defined by two parameters, the generating function 
of a tight Gabor frame 
, indexed by a lattice 
, and a domain 
whose boundary consists of line segments connecting certain points of . We provide an explicit formula for the boundary form 
, the normalized limit of the projection functional 
, where 
are the eigenvalues of the localization operators 
applied to dilated domains 
, R is an integer and is
the area of the fundamental domain. The boundary form expresses quantitatively the asymptotic interactions between the generating function 
and the oriented boundary 
from the point of view of the projection functional, which measures to what degree a given trace class operator fails to be an orthogonal projection. Keeping the area of the localization domain 
bounded above corresponds to controlling the relative dimensionality of the localization problem.
[1] Gröchenig, K. and Stöckler, J. (2013) Gabor Frames and Totally Positive Functions. Duke Mathematical Journal, 162, 1003-1031.
http://dx.doi.org/10.1215/00127094-2141944
[2] De Mari, F. and Nowak, K. (2001) Analysis of the Affine Transformations of the Time-Frequency Plane. Bulletin of the Australian Mathematical Society, 63, 195-218.
http://dx.doi.org/10.1017/S0004972700019274
[3] Daubechies, I. (1992) Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series, No. 6, SIAM, Philadelphia.
[4] Landau, H.J. and Widom, H. (1980) Eigenvalue Distribution of Time and Frequency Limiting. Journal of Mathematical Analysis and Applications, 77, 469-481.
http://dx.doi.org/10.1016/0022-247X(80)90241-3
[5] Landau, H.J. (1993) On the Density of Phase-Space Expansions. IEEE Transactions on Information Theory, 39, 1152-1156.
http://dx.doi.org/10.1109/18.243434
[6] Maz’ya, V.G. and Shaposhnikova, T.O. (2011) Sobolev Spaces: With Applications to Elliptic Partial Differential Equations (Grundlehren der Mathematischen Wissenschaften 342). Springer, Berlin.
[7] Folland, G. (1989) Harmonic Analysis in Phase Space. Princeton University Press, Princeton.
[8] Zhu, K. (2012) Analysis on Fock Spaces. Graduate Texts in Mathematics, Volume 263, Springer, New York.
[9] Feichtinger, H.G. and Nowak, K. (2001) A Szegö-Type Theorem for Gabor-Toeplitz Localization Operators. The Michigan Mathematical Journal, 49, 13-21.
http://dx.doi.org/10.1307/mmj/1008719032
[10] De Mari, F., Feichtinger, H.G. and Nowak, K. (2002) Uniform Eigenvalue Estimates for Time-Frequency Localization Operators. Journal of the London Mathematical Society, 65, 720-732.
http://dx.doi.org/10.1112/S0024610702003101
[11] Nowak, K. Optimal Localization Domains for Gabor-Toeplitz Operators. Unpublished Manuscript.
[12] Dörfler, M. and Gröchenig, K. (2011) Time-Frequency Partitions and Characterizations of Modulation Spaces with Localization Operators. Journal of Functional Analysis, 260, 1903-1924.
http://dx.doi.org/10.1016/j.jfa.2010.12.021
[13] Gröchenig, K.-H. and Toft, J. (2011) Isomorphism Properties of Toeplitz Operators in Time-Frequency Analysis. Journal d’Analyse Mathematique, 114, 255-283.
[14] Gröchenig, K. and Toft, J. (2013) The Range of Localization Operators and Lifting Theorems for Modulation and Bargmann-Fock Spaces. Transactions of the American Mathematical Society, 365, 4475-4496.
http://dx.doi.org/10.1090/S0002-9947-2013-05836-9
[15] Döpfner, K.A. (2012) Quality of Gabor Multipliers for Approximation of Hilbert-Schmidt Operators. Master’s Thesis, Department of Mathematics, University of Vienna, Vienna.
[16] Gröchenig, K. (2011) Representation and Approximation of Pseudodifferential Operators by Sums of Gabor Multipliers. Applicable Analysis, 90, 385-401.
http://dx.doi.org/10.1080/00036811.2010.499507
[17] Dörfler, M. and Torrésani, B. (2010) Representations of Operators in the Time-Frequency Domain and Generalized Gabor Multipliers. Journal of Fourier Analysis and Applications, 16, 261-293.
http://dx.doi.org/10.1007/s00041-009-9085-x
[18] Cordero, E., Nicola, F. and Rodino, L. (2010) Time-Frequency Analysis of Fourier Integral Operators. Communications on Pure and Applied Analysis, 9, 1-21.
[19] Cordero, E., Gröchenig, K. and Nicola, F. (2012) Approximation of Fourier Integral Operators by Gabor Multipliers. Journal of Fourier Analysis and Applications, 18, 661-684.
http://dx.doi.org/10.1007/s00041-011-9214-1
[20] Feichtinger, H.G., Nowak, K. and Pap, M. (2014) Spectral Properties of Toeplitz Operators Acting on Gabor Type Reproducing Kernel Hilbert Spaces. In: Rassias, T.M. and Pardalos, P.M., Eds., Mathematics without Boundaries: Surveys in Pure Mathematics, Springer, New York, 163-180.
[21] Nowak, K. (1996) Local Toeplitz Operators Based on Wavelets: Phase Space Patterns for Rough Wavelets. Studia Mathematica, 119, 37-64.
[22] Feichtinger, H.G. and Nowak, K. (2003) A First Survey of Gabor Multipliers. In: Fiechtinger, H.G. and Strohmer, T., Eds., Advances in Gabor Analysis, Birkhäuser, Boston, 99-128.
[23] Martinet, J. (2003) Perfect Lattices in Euclidean Spaces (Grundlehren der mathematischen Wissenschaften, Volume 327). Springer, Berlin.
http://dx.doi.org/10.1007/978-3-662-05167-2
[24] Stein, E.M. (1993) Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton.
[25] Nowak, W.G. (2004) Lattice Points in a Circle: An Improved Mean-Square Asymptotics. Acta Arithmetica, 113, 259-272.
http://dx.doi.org/10.4064/aa113-3-4
[26] Christensen, O. (2003) An Introduction to Frames and Riesz Bases. Birkhäuser, Boston.
http://dx.doi.org/10.1007/978-0-8176-8224-8
[27] Flandrin, P. (1999) Time-Frequency/Time-Scale Analysis. Academic Press, San Diego.
[28] Gröchenig, K. (2001) Foundations of Time-Frequency Analysis. Birkhäuser, Boston.
[29] Wojtaszczyk, P. (1997) A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511623790
[30] Feichtinger, H.G. and Strohmer, T. (1998) Gabor Analysis and Algorithms. Theory and Applications. Birkhäuser, Boston.
[31] Feichtinger, H.G. and Strohmer, T. (2003) Advances in Gabor Analysis. Birkhäuser, Boston.
[32] Balan, R. (2008) The Noncommutative Wiener Lemma, Linear Independence, and Spectral Properties of the Algebra of Time-Frequency Shift Operators. Transactions of the American Mathematical Society, 360, 3921-3941.
http://dx.doi.org/10.1090/S0002-9947-08-04448-6
[33] Casazza, P.G. (2000) The Art of Frame Theory. Taiwanese Journal of Mathematics, 4, 129-201.
[34] Heil, C. (2007) History and Evolution of the Density Theorem for Gabor Frames. Journal of Fourier Analysis and Applications, 13, 113-166.
http://dx.doi.org/10.1007/s00041-006-6073-2
[35] Gröchenig, K. and Leinert, M. (2003) Wiener’s Lemma for Twisted Convolution and Gabor Frames. Journal of the American Mathematical Society, 17, 1-18.
[36] Luef, F. Private Communication.
[37] Feichtinger, H.G. and Kaiblinger, N. (2003) Varying the Time-Frequency Lattice of Gabor Frames. Transactions of the American Mathematical Society, 356, 2001-2023.
[38] Lang, S. (1998) SL2(R), Graduate Texts in Mathematics. Volume 105, Springer, New York.
[39] Katznelson, Y. (1976) An Introduction to Harmonic Analysis. Dover Publications, New York.