Abstract
Nikol’skii–type inequalities, that is inequalities between different metrics of trigonometric polynomials on the torus Td for the Lorentz–Zygmund spaces, are obtained. The results of previous paper “Nikol’skii inequalities for Lorentz–Zygmund spaces” are extended. Applications to approximation spaces in Lorentz–Zygmund spaces and to Besov spaces are given.
Highlights
PreliminariesWe write 푋 ⊂ 푌 for two quasi-normed spaces and to indicate that is continuously embedded in . e notation 푋 ≅ 푌 means that 푋 ⊂ 푌 and 푌 ⊂ 푋
Introduction e classicalNikol’skii inequality for the trigonometric polynomials on [0,1] of degree at most can be written as [1, 2]儩儩儩儩푇푛儩儩儩儩푝≺ 푛((1/푞)−(1/푝))儩儩儩儩푇푛儩儩儩儩푞, (1)where1 ≤ 푞 < 푝 ≤ ∞ and ‖∗‖ is the usual norm on the Lebesgue spaces
Where1 ≤ 푞 < 푝 ≤ ∞ and ‖∗‖ is the usual norm on the Lebesgue spaces
Summary
We write 푋 ⊂ 푌 for two quasi-normed spaces and to indicate that is continuously embedded in . e notation 푋 ≅ 푌 means that 푋 ⊂ 푌 and 푌 ⊂ 푋. We write 푋 ⊂ 푌 for two quasi-normed spaces and to indicate that is continuously embedded in . We consider (equivalence classes of) complex-valued measurable functions on and bounded complex-valued sequences. E Lorentz–Zygmund space 퐿푝,푏;훼 ≡ 퐿푝,푏;훼 푑 can be defined as follows:. Where ‖∗‖ is the usual (quasi-)norm on the Lebesgue space (0,1). Where ‖∗‖ is the usual (quasi-)norm on the Lebesgue sequence space. We use the same notation ‖∗‖푝,푏;훼 for both (quasi-)norms. [23, eorem 7.4], on 푝,푏;훼 there exists a norm, equivalent to. In this case, it is a Banach function space. En, the space 푝,푏;훼 is -normed, that is, there exists 퐶 = 퐶 푝, 푏; 훼 > 0 such that.
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