Hans G. Feichtinger

Professional Memberships: AMS, SIAM, DMV, ÖMG, SPIE.

Core Links

Research Themes

Harmonic Analysis: Segal algebras, Wiener Amalgam spaces, irregular sampling;
Time-Frequency Analysis: STFT, spreading representation and Kohn-Nirenberg symbol;
Gabor Analysis: Gabor frames and Riesz projection bases, Janssen formula;
Modulation Spaces: using the STFT or BUPUs on the Fourier transform side;
Banach Gelfand Triples (BGTs): The Triple based on the Feichtinger algebra $S_0(G)$
Phase-Space Methods: operators understood by their phase space behaviour
Numerical Work: mostly iterative algorithms, Gabor Analysis,
Coorbit Theory: integrable group representations, wavelets, Gabor expansions;
Decompositions spaces: intermediate between Besov and modulation spaces;
Mild distributions are signals or observables (new viewpoint);
Hermite Functions: Approximation by finite dimensional models
Quantum Harmonic Analysis: Convolution of operators, using the BGT approach;
Choosing Function Spaces: symbols and motivation for their choice;
Frames, Banach frames, so-called continuous orthonormal basis.
Wiener's Third Tauberian Theorem, functions of bounded p-th means;
Transformable measures (Argabright/Lamadrid), quasicrystals and mild distributions;
The metaplectic group, fractional Fourier transforms and modulation spaces, cf. LCT, SAFT;
Teaching Fourier Analysis to Engineers and Physicists. Interpreting their formulas;
Convolution and Fourier Transform without measure theory! (an alternative approach);
Double modules, L1-convolution and FL1-multiplication, Fourier Standard Spaces;
Structure preserving approximation of continuous models by finite-dimensional ones.

Research Themes illustrated combined with links from www.nuhag.eu/feitalks

Harmonic Analysis: Segal algebras, Wiener amalgam spaces, irregular sampling.
- Localized Fourier analysis on locally compact groups, emphasizing Segal algebras, Wiener-type spaces, and sampling questions.
- Segal algebras and the ideal theorem
- Wiener Amalgam spaces and their applications
- Time-Frequency Analysis, Gabor Multipliers, and Banach Gelfand Triples
Kernel Theorems: operator representations via kernels; outer and inner kernel theorems in the S₀-framework.
- Representation of bounded linear operators by kernels in suitable tensor product spaces. In the setting of the Feichtinger algebra S₀, this leads to a particularly transparent version of the kernel theorem, including an “inner kernel theorem” describing operators acting within the Banach Gelfand Triple (S₀, L², S₀^').
- The Kernel Theorem for the Feichtinger Algebra
- The Kernel Theorem and Tensor Factorization of Gabor Frames
- Quantum Harmonic Analysis (QHA) via the Banach Gelfand Triple
Time-Frequency Analysis: STFT, spreading representation and Kohn-Nirenberg symbol.
- The STFT as the basic tool for describing signals and operators in phase space, including symbol and spreading viewpoints.
- Time-Frequency Analysis, Gabor Multipliers, and Banach Gelfand Triples
- How abstract mathematical analysis can be useful for engineers
- Gabor Analysis, Wigner Transform and the Metaplectic Group
Gabor Analysis: Gabor frames and Riesz projection bases, Janssen formula.
- Theory and applications of Gabor systems, with emphasis on frames, Riesz-type structures, and operator-theoretic viewpoints.
- Mathematical Foundations of Gabor Analysis
- What do we know about Gabor multipliers?
- WIENER AMALGAMS and GABOR ANALYSIS
Modulation Spaces: using the STFT or BUPUs on the Fourier transform side.
- Modulation spaces as natural function spaces for time-frequency analysis, defined via STFT methods or decomposition techniques.
- Modulation Spaces and Gabor Analysis
- Beyond modulation spaces (Alpha-modulation spaces and general decomposition spaces)
- Segal algebras and the ideal theorem
Banach Gelfand Triples (BGTs): The Triple based on the Feichtinger algebra S₀(G).
- The triple (S₀, L², S₀^') as a Banach-space version of a rigged Hilbert space for Fourier and operator analysis.
- The usefulness of certain Banach Gelfand Triples
- Foundation of coorbit theory
- Time-Frequency Analysis, Gabor Multipliers, and Banach Gelfand Triples
Phase-Space Methods: operators understood by their phase space behaviour.
- Signals and operators described via their localization in the time-frequency or phase-space plane.
- Gabor Analysis, Wigner Transform and the Metaplectic Group
- Quantum Harmonic Analysis (QHA) via the Banach Gelfand Triple
Numerical Work: mostly iterative algorithms, Gabor Analysis.
- Computational and algorithmic aspects of time-frequency analysis and Gabor methods.
- The Role of Finite-dimensional Approximation for Applications
- Finite dimensional approximation of continuous problems in Gabor analysis: the case of discrete Hermite functions
- Mathematical Foundations of Gabor Analysis
Coorbit Theory: integrable group representations, wavelets, Gabor expansions.
- Coorbit theory as a unifying framework for wavelets, Gabor systems, and representation-based Banach spaces.
- Foundation of coorbit theory
- Localized Frames via Coorbit Theory
- Modulation Spaces and Gabor Analysis
Decomposition spaces: intermediate between Besov and modulation spaces.
- General decomposition-space methods bridging classical Besov scales and modulation-space techniques.
- Beyond modulation spaces (Alpha-modulation spaces and general decomposition spaces)
- Wiener's third Tauberian Theorem (a short survey)
Mild distributions: signals or observables (new viewpoint).
- The viewpoint that mild distributions form the natural ambient space for signals, observables, and operator-theoretic constructions.
- Mild Distributions: A Simple and Universal Tool for Fourier and Time-Frequency Analysis
- Mild distributions: A new setting making distributions more accessible to engineers
- Quantum Harmonic Analysis (QHA) via the Banach Gelfand Triple
Hermite Functions: approximation by finite dimensional models.
- Discrete Hermite functions as a bridge between continuous models and finite-dimensional approximations.
- Discrete Hermite Functions and Computational Aspects of the Fractional Fourier Transform
- Finite dimensional approximation of continuous problems in Gabor analysis: the case of discrete Hermite functions
Quantum Harmonic Analysis: Convolution of operators, using the BGT approach.
- Operator convolution and phase-space analysis developed in the Banach Gelfand Triple setting.
- Quantum Harmonic Analysis (QHA) via the Banach Gelfand Triple
- Gabor Analysis, Wigner Transform and the Metaplectic Group
Choosing Function Spaces: symbols and motivation for their choice.
- Why Wiener amalgams, modulation spaces, and related Banach spaces are the right setting for Fourier and time-frequency questions.
- Wiener Amalgam spaces and their applications
- Wiener amalgams and Gabor Analysis
- Modulation Spaces and Gabor Analysis
Frames, Banach frames, so-called continuous orthonormal basis.
- From redundant discrete expansions to continuous basis-type systems and Banach-frame constructions.
- Localized Frames via Coorbit Theory
- Foundation of coorbit theory
- Mild Distributions: A Simple and Universal Tool for Fourier and Time-Frequency Analysis
Wiener’s Third Tauberian Theorem: functions of bounded p-th means.
- Tauberian questions, decomposition methods, and convolution algebras in the spirit of Norbert Wiener.
- Wiener’s third Tauberian Theorem (a short survey)
Transformable measures: Argabright/Lamadrid, quasicrystals and mild distributions.
- Fourier-transformable generalized objects and their relation to modern distributional viewpoints.
- Mild Distributions: A Simple and Universal Tool for Fourier and Time-Frequency Analysis
- The Role of Finite-dimensional Approximation for Applications
The metaplectic group, fractional Fourier transforms and modulation spaces, cf. LCT, SAFT.
- Metaplectic symmetries, chirps, Wigner methods, and fractional Fourier transforms in a time-frequency setting.
- Gabor Analysis, Wigner Transform and the Metaplectic Group
- Discrete Hermite Functions and Computational Aspects of the Fractional Fourier Transform
Teaching Fourier Analysis to Engineers and Physicists: interpreting their formulas.
- A conceptually clear, application-oriented presentation of Fourier analysis for engineers and physicists.
- How abstract mathematical analysis can be useful for engineers
- Mild distributions: A new setting making distributions more accessible to engineers
- An Alternative Approach to Convolution and the Fourier Transform
Convolution and Fourier Transform without measure theory! An alternative approach.
- A functional-analytic approach to convolution and Fourier transform avoiding the usual measure-theoretic entry point.
- An Alternative Approach to Convolution and the Fourier Transform
Structure preserving approximation of continuous models by finite-dimensional ones.
- Finite-dimensional models that preserve the essential algebraic and analytic structure of continuous Fourier-analytic problems.
- Approximation of continuous problems in Fourier Analysis by finite dimensional ones: The setting of the Banach Gelfand Triple
- Finite dimensional approximation of continuous problems in Gabor analysis: the case of discrete Hermite functions
- The Role of Finite-dimensional Approximation for Applications
Double Modules: function spaces stable under convolution and pointwise multiplication.
- A structural viewpoint on Fourier-analytic function spaces: Banach spaces that carry both a convolution module structure over L¹ and a multiplication module structure over the Fourier algebra.
- Invariant function spaces as double modules illustrated via Fourier Standard Spaces
- The Banach Gelfand Triple and Fourier Standard Spaces

Friends and Cooperation Partners (just started, rather incomplete)

Role Models and In Memoriam

Role Models
- Paul Butzer (1928–xxxx): Pioneer in approximation theory, Fourier analysis, and history of mathematics. Paul Butzer's WIKI
- Hans Triebel (1936–xxxx): Founder of modern function space theory (Besov, Triebel–Lizorkin spaces). Hans Triebel's WIKI
- Yves Meyer (1939–xxxx): Pioneer of wavelet theory; Abel Prize 2017; fundamental contributions to harmonic analysis and multiresolution analysis. Yves Meyer's WIKI
In Memoriam
- Hans Reiter:(1921–1992): Book on Classical Harmonic Analysis and Locally Compact Groups, LN on Segal algebras and the Metaplectic group. Hans Reiter's WIKI
- Norbert Wiener (1894–1964): Founder of cybernetics and a pioneer of modern harmonic analysis. Known for Wiener's Tauberian theorem and the introduction of Wiener's algebra, as well as foundational work on stochastic processes (Brownian motion) and prediction theory. His ideas laid the groundwork for generalized harmonic analysis and, indirectly, for Wiener amalgam spaces, with lasting impact on signal processing and time-frequency analysis. Wiener's WIKI
- Arne Beurling (1905–1986): One of the leading analysts of the 20th century, with deep contributions to harmonic and complex analysis. Known for Beurling algebras (weighted convolution algebras), Beurling’s theorem on invariant subspaces of the Hardy space, and influential work on spectral synthesis and generalized primes. His ideas have had lasting impact on abstract harmonic analysis and the development of function spaces. Beurling's WIKI
- Laurent Schwartz (1915–2002): Founder of the theory of distributions, providing a rigorous framework for generalized functions and Fourier analysis beyond classical functions. His introduction of Schwartz spaces and tempered distributions laid the foundation for modern harmonic analysis, partial differential equations, and time-frequency methods. Awarded the Fields Medal in 1950. Schwartz's WIKI
- John R. Klauder (1932–2025): Pioneer of coherent states, affine quantization, and rigorous approaches to quantum field theory; strong advocate of phase-space methods and continuous representation theory with deep connections to harmonic analysis. Author of influential monographs on coherent states and their applications in physics and mathematics. John Klauder's WIKI
- Pete Casazza (1948–2025): Fundamental contributions to frame theory and Banach space theory. Establishing connections between Kadison-Singer and Feichtinger Conjecture (Thanks Pete!) Pete's WIKI plus link to homepage of 2017
- Nicolai Vasilevski (1948–2023): Major contributions to the theory of Toeplitz operators, in particular on Bergman and Fock spaces, including the structure of commutative C*-algebras generated by such operators and their role in quantization. Author of the monograph Commutative Algebras of Toeplitz Operators on the Bergman Space (Birkhäuser, 2008), and a central figure in connecting operator theory with complex analysis and mathematical physics.
- Kurt Bernardo Wolf: (1948–2023): Fundamental contributions to frame theory and Banach space theory. Establishing connections between Kadison-Singer and Feichtinger Conjecture (Thanks Pete!)

Kaleidoscope (legacy)