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Absolutely convergent Fourier series and generalized Lipschitz classes of functions

Ferenc Móricz (2008)

Colloquium Mathematicae

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 | f ( x + h ) - f ( x ) | C h α L ( 1 / h ) for all x ∈ , h >...

Completely continuous multipliers from L 1 ( G ) into L ( G )

G. Crombez, Willy Govaerts (1984)

Annales de l'institut Fourier

For a locally compact Hausdorff group G we investigate what functions in L ( G ) give rise to completely continuous multipliers T g from L 1 ( G ) into L ( G ) . In the case of a metrizable group we obtain a complete description of such functions. In particular, for G compact all g in L ( G ) induce completely continuous T g .

Exponential sums with coefficients 0 or 1 and concentrated L p norms

B. Anderson, J. M. Ash, R. L. Jones, D. G. Rider, B. Saffari (2007)

Annales de l’institut Fourier

A sum of exponentials of the form f ( x ) = exp 2 π i N 1 x + exp 2 π i N 2 x + + exp 2 π i N m x , where the N k are distinct integers is called an idempotent trigonometric polynomial (because the convolution of f with itself is f ) or, simply, an idempotent. We show that for every p > 1 , and every set E of the torus 𝕋 = / with | E | > 0 , there are idempotents concentrated on E in the L p sense. More precisely, for each p > 1 , there is an explicitly calculated constant C p > 0 so that for each E with | E | > 0 and ϵ > 0 one can find an idempotent f such that the ratio E | f | p / 𝕋 | f | p 1 / p is greater than C p - ϵ . This is in fact...

Integrability theorems for trigonometric series

Bruce Aubertin, John Fournier (1993)

Studia Mathematica

We show that, if the coefficients (an) in a series a 0 / 2 + n = 1 a n c o s ( n t ) tend to 0 as n → ∞ and satisfy the regularity condition that m = 0 j = 1 [ n = j 2 m ( j + 1 ) 2 m - 1 | a n - a n + 1 | ] ² 1 / 2 < , then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series n = 1 b n s i n ( n t ) tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if n = 1 | b n | / n < . These conclusions were previously known to hold under stronger restrictions on the sizes of the differences...

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