no. 2
Fourier coefficients of continuous functions and a class of multipliers
Annales de l'Institut Fourier, Tome 38 (1988) no. 2, pp. 147-183.

Soit x une fonction bornée sur Z ; on définit le multiplicateur avec un symbole x (noté par M x ) par (M x f) ^=xf ^, f∈L 2 (T). On étudie des conditions sur x qui garantissent “l’inégalité interpolationnelle” ∥M x f∥ L p ≤C∥f∥ L 1 α ∥M x f∥ L q 1-α (ici 1<p<q, α=α(p,q,x) est entre 0 et 1 et C ne dépend pas de f). Cette inégalité exprime une sorte de régularité de M x sur L 1 (T). (Pour la plupart les multiplicateurs en question ne sont pas de type faible (1,1).) On utilise ces résultats pour démontrer qu’il y a bien des sous-ensembles E de Z tels que chaque suite positive dans l 2 (E) puisse être majorée par la suite {|f ^(n)|} n∈E pour une fonction continue f dont le spectre soit inclus dans E.

If x is a bounded function on Z, the multiplier with symbol x (denoted by M x ) is defined by (M x f) ^=xf ^, f∈L 2 (T). We give some conditions on x ensuring the “interpolation inequality” ∥M x f∥ L p ≤C∥f∥ L 1 α ∥M x f∥ L q 1-α (here 1<p<q and α=α(p,q,x) is between 0 and 1). In most cases considered M x fails to have stronger L 1 -regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets E⊂Z every positive sequence in ℓ 2 (E) can be majorized by the sequence { |f ^(n)|} n∈E for some continuous funtion f with spectrum in E.

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     author = {Kislyakov, Serguei V.},
     title = {Fourier coefficients of continuous functions and a class of multipliers},
     journal = {Annales de l'Institut Fourier},
     pages = {147--183},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {2},
     year = {1988},
     doi = {10.5802/aif.1138},
     zbl = {0607.42004},
     mrnumber = {89j:42004},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1138/}
}
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Kislyakov, Serguei V. Fourier coefficients of continuous functions and a class of multipliers. Annales de l'Institut Fourier, Tome 38 (1988) no. 2, pp. 147-183. doi : 10.5802/aif.1138. https://aif.centre-mersenne.org/articles/10.5802/aif.1138/

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