The World According to Wavelets: The Story of a Mathematical Technique in the Making, Second Edition

Part III: Appendices

Appendix List

Appendix A: Mathematical Symbols
Appendix B: A Review of Some Elementary Trigonometry
Appendix C: Integrals
Appendix D: The Fourier Transform: The Different Conventions
Appendix E: The Sampling Theorem: A Proof
Appendix F: The Heisenberg Uncertainty Principle: A Proof
Appendix G: The Fourier Transform of a Periodic Function
Appendix H: An Example of an Orthonormal Basis and a Proof of Fourier's Result

?, ?

delta

?

epsilon (commonly used to represent small numbers)

?

theta (a variable angle)

?

mu (often represents measures)

?

xi (a variable representing the wave number, or spatial frequency, for Fourier transforms of signals that depend on x, representing space)

?

sigma (standard deviation)

?

tau (a variable representing frequency, for Fourier transforms of signals that depend on t, representing time.)

Note: k often replaces ? or ? in formulas for Fourier series, according to the mathematical convention that k represents integer variables, while ? and ? represent continuous variables.

? or ?

phi (the scaling function; also used to represent the phase angle)

?

psi (the wavelet; in quantum mechanics, the wave function)

?

sum

?

integral

?

product

?

union

?

intersection

?

infinity

Overview

The trigonometric functions we use in this book are sine (abbreviated sin) and cosine (abbreviated cos). Consider a unit circle (a circle with radius 1) centered at 0: x 2+ y 2 = 1. Starting from the point (1,0), go a distance ? counterclockwise,...

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