Abstract
In this paper, we study the spectrality and frame-spectrality of exponential systems of the type \(E(\Lambda ,\varphi ) = \{e^{2\pi i \lambda \cdot \varphi (x)}: \lambda \in \Lambda \}\) where the phase function \(\varphi \) is a Borel measurable which is not necessarily linear. A complete characterization of pairs \((\Lambda ,\varphi )\) for which \(E(\Lambda ,\varphi )\) is an orthogonal basis or a frame for \(L^{2}(\mu )\) is obtained. In particular, we show that the middle-third Cantor measures and the unit disc, each admits an orthogonal basis with a certain non-linear phase. Under a natural regularity condition on the phase functions, when \(\mu \) is the Lebesgue measure on [0, 1] and \(\Lambda = {{\mathbb {Z}}},\) we show that only the standard phase functions \(\varphi (x) = \pm x\) are the only possible functions that give rise to orthonormal bases. Surprisingly, however we prove that there exist a greater degree of flexibility, even for continuously differentiable phase functions in higher dimensions. For instance, we were able to describe a large class of functions \(\varphi \) defined on \({{\mathbb {R}}}^{d}\) such that the system \(E(\Lambda ,\varphi )\) is an orthonormal basis for \(L^{2}[0,1]^{d}\) when \(d\ge 2.\) Moreover, we discuss how our results apply to the discretization problem of unitary representations of locally compact groups for the construction of orthonormal bases. Finally, we conclude the paper by stating several open problems.
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References
Bernier, D., Taylor, K.F.: Wavelets from square-integrable representations. SIAM J. Math. Anal. 27(2), 594–608 (1996)
Bohnstengel, J., Kesseböhmer, M.: Wavelets for iterated function systems. J. Funct. Anal. 259(3), 583–601 (2010)
Christensen, O.: An Introduction to Frames and Riesz Bases, vol. 7. Springer, New York (2003)
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Dutkay, D.E., Lai, C.-K.: Uniformity of measures with fourier frames. Adv. Math. 252, 684–707 (2014)
Dutkay, D.E., Lai, C.-K., Wang, Y.: Fourier bases and Fourier frames on self-affine measures. In: Recent Developments in Fractals and Related Fields, Trends Math., pp. 87–111. Birkhäuser/Springer, Cham (2017)
Feichtinger, H.G., Gröbner, P.: Banach spaces of distributions defined by decomposition methods. I. Math. Nachr. 123, 97–120 (1985)
Foschini, G.J.: Almost everywhere one-to-one functions and an \(n\)-cube decomposition. J. Math. Anal. Appl. 31, 314–317 (1970)
Fuglede, B.: Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16, 101–121 (1974)
Gabardo, J.-P., Lai, C.-K., Wang, Y.: Gabor orthonormal bases generated by the unit cubes. J. Funct. Anal. 269(5), 1515–1538 (2015)
Gröchenig, K.: Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc, Boston, MA (2001)
Gröchenig, K., Rottensteiner, D.: Orthonormal bases in the orbit of square-integrable representations of nilpotent lie groups. arXiv preprint arXiv:1706.06034 (2017)
He, X.-G., Lai, C.-K., Lau, K.-S.: Exponential spectra in \(l^2\) (\(\mu \)). Appl. Comput. Harmon. Anal. 34(3), 327–338 (2013)
Heil, C.: A Basis Theory Primer: Expanded Edition. Springer, New York (2010)
Holhoş, A.: Two area preserving maps from the square to the \(p\)-ball. Math. Model. Anal. 22(2), 157–166 (2017)
Jorgensen, P.E., Pedersen, S.: Dense analytic subspaces in fractall 2-spaces. Journal d’Analyse Mathematique 75(1), 185–228 (1998)
Kechris, A.S.: Classical Descriptive Set Theory, volume 156 of Graduate Texts in Mathematics. Springer, New York (1995)
Kozma, G., Nitzan, S.: Combining Riesz bases. Invent. Math. 199(1), 267–285 (2015)
Kozma, G., Nitzan, S.: Combining Riesz bases in \({\mathbb{R}}^d\). Rev. Mat. Iberoam. 32(4), 1393–1406 (2016)
Lev, N., Matolcsi, M.: The fuglede conjecture for convex domains is true in all dimensions. arXiv:1904.12262
Oussa, V.S.: Regular sampling on metabelian nilpotent lie groups: The multiplicity-free case. In: Frames and Other Bases in Abstract and Function Spaces, pp. 377–411. Springer (2017)
Oussa, V.: Frames arising from irreducible solvable actions i. J. Funct. Anal. 274(4), 1202–1254 (2018)
Oussa, V.: Compactly supported bounded frames on Lie groups. J. Funct. Anal. 277(6), 1718–1762 (2019)
Strichartz, R.S.: Mock Fourier series and transforms associated with certain Cantor measures. J. Anal. Math. 81, 209–238 (2000)
Tao, T.: Fuglede’s conjecture is false in 5 and higher dimensions. Math. Res. Lett. 11(2–3), 251–258 (2004)
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Gabardo, JP., Lai, CK. & Oussa, V. On Exponential Bases and Frames with Non-linear Phase Functions and Some Applications. J Fourier Anal Appl 27, 9 (2021). https://doi.org/10.1007/s00041-021-09814-5
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DOI: https://doi.org/10.1007/s00041-021-09814-5