MULTIDIMENSIONAL GENERALIZATION OF KOLMOGOROV-SELIVERSTOV-PLESSNER TEST SATBIR SINGH MALHI. (Under the Direction of Alexander Stokolos) ABSTRACT

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1 MULTIDIMENSIONAL GENERALIZATION OF KOLMOGOROV-SELIVERSTOV-PLESSNER TEST by SATBIR SINGH MALHI (Under the Direction of Alexander Stokolos) ABSTRACT In this thesis, we give the brief description of two most important tests of Fourier series namely Dini-Dirichel and Kolmogorov-Seliverstov-Plessner test. We develop further some results of L.V. Zhizhiashvili to obtain a multidimensional anisotropic version of the Kolmogorov-Seliverstov-Plessner test on convergence almost everywhere and negative C-summabilities of the Fourier series. Index Words: Latex, Thesis, Mathematics

2 MULTIDIMENSIONAL GENERALIZATION OF KOLMOGOROV-SELIVERSTOV-PLESSNER TEST by SATBIR SINGH MALHI A Thesis Submitted to the Graduate Faculty of Georgia Southern University in Partial Fulfillment of the Requirement for the Degree MASTER OF SCIENCE STATESBORO, GEORGIA 3

3 c 3 SATBIR SINGH MALHI All Rights Reserved iii

4 MULTIDIMENSIONAL GENERALIZATION OF KOLMOGOROV-SELIVERSTOV-PLESSNER TEST by SATBIR SINGH MALHI Major Professor: Alexander Stokolos Committee: Goran Lesaja Shijun Zheng Electronic Version Approved: July, 3 iv

5 Dedication This thesis is dedicated to my parents and teachers who have supported me all the way since the beginning of my studies. v

6 ACKNOWLEDGMENTS I am proud to express my gratitude to my thesis adviser Dr. Alexander Stokolos, for his guidance and constant supervision as well as providing necessary information regarding the research. Without his guidance and persistent help this thesis would not have been possible. I can not say thanks enough for his tremendous support and help. In addition to my adviser, I would like to thank the rest of thesis committee, Dr. Goran Lesaja and Dr. Shijun Zheng for their encouragement, insightful comments, hard questions and many valuable suggestions. I would also like to express my special gratitude and thanks to Dr. Shijun Zheng for offering me direct study classes and working with me on diverse and exciting projects. I felt motivated and encouraged every time, I attend his meeting. I would also like to thank Georgia Southern University s School of Graduate studies for their financial support and to the Department of Mathematics for providing a facility conductive to research. I am also very grateful to Dr. Stokolos for securing for me the Research Assistantship from the Office of the Vice President for Research of Georgia Southern University. The author wishes to express his love and gratitude to his family members for their understanding and endless support throughout the duration of his studies. vi

7 TABLE OF CONTENTS Page ACKNOWLEDGMENTS vi CHAPTER Introduction Dini-Dirichlet Test Dini Pointwise Test Dini Integral Test Kolmogorov-Seliverstov Test Introduction Multidimensional Fourier series Introduction Zhizhiashvili Theorem Generalization of Kolmogorov-Seliverstov-Plessner Test Introduction Two Dimensional Case Main results Corollaries N-dimensional case Corollaries Conclusion vii

8 CHAPTER INTRODUCTION Let us consider a π-periodic function f. The Fourier series of the function f is defined as where c n = π π π f(x)e inx dx. n= c n e inx, One can define the partial sum of the Fourier series, which is usually denoted by S N (f), as S N (f)(x) = N n= N c n e inx. The convergence of the Fourier series means that S N (f) f as N. A fundamental question about Fourier series is whether the Fourier series of a continuous function converges pointwise to the function. Lejeune Dirichlet was the first to prove that the Fourier series of a continuously differentiable function converges to it everywhere. After Dirichlet s result, many mathematicians, including Dirichlet, B. Riemann, K. Weierstrass and R. Dedekind, stated their belief that the Fourier series of any continuous function would converge everywhere. This was disproved by Paul du Bois- Reymond, who showed in 876 that there is a continuous function whose Fourier series diverges at one point. In 93, Andrey Kolmogorov [5] gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere (c.f. []). Therefore some additional restrictions on functions are indeed required in order to obtain convergence almost everywhere. A natural question to ask is, to what extent one can relax Dirichlet s smoothness

9 condition mentioned above. In the most weak form, it might be stated as pointwise Dini-Dirichlet test.

10 CHAPTER DINI-DIRICHLET TEST In this chapter, we will review Dini test and pointwise convergence of Fourier series. There are many known sufficient conditions for the Fourier series of a function to converge at a given point x. The pointwise convergence of Fourier series is a tricky business, and there is no ultimate theorem, that resolve the issue, just a collection of scattered results useful in different settings. In this chapter we concentrate on a criterion called Dini s test. For a π-periodic function f on the torus T = [, π), we denote the L p -norm f p = ( T f p ) /p p and the symmetric difference by (t; f)(x) = f(x + t) f(x t). The following observation will be widely used in the sequel : f( + t) f( t) p f( + t) p + f( t) p = f p. (.) The (symmetric) L p modulus of continuity is defined by Equation (.) implies ω(f, δ) p = sup (t; f) p. t δ ω(f, δ) p f p.. Dini Pointwise Test Theorem... [Dini Pointwise test(c.f. [] p.3)] Let f L [ π, π] be a πperiodic function, Assume there is some δ > and some fixed point x such that δ δ then the Fourier series of f converges to f at x. f(x t) + f(x + t)) f(x) dt <, (.) t

11 4. Dini Integral Test Theorem... [Dini integral test ]: Let f be defined on [, π) such that π t ω(f, t) dt <, then the Fourier Series of the function f converges to f almost everywhere (a.e) on [, π).

12 CHAPTER 3 KOLMOGOROV-SELIVERSTOV TEST 3. Introduction In this chapter, we will analyze the conditions under which the Fourier series converges almost everywhere using another important test, Kolmogorov-Seliverstov-Plessner test. The Dini test is simple enough for applications and is important because in general Fourier series of some summable functions can diverge as Kolmogorov s results demonstrates. Since the L p norm of a function increases with p (unless a function is a constant) one can replace the L -modulus of continuity in the Dini s test by any L p modulus. However, several improvements of Dini s test with L p modulus were suggested. Theorem 3... [Kolmogorov-Seliverstov (c.f. [] p. 363)] If (a n + b n) ln n <, (3.) then the series a converges almost everywhere. n= + (a n cos nx + b n sin nx) n= Theorem 3... [Plessner Theorem (c.f. [] p. 37)] The convergence of the integral ω(f,δ) dδ implies the convergence of the series δ n= (a n + b n) ln n. Corollary 3..3 (Kolmogorov-Seliverstov -Plessner). If π ω (f, t) dt <, t then the Fourier Series of the function f converges almost everywhere on T.

13 p. 38) We remark that the above condition is often written in a discrete form (c.f. [] n= n ω (f, n ) <. Theorem 3..4 (M. I. Dyachenko[3]). If f L (T ) and ω (f, t) t dt <, then the Fourier Series of f u-converges almost everywhere. 6 One can find the definition of u-convergence in [3]. Note that if the series is u-convergent almost everywhere, then it is convergent pointwise almost everywhere over rectangles. Theorem 3..5 (Marcinkiewicz [6]). If π ω p (f, t) p dt <, < p <, t then the Fourier Series of the function f converges to f almost everywhere on T. In 965, a great discovery came up, L. Carleson [] proved that in one dimension the Fourier series of square-integrable (in particular continuous) functions converge almost everywhere, thus proving Luzin s famous conjecture of 95. Carleson s theorem is a generalization of the Kolmogorov-Seliverstov-Plessner Theorem, although it does not replace the Dini test. After Carleson result, there has not been much more to explore in one dimensional case. In depth inclusion of convergence of Fourier series in one dimension can be found in classical, and quite extensive, monographs of N. Bary [] and A. Zymund []. However, the situation in two and more dimensions is still undeveloped and more complicated. There are fewer results and papers that consider multidimensional case. One of the very few and most comprehensive treatment to date of multidimensional case is the monograph of L.V. Zhizhiashvili []. In the following chapters, we discuss the multidimensional case of Fourier series.

14 CHAPTER 4 MULTIDIMENSIONAL FOURIER SERIES 4. Introduction Let f L (T N ) be a periodic function on the N-dimensional torus T N = [, π] [, π], then the Fourier series of the function f is defined as where a m = (π) N m Z N a m e im x, T N f(y)e im y dy. We denote the partial sums of the Fourier series by S m f. Then the convergence a.e. of these sums in Pringsheim sense means that lim S mf = f a.e. on T N. min(m,...,m N ) In 97 C. Fefferman [4] gave an example of continuous function with almost everywhere divergent double Fourier series. Thus we need to impose some restrictions, for example on the smoothness even for a continuous function. A nice improvement of Fefferman s theorem was done by M. Babuch and E.M. Nikishin [7] in 973. They demonstrated that Fefferman s function has C-modulus of continuity ( ω C (f, t) = O ). t Here ω C (f, t) = sup f( + h) f( ) C. h t One year later, K.I. Oskolkov [8] proved that the above estimate is close to being sufficient. If f C(T ) and ω C (f, t) = o ( t t ),

15 8 then the rectangular Fourier sums converge almost everywhere. The exact condition that may close this gap is still an open question. For a function f that is π-periodic with respect to each variable and integrable on T n, we denote f p = ( f p ) /p. For j =,..., n T n (t j ; f)(x) =f(x,..., x j, x j + t j, x j+..., x n ) f(x,..., x j, x j t j, x j+..., x n ), denotes the symmetric difference in j-th direction with the increment t j. Let (t j,..., t jk ; f) = (t j ; (t j ; (t j3 ;... (t jk ; f)))) be the composition difference taken in the directions of t j,..., t jk. Denote B = {j,..., j k }, (f, x, t B ) = (t j,..., t jk ; f)(x) and T n (B) or I n (B) the k-dimensional sub-torus of T n or I n = (, ) n respectively in variables (t j,..., t jk ). Zhizhiashvili proved the following theorem (see [] p.68), which is the generalization of Theorem... Theorem 4... Let f L(T n ). If [ ( B M T n I n (B) i B t i ) (f, x, t B ) dt B ] dx <, (4.) then the Fourier series of the function f converges almost everywhere. Here M = {,,..., n}(n N, n ) and B is an arbitrary subset of M. The following statement is claimed in [] as a corollary of the above theorem. Corollary 4... Let f L(T n ) and η (, + ). If ω i (t, f) ( π t ) (n+η) t (, π), i n, then the Fourier series of the function f converges almost everywhere.

16 9 We write a(t) b(t), meaning there is a constant C independent of t, so that a(t) Cb(t). In the sequel, the functions, ω i (t) will be called the modulus of continuity, satisfying the conditions ω i () =, ω i () =, and each ω i (t) is concave. To balance the influence of different moduli, we introduce some weights. For this, we translate the information on the system of moduli ω i (t) in terms of functions φ i,j and numbers θk i, defined by ω i (φ ij (t)) = ω j (t) i, j =,..., n t (, ) ω i (θ i k) = k, i =,..., n, k =,,... (4.) Next, we denote the anisotropic Hölder classes by H ω,...,ω n p (T n ) = {f L p (T n ) : ω i (f, t) p ω i (t), i =,..., n}, and the isotropic Hölder classes by H ω p (T n ) = {f L p (T n ) : ω i (f, t) p ω(t), i =,..., n}. In the anisotropic case, A.M. Stokolos proved following results which contain the information directly in terms of ω i (t). Theorem [9] The convergence of the series N k θk i k= i= <, is sufficient for the convergence almost everywhere of the Fourier series of functions from H ω,...,ω n (T n ). In the isotropic case the above theorem turns into the following.

17 Corollary The convergence of the integral ω(t) ( t t )n dt <, is sufficient for the convergence almost everywhere of the Fourier series of functions from H ω (T n ). Theorem 4..3 may be considered as an anisotropic version of the Dini integral test. A natural question arises about a multidimensional generalization of Kolmogorov-Seliverstov-Plessner s results. Theorem provides the answer to this question. Moreover, due to the results of Zhizhiashvili, we were able to consider not only convergence but much stronger results - negative C - summabilities. The definition of C- summabilities for arbitrary series of numbers is as follow: We define the Cesaro numbers as A α =, A α n = (α + )(α + )... (α + n) n! n, α,,..., i.e A α n are the coefficients of Taylor expansion of the function ( z) = A α nz n z <. +α n= Then for any series of numbers a n with partial sum S n, the linear means is defined as σ α n = A α n n k= A α n k S k = A α n n A α n ka k. Then we say the series whose partial sums are S n is summable or, briefly, summable (C, α) to the limit(sum) S, if lim n σα n = S. k=

18 4. Zhizhiashvili Theorem In this section we state some more important results of L.Zhizhiashvili, which will be useful in the sequel. Theorem 4... [] If α [, ) n, f L (T n ), and (f, x, t B ) dt B dx <, (4.3) t i +α i B M, B T n I n (B) i B then the Fourier series of the function f is (C, α) - summable almost everywhere. In [], it is stated without proof, that above theorem implies the following corollaries. Corollary 4... Let η (, + ). If f L (T n ) and ω i (t, f) ( π t ) ( n +η) t (, π), i n, then the Fourier series of the function f converges almost everywhere. Corollary Suppose that η (, + ). If f L (T n ), α (, ) and ω i (t, f) t α ( π t ) ( n +η) t (, π), i n, then the Fourier series of the function f is (C, β) - summable almost everywhere, where β = ( α n,, α n ).

19 CHAPTER 5 GENERALIZATION OF KOLMOGOROV-SELIVERSTOV-PLESSNER TEST 5. Introduction In this chapter, we give a new test for the almost everywhere convergence of the multidimensional Fourier series, which is a simplified version of Zhizhiashvili test. We claim that our test is simpler and easier to apply. We will begin with two dimensions case and generalize our results to N- dimensions. Lemma 5... [9] Suppose that ω i (t) are the moduli of continuity, ω i () =, and ϕ ij is the inverse function to φ ij. Then φ ij (φ jk (t)) = φ ik (t), ϕ ij (t) = φ ji (t), φ ij (θ j k ) = θi k, φ ii (t) = t, φ ij (t). For a square integrable function f on a torus T m, we define f = f(x) dx. T m Lemma 5... [9] For any s k N the following inequality is valid : (t i,..., t ik ; f) 4 (t i,..., t is, t is+, t ik ; f). The above lemma is an extension of (.).

20 3 5. Two Dimensional Case The following theorem is a corollary of Theorem 4.. where we change the order of integration. Theorem 5... Let f be π-periodic function defined on T such that + (t ; f) dt + t (t ; f) t dt (t, t ; f) dt dt t t <. Then the Fourier Series of the function f converges to f a.e on T. 5.. Main results In this section, we give the main results of the thesis, which are the generalization of Kolmogorov-Seliverstov-Plessner test in two dimensional case. We will begin with series of lemmas, which will be useful in proving the main theorems. The case of C(, ), C(, α) and C( α, α )- summabilities will be considered. Lemma 5... Let f be π-periodic function defined on T. Then Proof. We have (t, t, f) dt dt t t (t ; f) + t (t ; f) t φ, (t ) dt. φ, (t ) dt (t, t, f) dt dt t t = φ, (t ) (t, t, f) dt dt t t + φ, (t ) (t, t, f) dt dt t t = I + I.

21 4 For I, applying Lemma 5.. and integrating with respect to t, we get (t ; f) I 4 t φ, (t ) dt. For I, we interchange the the order of integration. Then taking into account that ϕ, = φ,, we get I = = φ, (t ) (t, t ; f) dt dt t t ( ) (t, t ; f) dt φ, (t ) t 4 (t ; f) ( φ, (t ) dt t ) dt t dt t 4 (t ; f) dt. φ, (t ) t Lemma Let f be π-periodic function defined on T. Then Proof. We have (t, t ; f) dt dt t +α t + (t, t ; f) dt dt t +α t (t ; f) t +α (t ; f) t (φ, (t )) α dt. φ, (t ) dt = φ, (t ) (t, t ; f) dt dt t +α t + φ, (t ) (t, t ; f) dt dt t +α t = I + I.

22 5 For I, applying Lemma 5.. and integrating with respect to t, we get I 4 (t ; f) t (φ, (t )) α dt. For I, we interchange the order of integration. Then taking into account that ϕ, = φ,, we get I = = = 4 4 φ, (t ) (t, t ; f) dt dt t +α t ( ) (t, t ; f) dt dt φ, (t ) t t +α ( ) 4 (t ; f) dt dt φ, (t ) t t +α (t ; f) (t ; f) t +α ( φ, (t ) dt t ) φ, (t ) dt. dt t +α Lemma Let f be π-periodic function defined on T. Then (t, t, f) + dt dt t +α t +α (t ; f) t +α (t ; f) t +α (φ, (t )) dt. α (φ, (t )) dt α

23 6 Proof. We have (t, t, f) dt dt t +α t +α = + φ, (t ) φ, (t ) (t, t, f) (t, t, f) dt dt t +α t +α dt dt t +α t +α = I + I. For I, applying Lemma 5.. and integrating with respect to t, we get I 4 (t ; f) t +α (φ, (t )) dt. α For I, we interchange the order of integration. Then taking into account that ϕ, = φ,, we get I = = = 4 φ, (t ) ( ( φ, (t ) φ, (t ) 4 (t ; f) (t ; f) t +α (t, t, f) dt dt t +α t +α (t, t ; f) dt t +α 4 (t ; f) dt t +α ( φ, (t ) dt t +α ) ) ) (φ, (t )) dt. α dt t +α dt t +α dt t +α In the following theorem, we consider the case of C(, )- summability, i.e almost everywhere convergence of the Fourier series.

24 7 Theorem If f H ω,ω (T ), then the convergence of the series k θ k= k θk implies the convergence of the Fourier Series of f almost everywhere on T. Proof. By Theorem 4.., changing the limits of integration in (4.) and making use of the Lemma 5.., we have the following derivation: (t ; f) (t ; f) dt + dt + t t (t, t ; f) dt dt t t (t ; f) (t ; f) dt + dt t t (t ; f) + t φ, (t ) dt (t ; f) + t φ, (t ) dt (t ; f) t φ, (t ) dt + (t ; f) t φ, (t ) dt θ k k= θk+ (ω (t )) t φ, (t ) dt + θ k k= θk+ (ω (t )) t φ, (t ) dt = k= θ k θ k+ k t k φ, (θk+ ) k= k= φ, (θk+ )dt + θ k+ k + θk+ θk+ Thus the convergence of the series is sufficient for (4.) to be valid. t dt + k= k= θ k k= k θ k= k θk θ k+ k t φ, (θ k+ )dt k φ, (θk+ ) k. θk+ θk+ θ k+ t dt

25 In the following theorem, we consider the case of C(, α)- summability of the Fourier series. 8 Theorem If f H ω,ω (T ), then convergence of the series k= k (θ k )α θ k implies the (C, α, ) - summability of the Fourier Series of f almost everywhere on T. Proof. By Theorem 4.., changing the limits of integration in (4.3) and making use of the Lemma 5..3, we have the following derivation: (t ; f) (t ; f) t +α dt + dt + t (t, t, f) dt dt t +α t (t ; f) (t ; f) t +α dt + dt t (t ; f) + t +α φ, (t ) dt + (t ; f) t (φ, (t )) α dt (t ; f) t +α φ, (t ) dt + (t ; f) t (φ, (t )) α dt θ k k= θk+ (ω (t )) t +α φ, (t ) dt + θ k k= θk+ (ω (t )) t (φ, (t )) α dt = k= θ k θ k+ k k= k= k t +α k θ k+ φ, (θ k+ ) φ, (θk+ )dt + θ k+ t +α (θ + k+ )α k= k= dt + θ k θ k+ k= k t (φ, (θ k+ ))α dt k (φ, (θ k+ ))α k. (θk+ )α θk+ θ k+ t dt

26 9 Thus, the convergence of the Series is sufficient for (4.3) to be valid. k= k (θ k )α θ k In the following theorem, we consider the case of C( α, α )- summability of Fourier series. Theorem If f H ω,ω (T ), then convergence of the series k= k (θ k )α (θ k )α implies the (C, α, α ) - summability of the Fourier Series of f almost everywhere on T. Proof. By Theorem 4.., changing the limits of integration in (4.3) and making use

27 of the Lemma 5..4, we have the following derivation: (t ; f) t +α dt + (t ; f) t +α dt + (t, t, f) dt dt t +α t +α + (t ; f) t +α dt + (t ; f) t +α (t ; f) t +α dt (φ, (t )) dt α + (t ; f) t +α (φ, (t )) dt α (t ; f) t +α (φ, (t )) dt α + (t ; f) t +α (φ, (t )) dt α θ k k= θk+ (ω (t )) t +α (φ, (t )) α dt + θ k k= θk+ (ω (t )) t +α (φ, (t )) α dt = k= k= k= θ k θ k+ k t +α k (φ, (θ k+ ))α k (θ k+ )α (φ, (θ k+ ))α dt + θ k+ t +α (θ + k+ )α Thus, the convergence of the Series is sufficient for (4.3) to be valid. k= k= k (θ k )α k= dt + θ k θ k+ k= k (θ k+ )α (θ k )α k t +α (φ, (θ k+ ))α dt k (φ, (θ k+ ))α (θ k+ )α. θ k+ t +α dt 5.. Corollaries The following corollaries state three major cases of convergence almost everywhere of the Fourier series in two dimensional case. (I) C(, ) - summability;

28 (II) C( α, ) - summability; (III) C( α, α ) - summability. For each of the above cases, we will also state and prove a particular case. Corollary Let ω (t) = ( /t) α, ω (t) = ( /t) α, α i >. Then the condition /α + /α < is sufficient for the convergence of the Fourier series of f, f H ω, ω (T ) almost everywhere on T. Note, that in the isotropic case, the condition in the above corollary should be /α i <, i =,. Proof. We have ω (t) = ( /t) α, ω (t) = ( /t) α, α >, α >. From the equation (4.), we have ω i (θ i k) = k, i =,..., n, k =,,.... By using the above balance equation, we can transfer our information in terms of θ k and θ k and we obtain ( ) ( ) = k/α, = k/α, k =,,... θk θk k θ k= k θk = = k k/α k/α k= k= ( ) + α α k. The series on the right hand side is a geometric series, which is convergent only if α + α <.

29 Therefore, from Theorem 5..5, it follows that the condition /α + /α < is sufficient for convergence of the Fourier series of f, f H ω, ω (T ) almost everywhere on T. Corollary Let ω (t) = ( /t) γ, ω (t) = t β, γ >, β >. Then the condition γ + α β < is sufficient for (C, α, ) - summability of the Fourier series of f, f H ω, ω (T ) almost everywhere on T. Proof. we have ω (t) = ( t ) γ, ω (t) = t β, γ >, β >. From equation (4.), we have ω i (θ i k) = k, i =,..., n, k =,,.... By using the above balance equation, we can transfer our information in terms of θ k and θ k. θ k = k/γ, θ k = k β, k =,,... and we obtain k θ k= k (θ = k k/γ (α/β)k k )α k= = ( γ + α β )k. The series on the right hand side is a geometric series, which is convergent only if + α <. γ β k= Therefore, from Theorem 5..6, it follows that the condition γ + α β < is sufficient for convergence of the Fourier series of f, f H ω, ω (T ) almost everywhere on T.

30 Corollary 5... Let ω (t) = t β, ω (t) = t β, β >, β >. Then the condition α β + α β < is sufficient for the (C, α, α ) - summability of the Fourier series of f, f H ω, ω (T ) almost everywhere on T. Proof. We have From equation (4.), we know that ω (t) = t β, ω (t) = t β, β >, β >. ω i (θ i k) = k, i =,..., n, k =,, By using the above balance equation, we can transfer our information in terms of θ k and θ k. and we obtain k= θ k = k/β, k (θ k )α θ k = k/β, k =,,... (θ = k (α /β )k (α /β )k k )α = k= k= ( ) α + β α β k. The series on the right hand side is a geometric series, which is convergent only if α β + α β <. Therefore, from Theorem 5..7, it follows that the condition α β + α β < is sufficient for the convergence of the Fourier series of f, f H ω, ω (T ) almost everywhere on T. 5.3 N-dimensional case In this section, we state and prove the main results of the thesis in N- dimension, which are the multidimensional extension of Kolmogorov-Seliverstov-Plessner test on convergence almost everywhere and negative C- summabilities of the Fourier series. We will begin with series of lemmas, which will be useful in proving the main theorems.

31 Lemma Suppose that ω i (t) are moduli of continuity, ω i () =, N, k N. Then J T n (B) s B ( ) (f,, t B ) t j dt B s tf t j B j B,j s φ j,s (t s ) dt. Proof. We will prove the lemma by induction. Without loss of generality, we can assume that B = (,,..., ). Here we denote (f,, t B ) = (t,..., t k ; f). The first step of the induction gives : 4 J = + dt t dt t J + J. dt k t k dt k t k φ k,k (t k ) φk,k (t k ) (t,..., t k ; f) dt k t k (t,..., t k ; f) dt k t k Let us estimate J. Applying Lemma 5.. and integrating with respect to t k, we obtain J 4 dt dt k t t k ( (t,..., t k, t k ; f) φ k,k (t k ) ) dtk t k. For J, we interchange the limits of integration with respect to t k and t k. Then taking into account that ϕ k,k = φ k,k, we get J = dt t t dt k t k dt k t k φ k.k(tk ) (t.... t k ; f) dt k t k. Applying Lemma 5.. with v = k and integrating with respect to t k, we have dt dt ( ) k dtk J 4 (t,..., t k, t k ; f). t t k φ k.k (t k ) t k As a result of the first step, we obtain the estimate k dt dt k J (t,..., t k, t s ; f) s=k t t k k ( ) dts. φ j,s (t s ) t s j=k,j s

32 5 Suppose that the s-th step of the induction, we have the following relation: J k i=k s dt dt k s t t k s (t,..., t k s, t i ; f) k j=k s,j i We will make the (s + ) th step. Let k s i k, then dt i φ j,i (t i ) t i k i=k s dt dt k s J i 4 (t,..., t k s ; f) t t k s φi,k s (t k s ) k dt i φ j,i (t i ) t i j=k s,j i dt dt φi,k s (t k s ) k s +4 (t,..., t k s, t i ; f) t t k s k j=k s,j i J i. dt i 4(Ji + Ji ). φ j,i (t i ) t i Lets us estimate J i. Taking into account the monotonicity of φ j,i (t), we get J i dt t (t,..., t k s ; f) k j=k s,j t φ j,i (φ i.k s (t k s )) φ i.k s (t k s ) dt i t i dt k s t k s. Since by Lemma 5.., φ j,i (φ i,k s (t k s )) = φ j,k s (t k s ), we have J i dt t (t,..., t k s ; f) k j=k s,j i φ i,k s (t k s ) φ j,k s (t k s ) dt k s t k s. Now let us estimate J i. Having changed the order of integration with respect to

33 6 t k s and t i and taking into account of the Lemma 5.., we obtain J i dt t φ k s,i (t i ) (t,..., t k s, t i ; f) dt k s t k s = j=k s,j i k j=k s,j i dt i φ j,i (t i ) t i dt (t,..., t k s, t i ; f) t k φ j,i (t i ) dt i. φ k s,i (t i ) t i then it is obvious that J k i=k s J i k i=k s dt dt k s (t,... t k s, t i ; f) t t k s k j=k s,j i dt i. φ j,i (t i ) t i Thus the passage from s to s+ is finished, which completes the proof of lemma. Lemma Suppose that ω i (t) are moduli of continuity, ω i () =, N, k N. Then J = T n (B) s B ( (f,, t B ) (t j ) i) +α dt B s tf t +α i j B j B,j s (φ j,s (t s )) α j dt. Proof. We will prove the lemma by induction. Without loss of generality we can assume that B = (,..., ). The first step of the induction gives : J = + dt t +α dt t +α J + J. dt k t +α k k dt k t +α k k φ k,k (t k ) φk,k (t k ) (t,..., t k ; f) dt k t +α k (t,..., t k ; f) dt k t +α k k k

34 Let us estimate J. Applying Lemma 5.. and integrating with respect to t k, we obtain dt J 4 +α t dt k t k +α k ( (t,..., t k, t k ; f) φ k,k (tk ) ) αk dt k t k +α k. For J, we interchange the limits of integration with respect to t k and t k. Then taking into account that ϕ k,k = φ k,k, we get J = dt t t +α φ k.k(tk ) dt k t k +α k dt k (t k ) +α k dt k (t.... t k ; f) +α t k. k Applying Lemma 5.. with v = k and integrating with respect to t k, we have dt J 4 +α t dt k t k +α k ( As a result of the first step, we obtain the estimate J k s=k dt +α t dt k t k +α k (t,..., t k, t k ; f) φ k.k (t k ) k j=k,j s ) αk dt k t k +α k. (t,..., t k, t s ; f) ( ) αj dt s +α φ j,s (t s ) t s. s Suppose that in the s th step of the induction, we have the following relation: J k i=k s dt +α t dt k s t k s +α k s k j=k s,j i (t,..., t k s, t i ; f) ( ) αj dt i φ j,i (t i ) +α t i i k J i. i=k s 7

35 8 We will make the (s + ) th step. Let k s i k. Then dt J i 4 +α t dt +4 +α t dt k s t k s +α k s dt k s t k s +α k s φi,k s (t k s ) k j=k s,j i φi,k s (t k s ) k j=k s,j i (t,..., t k s ; f) ( ) αj dt i φ j,i (t i ) +α t i i (t,..., t k s, t i ; f) ( ) αj dt i +α φ j,i (t i ) t i 4(Ji + Ji ). i Lets us estimate J i. Taking into account the monotonicity of φ j,i (t), we get J i dt t +α (t,..., t k s ; f) k ( j=k s,j t φ j,i (φ i.k s (t k s )) φ i.k s (t k s ) Since, by Lemma 5.., φ j,i (φ i,k s (t k s )) = φ j,k s (t k s ), we have J i dt t +α (t,..., t k s ; f) ( k j=k s,j i φ i,k s (t k s ) ) αj dt i t i +α i dt k s t k s. ( φ j,k s (t k s ) ) αj ) αi dt k s +α t k s. k s Now, let us estimate J i. Having exchanged the order of integration with respect to t k s and t i and taking into account of the Lemma 5.., we obtain J i dt +α t φ k s,i (t i ) (t,..., t k s, t i ; f) dt k s t k s +α k s = k j=k s,j i ( φ j,i (t i ) dt α t ) αj ( k j=k s,j i ( ) αj dt i φ j,i (t i ) +α t i i (t,..., t k s, t i ; f) φ k s,i (t i ) ) αk s dt i t i +α i.

36 9 Then it is obvious that J k i=k s J i k i=k s dt +α t dt k s t k s +α k s (t,... t k s, t i ; f) k j=k s,j i ( ) αj dt i +α φ j,i (t i ) t i. i Thus, the passage from s to s + is finished, which completes the proof. In the following theorem, we consider the case of C(,,..., )- summability i.e almost everywhere convergence of the Fourier series. Theorem Suppose that ω i (t) are moduli of continuity ω i () =, and θ i k are defined by ω i (θ i k) = k, k =,,... i =,..., N. Then for f H ω,...,ω N (T N ) the convergence of the series N k θk i k= implies the convergence of the Fourier Series of f almost everywhere on T N. i= Proof. By the Theorem 4.., exchanging of the limits of integration in (4.) gives the term T N (B) which by Lemma 5.3. is dominated by The above term is smaller than (f,, t B ) dt B, t i i B (t, t,..., t s ; f) t (t, t,..., t s ; f) t j B,j s N j=b,j s φ j,s (t s ) dt. φ j,s (t s ) dt

37 3 Let us estimate the above integral. For a fixed j, we have (t, t,..., t j ; f) t k= θ j k θ j k+ (ω j (t)) N k k= t i=,i j N i=,i j N i=,i j φ i,j (θ j k+ ) φ i,j (t) dt φ i,j (t) dt θ j k+ dt t N k. θk i Naturally, the constant of the sign depends on the dimension and the constants of the definition of H ω,...,ω N. Thus, the convergence of the series k= i= is sufficient for (4.) to be valid. N k θk i k= i= < In the following theorem, we consider the case of (C, α, α..., α N ) - summability of Fourier Series. Theorem Suppose that ω i (t) are moduli of continuity ω i () =, and θ i k are defined by ω i (θ i k) = k, k =,,... i =,..., N Then for f H ω,...,ω N (T N ), the convergence of the series N k k= (θ i i= k )α k implies the (C, α, α..., α N ) - summability of the Fourier Series of f almost everywhere on T N, < α i < Proof. By Theorem 4.., changing the limits of integration in (4.3) gives the term T n (B) (f,, t B ) dt B, t i +α i i B

38 3 which by Lemma 5.3. is dominated by The above term is smaller than s tf t +α s tf t +α j B,j s N j= (φ j,s ) +α j dt (φ j,s ) +α j dt. Let us estimate the above integral. For a fixed j, we have j tf t +α k= θ j k θ j k+ N i=,i j N k k= (ω j (t)) t +α i= (φ i,j ) α i dt N i= (φ i,j (θ j k+ ))α i (φ i,j (t)) α i dt θ j k+ dt t N k +α k= (θ i i= k )α k Naturally, the constant of the sign depends on the dimension and the constants of the definitions of H ω,...,ω N Thus, the convergence of the series. is sufficient for 4.3 to be valid. N k (θ i k= i= k )α k < Define Note that α = P α (t) = t /(αt α ) α >. ξ dt t +α P α(ξ) α. Theorem Suppose that ω i (t) are moduli of continuity, ω i () =, and θ i k are defined by ω i (θ i k) = k, k =,,... i =,..., N.

39 Then for f H ω,...,ω n (T n ), the convergence of the series N k P αi (θk), i k= i= implies (C, α,..., α n ) - summability almost everywhere of the Fourier Series of f on T n, α i <. We remark that the theorem above can be stated in direct terms as follows: Theorem Suppose that ω i (t) are moduli of continuity, ω i () =, and θ i k are defined by ω i (θ i k) = k, k =,,... i =,..., N. Then, for f H ω,...,ω N (T n ), the convergence of the series v k θk i k= i= N (θ i i=v+ k )α k implies the (C,,..., α v+,..., α N ) - summability almost everywhere of the Fourier series of f on T N, α i < Corollaries Corollary Let ω i (t) = ( ( t )) α i, α i >, i =,,... N. Then the condition α + α α N < is sufficient for the convergence of the Fourier series of f, f H ω,ω,...,ω N (T N ) almost everywhere on T N. Proof. From equation (4.), we have ω i (θ i k) = k, i =,..., n, k =,,.... By using the above balance equation, we can transfer our information in terms of θk i, ω i (t) = (( t ) αi ) = k/α i, i =,..., n, k =,,..., θk i

40 33 and we obtain N k = θk i k= i= = N k k= k= i= k/α i ( α + α α N )k. The series on the right hand side is a geometric series, which is convergent only if α + α α N <. Therefore, from Theorem 5.3.4, it follows that the condition α + α α N <. is sufficient for the convergence of the Fourier series of f, f H ω,ω,...,ω N (T N ) almost everywhere on T N. Corollary Let ω i (t) = t β i β i >, i =,,... N. Then the condition α β + α β α N β N < is sufficient for (C, α, α,..., α N )- summability of the Fourier series of f, f H ω,ω,...,ω N (T N ) almost everywhere on T N. Proof. From equation (4.), we have ω i (θ i k) = k, i =,..., n, k =,,.... By using the above balance equation, we can transfer our information in terms of θ i k, ω i (t) =t β i θ i k = k/β i, i =,..., n, k =,,..., and we obtain N k k= (θ i i= k )α k = = N k k= k= i= α ik/β i ( α β + α β α N βn )k. The series on the right hand side is a geometric series, which is convergent only if α β + α β α N β N <.

41 Therefore, from theorem 5.3.4, it follows that the condition α β + α β α N β N < is sufficient for (C, α, α,..., α N )- summability of the Fourier series of f, f H ω,ω,...,ω N (T N ) almost everywhere on T N Conclusion We have suggested the multidimensional extension of Kolfmogorov-Seliverstov-Plessner test on almost everywhere convergence of the Fourier Series. Both isotropic and anisotropic situations were considered as well as a stronger case of a negative (C, α) summability.

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