diff --git a/Project.toml b/Project.toml
index 02a02f68..6ce50610 100644
--- a/Project.toml
+++ b/Project.toml
@@ -12,6 +12,7 @@ FillArrays = "1a297f60-69ca-5386-bcde-b61e274b549b"
 ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
 GPLikelihoods = "6031954c-0455-49d7-b3b9-3e1c99afaf40"
 IrrationalConstants = "92d709cd-6900-40b7-9082-c6be49f344b6"
+KernelFunctions = "ec8451be-7e33-11e9-00cf-bbf324bd1392"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 LogExpFunctions = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
 PDMats = "90014a1f-27ba-587c-ab20-58faa44d9150"
diff --git a/src/ApproximateGPs.jl b/src/ApproximateGPs.jl
index 029be317..41b7c116 100644
--- a/src/ApproximateGPs.jl
+++ b/src/ApproximateGPs.jl
@@ -19,6 +19,10 @@ include("LaplaceApproximationModule.jl")
 @reexport using .LaplaceApproximationModule:
     build_laplace_objective, build_laplace_objective!
 
+include("FiniteBasisModule.jl")
+@reexport using .FiniteBasisModule:
+    FFApprox, FiniteBasis, DegeneratePosterior
+
 include("deprecations.jl")
 
 include("TestUtils.jl")
diff --git a/src/FiniteBasisModule.jl b/src/FiniteBasisModule.jl
new file mode 100644
index 00000000..a1d777b5
--- /dev/null
+++ b/src/FiniteBasisModule.jl
@@ -0,0 +1,76 @@
+module FiniteBasisModule
+
+using KernelFunctions,LinearAlgebra, AbstractGPs, Random
+import AbstractGPs: AbstractGP, FiniteGP
+import Statistics
+import ChainRulesCore
+
+struct FiniteBasis <: KernelFunctions.SimpleKernel end
+
+KernelFunctions.kappa(::FiniteBasis, d::Real) = d
+KernelFunctions.metric(::FiniteBasis) = KernelFunctions.DotProduct()
+
+struct DegeneratePosterior{P,T,C} <: AbstractGP
+	prior::P
+	w_mean::T
+	w_prec::C
+end
+
+weight_form(A::KernelFunctions.ColVecs) = A.X'
+weight_form(A::KernelFunctions.RowVecs) = A.X
+
+function AbstractGPs.posterior(fx::FiniteGP{GP{M, B}}, y::AbstractVector{<:Real}) where {M, B <: FiniteBasis}
+	kern = fx.f.kernel
+	δ = y - mean(fx)
+	X = weight_form(fx.x)
+	X_prec = X' * inv(fx.Σy)
+	Λμ = X_prec * y
+	prec = cholesky(I + Symmetric(X_prec * X))
+	w = prec \ Λμ
+	DegeneratePosterior(fx.f, w, prec)
+end
+
+function Statistics.mean(f::DegeneratePosterior, x::AbstractVector)
+	w = f.w_mean
+	X = weight_form(x)
+	X * w
+end
+
+function Statistics.cov(f::DegeneratePosterior, x::AbstractVector)
+	X = weight_form(x)
+	AbstractGPs.Xt_invA_X(f.w_prec, X')
+end
+
+function Statistics.cov(f::DegeneratePosterior, x::AbstractVector, y::AbstractVector)
+	X = weight_form(x)
+	Y = weight_form(y)
+	AbstractGPs.Xt_invA_Y(X', f.w_prec, Y')
+end
+
+function Statistics.var(f::DegeneratePosterior, x::AbstractVector)
+	X = weight_form(x)
+	AbstractGPs.diag_Xt_invA_X(f.w_prec, X')
+end
+
+function Statistics.rand(rng::AbstractRNG, f::DegeneratePosterior, x::AbstractVector)
+	w = f.w_mean
+	X = weight_form(x)
+	X * (f.w_prec.U \ randn(rng, length(x)))
+end
+
+struct RandomFourierFeature
+	ws::Vector{Float64}
+end
+
+RandomFourierFeature(kern::SqExponentialKernel, k::Int) = RandomFourierFeature(randn(k))
+RandomFourierFeature(rng::AbstractRNG, kern::SqExponentialKernel, k::Int) = RandomFourierFeature(randn(rng, k))
+
+FFApprox(kern::Kernel, k::Int) = FiniteBasis() ∘ FunctionTransform(RandomFourierFeature(kern, k))
+FFApprox(rng::AbstractRNG, kern::Kernel, k::Int) = FiniteBasis() ∘ FunctionTransform(RandomFourierFeature(rng, kern, k))
+
+
+function (f::RandomFourierFeature)(x)
+	Float64[cos.(f.ws .* x); sin.(f.ws .* x)] .* sqrt(2/length(f.ws))
+end
+
+end
\ No newline at end of file
diff --git a/test/FiniteBasisModule.jl b/test/FiniteBasisModule.jl
new file mode 100644
index 00000000..2dc3e867
--- /dev/null
+++ b/test/FiniteBasisModule.jl
@@ -0,0 +1,35 @@
+@testset "finite_basis" begin
+    rng = MersenneTwister(123456)
+    N = 50
+    x = rand(rng, 2, N);
+    y = sin.(norm.(eachcol(x)))
+
+    @testset "Verify equivalence of weight space and function space posteriors" begin
+        kern = FiniteBasis()
+        x2 = ColVecs(rand(2, N))
+
+        # Predict mean and covariance using weight space view
+        f = GP(kern)
+        fx = f(x, 0.001)
+        opt_pred = mean_and_cov(posterior(fx, y)(x2))
+
+        # Predict mean and covariance as normal
+        fx2 = GP(kern + ZeroKernel())(x, 0.001)
+        pred = mean_and_cov(posterior(fx2, y)(x2))
+
+        # The two approaches should be the same
+        @test all(opt_pred .≈ pred)
+    end
+
+    @testset "Verify that the RFF approximation matches the exact posterior" begin
+        rng = MersenneTwister(12345)
+        rbf = SqExponentialKernel()
+        flat_x = rand(rng, N)
+        flat_x2 = rand(rng, N)
+        ffkern = FFApprox(rng, rbf, 200)
+
+        opt_pred = mean_and_cov(posterior(GP(ffkern)(flat_x, 0.001), y)(flat_x2))
+        pred = mean_and_cov(posterior(GP(rbf)(flat_x, 0.001), y)(flat_x2))
+        @test all(isapprox.(opt_pred, pred; atol=1e-2))
+    end
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index fa26a951..46009ddd 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -61,6 +61,10 @@ include("test_utils.jl")
         include("LaplaceApproximationModule.jl")
         println(" ")
         @info "Ran laplace tests"
+
+        include("FiniteBasisModule.jl")
+        println(" ")
+        @info "Ran finite basis tests"
     end
 
     if GROUP == "All" || GROUP == "CUDA"