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## On p-Λ-bounded variation

Venue: | Bull. Iranian Math. Soc |

Citations: | 3 - 3 self |

### Citations

457 | Frame Analysis - Goffman - 1974 |

15 |
On convergence of Fourier series of functions of generalized bounded variation
- Waterman
- 1972
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Citation Context ... is found. 1. Introduction In 1980 M. Shiba [13] introduced the class ΛBV (p) (1 ≤ p < ∞) expanding a fundamental concept of bounded Λ-variation formulated and usefully applied by D. Waterman in 1972 =-=[18]-=-. The main objective of this note is to find a necessary and sufficient condition for the embedding ΛBV (p) ⊂ Hω1 . However, some estimates on the modulus of variation and on the integral modulus of c... |

12 |
On the magnitude of Fourier coefficients
- Schramm, Waterman
- 1982
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Citation Context ...Some of the basic properties of functions of class ΛBV (p) were discussed by R. G. Vyas in [16] recently. More results concerned with the Waterman-Shiba classes and their applications can be found in =-=[2, 3, 10, 11, 12, 14]-=- and [15]. ΛBV (p) equipped with the norm ‖f‖Λ, p := |f(0)|+ V (f) is a Banach space. Given a function f ∈ ΛBV (p), we will also consider the p-Λ-variation function of f v(x) := V (f ; [0, x]) defined... |

11 |
On the absolute convergence of Fourier series of functions of class ΛBV (p
- Shiba
- 1980
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Citation Context ...lasses into classes of functions with given integral modulus of continuity is given. A useful estimate on modulus of variation of functions of class ΛBV (p) is found. 1. Introduction In 1980 M. Shiba =-=[13]-=- introduced the class ΛBV (p) (1 ≤ p < ∞) expanding a fundamental concept of bounded Λ-variation formulated and usefully applied by D. Waterman in 1972 [18]. The main objective of this note is to find... |

8 |
Absolute convergence of Fourier series of functions of ΛBV (p
- Schramm, Waterman
- 1982
(Show Context)
Citation Context ...Some of the basic properties of functions of class ΛBV (p) were discussed by R. G. Vyas in [16] recently. More results concerned with the Waterman-Shiba classes and their applications can be found in =-=[2, 3, 10, 11, 12, 14]-=- and [15]. ΛBV (p) equipped with the norm ‖f‖Λ, p := |f(0)|+ V (f) is a Banach space. Given a function f ∈ ΛBV (p), we will also consider the p-Λ-variation function of f v(x) := V (f ; [0, x]) defined... |

7 | On the absolute convergence of small gaps Fourier series of fuction of ΛBV (p
- Vyas
(Show Context)
Citation Context ...Some of the basic properties of functions of class ΛBV (p) were discussed by R. G. Vyas in [16] recently. More results concerned with the Waterman-Shiba classes and their applications can be found in =-=[2, 3, 10, 11, 12, 14]-=- and [15]. ΛBV (p) equipped with the norm ‖f‖Λ, p := |f(0)|+ V (f) is a Banach space. Given a function f ∈ ΛBV (p), we will also consider the p-Λ-variation function of f v(x) := V (f ; [0, x]) defined... |

5 |
On the singularities of certain families of nonlinear mappings
- Breckner, Trif
- 1995
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Citation Context ...equence of Theorem 3.1. Theorem 3.4. Let f in ΛBV (p) be 1-periodic. Then, ω1(δ; f) ≤ 2V (f)(∑b 1 δ c i=1 1 λi ) 1 p . Proof. Given a positive integer n, we get (3.2) ν(n, f, [0, 1 + 1 n ]) ≤ ν(n, f, =-=[0, 2]-=-) ≤ 2ν(n, , f, [0, 1]), because f is 1-periodic and because the modulus of variation is subadditive with respect to intervals. Therefore, ω1( 1n , f) ≤ sup 0<h≤ 1 n ∫ 1 0 |f(x+ h)− f(x)|dx = sup 0<h≤ ... |

5 |
Varga,Some applications of the condensation of the singularities of families of nonnegative functions, Anal
- Breckner, Trif, et al.
- 1999
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Citation Context |

4 |
The modulus of variation and its application in the theory of Fourier series(Russian), Dokl. Akad. Nauk 214
- Chanturia
- 1974
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Citation Context ...f) = ν(n, f, [0, 1]) := sup n∑ k=1 |f(Ik)|, the supremum is taken over all n-element families {Ik} of nonoverlapping subintervals of [0, 1]. The concept was introduced by Z. A. Chanturia in 1974 (see =-=[4]-=-) and many interesting applications of it to Fourier series are well-known. It is easy to see that the modulus of variation is subadditive On p-Λ-bounded variation 37 with respect to the underlying in... |

4 |
A note on embedding of classes Hω
- Leindler
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4 | On the convolution of functions of generalized bounded variations
- Vyas
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Citation Context ...s of functions of class ΛBV (p) were discussed by R. G. Vyas in [16] recently. More results concerned with the Waterman-Shiba classes and their applications can be found in [2, 3, 10, 11, 12, 14] and =-=[15]-=-. ΛBV (p) equipped with the norm ‖f‖Λ, p := |f(0)|+ V (f) is a Banach space. Given a function f ∈ ΛBV (p), we will also consider the p-Λ-variation function of f v(x) := V (f ; [0, x]) defined for x ∈ ... |

4 | Properties of functions of generalized bounded variation,Mat. Vesnik 58
- Vyas
- 2006
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Citation Context ...oduced in 1980 by M. Shiba in [13] and it clearly is a generalization of the well-known Waterman class ΛBV . Some of the basic properties of functions of class ΛBV (p) were discussed by R. G. Vyas in =-=[16]-=- recently. More results concerned with the Waterman-Shiba classes and their applications can be found in [2, 3, 10, 11, 12, 14] and [15]. ΛBV (p) equipped with the norm ‖f‖Λ, p := |f(0)|+ V (f) is a B... |

4 | Some properties of the functions of Λ-bounded variation - Wang - 1982 |

3 | On Λ-bounded variation, Studia Math. 57 - Waterman - 1976 |

2 | The optimal form of selection principles for functions of a real variable - Chistyakov |

2 | On the embedding of Waterman class in the class Hωp - Goginava - 2005 |

1 |
On the classes ΛBV and V [ν
- Avdispahić
- 1985
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Citation Context ...s of P and where f(Ii) := f(sup Ii) − f(inf Ii) is the change of the function f over the interval Ii. The symbol ΛBV (p) denotes the linear space of all functions of bounded p-Λ-variation with domain =-=[0, 1]-=-. We will write V (f) instead of V (f, P ) , if P = [0, 1]. Also, we will write VΛ(f) instead of V (f) sometimes when it is important to emphasize for which particular Λ-sequence the p-Λ-variation is ... |

1 |
Absolute convergence of Fourier series (Russian),Mat. Zametki 18
- Chanturiya
- 1975
(Show Context)
Citation Context ...an be [0, 2pi] instead of [0, 1]. Theorem 3.3. If f ∈ ΛBV (p), then |f̂(n)| ≤ V (f) 2 (∑n i=1 1 λi ) 1 p . Proof. It follows directly from Theorem 3.1 and from the following result of Z. A. Chanturia =-=[5]-=-: if f ∈ V [h], then |f̂(n)| ≤ 12 ν(n, f)n (see also [7, Theorem 11.19]). 44 Hormozi, Ledari and Prus-Wísniowski It is worth noticing that the inequality of Theorem 3.1 combined with [4, Theorem 5]... |

1 | On the embedding of Waterman, Chanturia and generalized Wiener classes - Goginava |