×

zbMATH — the first resource for mathematics

Functions with bounded spectrum. (English) Zbl 0828.42009
Summary: Let \(0< p\leq \infty\), \(f(x)\in L_p(\mathbb{R}^n)\), and \(\text{supp } Ff\) be bounded, where \(F\) is the Fourier transform. We prove in this paper that the sequence \(|D^\alpha f|^{1/|\alpha|}_p\), \(\alpha\geq 0\), has the same behavior as the sequence \(\sup_{\xi\in \text{supp }Ff} |\xi^{\alpha}|^{1/|\alpha|}\), \(\alpha\geq 0\). In other words, if we know all “far points” of \(\text{supp } Ff\), we can wholly describe this behavior without any concrete calculation of \(|D^\alpha f|_p\), \(\alpha\geq 0\). A Paley-Wiener-Schwartz theorem for a nonconvex case, which is a consequence of the result, is given.

MSC:
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
26D10 Inequalities involving derivatives and differential and integral operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030
[2] Ha Huy Bang, A property of infinitely differentiable functions, Proc. Amer. Math. Soc. 108 (1990), no. 1, 73 – 76. · Zbl 0707.26015
[3] Kha ZuĭBang, Some embedding theorems for spaces of periodic functions of infinite order, Mat. Zametki 43 (1988), no. 4, 509 – 517, 574 (Russian); English transl., Math. Notes 43 (1988), no. 3-4, 293 – 298. · Zbl 0825.42001 · doi:10.1007/BF01139134 · doi.org
[4] -, On imbedding theorems for Sobolev spaces of infinite order, Mat. Sb. 136 (1988), 115-127. · Zbl 0673.46016
[5] Ha Huy Bang, Imbedding theorems of Sobolev spaces of infinite order, Acta Math. Vietnam. 14 (1989), no. 1, 17 – 27. · Zbl 0734.46024
[6] Ju. B. Egorov, Lectures on partial differential equations, Moscow State Univ. Press., Moscow, 1975.
[7] Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. Lars Hörmander, The analysis of linear partial differential operators. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients.
[8] P. I. Lizorkin, Bounds for trigonometrical integrals and an inequality of Bernstein for fractional derivatives, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 109 – 126 (Russian).
[9] R. J. Nessel and G. Wilmes, Nikolskii-type inequalities for trigonometric polynomials and entire functions of exponential type, J. Austral. Math. Soc. Ser. A 25 (1978), no. 1, 7 – 18. · Zbl 0376.42001
[10] S. M. Nikolsky, Approximation of functions of several variables and imbedding theorems, ”Nauka”, Moscow, 1977.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.