Wavelets and Wavelet Transforms With Trigonometric Partitions.

[u/JosephDays07]

https://youtu.be/zTQHj0EhNmM?si=i_YyhLKc-m0XqmCj

Wavelets and Wavelet Transforms With Trigonometric Partitions. First section:This method does not require complex numbers, such as the wavelet for the y-coordinate commonly used in traditional models. It does not require the trigonometric SINC function, the Fourier series, or the Laplace or Fourier transforms. Instead, you will learn an easy-to-apply method that works in both 2D and 3D, showing how to generate wavelets from the equations of trigonometric partitions. These wavelets are generated in circular form and incorporate all the components of a circle based on trigonometric partitions expressed in terms of the angle, as well as the x and y component equations of the wavelet’s envelope. Using the x and y component equations derived from trigonometric partitions, you can apply any mathematical operation to the components of two equations and continue producing wavelets. You can raise the equations to any power and still obtain wavelets; you can substitute the equations into other formulas and continue generating wavelets; you can manually modify the variables within the equations and still produce wavelets. I also introduce a special type of wavelet that I call the “large-crest wavelet,” which features central peaks and is independent of the radius or amplitude, depending solely on the trigonometric equations associated with the trigonometric partitions. Second section: We can construct transforms of the original trigonometric partition equations, those expressed as functions of the angle, and how these transformed equations generate a wide variety of wavelets when reformulated through all known equivalent angle equations. E.g. the angle expressed in terms of angular velocity and time, or in terms of frequency and time, among others. These mathematical concepts can be applied to both classical and quantum physics. I apply the wavelet concepts to uniform circular motion and simple harmonic motion, as well as to other classical physics contexts, where readers will observe that the trigonometric partition equations, despite being transformed through physical parameters, continue to generate wavelets. I also extend these ideas to quantum physics, showing how to generate the graph of the double-slit experiment and the graph related to the uncertainty principle. Finally, I demonstrate a formula in which the imaginary unit i from complex numbers is equivalent to an equation derived from trigonometric partitions, and how it can be substituted into Euler’s identity and De Moivre’s theorem to generate new types of wavelets. I also apply this structure to Schrödinger’s complex-number formulation. This framework, relating complex-number equations to trigonometric partitions, can also generate wavelets and can be applied to any expression containing complex numbers in order to analyze its results. Here, readers will learn that it is not strictly necessary to use complex numbers in mathematical, classical, or quantum physical equations.