On the uniform summability of multiple Walsh- Fourier series

Transcription

1 From the SelectedWorks of Ushangi Goginava 000 On the uniform summability of multiple Walsh- Fourier series Ushangi Goginava Available at:

2 Dedicated to the 60th anniversary of F. Moricz ON THE UNIFORM SUMMAILITY OF MULTIPLE WALSH-FOURIER SERIES U. GOGINAVA Abstract. In this paper we prove that if f C ρ [0 ] N and the function f is of bounded partial variation then the N-dimensional Walsh-Fouerie series of the function f is uniformly C α summable α α N < α i > 0 i =... N in the sense of Pringsheim. If α α N = α i > 0 i =... N then on [0 ] N there exists the continuous function f 0 of bounded partial variation for which Cesaro C α means σm α over cubes. f 0 0 of N-dimmensional Walsh-Fourier series of f 0 will diverge. Introduction In one-dimensional cases Onnewer [] has proved that if f C ρ [0 ] and the function f is of bounded variation then the Walsh-Fourier series of this function uniformly converge. The problems of uniform convergence and summability of Cesaro means of negative order for simple and multiple Walsh-Fourier series were studied in the works []-[8]. For double Walsh-Fourier series F. Moricz has proved [9] that if f C ρ [0 ] and the function f is of bounded variation in Hardy s sense [0] then the two-dimensional Walsh-Fourier series is uniformly converge to f. For N-dimensional Walsh-Fourier series the author [7][8] has proved that if f C ρ [0 ] N and the function f is of bounded partial variation then the N- dimensional Walsh-Fourier series of the function f uniformly converge to f. The analogous result for the N-dimensional trigonometric Fourier series is verified by the author []. In this paper we study the convergence and divergence of N-dimensional Walsh- Fourier series by the Cesaro method. In particular we prove that if f C ρ [0 ] N and the function f is of bounded partial variation then the N-dimensional Walsh-Fourier series of the function f is uniformly C α summable α α N < α i > 0 i =... N in the sense of Pringsheim. If α α N = α i > 0 i =... N then on [0 ] N there exists the continuous function f 0 of bounded partial variation for which Cesaro C α means σm α f 0 0 of the N-dimensional Walsh-Fourier series will diverge over cubes.. Definitions and Notation Let R N be the N-dimensional Euclidean space. The elements of R N will be denoted addition modulo in each coordinate. Let M = {... N} = {s... s r } = { s }... s r sk < s k s i < s i k =... r i =... r M = M\ = M\

3 =... α = α... α N. For the integer m the vector m... m of the space R N we denote by m. Let x be a vector of the R N space whose coordinates with indices from the set coinside with the coordinate of the vector x and the coordinates with indices from the set are zero i.e. x = x... x N where x i = x i if i and x i = 0 if i. Denote x M = x x = 0. elow we shall identify the symbols m ν =p and m s ν s =p s m sr ν sr =p sr du and du s du sr k and k... k N q q k and... q N. k k N Denote by C ρ [0 ] N the space of uniformly ρ-continuous on [0 ] N -periodic with respect to each variable functions with the norms Let f Cρ = sup f x. x [0] N {s i} f x h {si } = f x h {si } f x i =... r. The expression we shall get by successive applications of we denote by f x h i.e. {s } f x h {s }... {s r} f x h {sr} f x h = {sr} \{sr} x h {sr}. Let the function f be bounded on [0 ] N -periodic with respect to each variable. We say that the function f is of bounded partial variation if for any i =... N V i f = sup x M\{i} sup Π n f x... x i x k i x i... x N k=0 f x... x i x k i x i... x N < n =... where Π is an arbitrary system of disoint intervals x k i x k i k = 0... n on [0 ] i.e. 0 x 0 i < x i x i < x n i < x n i. We consider the Walsh orthonormal system {w x : 0} defined on the unit interval [0 in the Paley enumeration. To be more specific let r 0 x = { if x [0 / if x [/ r 0 x = r 0 x

4 3 r x = r 0 x and x [0 be the well-known Rademacher functions. For = 0 w 0 x = ;if and if = i i i = 0 or i=0 is the dyadic representation of an integer then w x = [r i x] i. i=0 Given a function f integrable in the Lebesgue sense on [0 ] N -periodic with respect to each variable its N dimensional Fourier series relatively the Walsh system is defined by a ν w νi x i. ν =0 ν N =0 i M where a ν = a ν...ν N f = f x w νi x i dx M. [0] N i M The rectangular partial sums of series. are defined as follows: m s m f x = a ν w νi x i ν= 0 i M the Cesaro C α means of series. are defined by σm α m f x = a A α ν A α i m i i ν i w νi x i m i i M ν= 0 i M where A α α α m m = α >. m! It is well-known that N σm α f x = f u K α i m i x u du [0] N i= where K α i m i x i = A α i m i m i A α i m i ν i w νi x i.

5 4 3. Main Results Theorem 3.. Let f C ρ [0 ] N and f be of bounded partial variation. Then the N-diemensional Walsh-Fourier series of the function f is uniformly C α summable α α N < α i > 0 i =... N in Pringsheim s sense. Theorem 3.. Let α α N = α i > 0 i =... N. Then on [0 ] N there exists the continuous function f 0 of bounded partial variation for wich Cesaro C α means σm α f0 0 will diverge over cubes. 4. Auxiliary Propositions Lemma 4.. [4] For any α 0 and t 0 Kn α t = O. A α n t α Lemma 4.. [4] If α 0 and p m then m sgn A α p νw ν t = sgnw m t t 0. ν=0 Lemma 4.3. [4] Let α 0. Then there exists a positive integer p such that i m i m is fulfilled if i < m p for great m. K α m t dt C α iα Lemma 4.4. Let a i... a in and b i... in be real numbers then b k...k N. i= For N = this equality is called Abel s Transformation while for N = Hardy s Transformation. The validity of the above equality can be easily verified by induction for any N 3. Lemma 4.5. [8] Let k U ka A k f u α = q k A q A = A q If for any A M A = k A k k A q = A q α U ka A k f u α 0 k A f u uniformly with respect to u i i M as k i i A then the N-dimensional Walsh-Fouriers seris of the function f is uniformly C α summable 0 α i < i =... N in the sense of Pringsheim..

6 Lemma 4.6. [8] Let f C ρ [0 ] N m i = k i m i 0 m i < k i α i 0 i M. Then A α i k [ m i m i ν i ω νi x i f x ] u f x dx = o i M [0] N ν= 0 i M uniformly with respect to u i as m i i M. 5. Proofs Proof of Theorem 3.. To prove the theorem on the basis of Lemma 4.5 it suffices to show that for any A M A α α N < α i > 0 i =... N U ka A k f u α 0 uniformly with respect to u i i M as k A. Let M. Since α i < there exists η > 0 such that i α i = η. i Let ε i = α i η i \{s r } and ε i = where η 0 η /N. i It is evident that α i < ε i i. 5. Let A = M then from Lemma 4.4 we obtain = k q = U k f u α = U k f u α f u q q α = i \ k \ k k i α i q l \ = \ l = q = k q α k q α f u l k k = I. 5. Let card = then from the condition of the theorem under consideration and by 5. we obtain I 5

7 6 = i k k i α i l = C i f u uniformly with respect to u i i M as k i i. Let card. Since : we have C i \{} k l = l k k k i α i = o 5.3 f u l k k k l = {} f u l k k i max u ss M\{} {} k f u l l = k k = k f u i l = = C 3 k i ε k i \{i} max {i} f u u i ss M\{i} l i = l k {i} ε i k i y and from the condition of the theorem we obtain = i k k i α i max u ss M\{i} l = I f u l C 4 α i i k i k i f u l l i = {i} C 5 i k k. 5.4 ε \{i} k uniformly with respect to u i i M as k i i. {i} k ε i k i k i ε i α i = o 5.5

8 7 From 5.3 and 5.5 it follows that : M uniformly with respect to u i i M as k i i. For I we have I = k i \ k i α i k \ q l \ = \ l = C 6 α i \ I = o 5.6 q = q α q α f u l k k k k i α i q = q α Let. Since k \ q l \ = \ l = f u l k k 5.7 k \ q f u l l \ = \ l = k k = k ε i \ q f u l i l \ = \ l = k k C 7 k ν q µ i \ ν \ \{i} µ k i ε i max u ss M\{i} l i = {i} f u l k {i} k i q m k n m \{} n \ q ε max u ss M\{} l = {} f u l k {} k = C 7 k i ε n n \{i} i \

9 8 q max u ss M\{i} ε m m \{} k i l i = {i} f u q max u ss M\{} l = {} l k f u {i} l k ε i k i {} ε k by virtue of and from the condition of the theorem we obtain I C 8 α α i ε n i \ k i n \{i} k q = q ε m α m \{} k = C 8 α i \ k i ε i α i q = uniformly with respect to u i i M as k i i. Let =. Analogously we have q ε α = o 5.8 I = k C 9 α k = C 9 α k q = q l = q q = q α l = q q = q α i l = k C 9 α q = q i max u ss M\{i} l i = {i} = C 9 α i q α q α f u l k k i f u k i q i = q f u f u q α i i n \{i} l k {i} l k k ε i l k k q ε i n α i ε n n \{i} i ε i k i

10 Let = min q i max u ss M\{i} l i = {i} = C 9 α i q i max u ss M\{i} l i = {i} sup u ss M η< k i 3 sup h i k i sup u ss M f u k i q i = f u l k {i} q ε i α i i l k {i} f u ε i ηk i h {i} f u q i = sup h i k i k i q i =ηk i k i q i =η q ε i α i i q ε i α i i ε i k i ε i k i q α i i f u ε i η h {i} f u q i =. Then by 5. and from the condition of the theorem we have C 0 k q q = q ε α max {} f u u ss M\{} l = ε l k {} k ηk q = q = q ε α max {} f u u ss M\{} l = ε l k {} k k q q =ηk q ε α max {} f u u ss M\{} l = ε l k {} k sup f u ε ηk h {} f u u i i M q = sup h k q α i i q α 9 5.9

11 0 k q =ηk q ε α uniformly with respect to u i i M as k. On the basis of 5.9 and 5.0 we find that uniformly with respect to u i i M as k. It follows from and 5. that = o 5.0 I = o 5. U k f u α = o uniformly with respect to u i i M as k. Analogously when A M A A = we obtain U ka A k f u α 0 uniformly with respect to u i i M as k i i A. The proof of Theorem 3. is complete. Proof of Theorem 3.. Let l 0 > p where p is as in Lemma 4.3. Consider the function ϕ α i 0 x = ϕ 0 x sgnk α i x i = l 0... N. Let there be constructed l 0... l k and -periodic functions ϕ 0... ϕ k. Then we can define integer l k satisfying the following properties: l k > l k l k l k k ; card i α i k l n= k Consider the function ϕ k x defined by l n lncard : M. k if x = r r =... l k ; ϕ k x = 0 if x [ ] [ 0 l k ] x = r r =... l k ; is linear and continuous for other x [0 ]. Let ϕ k x l = ϕ k x l Z. where f 0 x = n= ln l n N i= ϕ α i n x i f 0 0 = 0 ϕ α i k x = ϕ kxsgnk α i x i =... N. It is evident that the function f 0 is continuous on [0 ] N -periodic with respect to each variable and V f < =... N.

12 We show that the C α means of the N-dimensional Walsh-Fourier series of the function f 0 x diverge over cubes for x = 0 Indeed σ α... α N... f0 0 f 0 0 = = M [0/ ] card [/ ] card = M [0] N f 0 x f 0 x N i= N i= K α i x i dx M K α i x i dx dx II. 5. Since M f 0 x = 0 if x [ 0 / ] card [ / l k ] card it is evident that if and M then II = If = M then from the construction of the function f 0 we have N II M = f 0 x K α i N x i dx M f 0 x K α i x i dx M [0/ ] N i= [0/ ] N i= C α N l k Nl k max x [0/ ] N f 0 x = o k. If = then we obtain 5.4 = II = [/ ] N f 0 x N i= M [/ / ] card [/ ] card = M K α i x i dx M f 0 x N i= K α i x i dx dx II 5.5 From the construction of the function f 0 it is clear that if and M then Let =. Then we have II =

13 II N = f 0 x K α i x i dx M [/ ] cardm i= N C α α i i= f 0 x M [/ ] N ν i w νi x i ν w ν x dx M i Let = M. Then we have = M A αi ν = II 5.7 II M = C α l k f 0 x ν i w νi x i dx M [/ ] N i M C 3 α l k f 0 x 3 M [0] card 3 [0/ ] card 3 i M ν i w νi x i dx 3 dx If 3 = M then by Lemma 4.6 we obtain f 0 x [0] N i M ν i w νi x i dx M = o as l k. 5.9 Let 3 M 3 M. Then by Lemma 4. and from the condition of the theorem we have [0] card 3 max f 0 x x [0/ ] N f 0 x [0/ ] card 3 i M [0] card 3 [0/ ] card 3 i M [0] N i M max f 0 x x [0/ ] N ν i w νi x i dx 3 dx 3 l k ν i w νi x i dx 3 dx 3 ν i w νi x i dx M

14 = max x [0/ ] N f 0 x i M [0] N ν i w νi x i w x i dx i C 4 max f 0 x l kα i = o as k. 5.0 x [0/ ] N i M y virtue of we get II M = o k. 5. Let M M. Then by Lemma 4. we obtain l k f 0 x ν i w νi x i A αi ν w ν x dx M [/ ] N i ν = = l k [/ ] card i l k ν i w νi x i w l k x i f 0 x A α ν w ν x w x dx dx [/ ] card ν =0 = k n= ln l n [/ ] card i l k ν i w νi x i w l k x i ϕ α i n x i [/ ] card = k n= ν =0 A α ν w ν x w x ϕ α ln l n i [/ ] [/ ν =0 ] k n= ln l n i [0] [/ ν =0 ] n x dx dx ν i w νi x i w l k x i ϕ α i n x i dx i A α ν w ν x w x ϕ α n x dx ν i w νi x i w x i ϕ n x i dx i A α ν w ν x w x ϕ α 3 n x dx. 5.

15 4 Obviously [0] ν i w νi x i w x i ϕ n x i dx i C 5 α α il k. 5.3 From the construction of the function ϕ α i n x it is not difficult to see that n =... k Since ϕα i n x ϕ α i n x ln C l k w νi x = w ν i xw x 0 ν i < d w x = if x d d w x = if x d by 5.4 and from Lemma 4. we obtain = = = [/ ν =0 ] d = l k d = l k d = l k d d d l d k d l d k d = l k d l d ν =0 k A α [0] ν =0 A α ν w ν x w x ϕ α ν =0 ν =0 ν =0 n x dx n x dx A α ν w ν x w x ϕ α A α ν w ν x w x ϕ α n x dx A α ν w ν x w x ϕ α n x dx [ A α ν w ν x ϕ α n x ϕ α n x ] ν w ν x ϕα n x ϕ α n x dx l k dx

16 5 C 7 ln [0] x α dx C 8 α ln. 5.5 On the basis of and 5.5 for M M we have I C 9 α k n= ln l n α il k l n card i = C 9 α l k α i card i It follows from and 5.6 that II C 0α k n= l n lncard. 5.6 M M l k α i card n= i Let = M. Then from the constuction of the function f 0 by Lemma 4.3 we obtain = = = C α II M = l k [/ / ] N f 0 x [/ / ] N N N i= l k i= [/ / ] N N l k i= C α l k l k d i = l k i= N l k i= N l k l k d i = [ ] d i d i l k l k d i = N i= k ϕ k x i l n lncard o 5.7 K α i x i dx M x i dx M K α i ϕ k x i K α i x i dxi ϕ k x i K α i x i dxi [ ] d i d i K α i x i dx i N α id i C 3 α lk lk l i= k l k Owing to and 5.8 we arrive at α i = C 3 α >

17 6 C 4 α C 5 α M M σ α... α N... f 0 f 0 card i α i k l n= k C 4 α C 6 α k o lim σ α... α N k... f 0 f 0 C4 α > 0 The proof of Theorem 4.3 is complete. l n lncard o

18 References. G.W. ONNEWER. On uniform convergence for Walsh-Fourier series Pasific. Math N.I. FINE. On the Walsh functions. Trans. Amer. Math. Soc V.I. TEVZADZE. Uniform convergence of Walsh-Fourier series. Soobsh. AN GSSR in Russian 4. V.I. TEVZADZE. Uniform convergence of Cesaro means of negative order of Fourier-Walsh series Soobsh. AN GSSR in Russian 5. U.K. GOGINAVA. About uniform convergence and summability of multiple Fourier-Walsh-Paley series. ulletin of Georg. Acad. Sci U.K. GOGINAVA. About uniform convergence of Cesaro means of negative order of Fourier-Walsh-Paley series. Seminar of I. Vekua s Institute of Applied Mathematics U.K. GOGINAVA. On the convergence and divergence of multiple Fourier-Walsh series. ulletin of Georg. Acad. Sci U.K. GOGINAVA. On the convergence and summability of multiple Fourier series with respect to the Walsh-Paley system in L p [0 ] N p [ ] spaces.georg. Math. J to appear. 9. F. MORICZ. On the uniform convergence and L -convergence of double Walsh- Fourier series. Stud. Math G.H. HARDY. On double Fourier series and especially those which represent the double zeta function with real and incommeasurable parametrs. Quart.. Math U.K. GOGINAVA. On the uniform convergence of multiple Fourier series with respect to the trigonometric system. ull. of Georg. Acad Sci Department of Mechanics and Mathematics Tbilisi State University University St. Tbilisi Georgia 7