42Bxx Fourier analysis in several variables
This subtopic studies Fourier analysis in several variables, emphasizing multi-dimensional transforms, product structure, and the geometry of frequency domains.
Specific topics
42B05 Fourier series and coefficients in several variables
Overview
This topic studies multidimensional Fourier series, coefficient behavior, and convergence structures on tori and related domains.
Related Wikipedia Page
Multiple Fourier series (Wikipedia)
Useful Links
Key Ideas
- Multi-index harmonic expansions
- Coefficient decay versus mixed smoothness
- Convergence issues specific to higher dimensions
Typical Uses
Used in multidimensional periodic modeling and spectral discretizations for PDEs.
Applications
- Spectral PDE solvers
- Image/volume analysis
- Multidimensional signal processing
References
Recommended Textbooks
42B08 Summability in several variables
Overview
This topic addresses summability methods for multidimensional Fourier expansions, where partial-sum geometry creates challenges absent in one variable.
Related Wikipedia Page
Summability method (Wikipedia)
Useful Links
Key Ideas
- Rectangular/spherical summation schemes
- Regularization of divergent multidimensional series
- Kernel methods and convergence improvement
Typical Uses
Used to stabilize Fourier reconstruction and prove convergence in higher-dimensional harmonic settings.
Applications
- Multidimensional spectral analysis
- Numerical harmonic reconstruction
- PDE approximation
References
Recommended Textbooks
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type, several variables
Overview
This topic studies multidimensional Fourier and Fourier-Stieltjes transforms, including transform-domain methods for functions, distributions, and measures in several variables.
Related Wikipedia Page
Fourier transform (Wikipedia)
Useful Links
Key Ideas
- Multi-dimensional transform theory for functions and measures
- Decay-regularity duality in frequency space
- Transform methods for PDE and inverse problems
Typical Uses
Used to analyze high-dimensional oscillatory behavior and solve PDE/inverse tasks via frequency-domain methods.
Applications
- PDE analysis
- Imaging and tomography
- Multivariate signal processing
References
Recommended Textbooks
42B15 Multipliers for harmonic analysis in several variables
Overview
This topic studies multiplier operators in multiple dimensions, including symbol conditions for boundedness and mapping properties across Lp-type spaces.
Related Wikipedia Page
Fourier multiplier (Wikipedia)
Useful Links
Key Ideas
- Symbol smoothness and homogeneity conditions
- Boundedness on Lp and Sobolev scales
- Links with pseudo-differential operators
Typical Uses
Used to justify frequency-domain filters and regularity transfers in PDE and harmonic-analysis pipelines.
Applications
- PDE regularity theory
- Multichannel filtering
- Time-frequency analysis
References
Recommended Textbooks
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Overview
This topic studies singular and oscillatory integral operators in several variables, including Calderon-Zygmund theory and modern extensions.
Related Wikipedia Page
Singular integral (Wikipedia)
Useful Links
Key Ideas
- Kernel cancellation and principal-value operators
- Oscillation-based decay and stationary phase methods
- Lp boundedness and endpoint phenomena
Typical Uses
Used in modern harmonic analysis and PDE to quantify regularity and dispersive effects of integral operators.
Applications
- Elliptic and dispersive PDEs
- Microlocal analysis
- Advanced signal analysis
References
Recommended Textbooks
42B25 Maximal functions, Littlewood-Paley theory
Overview
This topic studies maximal functions, littlewood-paley theory within fourier analysis in several variables, emphasizing structural theorems, convergence behavior, and computational implications.
Related Wikipedia Page
Harmonic analysis (Wikipedia)
Useful Links
Key Ideas
- Function-space and operator-theoretic viewpoints
- Convergence, boundedness, and representation properties
- Connections between abstract theory and applied analysis
Typical Uses
Used in research and applications involving fourier analysis in several variables, including rigorous analysis of transforms, expansions, and operators.
Applications
- Fourier/harmonic analysis
- Partial differential equations
- Signal and data modeling
References
Recommended Textbooks
42B30 $H^p$-spaces in several variables
Overview
This topic studies $h^p$-spaces in several variables within fourier analysis in several variables, emphasizing structural theorems, convergence behavior, and computational implications.
Related Wikipedia Page
Harmonic analysis (Wikipedia)
Useful Links
Key Ideas
- Function-space and operator-theoretic viewpoints
- Convergence, boundedness, and representation properties
- Connections between abstract theory and applied analysis
Typical Uses
Used in research and applications involving fourier analysis in several variables, including rigorous analysis of transforms, expansions, and operators.
Applications
- Fourier/harmonic analysis
- Partial differential equations
- Signal and data modeling
References
Recommended Textbooks
42B35 Function spaces arising in harmonic analysis
Overview
This topic studies function spaces arising in harmonic analysis within fourier analysis in several variables, emphasizing structural theorems, convergence behavior, and computational implications.
Related Wikipedia Page
Harmonic analysis (Wikipedia)
Useful Links
Key Ideas
- Function-space and operator-theoretic viewpoints
- Convergence, boundedness, and representation properties
- Connections between abstract theory and applied analysis
Typical Uses
Used in research and applications involving fourier analysis in several variables, including rigorous analysis of transforms, expansions, and operators.
Applications
- Fourier/harmonic analysis
- Partial differential equations
- Signal and data modeling
References
Recommended Textbooks
42B37 Harmonic analysis and PDEs
Overview
This topic studies harmonic analysis and pdes within fourier analysis in several variables, emphasizing structural theorems, convergence behavior, and computational implications.
Related Wikipedia Page
Harmonic analysis (Wikipedia)
Useful Links
Key Ideas
- Function-space and operator-theoretic viewpoints
- Convergence, boundedness, and representation properties
- Connections between abstract theory and applied analysis
Typical Uses
Used in research and applications involving fourier analysis in several variables, including rigorous analysis of transforms, expansions, and operators.
Applications
- Fourier/harmonic analysis
- Partial differential equations
- Signal and data modeling
References
Recommended Textbooks