Geodesic
On the space of periodic ultradifferentiable functions of Roumieu type and its dual
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2023), pp. 89-95 Cet article a éte moissonné depuis la source Math-Net.Ru

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With a help of a family ${\mathcal H}$ of convex nondecreasing functions on $[0, \infty)$ we define the space $J({\mathcal H})$ of $2 \pi$-periodic infinitely differentiable functions on the real line with given estimates for all derivatives. It belongs to the class of spaces of ultradifferentiable functions of Roumieu type. A description of the space $G({\mathcal H})$ is obtained in terms of the best trigonometric approximations and the rate of decrease of the Fourier coefficients. A general form of linear continuous functionals on $J({\mathcal H})$ is found. It is shown that some well-known classes of $2 \pi$-periodic functions of Gevrey type are special cases of the spaces $J({\mathcal H})$.
Keywords: Fourier series, trigonometric polynomials.
Mots-clés : Fourier coefficients 
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I. Kh. Musin. On the space of periodic ultradifferentiable functions of Roumieu type and its dual. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2023), pp. 89-95. http://geodesic.mathdoc.fr/item/IVM_2023_4_a7/

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