Let α, β ∈ R and s ∈ N be given. Let δx denote the Dirac measure at x ∈ R, and let ∗ denote convolution. If µ is a measure, µ ⋆ is the measure that assigns to each Borel set A the value µ(−A). If u ∈ R, we put µα,β,u = e iu(α−β)/2 δ0 − e iu(α+β)/2 δu. hen we call a function g ∈ L 2 (R) a generalized (α, β)-diòerence of order 2s if for some u ∈ R and h ∈ L 2 (R) we have g = [µα,β,u + µ ⋆ α,β,u ] s ∗ h. We denote by Dα,β,s(R) the vector space of all functions f in L 2 (R) such that f is a ûnite sum of generalized (α, β)-diòerences of order 2s. It is shown that every function in Dα,β,s(R) is a sum of 4s + 1 generalized (α, β)-diòerences of order 2s. Letting ̂f denote the Fourier transform of a function f ∈ L 2 (R), it is shown that f ∈ Dα,β,s(R) if and only if ̂f “vanishes” near α and β at a rate comparable with (x − α) 2s (x − β) 2s . In fact, Dα,β,s(R) is a Hilbert space where the inner product of functions f and g is ∫ ∞ −∞(1 + (x − α) −2s (x − β) −2s )̂f (x)̂g(x) dx. Letting D denote diòerentiation, and letting I denote the identity operator, the operator (D 2 − i(α + β)D − αβI) s is bounded with multiplier (−1) s (x−α) s (x−β) s , and the Sobolev subspace of L 2 (R) of order 2s can be given a norm equivalent to the usual one so that (D 2 −i(α+β)D−αβI) s becomes an isometry onto the Hilbert space Dα,β,s(R). So a space Dα,β,s(R) may be regarded as a type of Sobolev space having a negative index.

Let α, β ∈ R and s ∈ N be given. Let δx denote the Dirac measure at x ∈ R, and let ∗ denote convolution. If µ is a measure, µ ⋆ is the measure that assigns to each Borel set A the value µ(−A). If u ∈ R, we put µα,β,u = e iu(α−β)/2 δ0 − e iu(α+β)/2 δu. hen we call a function g ∈ L 2 (R) a generalized (α, β)-diòerence of order 2s if for some u ∈ R and h ∈ L 2 (R) we have g = [µα,β,u + µ ⋆ α,β,u ] s ∗ h. We denote by Dα,β,s(R) the vector space of all functions f in L 2 (R) such that f is a ûnite sum of generalized (α, β)-diòerences of order 2s. It is shown that every function in Dα,β,s(R) is a sum of 4s + 1 generalized (α, β)-diòerences of order 2s. Letting ̂f denote the Fourier transform of a function f ∈ L 2 (R), it is shown that f ∈ Dα,β,s(R) if and only if ̂f “vanishes” near α and β at a rate comparable with (x − α) 2s (x − β) 2s . In fact, Dα,β,s(R) is a Hilbert space where the inner product of functions f and g is ∫ ∞ −∞(1 + (x − α) −2s (x − β) −2s )̂f (x)̂g(x) dx. Letting D denote diòerentiation, and letting I denote the identity operator, the operator (D 2 − i(α + β)D − αβI) s is bounded with multiplier (−1) s (x−α) s (x−β) s , and the Sobolev subspace of L 2 (R) of order 2s can be given a norm equivalent to the usual one so that (D 2 −i(α+β)D−αβI) s becomes an isometry onto the Hilbert space Dα,β,s(R). So a space Dα,β,s(R) may be regarded as a type of Sobolev space having a negative index.