The projection π_θ f : ℝ → ℂ of an L² function f : ℝ^d → ℂ along the
direction θ ∈ S^{d−1}: at the point t ∈ ℝ it returns the integral of f over
the affine hyperplane t·θ + θ^⊥, with respect to the Haar measure on θ^⊥.
Instances For
Sobolev embedding (one dimension). If s > 1/2 + k, then the Sobolev
H^s norm controls the C^k norm: there is a constant C > 0 such that
‖g‖_{C^k}² ≤ C · ‖g‖_{H^s}² for every g : ℝ → ℂ.
Plancherel's identity on ℝ^d. For f ∈ L²(ℝ^d),
∫ |f|² = ∫ |𝓕 f|².
Polar coordinate decomposition of ‖𝓕 f‖_{L²}². There exists C_d > 0
(the polar Jacobian constant) such that
∫ |𝓕 f|² dξ = C_d ∫_{S^{d−1}} ∫_ℝ |r|^{d−1} |𝓕 f(r θ)|² dr dσ(θ).
Fubini step used in the Fourier slice theorem: rewriting the Fourier integral
of f against the character at r θ as a one-dimensional Fourier integral of
the θ-slice integral of f.
Fourier slice theorem. For f ∈ L²(ℝ^d) and θ ∈ S^{d−1}, the 1-D Fourier
transform of the projection π_θ f at r equals the d-dimensional Fourier
transform of f evaluated at r·θ:
𝓕(π_θ f)(r) = 𝓕f(r θ).
Combining polar coordinates with the Fourier slice theorem: the L² norm
of 𝓕 f equals (up to a dimensional constant) the average over θ ∈ S^{d−1} of
the homogeneous Sobolev Ḣ^{(d−1)/2} squared norm of the projection π_θ f.
Bounded version: combining Plancherel on ℝ^d with the polar/Fourier slice
identity, the spherical average of the homogeneous Sobolev Ḣ^{(d−1)/2} norms of
the projections is bounded by a constant times ‖f‖_{L²}².
1-D Plancherel for the projection: ‖𝓕(π_θ f)‖_{L²(ℝ)}² = ‖π_θ f‖_{L²(ℝ)}².
Fubini for the projection-norm integrand: the slice integral of ‖f‖ is
integrable in t and its integral over ℝ equals ‖f‖_{L¹(ℝ^d)}.
L¹-bound on the projection: ‖π_θ f‖_{L¹(ℝ)} ≤ ‖f‖_{L¹(ℝ^d)}.
For a nonnegative function f, any slice integral
∫_{θ^⊥} f(tθ + z) dz is bounded above by the full integral ∫_{ℝ^d} f.
Pointwise L^∞-bound on the projection by the L¹ norm of f:
|π_θ f(t)| ≤ ‖f‖_{L¹(ℝ^d)} for every t.
The norm t ↦ ‖π_θ f(t)‖ of the projection is integrable on ℝ.
L² Fourier bound: ‖𝓕(π_θ f)‖_{L²(ℝ)}² ≤ ‖f‖_{L¹(ℝ^d)}². Combines the
1-D Plancherel identity with the pointwise L^∞ bound.
Low-frequency bound: the spherical average of ∫ |𝓕(π_θ f)|² is bounded
by a constant times ‖f‖_{L¹(ℝ^d)}².
The Sobolev squared norm is nonnegative.
The map θ ↦ ∫ |𝓕(π_θ f)|² is AEStronglyMeasurable on S^{d−1}.
The map θ ↦ ∫ |𝓕(π_θ f)|² is integrable on S^{d−1}.
The map θ ↦ ‖π_θ f‖_{Ḣ^{(d−1)/2}}² is AEStronglyMeasurable on S^{d−1}.
The map θ ↦ ‖π_θ f‖_{Ḣ^{(d−1)/2}}² has finite integral on S^{d−1}.
The map θ ↦ ‖π_θ f‖_{Ḣ^{(d−1)/2}}² is integrable on S^{d−1}.
Auxiliary integrability statement used in sobolev_norm_split: integrability
of the (1+|ξ|²)^s · |ĝ|² integrand transfers to the majorant
2^{|s|}(|ĝ|² + |ξ|^{2s}|ĝ|²).
Splitting the integral of |ĝ|² + |ξ|^{2s}|ĝ|² as the sum of two integrals,
given integrability of the joint integrand.
Pointwise control of the inhomogeneous Sobolev norm by the L² and homogeneous
Sobolev parts: ‖g‖_{H^s}² ≤ 2^{|s|}(‖ĝ‖_{L²}² + ‖g‖_{Ḣ^s}²).
The map θ ↦ ‖π_θ f‖_{H^{(d−1)/2}}² is AEStronglyMeasurable on S^{d−1}.
The map θ ↦ ‖π_θ f‖_{H^{(d−1)/2}}² is integrable on S^{d−1}.
Spherical-averaged version of sobolev_norm_split: the average inhomogeneous
Sobolev norm of the projections is controlled by the sum of the averaged
L²-Fourier norm and the averaged homogeneous Sobolev norm.
Combined bound: the spherical average of ‖π_θ f‖_{H^{(d−1)/2}}² is integrable
and bounded by a constant times ‖f‖_{L²}² + ‖f‖_{L¹}².
Cauchy–Schwarz for a function supported in the unit closed ball:
‖f‖_{L¹}² ≤ vol(B_1) · ‖f‖_{L²}².
For f ∈ L²(ℝ^d) supported in the unit ball, the spherical average of
‖π_θ f‖_{H^{(d−1)/2}}² is controlled by a constant times ‖f‖_{L²}².
Theorem 6.1 (Projection smoothing). If f ∈ L²(ℝ^d) is supported in the
unit ball and (d−1)/2 > 1/2 + k, then
$$\int_{S^{d-1}} \|\pi_\theta f\|_{C^k}^2\, d\theta \lesssim \|f\|_{L^2}^2.$$
Projection along a random direction θ smooths an L² function into a C^k
function on average.