Math 254A - Topics in real analysis

Summary of stationary phase estimates in 1D

ϕ is a smooth real-valued function, ψ is a smooth compactly supported function. Fix real numbers a < b, and let I(λ) denote the integral

I(λ) = ∫_{a}^{b} e^{iλϕ(x)} dx.

Trivial estimate: |I(λ)| ≤ b − a. If ϕ is completely stationary then this is the best one can do.

Van der Corput (for 1 derivative): If |ϕ′(x)| ≥ 1 for all x ∈ [a, b] and ϕ′ is monotone, then |I(λ)| ≤ Cλ^{−1}. Van der Corput (for 2+ derivatives): If |ϕ^{(k)}(x)| ≥ 1 for all x ∈ [a, b] for some integer k > 2, then

|I(λ)| ≤ C_{k}λ^{−1/k}.

Any estimate for ∫_{a}^{b} e^{iλϕ(x)} dx which is independent of a and b implies a corresponding estimate for ∫ e^{iλϕ(x)}ψ(x) dx as long as ψ has bounded variation, i.e. ∫ |ψ′(x)| dx ≤ C. (Integration by parts).

Principle of non-stationary phase: If ϕ, ψ are smooth and ϕ′ is never vanishing, then

| ∫ e^{iλϕ(x)}ψ(x) dx| ≤ C_{N,ϕ,ψ}λ^{−N}

for any integer N. (Repeated integration by parts).

Asymptotics: If ϕ, ψ are smooth, the only stationary point of ϕ is at x_{0}, and ϕ′′(x_{0}) ≠ 0, then we have the asymptotic expansion

∫ e^{iλϕ(x)}ψ(x) dx ∼ e^{iλϕ(x_{0})}[a_{−1/2}λ^{−1/2} + a_{−1}λ^{−1} + …]

where

a_{−1/2} = \sqrt{\frac{πi}{ϕ′′(x_{0})}}ψ(x_{0}).

If instead we have ϕ′′(x_{0}) = … = ϕ^{(k−1)}(x_{0}) = 0 but ϕ^{(k)}(x_{0}) ≠ 0, then we get an asymptotic expansion of the form

∫ e^{iλϕ(x)}ψ(x) dx ∼ e^{iλϕ(x_{0})}[a_{−1/k}λ^{−1/k} + a_{−2/k}λ^{−2/k} + …]

If there are multiple stationary points, then there is a contribution from each term.

Principle of stationary phase: if the only stationary point of ϕ(x) is at x_{0}, then the size of ∫ e^{iλϕ(x)}ψ(x) dx is roughly equal to the length of the longest interval around x_{0} on which e^{iλϕ(x)} does not oscillate, and the phase of ∫ e^{iλϕ(x)}ψ(x) dx is e^{iλϕ(x_{0})}.

Example: the Bessel function of order m

J_{m}(λ) = \frac{1}{2π} ∫_{0}^{2π} e^{iλ sin θ}e^{−imθ} dθ

has an expansion

J_{m}(λ) ∼ e^{iλ}[a_{−1/2}λ^{−1/2} + …] + e^{−iλ}[b_{−1/2}λ^{−1/2} + …]

where

a_{−1/2} = \sqrt{\frac{1}{2π}} e^{− \frac{2m+1}{4} πi}, b_{−1/2} = \sqrt{\frac{1}{2π}} e^{\frac{2m+1}{4} πi}.