Almost Everywhere Strong C ,1,0 Summability of 2-Dimensional Trigonometric Fourier Series
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DOI: 10.1007/s13226-020-0457-x
ALMOST EVERYWHERE STRONG C,1,0 SUMMABILITY OF 2-DIMENSIONAL TRIGONOMETRIC FOURIER SERIES1 Ushangi Goginava I. Vekua Institute of Applied Mathematics and Faculty of Exact and Natural Sciences of I. Javakhishvili Tbilisi State University, Tbilisi 0186, 2 University str., Georgia e-mail: [email protected] (Received 28 September 2016, accepted 21 June 2019) In this paper we study the a. e. exponential strong (C, 1, 0) summability of of the 2-dimensional ¡ ¢2 trigonometric Fourier series of the functions belonging to L log+ L . Key words : 2-Dimensional Fourier series; strong summability; exponential means. 2010 Mathematics Subject Classification : 42B08.
1. I NTRODUCTION We shall denote the set of all non-negative integers by N. Let T := [−π, π) = R/2π and R := (−∞, ∞). We denote by Lp (T) the class of all measurable functions f on R that are 2π-periodic and satisfy
1/p Z kf kp := |f |p < ∞, 1 ≤ p < ∞. T
Suppose f ∈ L1 (T) and put M (x; f ) := sup I
1 |I|
Z |f (y)| dy, I
where the supremum is taken over the collection {I} of those open intervals I which contains x and of lenght ≤ 2π. 1
The author supported by Shota Rustaveli National Science Foundation grant 217282 (Operators of Fourier analysis in
some classical and new function spaces).
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USHANGI GOGINAVA
The Fourier series of the function f ∈ L1 (T) with respect to the trigonometric system is the series
∞ X
fb(n) einx ,
(1)
n=−∞
where
1 fb(n) := 2π
Z f (x) e−inx dx T
are the Fourier coefficients of f . We shall write a . b, if a < c · b and c > 0 is an absolute constant. Denote by Sn (x, f ) the partial sums of the Fourier series of f and let n
1 X Sk (x, f ) σn (x, f ) = n+1 k=0
be the (C, 1) means of (1). Fej´ez [2] proved that σn (f ) converges to f uniformly for any 2π-periodic continuous function. Lebesgue in [18] established almost everywhere convergence of (C, 1) means if f ∈ L1 (T). The strong summability problem, i.e. the convergence of the strong means n
1 X |Sk (x, f ) − f (x)|p , n+1
x ∈ T,
p > 0,
(2)
k=0
was first considered by Hardy and Littlewood in [11]. They showed that for any f ∈ Lr (T) (1 < r < ∞) the strong means tend to 0 a.e., if n → ∞. The trigonometric Fourier series of f ∈ L1 (T) is said to be (H, p)-summable at x ∈ T, if the values (2) converge to 0 as n → ∞. The (H, p)-summability problem in L1 (T) has been investigated by Marcinkiewicz [19] for p = 2, and later by Zygmund [34] for the general case 1 ≤ p < ∞. Oskolkov in [20] proved the following Theorem Os — (Oskolkov). Let f ∈ L1 (T) and let Φ be a continuous positive convex function on [0, +∞) with Φ (0) = 0 and ln Φ (t) = O (t/ ln ln t)
(t → ∞) .
(3)
Then for almost all x n
1 X Φ (|Sk (x, f ) − f (x)|) = 0. lim n→∞ n + 1
(4)
k=0
It was noted in [20] that Totik announced the conjecture that (4) holds almost everywhere for any f ∈ L1 (T), provided ln Φ (t) = O (t)
(t → ∞) .
(5)
ALMOST EVERYWHERE EXPONENTIAL SUMMABILITY
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In [21] Rodin proved Theorem R — (Rodin). Let f ∈ L1 (T). Then for any A > 0 n
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