Convolution with an integrable kernel as an operator on Lᵖ #
Let k be an integrable real function on a finite-dimensional real normed space E with an
additive Haar measure μ, and let 1 ≤ p < ∞. Convolution with k acts on Lᵖ(μ; F) by the
Lᵖ-valued Bochner integral
f ↦ ∫ y, k(y) • f(· - y) dμ(y),
which is an average of translates of f. Taking the integral in Lᵖ rather than pointwise
avoids any choice of representative: translation is a strongly continuous family of linear
isometries of Lᵖ for p < ∞, so the integrand is Bochner integrable as soon as k is.
This file bundles that integral as a continuous linear operator, identifies it almost everywhere with the classical pointwise convolution, and proves the two facts on which a convolution semigroup rests:
- Young's inequality for an
L¹kernel: the operator norm is at most∫ ‖k‖, that is,‖k ⋆ f‖_p ≤ ‖k‖₁ ‖f‖_p; - associativity: convolving with
k₁ ⋆ k₂is convolving withk₂and then withk₁.
Main declarations #
TauCeti.convolutionLp: convolution with a real kernel as a continuous linear operator onLᵖ; it is zero when the kernel is not integrable.TauCeti.convolutionLp_apply: the definingLᵖ-valued Bochner integral.TauCeti.norm_convolutionLp_le: Young's inequality‖k ⋆ f‖_p ≤ ‖k‖₁ ‖f‖_p.TauCeti.convolutionLp_convolution:(k₁ ⋆ k₂) ⋆ f = k₁ ⋆ (k₂ ⋆ f).TauCeti.convolutionLp_ae_eq_convolution: the operator is represented almost everywhere by the classical pointwise convolution(k ⋆ f)(x) = ∫ y, k(y) • f(x - y) dμ(y).TauCeti.integrable_convolution_integrand_restrict: on a set of finite measure, the convolution integrand(x, y) ↦ k(y) • f(x - y)of anL¹kernel and anLᵖfunction is integrable, which justifies local Fubini arguments.
References #
- G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Section 8.2.
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Theorem 4.15.
For p < ∞, weighting the translates f(· - y) of an Lᵖ function by an integrable kernel
gives a Bochner-integrable Lᵖ-valued function of y.
Convolution with a real kernel k, as a continuous linear operator on Lᵖ for p < ∞:
for integrable k it sends f to the Lᵖ-valued Bochner integral ∫ y, k y • f(· - y) dμ
(TauCeti.convolutionLp_apply). It is defined to be zero when k is not integrable.
As with MeasureTheory.integral, completeness of F is not part of the definition: the
Bochner integral vanishes unless Lᵖ(μ; F) is complete, so the operator is the advertised
convolution precisely when F is a Banach space. Its norm is at most ∫ ‖k‖ in either case
(TauCeti.norm_convolutionLp_le).
Instances For
The defining Lᵖ-valued Bochner integral of convolution with an integrable kernel.
Convolution with a non-integrable kernel is the zero operator by convention.
Young's inequality for convolution with an integrable kernel:
‖k ⋆ f‖_p ≤ ‖k‖₁ ‖f‖_p for 1 ≤ p < ∞, as a bound on the operator norm.
Associativity of convolution on Lᵖ. For integrable kernels k₁ and k₂, convolving
an Lᵖ function with k₁ ⋆ k₂ is convolving it with k₂ and then with k₁.
On a set s of finite measure, the convolution integrand (x, y) ↦ k(y) • f(x - y) of an
integrable kernel k and an Lᵖ function f is integrable: ∫_s ‖f(x - y)‖ dx is bounded
uniformly in y, by the Lᵖ norm of f.
The pointwise representative of convolutionLp. For an integrable kernel k and
f ∈ Lᵖ, the Lᵖ class convolutionLp hp k μ f is represented almost everywhere by the
classical convolution (k ⋆ f)(x) = ∫ y, k(y) • f(x - y) dμ(y).