Abstract
A semidiscrete multiplier is an operator between a space of functions or distributions on a locally compact Abelian group G on the one hand, and a space of sequences on a discrete subgroup H of G on the other hand, with the property that it commutes with shifts by H. We describe the basic form of such operators and show a number of representation theorems for classical spaces like L p, C 0, etc.
We also point out parallels to representation theorems for multipliers.
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References
A. Benedek and R. Panzone, The spaces L P, with mixed norm, Duke Math. J., 28 (1961), pp. 301–324.
J. J. Benedetto, Irregular sampling and frames, in: Wavelets: A Tutorial in Theory and Applications, C. Chui, ed., Academic Press, Boston, 1992, pp. 445–507.
J. J. Benedetto and G. Zimmermann, Sampling multipliers and the Poisson Summation Formula, J. Fourier Anal. Appl., 3 (1997), pp. 505–523.
C. de Boor and G. Fix, Spline approximation by quasi-interpolants, J. Approx. Theory, 8 (1973), pp. 19–45.
R. E. Edwards and G. I. Gaudry, Littlewood-Paley and Multiplier Theory, Ergeb. Math. Grenzgeb., 90, Springer, Berlin, 1977.
W. Gautschi, Attenuation factors in practical Fourier analysis, Numer. Math., 18 (1972), pp. 373–400.
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Grundlehren Math. Wiss., 115, Springer, Berlin, 1963.
R.-Q. Jia and C. A. Micchelli, Using the refinement equations for the construction of pre-wavelets. II: Powers of two, in: Curves and Surfaces, P. J. Laurent, A. Le Méhauté, and L. L. Schumaker, eds., Academic Press, Boston, 1991, pp. 209–246.
R. Larsen, An Introduction to the Theory of Multipliers, Grundlehren Math. Wiss., 175, Springer, Berlin, 1971.
J. Lei, R.-Q. Jia, and E. W. Cheney, Approximation from shift-invariant spaces by integral operators. SIAM J. Math. Anal., 28 (1997), pp. 481–498.
S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L 2(ℝ), Trans. Amer. Math. Soc., 315 (1989), pp. 69–87.
H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, Quasi-interpolation in shift invariant spaces, J. Math. Anal. Appl., 251 (2000), pp. 356–363.
W. Sweldens, Construction and Applications of Wavelets in Numerical Analysis, Ph.D. Thesis, Department of Computer Science, Katholieke Universiteit Leuven, Belgium, 1994.
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Dedicated to my academic teacher Prof. John J. Benedetto with many thanks for all the beauty in mathematics he taught me to see.
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© 2006 Birkhäuser Boston
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Zimmermann, G. (2006). Semidiscrete Multipliers. In: Heil, C. (eds) Harmonic Analysis and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4504-7_3
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DOI: https://doi.org/10.1007/0-8176-4504-7_3
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-3778-1
Online ISBN: 978-0-8176-4504-5
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