f x u, y v 2 dudv 2 mn I m I n 2 n r= τ n r n r f x u, y v 2 dudv, consequently, II C W n 2 f C. 8 2 The estimation of III is analogous to the estimation of II and we have III C W m f C. 9 2 Following a similar patteern to the case of II, by a-d we obtain IV = 2 m s= 2 n r= I m 2s I n 2r D i τ n r D j τ m s I m 2s I n 2r I m 2s I n 2r I m 2s I n 2r [f x u, y v f x, y] dudv

13 = 2m 2 n s= r= D i τ n r D j τ m s I m 2s I n 2r [f x u 2 m, y v 2 n f x u 2 m, y v f x u, y v 2 n f x u, y v] dudv = 2 m s= 2 n r= D i τ n r D j τ m s I m I n [ f x u 2s 2 m, y v 2r 2 n f x u 2s 2 m, y v 2r2 n f x u 2s2 m, y v 2r 2 n f x u 2s2 m, y v 2r2 n ] dudv. We obtain by 5 that IV ij I m I n mn f x u, y v dudv j2 n i2 m 2 nm I m I n I m I n 2 n I m I n r= 2 nm 2 n r= 2 m s= 2 m s= I m I n τ n r mn r f x u, y v dudv τ m s mn s f x u, y v dudv τ n r τ m s mn sr f x u, y v dudv W mn f f; x u, y v dudv, 2

14 consequently, IV C 4 W mn f C. and Combining 6- we complete the proof of Theorem. Proof of Theorem 2. Since 2 n r= λ n r = 2 n r= 2n τ n r = 2 W m f; x, y ω f; m 2 m C W 2 n f; x, y ω 2 f; 2 n W mn f; x, y ω,2 f; 2, m 2 n C C 2 n r= r= 2 n i= 2 m i= the validity of Theorem 2 follows from Theorem. r= r cn λ m r cω f; m, 2 m λ n i cω 2 f; 2 n n, λ n r λ m i cω,2 f; 2, nm, m 2 n Calculations, similar to those that were performed in the proofs of Theorems, 2 and an application of the Minkowski inequality yield the validity of Theorem 3. Proof of Theorem 4. On the basis of Theorem, it suffices to show that Let W m f as m, C W 2 n f C as n, W mn f C as n, m. A m r = { j : j =, 2 m, 2 r τ m j < 2 r}, r =,,..., m. Then it is evident that A m r < 2 r 3

15 and where m r= A m r = {, 2,..., 2 m }. 2 From the condition of the theorem and by, 2 we get 2 W m f; x, y = m m f x, y j= ω f; consequently, ω f; ω f; ω f; 2 m C c {ω f; min {ω f; η 2 m m 2 m C r= j A m r m 2 m C r= ηm 2 m C r= m r=ηm ω f; 2 r j A m r τ m j m j f x, y m j f x, y τ m j 2 r j A m r 2 r j A m r m j f x, y m j f x, y m j f x, y ηm 2 m C r= 2 r Am r V f } η m 2 m C 2 ηm η } = ω 2 m C 2 η f; W m m r=ηm as m, 2 r η m 2 m C 2, ηm f as m, 3 C analogously W n 2 f as n. 4 C 4

16 We write 2 m j= 2 W mn f; x, y = mn n f x, y i= τ m j It is evident that Since from 3 we get j f x, y mn 2 m j= 2 n i= τ n i mn i f x, y τ n i τ m j ji f x, y mn = I II III IV. 5 mn f x, y ω,2 f; 2, as n, m. 6 m 2 n C mn j f x, y m j f x, y m j f x, y 2 n, II C 2 W m f C as m. 7 Analogously, III C as m. 8 From and 2 we get IV = m n r= s= j A m r i A n s mn ji f x, y τ n i τ m j Since j A m r i A n s m r= n 2 r s= 2 s j A m r i A n s mn ji f x, y 2 A m r sup mn ji f x, y, 9 x [,] i A n s n i f x, y 2, 5

17 we have j A m r j A m r i A n s i A n s mn ji f x, y 2 A n s sup mn ji f x, y = j A m r y [,] j A m r m j f x, y, /2 mn ji f x, y i A n s /2 mn ji f x, y 2 [ A m r A n s j A m r i A n s /2 sup n i f x, y 2 sup m j f x, y. 2 x [,] i A n y [,] s j A m r After substituting 2 in 9 we obtain by and from the condition of the theorem that m IV 2 sup r= 2 r/2 = 2 y [,] j A m r n sup s= 2 s/2 x [,] i A n s ϕm sup r= 2 r/2 y [,] j A m r m r=ϕm ψn s= sup 2 r/2 y [,] j A m r sup 2 s/2 x [,] i A n s m j f x, y n i f x, y 2 /2 /2 m j f x, y m j f x, y n i f x, y 2 /2 /2 /2 /2 n sup n 2 s=ψn r/2 i f x, y 2 x [,] i A n s 6

18 where and min ϕ 2 m min ψ 2 n c { ω f; ϕ m 2 m C { ω 2 f; 2 n { ω f; 2 m C { ω 2 f; 2 n ψ n C 2 ψn/2 Combining 5-8 and 2 we have C } 2 ϕm/2 } ϕ } = ω 2 ϕ/2 f; 2 m C ψ } = ω 2 ψ/2 2 f; 2 n C as m, n, 2 ϕ m 2 ϕm/2 ψ n 2 ψn/2. W mn f C as n, m. 22 From 3, 4 and 22 we complete the proof of Theorem 4. References [] L. A. Balashov, Series with respect to the Walsh system with monotone coefficients. Sibirsk. Math. J. 297, [2] R. D. Getzadze, Convergence and divergence of multiple Fourier series with respect to the Walsh-Paley system in the spaces C and L, Anal. Math. 3987, [3] U. Goginava, On the uniform convergence of multiple trigonometric Fourier series, East J. Approx. 5999, [4] U. Goginava, On the convergence and summability of N-dimensional Fourier series with respect to the Walsh-Paley systems in L p [, ] N, p [, ] spaces, Georg. Math. J. 72,

19 [5] B. I. Golubov, A.V. Efimov, and V.A. Skvortsov. Series and transformations of Walsh, Moscow, 987 Russian; English translation, Kluwer Academic, Dordrecht, 99. [6] G. H. Hardy, On double Fourier series, Quart. J. Math. 3796, [7] C. Jordan, Sur le series de Fourier. C. R. Acad. Sci. Paris, 9288, [8] F. Moricz, On the uniform convergenve and L -convergence of double Walsh-Fourier series, Stud. Math. 2992, [9] F. Schipp, W. R. Wade, P. Simon, and J. Pàl, Walsh series, Introduction to dyadic harmonic analysis, Adam Hilger, Briston and New York, 99. [] F. Schipp, Certain reargaments of series in the Walsh system. Mat. Zametki 8975, [] V. A. Skvorcov, On Fourier series with respect to the Walsh-Kaczmarz system. Anal. Math. 798, 4-5. [2] W. S. Young, On the a. e. convergence of Walsh-Kaczmarz-Fourier series, Proc. Amer. Math. Soc , [3] L. V. Zhizhiashvili, Trigonometric Fourier series and their conjugates, Tbilisi, 993 Russian; English transl.: Kluwer Acad. publ; 996. Author s address: Department of Mechanics and Mathematics Tbilisi State University Chavchavadze str., Tbilisi, 3828, Georgia. 8