1. Primer on Fourier Analysis
Dana Moshkovitz
Princeton University and
The Institute for Advanced Study

2. Fourier Analysis in Theoretical Computer Science

3. Fourier Analysis in Theoretical Computer Science (Unofficial List)
Polynomials multiplication (FFT)
List Decoding [AGS03]
Collective Coin Flipping [BL,KKL]
Analysis of expansion/sampling (e.g., [MR06])
Computational Learning [KM]
Linearity testing [BLR]
Analysis of threshold phenomena
Hardness of Approximation (dictator testing) [H97]
Voting/social choice schemes …
Quantum Computing

4. “something that looks scary to analyze” “bunch of (in)equalities”
“The Fourier Magic”
Fourier Analysis
“something that looks scary to analyze”
“bunch of (in)equalities”

5. Today: Explain the “Fourier Magic”
Why is it useful?
What is it?
What does it do?
When to use it?
What do we know about it?

6. It’s Just a Different Way to Look at Functions

7. It’s Changing Basis
Background: Real/complex functions form vector space
Idea: Represent functions in Fourier basis, which is the basis of the shift operators (representation by frequency).
Advantage: Convolution (complicated “global” operation on functions) becomes simple (“local”) in Fourier basis
Generality: Here will only consider the Boolean case – very-special case

8. Fourier Basis (Boolean Cube Case)
Boolean cube: additive group Z2n
Space of functions: Z2n.
Inner product space where f,g=Ex[f(x)g(x)].
Characters: (x+y)=(x)(y)

9. Foundations
Claim (Characterization): The characters are the eigenvectors of the shift operators Ssf(x)→ f(x+s).
Corollary (Basis): The characters form an orthonormal basis.
Claim (Explicit): The characters are the functions S(x) = (-1)iSxi for S[n].
Shift operators occur “everywhere” [try to think of examples; some will be given in the sequel].
In particular, convolution involves shifting.
Analysis in the eigenvectors basis seems “natural”.
Notice: in the Boolean case, the characters correspond to the linear function <s,x> for the binary vectors s.
This is what is used for “linearity testing”.
[the (-1) is for addition <-->multiplication]

10. Fourier Transform = Polynomial Expansion
Fourier coefficients: f^(S) = f,S.
Note: f^()=Ex[f(x)]
Polynomial expansion: substitute yi=(-1)xi
f(y1,…,yn) = Sµ[n]f^(S)i2Syi
Fourier transform: f  f^

11. The Fourier Spectrum
level n
n-1 …
n/2
|S| … 1

12. Degree-k Polynomial n
n-1 …
n/2
|S| … 1 k

13. k-Junta n
n-1 …
n/2
|S| … 1
k-junta is a degenerate case – instead of considering n vars, we could have considered k k

14. Orthonormal Bases
Parseval Identity (generalized Pythagorean Thm): For any f, S(f^(S))2 = Ex[ (f (x))2] So, for Boolean f:{±1}n→{±1}, we have: x(f^(x))2 = 1 In general, for any f,g, f,g = 2nf^,g^

15. Convolution
Convolution: (f*g)(x) = Ey[f(y)g(x-y)] Example Weighted average: (f*w)(0) = Ey[f(y)w(y)]

16. Convolution in Fourier Basis
Claim: For any f,g, (f*g)^  f^·g^ Proof: By expanding according to definition.

17. Things You Can Do with Convolution

18. Parts of The Spectrum Variance:…
n/2 … 1
Variance:
Varx[f(x)] = Ex[f(x)2] - Ex[f(x)]2 = S≠; f^(S)2
Influence of i’th variable:
Infi(f) = Px[f(x)≠f(xei)] = S3i f^(S)2
Intuition:
Variance = the non-constant part of the function
Influence of i = the part of the spectrum that depends on i

19. Smoothening f Perturbation: x»±y : for each i, T±f(x) = Ex»±y[f(y)]
yi = xi with probability 1-±
yi = 1-xi otherwise
T±f(x) = Ex»±y[f(y)]
Convolution: T±f  f*P(noise=µ)
Fourier: (T±f)^  (1-2±)|S|·f^

20. Smoothed Function is Close to Low Degree!
Tail: Part of |T±f|22 on levels ¸ k is: · (1-2±)2k |f|22· e-c±k Hence, weight  on levels ¸ C · 1/ · log 1/ 

21. Hypercontractivity
Theorem (Bonami, Gross): For f, for ± · √(p-1)/(q-1), |T±f|q · |f|p Roughly, and incorrectly ;-): “T±f much [in a “tougher” norm] smoother than f”

22. Noise Sensitivity and Stability
NS±(f) = Px»±y (f(x)f(y))
Correlation:
NS±(f) = 2(E[f]-f,T±f)
Stability: Set  := 1/2-/2
S½(f) = f,T±f
Fourier:
S±(f) = f^, |S|f^
= §S |S| f^(S)2
NS±(f) = 2f^() – 2S(1-2±)|S|f^(S)2

23. Thresholds Are Stablest and Hardness of Approximation
What is it? Isoperimetric inequality on noise stability [MOO05].
Applications to hardness of approximation (e.g., Max-Cut [KKMO04]).
Derived from “Invariance Principle” (extended Central Limit Theorem), used by the [R08] extension of [KKMO04].
Isoperimetry = Largest area for specified boundary/[Here:]smallest boundary for given area.
For more applications – look at [MOO].
The application go far beyond hardness of approximation.

24. Thresholds Are Stablest
Theorem [MOO’05]: Fix 0<<1. For balanced f (i.e., E[f]=0) where Infi(f)≤ for all i,
Sρ(f) ≤ 2/π · arcsin ρ + O( (loglog 1/²)/log1/²)
≈ noise stability of threshold functions
t(x)=sign(∑aixi), ∑ai2=1
Balanced is [somewhat] necessary: constant function is stable!
Can “play” with that – consider somewhat non-constant function.
In hardness of approximation – usually easy to ensure the function is balanced.
Also, commonly in hardness constructions we consider negative rho.
- All influences are small = no variable determines the function to a large extent
Can “play” with that – first remove influential variables.
But in hardness construction this is exactly what we exploit: stable balanced functions have an influential variable!

25. More Material
There are excellent courses on Fourier Analysis available on the homepages of: Irit Dinur and Ehud Friedgut, Guy Kindler, Subhash Khot, Elchanan Mossel, Ryan O’Donnell, Oded Regev.