Fourier coefficients of continuous functions and a class of multipliers
If x is a bounded function on \({\mathbb{Z}}\), the multiplier with symbol \(x\) (denoted by \(M_ x)\) is defined by \((M_ xf){\hat{\;}}=x\hat f\), \(f\in L^ 2({\mathbb{T}})\). We give some conditions on \(x\) ensuring the ''interpolation inequality'' \(\| M_ xf\| _{L^ p}\leq C\| f\| ^{\alpha}_{L^ 1}\| M_ xf\| _{L^ q}^{1-\alpha}\) (here \(1<p<q\) and \(\alpha =\alpha (p,q,x)\) 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 \(E\subset {\mathbb{Z}}\) every positive sequence in \(\ell ^ 2(E)\) can be majorized by the sequence \(\{| \hat f(n)|\}_{n\in E}\) for some continuous function \(f\) with spectrum in \(E\).
scientific article; zbMATH DE number 3981761
scientific article