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Representation of linear functionals on the space \(H_ p^ \omega\) and its application in the theory of absolute convergence of series of Fourier coefficients. (English. Russian original) Zbl 0822.42003

Mosc. Univ. Math. Bull. 48, No. 1, 67-69 (1993); translation from Vestn. Mosk. Univ., Ser. I 1993, No. 1, 94-97 (1993).
Let \(H^ \omega_ p\) be a Banach space of 1-periodic functions on \(\mathbb{R}\) with modulus of continuity \(\omega\) and with norm \(\| f\|= \| f\|_ p+ \sup_{0< \delta\leq 1} {1\over \omega(\delta)} \sup_{| h|\leq \delta} \| f(x+ h)- f(x)\|_ p\), \(p\in [1, \infty]\) (\(\| f\|_ p\) is the norm in the Banach space \(L_ p(0, 1)\)). The author obtains the representation of continuous linear functionals on the space \(H^ \omega_ p\) and two-sided estimates for its norm.

MSC:

42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
42A20 Convergence and absolute convergence of Fourier and trigonometric series
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