# Spaces of sequences, sampling theorem, and functions of exponential type

Studia Mathematica (1991)

- Volume: 100, Issue: 1, page 51-74
- ISSN: 0039-3223

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topTorres, Rodolfo. "Spaces of sequences, sampling theorem, and functions of exponential type." Studia Mathematica 100.1 (1991): 51-74. <http://eudml.org/doc/215873>.

@article{Torres1991,

abstract = {We introduce certain spaces of sequences which can be used to characterize spaces of functions of exponential type. We present a generalized version of the sampling theorem and a "nonorthogonal wavelet decomposition" for the elements of these spaces of sequences. In particular, we obtain a discrete version of the so-called φ-transform studied in [6] [8]. We also show how these new spaces and the corresponding decompositions can be used to study multiplier operators on Besov spaces.},

author = {Torres, Rodolfo},

journal = {Studia Mathematica},

keywords = {Fourier transform; compact support; spaces of distributions; periodic distributions; sampling theorem; Besov spaces},

language = {eng},

number = {1},

pages = {51-74},

title = {Spaces of sequences, sampling theorem, and functions of exponential type},

url = {http://eudml.org/doc/215873},

volume = {100},

year = {1991},

}

TY - JOUR

AU - Torres, Rodolfo

TI - Spaces of sequences, sampling theorem, and functions of exponential type

JO - Studia Mathematica

PY - 1991

VL - 100

IS - 1

SP - 51

EP - 74

AB - We introduce certain spaces of sequences which can be used to characterize spaces of functions of exponential type. We present a generalized version of the sampling theorem and a "nonorthogonal wavelet decomposition" for the elements of these spaces of sequences. In particular, we obtain a discrete version of the so-called φ-transform studied in [6] [8]. We also show how these new spaces and the corresponding decompositions can be used to study multiplier operators on Besov spaces.

LA - eng

KW - Fourier transform; compact support; spaces of distributions; periodic distributions; sampling theorem; Besov spaces

UR - http://eudml.org/doc/215873

ER -

## References

top- [1] P. Ausscer and M. Carro, On the relations between operators on ${\mathbb{R}}^{n}$, ${}^{n}$ and ${\mathbb{Z}}^{n}$, Studia Math., to appear.
- [2] R. Boas, Entire Functions, Academic Press, New York 1954. Zbl0058.30201
- [3] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909-996. Zbl0644.42026
- [4] K. de Leeuw, On ${L}^{p}$ multipliers, Ann. of Math. 81 (1965), 364-379.
- [5] H. Feichtinger and K. Gröchenig, Banach spaces related to intergrable group representations and their atomic decompositions, I, J. Funct. Anal. 86 (1989), 307-340. Zbl0691.46011
- [6] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799. Zbl0551.46018
- [7] M. Frazier and B. Jawerth, The φ-transform and applications to distribution spaces, in: Function Spaces and Applications, M. Cwikel et al. (eds.), Lecture Notes in Math. 1302, Springer, 1988, 223-246. Zbl0648.46038
- [8] M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces. J. Funct. Anal., to appear.
- [9] M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conf. Ser. in Math., to appear. Zbl0757.42006
- [10] M. Frazier and R. Torres, The sampling theorem, φ-transform and Shannon wavelets for R, Z, T and ${Z}_{N}$, preprint.
- [11] C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), 628-666. Zbl0683.42031
- [12] M. Holschneider, Wavelet analysis on the circle, J. Math. Phys. 31 (1990), 39-44. Zbl0729.46005
- [13] Y. Katznelson, An Introduction to Harmonic Analysis, Dover, New York 1976.
- [14] S. Mallat, Multiresolution representations and wavelets, Ph.D. Thesis, Electrical Engineering Department, Univ. of Pennsylvania, 1988.
- [15] Y. Meyer, Wavelets and operators, Proc. of the Special Year in Modern Analysis at the University of Illinois, London Math. Soc. Lecture Note Ser. 137, Cambridge Univ. Press, Cambridge 1989, 256-364.
- [16] Y. Meyer, Ondelettes et opérateurs, Hermann, Paris 1990. Zbl0694.41037
- [17] C. Onneweer and S. Weiyi, Homogeneous Besov spaces on locally compact Vilenkin groups, Studia Math. 93 (1989), 17-39. Zbl0681.43006
- [18] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Ser. 1, Durham, N.C., 1976. Zbl0356.46038
- [19] V. Peller, Wiener-Hopf operators on a finite interval and Schatten-von Neumann classes, Proc. Amer. Math. Soc. 104 (1988), 479-486. Zbl0692.47027
- [20] B. Petersen, Introduction to the Fourier Transform and Pseudo-differential Operators, Pitman, Boston 1983.
- [21] R. Rochberg, Toeplitz and Hankel operators on the Paley-Wiener spaces, Integral Equations Operator Theory 10, (1987), 187-235. Zbl0634.47024
- [22] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. Zbl0207.13501
- [23] R. Torres, Boundedness results for operators with singular kernels on distribution spaces, Mem. Amer. Math. Soc. 442 (1991). Zbl0737.47041
- [24] H. Triebel, Theory of Function Spaces, Monographs Math. 78, Birkhäuser, Basel 1983.
- [25] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge Univ. Press, London 1968. Zbl0157.38204

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