Hans Georg Feichtinger

Hans Georg Feichtinger was born in Wiener Neustadt where he graduated from the Gymnasium and received the Matura "summa cum laude" in 1969. In the same year he started his studies in mathematics and physics. He finished his PhD at the University of Vienna in 1974 under the supervision of Hans Reiter in 1974 with a doctoral thesis on Subalgebras of L¹(G). Feichtinger attained professorship with the defense of his habilitation thesis on Banach convolution algebras of functions[1] in 1979. Feichtinger is author or co-author to roughly 200 scientific publications.

The University of Vienna being the center of his scientific life, Feichtinger still had several visiting positions across Europe and in the USA between 1980 and today, e.g. at the University of Maryland, College Park and the University of Connecticut, Storrs. He is married and the father of four children.[2]

The scientific work of Hans Georg Feichtinger includes, but is not limited to results on function spaces, irregular sampling, time-frequency analysis, Gabor analysis and frame theory. Some of his most notable contributions are listed below.

In the early 1980s, Feichtinger introduced modulation spaces, a family of function spaces defined by the behavior of the short-time Fourier transform with respect to a test function from the Schwartz space. They have become the standard spaces in time-frequency analysis. Also, while the concept of function spaces treating local and global behavior separately was already known earlier, Wiener amalgam spaces were introduced by Feichtinger in 1980, also his publications have contributed to the acknowledgment of amalgam spaces as a useful tool in various mathematical fields.

Around 1990 the joint research with Karlheinz Gröchenig lead to a series of papers, which is today referred to as coorbit theory. The theory provides a unified framework for different important transforms, e.g. the wavelet transform and the short-time Fourier transform.

Feichtinger also proposed the use of Banach Gelfand triples, especially the Banach Gelfand triple ${\displaystyle S_{0}\subset L^{2}\subset S_{0}'}$ that has proven very useful, e.g. in time-frequency analysis.

Feichtinger's conjecture

Prof. Feichtinger and Prof. Kadison at a meeting in 2011

Feichtinger once raised the question, whether

Every bounded frame can be written as a finite union of Riesz basic sequences.

The question is now widely referred to as Feichtinger's conjecture, a term first used by Peter G. Casazza.

This question was not only an important open problem in frame theory but was found to be equivalent to the famous and long-open Kadison–Singer problem in analysis (first stated in 1959).

Proofs were known for certain special cases since 2005,[8][9] and in 2013 an equivalent to the full conjecture was proved by Adam Marcus, Daniel A Spielman and Nikhil Srivastava.[10]

Hans Georg Feichtinger has published approximately 200 scientific articles,[11] a selection of which is presented below (in chronological order):

• Feichtinger, Hans G. (1981). "On a new Segal algebra". Monatshefte für Mathematik. Springer Science and Business Media LLC. 92 (4): 269–289. doi:10.1007/bf01320058. ISSN 0026-9255. S2CID 121292861.
• Feichtinger, Hans G. (1981). "Banach Spaces of Distributions of Wiener's Type and Interpolation". Functional Analysis and Approximation. Basel: Birkhäuser Basel. pp. 153–165. doi:10.1007/978-3-0348-9369-5_16. ISBN 978-3-0348-9371-8.
• Feichtinger, Hans G.; Gröchenig, Karlheinz (1988). "A unified approach to atomic decompositions via integrable group representations". Function Spaces and Applications. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/bfb0078863. ISBN 978-3-540-18905-3. ISSN 0075-8434.
• Feichtinger, Hans G.; Gröchenig, Karlheinz (1992). "Gabor Wavelets and the Heisenberg Group: Gabor Expansions and Short Time Fourier Transform from the Group Theoretical Point of View". Wavelets. Elsevier. pp. 359–397. doi:10.1016/b978-0-12-174590-5.50018-6. ISBN 9780121745905.
• Feichtinger, Hans G.; Gröchenig, Karlheinz (7 July 2021). "Theory and practice of irregular sampling". Wavelets. Boca Raton: CRC Press. doi:10.1201/9781003210450-10. ISBN 978-1-003-21045-0. S2CID 14607775.
• Feichtinger, Hans G.; Gr\"ochenig, Karlheinz; Strohmer, Thomas (1 February 1995). "Efficient numerical methods in non-uniform sampling theory". Numerische Mathematik. Springer Science and Business Media LLC. 69 (4): 423–440. doi:10.1007/s002110050101. ISSN 0029-599X. S2CID 15340037.
• Feichtinger, Hans G.; Gröchenig, K (1997). "Gabor Frames and Time-Frequency Analysis of Distributions". Journal of Functional Analysis. Elsevier BV. 146 (2): 464–495. doi:10.1006/jfan.1996.3078. ISSN 0022-1236.
  • Bolcskei, H.; Hlawatsch, F.; Feichtinger, H.G. (1998). "Frame-theoretic analysis of oversampled filter banks". IEEE Transactions on Signal Processing. Institute of Electrical and Electronics Engineers (IEEE). 46 (12): 3256–3268. doi:10.1109/78.735301. ISSN 1053-587X.
• Feichtinger, Hans G.; Strohmer, Thomas (1998). Gabor Analysis and Algorithms : Theory and Applications. Boston, MA: Birkhäuser Boston. ISBN 978-1-4612-2016-9. OCLC 840278297.
• Feichtinger, Hans G.; Strohmer, Thomas (2003). Advances in Gabor Analysis. Boston, MA: Birkhäuser Boston. ISBN 978-1-4612-0133-5. OCLC 840277709.
• Feichtinger, Hans G.; Kaiblinger, Norbert (12 November 2003). "Varying the time-frequency lattice of Gabor frames". Transactions of the American Mathematical Society. American Mathematical Society (AMS). 356 (5): 2001–2023. doi:10.1090/s0002-9947-03-03377-4. ISSN 0002-9947.

• Hans Georg Feichtinger at the Mathematics Genealogy Project
• Hans Georg Feichtinger's homepage at University of Vienna
• NuHAG homepage