Abstract
We consider the existence and structure properties of Parseval frames of kernel functions in vector valued de Branges spaces. We develop some sufficient conditions for Parseval sequences by identifying the main construction with Naimark dilation of frames. The dilation occurs by embedding the de Branges space of vector valued functions into a dilated de Branges space of vector valued functions. The embedding also maps the kernel functions associated with a frame sequence of the original space into a Riesz basis for the embedding space. We also develop some sufficient conditions for a dilated de Branges space to have the Kramer sampling property.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
D. Alpay, A. Dijksma, J. Rovnyak, H. de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, in Operator Theory: Advances and Applications, vol. 96 (Birkhäuser, Basel, 1997)
D. Alpay, A. Dijksma, J. Rovnyak, H.S.V. de Snoo, Realization and factorization in reproducing kernel Pontryagin spaces, in Operator Theory, System Theory and Related Topics (Beer-Sheva/Rehovot, 1997). Operator Theory: Advances and Applications, vol. 123 (Birkhäuser, Basel, 2001), pp. 43–65
S. al-Sa’di, E. Weber, Sampling in de Branges Spaces and Naimark Dilation. Compl. Anal. Oper. Theory 11(3), 583–601 (2017)
N. Aronszajn, Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)
D.Z. Arov, H. Dym, J-contractive matrix valued functions and related topics, in Encyclopedia of Mathematics and its Applications, vol. 116 (Cambridge University, Cambridge, 2008)
D.Z. Arov, H. Dym, Bitangential direct and inverse problems for systems of integral and differential equations, in Encyclopedia of Mathematics and its Applications, vol. 145 (Cambridge University Press, Cambridge, 2012)
D.Z. Arov, H. Dym, de Branges Spaces of Vector-Valued Functions. Oper. Theory, 721–752 (2015)
R. Balan, Density and redundancy of the noncoherent Weyl-Heisenberg superframes, in The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, TX, 1999). Contemporary Mathematics, vol. 247 (American Mathematical Society, Providence, 1999), pp. 29–41
J.A. Ball, A lifting theorem for operator models of finite rank on multiply-connected domains. J. Oper. Theory 1(1), 3–25 (1979)
G. Bhatt, B.D. Johnson, E. Weber, Orthogonal wavelet frames and vector-valued wavelet transforms. Appl. Comput. Harmon. Anal. 23(2), 215–234 (2007)
P.G. Casazza, The art of frame theory. Taiwanese J. Math. 4(2), 129–201 (2000)
L. de Branges, The comparison theorem for Hilbert spaces of entire functions. Integr. Equ. Oper. Theory 6(5), 603–646 (1983)
L. de Branges, J. Rovnyak, Canonical models in quantum scattering theory, in Perturbation Theory and its Applications in Quantum Mechanics (Proceedings of the Advance Seminar Mathematics Research Center, U.S. Army, Theoretical Chemical Institute, University of Wisconsin, Madison, Wisconsin, 1965) (Wiley, New York, 1966), pp. 295–392
R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
H. Dym, J contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, in CBMS Regional Conference Series in Mathematics, vol. 71 (American Mathematical Society, Providence, 1989). Published for the Conference Board of the Mathematical Sciences, Washington, DC
H. Dym, S. Sarkar, Multiplication operators with deficiency indices (p, p) and sampling formulas in reproducing kernel Hilbert spaces of entire vector valued functions. J. Funct. Anal. 273(12), 3671–3718 (2017)
D. Han, D.R. Larson, Frames, bases and group representations. Mem. Am. Math. Soc. 147(697), x+94 (2000)
D. Han, K. Kornelson, D. Larson, E. Weber, Frames for undergraduates, in Student Mathematical Library, vol. 40 (American Mathematical Society, Providence, 2007)
H.P. Kramer, A generalized sampling theorem. J. Math. Phys. 38(1–4), 68–72 (1959)
M. Neumark, Spectral functions of a symmetric operator. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4, 277–318 (1940)
M.A. Neumark, On a representation of additive operator set functions. C. R. (Doklady) Acad. Sci. URSS (N.S.) 41, 359–361 (1943)
K. Seip, Interpolation and sampling in spaces of analytic functions, in University Lecture Series, vol. 33 (American Mathematical Society, Providence, 2004)
E. Weber, Orthogonal frames of translates. Appl. Comput. Harmon. Anal. 17(1), 69–90 (2004)
Acknowledgements
Eric Weber was supported in part by NSF awards #1830254 and #1934884.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Al-Sa’di, S., Weber, E.S. (2021). On Parseval Frames of Kernel Functions in de Branges Spaces of Entire Vector Valued Functions. In: Alpay, D., Peretz, R., Shoikhet, D., Vajiac, M.B. (eds) New Directions in Function Theory: From Complex to Hypercomplex to Non-Commutative. Operator Theory: Advances and Applications(), vol 286. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-76473-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-76473-9_1
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-76472-2
Online ISBN: 978-3-030-76473-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)