Uniform convergence of N-dimensional Walsh Fourier series

Transcription

1 STUDIA MATHEMATICA Uniform convergence of N-dimensional Walsh Fourier series by U. Goginava Tbilisi Abstract. We establish conditions on the partial moduli of continuity which guarantee uniform convergence of the N-dimensional Walsh Fourier series of functions f from the class C W I N N i V i{pn} where pn as n.. Definitions and notation. Let I N [0 N be the unit cube in the N-dimensional Euclidean space R N. The elements of R N are denoted by x x... x N. For any x x... x N and y y... y N the vector x y... x N y N R N is denoted by x y where denotes dyadic addition. Let M {... N} {s... s r } {s r... s r } s k < s k+ s ri < s ri+ k... r i... M M \ M \. For an integer n the vector n... n R N is denoted by ñ. The cardinality of is denoted by. For any x x... x N and M let x denote the vector in R N whose coordinates with indices from coincide with the corresponding coordinates of x and the coordinates with indices from are zero. Note that x M x and x 0. For later convenience we introduce the following notation: m i p for q 2 k for m s m s r i s p s q 2 k... q N 2 k N du for du du N. i s r p sr Denote by CI N the space of all real-valued functions continuous on I N that can be extended to functions -periodic in each variable on R N. If f 2000 Mathematics Subect Classification: Primary 42C0. Key words and phrases: Walsh function N-dimensional Fourier series uniform convergence. []

2 2 U. Goginava CI N then the function ω i δ f x h i δ fx + h {i} fx i... N is called a partial modulus of continuity of f. Denote by C W I N the space of all real-valued functions uniformly W - continuous on I N that extend to functions -periodic in each variable with the norm f CW fx. x I N Let {si} f x h {si } fx h {si } fx i... r. Successive application of such partial difference operators leads to the definitions: f x h {sr} \{sr} f h \{sr } x h {sr } and ω δ f f h CW. 0 h i <δ i i Definition. Suppose that the function f is bounded on I N and extends to a function -periodic in each variable. Let p <. We say that f is of bounded partial p-variation written f P V p if for any i... N V i f x M\{i} n n π i k0 fx... x i x 2k i x i+... x N fx... x i x 2k+ i x i+... x N p < where π i is an arbitrary partition 0 x 0 i < x i x 2 i < x 2n 2 i < x 2n i. Let f be a function defined on R N which is -periodic relative to each variable. Π i is said to be a partition with period if Π i : < t i < ti 0 < t i < < t i m i < t i m i + < satisfies t i k+m i t i k + for k Z where m i is a positive integer. Definition 2. Let pn p as n where p. We say that a function f on R N belongs to the class V i{pn} if V i{pn} f x s s M\{i} n Π i { m i fx... x i t i x i+... x N fx... x i t i x i+... x N pn /pn : ϱπ i 2 n } <

3 where Walsh Fourier series 3 ϱπ i inf k ti k ti k. For N see [7]. When pn p for all n it is easy to see that N i V i{pn} coincides with P V p. Let r 0 be a function on R defined by { if x [0 /2 r 0 x r 0 x + r 0 x. if x [/2 The Rademacher system is defined by r n x r 0 2 n x n x [0. Let w 0 w... represent the Walsh functions i.e. w 0 x and if k 2 n n s is a positive integer with n > > n s then w k x r n x r ns x. The idea of using products of Rademacher functions to define the Walsh system comes from Paley [9]. The Walsh Dirichlet kernel is defined by n D n x w k x. k0 The rectangular partial sums of the N-dimensional Walsh Fourier series are defined as follows: where S m f x m ν 0 a ν w νi x i i M a ν a ν...ν N f fx w νi x i dx. I N i M 2. Formulation of the problem. Getzadze [ 2] considered the question of uniform convergence for N-dimensional Walsh Fourier series in terms of partial moduli of continuity. He proved the following and Theorem A. a Let f CI N. If there exists i 0 M such that N ω i0 δ f o as δ 0+ log/δ N ω i δ f O log/δ as δ 0+ i N i i 0

4 4 U. Goginava then the N-dimensional Walsh Fourier series of f converges uniformly in the sense of Pringsheim. b There exists a function f 0 CI N such that ω i δ f 0 O log/δ N as δ 0+ i... N and the N-dimensional Walsh Fourier cubic partial sums of f diverge in the metric of C. In 88 Jordan [6] introduced a class of functions of bounded variation and applying it to the theory of trigonometric Fourier series he proved that if a continuous function has bounded variation then its trigonometric Fourier series converges uniformly. In 906 G. Hardy [5] generalized the Jordan criterion to double Fourier series and introduced the notion of bounded variation for functions of two variables. He proved that if a continuous function of two variables has bounded variation in the sense of Hardy then its trigonometric Fourier series converges uniformly in the sense of Pringsheim. Móricz [8] proved that if f C W I 2 and the function f is of bounded variation in Hardy s sense [5] then its two-dimensional Walsh Fourier series is uniformly convergent to f. For N-dimensional Walsh Fourier series the author [4] proved that if f C W I N and the function f is of bounded partial p-variation f P V p for some p [ then the N-dimensional Walsh Fourier series is uniformly convergent to f. The analogous result for the N-dimensional trigonometric Fourier series was verifed by the author [3]. On the basis of the above facts we can formulate the following problem: Let pn as n and f C W I N N i V i{pn}. What conditions on the partial moduli of continuity ensure the uniform convergence in the Pringsheim sense of the N-dimensional Walsh Fourier series of the function f? is A solution of this problem is given in Theorems and Formulation of the main results. The main result of this paper Theorem. Let pn as n and f C W I N N i V i{pn}. If there exists i 0 M such that ω {i0 }/2 k f o pk + log pk + N as k An N-dimensional series is said to converge in the sense of Pringsheim if its rectangular partial sums converge.

5 Walsh Fourier series 5 and ω {i} /2 k N f O pk + log pk + as k i N i i 0 then the N-dimensional Walsh Fourier series of f converges uniformly in Pringsheim s sense. Corollary. Let pn as n and f CI N N i V i{pn}. If there exists i 0 M such that ω i0 /2 k f o pk log pk and ω i /2 k N f O pk log pk N as k as k i N i i 0 then the N-dimensional Walsh Fourier series of f converges uniformly in Pringsheim s sense. Corollary 2. Let pn as n and p2m cpm for all m where c > 0 is a constant and let f CI N N i V i{pn}. If there exists i 0 M such that N ω i0 δ f o as δ 0+ p[log/δ] log p[log/δ] and N ω i δ f O p[log/δ] log p[log/δ] as δ 0+ i N i i 0 then the N-dimensional Walsh Fourier series of f converges uniformly in Pringsheim s sense. Theorem 2. Let pn and pn log pn on as n and p2m cpm for all m where c > 0 is a constant. Then for any N 2 there exists a function f 0 CI N N i V i{pn} such that N ω i δ f 0 O p[log/δ] log p[log/δ] as δ 0+ i... N and the N-dimensional Walsh Fourier cubic partial sums of f 0 diverge at some point.

6 6 U. Goginava 4. Auxiliary propositions. We shall need the following. Lemma. Let f C W I N. Assume that for any nonempty M we have 2 k V k f u f 2q u 0 q q as k i uniformly with respect to u i i M. Then the N-dimensional Walsh Fourier series of f converges uniformly in Pringsheim s sense. For N 2 the proof can be found in [8]. Using the method of [8] we can easily extend this criterion to N-dimensional Walsh Fourier series. Lemma 2. Let a i... a in and b i...i N be real numbers. Then m M i M M M a i b i...i N M i a m m k k m i a i a i + b k...k N. For N this is the well known Abel transformation and for N 2 it is called the Hardy transformation. The validity of the above equality for any N 3 can be easily verified by induction. where Lemma 3. We have 2 i 2n 2 2 i 2n 3 D qn t dt c > 0 i... 2n + 2 q n 2 2n n n The proof can be found in [0]. 5. Proofs of the main results Proof of Theorem. y Lemma it suffices to show that for all nonempty M 2 k q f 2q u uniformly with respect to u i i M as k i i. q 0

7 From Lemma 2 we write 2 k q f 2q u q Walsh Fourier series 7 q 2 k 2 2 k i i \ 2 k \ l l \ \ + i 2 k i Since for all nonempty M f 2l 2 u we have 3 It is evident that 2 k l q f 2l u f 2l u I f u + I f u. I f u O ω. 2 k f 2 k 2 4 I f u O q 2 q q q + ω 2 k f q u i i l 2l Since for all nonempty M and all q 5 2l f u u i i l O q q i i \{} u i i M\{} l f 2l {} u {} f u. {}

8 8 U. Goginava we have 6 q u i i l [ q u i i l 2l f u [ O q / y 4 and 6 we obtain 7 I f u O Define 2l ] / f u q u i i M\{} l 2 k i 2 f {} u 2l q q +/ [ 2l {} q u i i M\{} l {} χk 4 pk + log 2 pk +. If we apply Hölder s inequality from 7 we get 8 I f u O + { χk 2 k 2 q q χk + q +/ [ 2l q +/ 2l [ {} ] / f {} u {} q u i i M\{} l {} q u i i M\{} l {} ] /. f {} u {} ] / f {} u {} ] / }

9 O + Walsh Fourier series 9 2l {} { / ω {} 2 k f log χk 2 k 2 q χk + {} q +/ [ pk +] /pk + q u i i M\{} l q /pk + y 8 and the assumption of the theorem we obtain { / 9 I f u O ω {} 2 k f log χk O + O { ω {} 2 k 2 q χk + V {pn} f / } q +/ pk + { / ω {} 2 k f log χk + pk + χk 2 k f / pk +} / pk + log pk + + f {} u / }. }. pk + Let M. Then by 9 and the assumption of the theorem we get N { /N 0 I M f M u O ω {} 2 k f pk + log pk + } + o as k M. pk + Let M and < N. Then by 9 and the assumption of the theorem we get I f u O ω {} 2 k f { /N ω {} 2 k f pk + log pk + / /N + } o as k M. pk +

10 0 U. Goginava Owing to 2 0 and the proof of the theorem is complete. Proof of Theorem 2. Let < pl log pl 2l +2. Define the following closed intervals: [ E 2 2l +2 + ] 2 2l... 2 [pl log pl ]. +2 Denote by ϕ the function equal to zero outside this interval at its center and linear on each half-interval. Let where ϕ x 2 [pl log pl ] ϕ x f x ϕ x sgn D ql x f x + l f x l Z q l 2 2l l Suppose that the integers l... l k and -periodic functions f f k are already defined. Then we define l k to be an integer with the following properties: 2 3 k s pl s log pl s l k > l k 2 [pl k log pl k ] 2 2l k+2 2 2l k +2 pl k log pl k l k N N f s x i i [/2 2l k +2 ] where Define [ E k w qlk q lk x i D qlk +x i dx i k q lk 2 2l k l k l k l k+2 ]... 2 [pl k log pl k ]. Denote by ϕ k the function equal to zero outside this interval at its center and linear on each half-interval. Let ϕ k x 2 [pl k log pl k ] ϕ k x f k x ϕ k x sgn D qlk x f k x + l f k x l Z.

11 Define where f 0 x Walsh Fourier series g k x f k N N g k x f k x i. pl k log pl k It is evident that f 0 CI N. First we prove that f 0 V i{pn} i... N. Let Π i : < t i < ti 0 < t i < < t i m i < be any partition with period and ϱπ i /2 n. For n 2l + 2 we can choose integers l k and l k for which 2 2lk +2 2 n < 2 2lk+2. Then i p2l k + 2 pn p2l k + 2. Let s > k. Then it is evident that m i 4 g s x... x i t x i+... x N g s x... x i t x i+... x N pn /pn N. pl k log pl k Let now s < k. Then from the construction of the function f 0 we obtain m i 5 g s x... x i t i x i+... x N N m i pl s log pl s g s x... x i t i x i+... x N pn /pn f s t i f st i N { } pls log pl s exp pl s log pl s 2 pl k N pl s <. pl s log pl s It is evident that m i 6 g k x... x i t i x i+... x N pn /pn q i f s x q g k x... x i t i x i+... x N pn /pn

12 2 U. Goginava N m i f k t i pl k log pl k f kt i pn /pn N 2 n /pn c pl k log pl k 2 2l exp 2{pl k k log pl k }. Let 2l k + 2 n < l k +. Then from 2 we get 7 2 n 2 2l k exp 2{pl k log pl k } q i f k x q exp 2{n + pl k log pl k } 2 2l exp 2{l k + + l k } k 2 2l 2. k Let now l k + n < 2l k + 2. Then we get 2 n /pn { } 8 2 2l exp plk log pl k 2{pl k k log pl k } 4 exp 2 4pl k. pl k + From 6 8 we have 9 V i{pn} g k <. Owing to 4 5 and 9 we obtain f 0 V i{pn}. Next we shall prove that { } N 20 ω i δ f O as δ 0+ p[log/δ] log p[log/δ] for i... N. Let /2 2l k h < /2 2l k. Then it is evident that p2l k p[log 2 /h] p2l k cpl k. Let s k. Then we get N 2 g s x... x i x i + h x i+... x N g s x pl s log pl s N } N O{. pl k log pl k p[log/h] log p[log/h] Let now s < k. Then from the assumption on f s we obtain 22 g s x... x i x i + h x i+... x N g s x... x N N f s x i + h f s x i f s x q pl s log pl s c h2 2l { s pl s log pl s N O From 2 and 22 we obtain 20. q i p[log/h] log p[log/h] } N.

13 Walsh Fourier series 3 Finally we show that the N-dimensional Walsh Fourier series of f 0 diverges at Indeed N 23 S qlk...q lk f 0 0 f 0 0 f 0 u D qlk u i du [0] N i N f 0 u D qlk u i du [02 2l k 2 ] N i N + f 0 u D qlk u i du [2 2l k 2 2 2l k 2 ] N i N + f 0 u D qlk u i du [2 2l k 2 ] N i I + II + III. From the construction of f 0 we obtain 24 Since for x [2 2l k 2 we obtain Then by 3 we get 25 I o as k. { 2 n if x [0 2 n D 2 nx 0 if x [2 n D qlk x w qlk q lk xd qlk +x. III o as k. From the construction of f 0 we have N 26 II f 0 u D qlk u i du [2 2l k 2 2 2l k 2 ] N i N N 2 2l k 2 f k u i D qlk u i du i pl k log pl k i 2 2l k 2 N N 2 [pl k log pl k ] 2l k 2 ϕ k u i D qlk u i du i pl k log pl k i 2 2l k 2 N N 2 [pl k log pl k ] 2l k 2 c D qlk u i du i. pl k log pl k i 2 2l k 2

14 4 U. Goginava From Lemma 3 we obtain 27 2 [pl k log pl k ] 2l k 2 2 2l k 2 D qlk u i du i [pl k log pl k ] 2 i 2l k 2 D qlk u i du i cpl k log pl k. i 2 i 2l k 3 Combining 26 and 27 we have N 28 II c pl k log pl k N c > 0. pl k log pl k Owing to and 28 we obtain lim S q lk...q k lk f 0 0 f 0 0 c > 0. The proof of Theorem 2 is complete. References [] R. D. Getzadze Convergence and divergence of multiple orthogonal Fourier series in C and L metrics Soobshch. Akad. Nauk Gruzin. SSSR in Russian. [2] On the convergence and divergence of multiple Fourier series with respect to the Walsh Paley system in the spaces C and L Anal. Math [3] U. Goginava On the uniform convergence of multiple trigonometric Fourier series East J. Approx [4] On the convergence and summability of N-dimensional Fourier series with respect to the Walsh Paley systems in L p [0 ] N p [ + ] spaces Georg. Math. J [5] G. H. Hardy On double Fourier series Quart. J. Math [6] C. Jordan Sur les séries de Fourier C. R. Acad. Sci. Paris [7] H. Kita and K. Yoneda A generalization of bounded variation Acta Math. Hungar [8] F. Móricz On the uniform convergence and L -convergence of double Walsh Fourier series Studia Math [9] R. E. A. C. Paley A remarkable series of orthogonal functions Proc. London Math. Soc [0] V. I. Tevzadze On the uniform convergence of Walsh Fourier series in: Some Problems of Function Theory Tbilisi in Russian. Department of Mechanics and Mathematics Tbilisi State University Chavchavadze St. Tbilisi 028 Georgia z goginava@hotmail.com Received June Revised version January