Exponential bases on two dimensional trapezoids

De Carli, Laura; Kumar, Anudeep

doi:10.1090/S0002-9939-2015-12329-5

Exponential bases on two dimensional trapezoids
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by Laura De Carli and Anudeep Kumar PDF

Proc. Amer. Math. Soc. 143 (2015), 2893-2903 Request permission

Abstract:

We discuss the existence and stability of Riesz bases of exponential type of $L^2(T)$ for special domains $T\subset \mathbb {R}^2$ called trapezoids, and we construct exponential bases on the finite union of rectangles with the same height. We also generalize our main theorems in dimension $d\ge 3$.

References

Avdonin, S. A. On the question of Riesz bases of exponential functions in $L^2$. (translated into English in: Vestnik Leningrad Univ. Ser. Mat., 13, pp 203–2011 (1979)
S. A. Avdonin and S. A. Ivanov, Riesz bases of exponentials and divided differences, Algebra i Analiz 13 (2001), no. 3, 1–17 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 3, 339–351. MR 1850184
Richard Bellman, Convergence of non-harmonic Fourier series, Duke Math. J. 10 (1943), 551–552. MR 8431
Harold E. Benzinger, Nonharmonic Fourier series and spectral theory, Trans. Amer. Math. Soc. 299 (1987), no. 1, 245–259. MR 869410, DOI 10.1090/S0002-9947-1987-0869410-0
Peter G. Casazza, The art of frame theory, Taiwanese J. Math. 4 (2000), no. 2, 129–201. MR 1757401, DOI 10.11650/twjm/1500407227
P. Casazza and G. Kutyniok, Finite Frames: Theory and Applications, Birkhauser, 2013.
R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366. MR 47179, DOI 10.1090/S0002-9947-1952-0047179-6
Erich Freund and Joachim Petzold, Nonharmonic Fourier series: a formalism for analyzing signals, Elektron. Informationsverarb. Kybernet. 20 (1984), no. 10-11, 575–592 (English, with French, German and Russian summaries). MR 777071
S. Grepstad and N. Lev, Multi-tiling and Riesz bases, arXiv:1212.4679 (2012)
A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z. 41 (1936), no. 1, 367–379. MR 1545625, DOI 10.1007/BF01180426
G. Kozma and S. Nitzan, Combining Riesz bases arXiv:1210.6383 (2012)
Nir Lev, Riesz bases of exponentials on multiband spectra, Proc. Amer. Math. Soc. 140 (2012), no. 9, 3127–3132. MR 2917085, DOI 10.1090/S0002-9939-2012-11138-4
Norman Levinson, On non-harmonic Fourier series, Ann. of Math. (2) 37 (1936), no. 4, 919–936. MR 1503319, DOI 10.2307/1968628
M. Ĭ. Kadec′, The exact value of the Paley-Wiener constant, Dokl. Akad. Nauk SSSR 155 (1964), 1253–1254 (Russian). MR 0162088
Yurii I. Lyubarskii and Alexander Rashkovskii, Complete interpolating sequences for Fourier transforms supported by convex symmetric polygons, Ark. Mat. 38 (2000), no. 1, 139–170. MR 1749363, DOI 10.1007/BF02384495
J. Marzo, Riesz basis of exponentials for a union of cubes in $\mathbb {R}^{d}$, arXiv:math/0601288 (2006)
Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142, DOI 10.1090/coll/019
Kristian Seip, On the connection between exponential bases and certain related sequences in $L^2(-\pi ,\pi )$, J. Funct. Anal. 130 (1995), no. 1, 131–160. MR 1331980, DOI 10.1006/jfan.1995.1066
Kristian Seip, A simple construction of exponential bases in $L^2$ of the union of several intervals, Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 1, 171–177. MR 1317335, DOI 10.1017/S0013091500006295
Wenchang Sun and Xingwei Zhou, On the stability of multivariate trigonometric systems, J. Math. Anal. Appl. 235 (1999), no. 1, 159–167. MR 1758675, DOI 10.1006/jmaa.1999.6386
Sun, W. and Zhou, X. On Kadec’s $\frac 14$-Theorem and the Stability of Gabor Frames. Applied and Comput. Harm. An. 7 (1999) no. 2, 239–242.
Robert M. Young, An introduction to nonharmonic Fourier series, 1st ed., Academic Press, Inc., San Diego, CA, 2001. MR 1836633