Transform Methods in Mathematics

  1. Specialized Integral Transforms
    1. Mellin Transform
      1. Definition and Basic Properties
        1. Relationship to Other Transforms
          1. Connection to Laplace Transform
            1. Connection to Fourier Transform
            2. Applications
              1. Asymptotic Analysis
                1. Number Theory
                  1. Probability Theory
                2. Hankel Transform
                  1. Definition with Bessel Function Kernel
                    1. Properties and Transform Pairs
                      1. Relationship to Fourier Transform
                        1. Cylindrical Coordinates
                          1. Radial Symmetry
                          2. Applications
                            1. Problems with Cylindrical Symmetry
                              1. Heat Conduction
                                1. Wave Propagation
                              2. Hilbert Transform
                                1. Definition as Principal Value Integral
                                  1. Properties
                                    1. Phase Shifting
                                      1. Causality Relations
                                      2. Analytic Signals
                                        1. Complex Envelope
                                          1. Instantaneous Frequency
                                          2. Applications
                                            1. Signal Processing
                                              1. Communications
                                                1. Amplitude Modulation
                                              2. Wavelet Transforms
                                                1. Continuous Wavelet Transform
                                                  1. Definition and Properties
                                                    1. Time-Frequency Localization
                                                      1. Wavelet Functions
                                                      2. Discrete Wavelet Transform
                                                        1. Multiresolution Analysis
                                                          1. Scaling Functions
                                                            1. Filter Bank Implementation
                                                            2. Comparison with Fourier Methods
                                                              1. Time-Frequency Resolution
                                                                1. Non-stationary Signal Analysis

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                                                            4. Discrete Transform Methods

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                                                            6. Advanced Applications