Introduction to Analysis on the Discrete Cube I

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" summary

Defines the discrete cube {-1,1}^n with normalized counting measure, L^p norms and Hamming distance, then builds the Walsh functions as an orthonormal basis of size 2^n and gives the Walsh–Fourier expansion with Plancherel. Shows Walsh functions are eigenfunctions of the simple neighbor-averaging operator with eigenvalue 1-2|S|/n, derives lazy-walk eigenvalue 1-λ+λ(1-2|S|/n), and explains the fast Walsh transform by splitting the cube into two (add/subtract, divide by 2) to cut naive quadratic work to O(n2^n). Presents continuous-time construction and the resulting Markov semigroup and explicit transition formulas.

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" chapters (15)

1. 0:00Start
2. 0:13Disclaimer
3. 1:15Discrete cube
4. 4:182 structure
5. 6:24Walsh system
6. 16:59Elementary argument
7. 19:24Simple consequences of the orthonormality
8. 22:08LCA setting
9. 30:20Computing the Walsh-Fourier transform
10. 38:23Discrete time random walk
11. 41:43Discrete time walk - spectral properties
12. 45:50Discrete time walk - estimates
13. 48:17Discrete random walk- ergodicity
14. 54:06Lazy random walk- ergodicity
15. 57:34Lazy walk - limit behaviour