scholarly journals Embedding relations of Besov classes under MVBV

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5221-5237
Author(s):  
Wen-Tao Cheng ◽  
Kan Yu ◽  
Ding-Yuan Chen
Keyword(s):  
Modulus Of Continuity ◽  
Fourier Coefficients ◽  
Besov Class ◽  
Lp Norm

In this paper, we strengthen some of Leindler?s results from [L. Leindler. Embedding relations of Besov classes. Acta Sci. Math. (Szeged), 73(2007) 133-149.] under MVBV condition. First, we discuss embedding relations between two Besov classes. Next, we give an equivalent estimate for the k-order modulus of continuity of f (x) in Lp norm under MVBV condition. Finally, we give the condition to ensure a function f ? Lp having Fourier coefficients of MVBV belongs to the Besov class.

scholarly journals Fourier series with small gaps

1989 ◽  
Vol 46 (2) ◽  
pp. 212-219
Author(s):  
P. Isaza ◽  
D. Waterman
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Bounded Variation ◽  
Modulus Of Continuity ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Order Condition ◽  
Circle Group ◽  
The Difference ◽  
Bounded Below

AbstractA trigonometric series has “small gaps” if the difference of the orders of successive terms is bounded below by a number exceeding one. Wiener, Ingham and others have shown that if a function represented by such a series exhibits a certain behavior on a large enough subinterval I, this will have consequences for the behavior of the function on the whole circle group. Here we show that the assumption that f is in any one of various classes of functions of generalized bounded variation on I implies that the appropriate order condition holds for the magnitude of the Fourier coefficients. A generalized bounded variation condition coupled with a Zygmundtype condition on the modulus of continuity of the restriction of the function to I implies absolute convergence of the Fourier series.

On the almost everywhere convergence of Fourier series

1967 ◽  
Vol 63 (3) ◽  
pp. 703-705 ◽  
Author(s):  
B. S. Yadav
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Modulus Of Continuity ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Modulus Of Smoothness ◽  
General Concept ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere ◽  
Image Position

Let f be a 2π-periodic function of the class L(−π,π). PutWe call, with Žuk(6), the quantity L(p)(h, f) the L-modulus of smoothness of order p of the function f. Žuk has recently obtained, in (5) and (6), generalizations of a number of classical results on the absolute convergence of Fourier series, as also on the order of Fourier coefficients by employing the concept of the L-modulus of smoothness which is obviously a more general concept than that of the modulus of continuity. It is the purpose of this note to prove a theorem on the almost everywhere convergence of Fourier series of f involving the concept of L(p)(h, f).

scholarly journals Estimates ofLpModulus of Continuity of Generalized Bounded Variation Classes

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Heping Wang ◽  
Zhaoyang Wu
Keyword(s):  
Bounded Variation ◽  
Modulus Of Continuity ◽  
Fourier Coefficients ◽  
Necessary Conditions ◽  
Sharp Estimates ◽  
Sufficient And Necessary Conditions ◽  
Special Case

Some sharp estimates of theLp1≤p<∞modulus of continuity of classes ofΛφ-bounded variation are obtained. As direct applications, we obtain estimates of order of Fourier coefficients of functions ofΛφ-bounded variation, and we also characterize some sufficient and necessary conditions for the embedding relationsHpω⊂ΛφBV. Our results include the corresponding known results of the classΛBVas a special case.

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