Mathematics Branches, Topics, and Sub-Topics

A structured visual guide to the major mathematical areas and their relationships.

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42Cxx Nontrigonometric harmonic analysis

This subtopic studies nontrigonometric harmonic analysis, including wavelets, orthogonal systems, and other bases used to represent functions in modern analysis.

Specific topics

42C05 Orthogonal functions and polynomials, general theory

Overview

This topic studies orthogonal functions and polynomials, general theory within nontrigonometric harmonic analysis, emphasizing structural theorems, convergence behavior, and computational implications.

Related Wikipedia Page

Orthogonal functions (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 nontrigonometric harmonic analysis, including rigorous analysis of transforms, expansions, and operators.

Applications

  • Fourier/harmonic analysis
  • Partial differential equations
  • Signal and data modeling

References

Recommended Textbooks

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

Overview

This topic studies fourier series in special orthogonal functions (legendre polynomials, walsh functions, etc.) within nontrigonometric harmonic analysis, emphasizing structural theorems, convergence behavior, and computational implications.

Related Wikipedia Page

Orthogonal functions (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 nontrigonometric harmonic analysis, including rigorous analysis of transforms, expansions, and operators.

Applications

  • Fourier/harmonic analysis
  • Partial differential equations
  • Signal and data modeling

References

Recommended Textbooks

42C15 General harmonic expansions, frames

Overview

This topic studies general harmonic expansions, frames within nontrigonometric harmonic analysis, emphasizing structural theorems, convergence behavior, and computational implications.

Related Wikipedia Page

Orthogonal functions (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 nontrigonometric harmonic analysis, including rigorous analysis of transforms, expansions, and operators.

Applications

  • Fourier/harmonic analysis
  • Partial differential equations
  • Signal and data modeling

References

Recommended Textbooks

42C20 Other transformations of harmonic analysis type

Overview

This topic studies other transformations of harmonic analysis type within nontrigonometric harmonic analysis, emphasizing structural theorems, convergence behavior, and computational implications.

Related Wikipedia Page

Orthogonal functions (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 nontrigonometric harmonic analysis, including rigorous analysis of transforms, expansions, and operators.

Applications

  • Fourier/harmonic analysis
  • Partial differential equations
  • Signal and data modeling

References

Recommended Textbooks

42C25 Uniqueness and localization for orthogonal series

Overview

This topic studies uniqueness and localization for orthogonal series within nontrigonometric harmonic analysis, emphasizing structural theorems, convergence behavior, and computational implications.

Related Wikipedia Page

Orthogonal functions (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 nontrigonometric harmonic analysis, including rigorous analysis of transforms, expansions, and operators.

Applications

  • Fourier/harmonic analysis
  • Partial differential equations
  • Signal and data modeling

References

Recommended Textbooks

42C30 Completeness of orthogonal systems

Overview

This topic studies completeness of orthogonal systems within nontrigonometric harmonic analysis, emphasizing structural theorems, convergence behavior, and computational implications.

Related Wikipedia Page

Orthogonal functions (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 nontrigonometric harmonic analysis, including rigorous analysis of transforms, expansions, and operators.

Applications

  • Fourier/harmonic analysis
  • Partial differential equations
  • Signal and data modeling

References

Recommended Textbooks

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems

Overview

This topic studies nontrigonometric harmonic analysis involving wavelets and other special systems within nontrigonometric harmonic analysis, emphasizing structural theorems, convergence behavior, and computational implications.

Related Wikipedia Page

Orthogonal functions (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 nontrigonometric harmonic analysis, including rigorous analysis of transforms, expansions, and operators.

Applications

  • Fourier/harmonic analysis
  • Partial differential equations
  • Signal and data modeling

References

Recommended Textbooks