On Lipschitz numbers
A.S. Besicovitch (1929)
Mathematische Zeitschrift
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A.S. Besicovitch (1929)
Mathematische Zeitschrift
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Walter R. Bloom (1988)
Mathematische Zeitschrift
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Tong-Seng Quek, Leonard Y.H. Yap (1983)
Mathematische Zeitschrift
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J.M. Anderson (1976)
Mathematische Zeitschrift
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Thomas M. MacRobert (1961)
Mathematische Zeitschrift
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Constantine Georgakis (1972)
Mathematische Zeitschrift
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R. Askey, R.P. jr. Boas (1967)
Mathematische Zeitschrift
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Roop Narain (1958)
Mathematische Zeitschrift
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O.Carruth McGehee (1979)
Mathematische Annalen
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R.K.S. Rathore (1975)
Mathematische Zeitschrift
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Constantine Georgakis (1973)
Mathematische Zeitschrift
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Graham H. Williams (1977)
Mathematische Zeitschrift
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Ferenc Móricz (2010)
Studia Mathematica
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We consider complex-valued functions f ∈ L¹(ℝ), and prove sufficient conditions in terms of f to ensure that the Fourier transform f̂ belongs to one of the Lipschitz classes Lip(α) and lip(α) for some 0 < α ≤ 1, or to one of the Zygmund classes zyg(α) and zyg(α) for some 0 < α ≤ 2. These sufficient conditions are best possible in the sense that they are also necessary in the case of real-valued functions f for which either xf(x) ≥ 0 or f(x) ≥ 0 almost everywhere.
Louis Pigno, Sadahiro Saeki (1975)
Mathematische Zeitschrift
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Ferenc Móricz (2008)
Colloquium Mathematicae
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We investigate the order of magnitude of the modulus of continuity of a function f with absolutely convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that f belong to one of the generalized Lipschitz classes Lip(α,L) and Lip(α,1/L), where 0 ≤ α ≤ 1 and L = L(x) is a positive, nondecreasing, slowly varying function such that L(x) → ∞ as x → ∞. For example, a 2π-periodic function f is said to belong to the class Lip(α,L) if for all...