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Polynomial singular decay with power and logarithmic source balls

Example

Assume Countable Choice. Put $I = N ∖ {0} = {1, 2, \dots}$ . On real $ℓ^{2} (I)$ , let $e_{j}$ be the coordinate vectors and define $K_{a} x = (j^{- a} x_{j})_{j \in I}$ for $a > 0$ . For $ν, q, ρ > 0$ , the power source ball $X_{ν, ρ} (K_{a}) : = {∣ K_{a} ∣^{ν} w : ∥ w ∥_{2} \leq ρ}$ has coordinates $x_{j} = j^{- a ν} w_{j}$ , while the logarithmic source ball $X_{\log, q, ρ} (K_{a}) : = {ψ_{q} (∣ K_{a} ∣) w : ∥ w ∥_{2} \leq ρ}$ , where $ψ_{q} (0) = 0$ and $ψ_{q} (t) = (1 + \log (1 / t))^{- q}$ for $0 < t \leq 1$ , has coordinates $x_{j} = (1 + a \log j)^{- q} w_{j}$ . The operator is compact and injective with $∥ K_{a} ∥ = 1$ . For the power ball the centered modulus has the bound in the conditional-modulus theorem, with equality at $δ_{j} = ρ j^{- a (ν + 1)}$ ; this does not assert equality for every $δ$ .

Example references

Spectral-index source balls

Power source ball

Conditional modulus on a power source ball

Compact Hilbert-space inverse model

Clason, Regularization of Inverse Problems, Section 4.2

Hofmann and Mathé, A Note on the Modulus of Continuity, Section 1

Facts & Assumptions

Given: Countable Choice, $I = N ∖ {0}$ , real $ℓ^{2} (I)$ , coordinate vectors $e_{j}$ for $j \in I$ , $a, ν, q, ρ > 0$ , the diagonal map $K_{a}$ , and the power/logarithmic source-ball conventions.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is the hypothesis of the compact norm-limit theorem, Hilbert adjoint, and compact square-root results used below.

[F1]

Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ and The real square-summable sequence space is complete: For any index set $I$ , $ℓ^{2} (I, R)$ has the coordinate inner product, norm squared $\sum_{j \in I} ∣ x_{j} ∣^{2}$ , and unit coordinate vectors $e_{j}$ . The real space $ℓ^{2} (N, R)$ is complete.

[F2]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs and The operator norm as the least bound and as the unit-sphere or unit-ball supremum: The inner product satisfies Cauchy–Schwarz and the operator norm is the supremum of $∥ T x ∥$ over $∥ x ∥ \leq 1$ .

[F3]

The Hilbert-space adjoint of a bounded operator, Hilbert-adjoint identities, and Self-adjoint, positive, unitary and normal operators: Every bounded operator on a Hilbert space has a unique adjoint satisfying $⟨ T x, y ⟩ = ⟨ x, T^{*} y ⟩$ ; self-adjointness is $T^{*} = T$ , and positivity is $⟨ T x, x ⟩ \geq 0$ for every $x$ .

[F4]

Bounded finite rank operators are compact and Norm limit of compact operators is compact: Bounded finite-rank maps are compact, and a norm limit of compact maps into a Banach space is compact under Countable Choice.

[F5]

Positive square root of a compact positive operator and Compact spectral filters, source conditions, and qualification: A compact self-adjoint positive operator has a unique compact positive square root, acting by $\sqrt{λ}$ on each positive eigenspace and by zero on the kernel; compact spectral multipliers act eigenvalue by eigenvalue.

[F6]

Spectral-index source balls for compact operators and Power source ball: The source ball is the image of the radius- $ρ$ vector ball under its spectral index multiplier; the power index is $t^{ν}$ , and the logarithmic candidate defines a source ball once it is bounded on the compact spectrum.

[F7]

Conditional modulus on a power source ball: The centered power-source modulus is at most $δ^{ν / (ν + 1)} ρ^{1 / (ν + 1)}$ and attains this value at the radii $ρ s_{j}^{ν + 1}$ .

[F8]

The natural logarithm as the inverse of the exponential function, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, Real powers for positive bases, with the zero-base positive-exponent convention, The real exponential function and the number $e$ by a power series, The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , Continuity and derivatives of positive-base real powers, The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents, For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ , Inverses of positives are positive, and reciprocation reverses order, and Dedekind-complete ordered real field: The real field is complete and ordered; $\log$ is strictly increasing with $\log 1 = 0$ , $\exp (0) = 1$ , and positive-base real powers are positive. For $a > 0$ , $(t^{a})^{'} = a t^{a - 1} > 0$ on $t > 0$ , so the mean value theorem makes $t \mapsto t^{a}$ increasing. For every $ε > 0$ , choose $N$ with $1 / N < ε$ by the reciprocal Archimedean property; if $j \geq N$ , reciprocal order gives $1 / j \leq 1 / N < ε$ .

Verification

1.1givenF6F8algebra

For $0 < t \leq 1$ , $1 / t \geq 1$ , so [F8] gives $\log (1 / t) \geq \log 1 = 0$ . Thus $1 + \log (1 / t) \geq 1$ ; since $q > 0$ , the positive power of this base is at least $1$ , and its reciprocal $ψ_{q} (t)$ lies in $(0, 1]$ . Together with $ψ_{q} (0) = 0$ , this verifies that the logarithmic index is bounded and is admissible under [F6].

1.2givenA1F1F2F3algebra

The bijection $U x = (x_{n + 1})_{n \in N}$ from $ℓ^{2} (I, R)$ to $ℓ^{2} (N, R)$ preserves the coordinate inner product, so [F1] transfers completeness to $ℓ^{2} (I, R)$ . For $x = (x_{j}) \in ℓ^{2} (I)$ , $∥ K_{a} x ∥_{2}^{2} = \sum_{j \in I} j^{- 2 a} ∣ x_{j} ∣^{2} \leq ∥ x ∥_{2}^{2}$ , so $K_{a}$ is a well-defined bounded linear map with $∥ K_{a} ∥ \leq 1$ ; $K_{a} e_{1} = e_{1}$ gives $∥ K_{a} ∥ \geq 1$ . For $x, y \in ℓ^{2} (I)$ , [F2] makes $\sum_{j \in I} j^{- a} x_{j} y_{j}$ absolutely convergent, and the coordinate inner product gives $⟨ K_{a} x, y ⟩ = \sum_{j \in I} j^{- a} x_{j} y_{j} = ⟨ x, K_{a} y ⟩$ . Uniqueness of the adjoint in [F3] yields $K_{a}^{*} = K_{a}$ , and $⟨ K_{a} x, x ⟩ = \sum_{j \in I} j^{- a} ∣ x_{j} ∣^{2} \geq 0$ , so $K_{a}$ is positive.

2.1givenA1step 1.2F1F2F4F8algebrachoose

For $N \geq 1$ define $(K_{a, N} x)_{j} = j^{- a} x_{j}$ for $j \leq N$ and $0$ for $j > N$ . The coordinate estimate from step 1.2 gives $∥ K_{a, N} x ∥_{2} \leq ∥ x ∥_{2}$ , so $K_{a, N}$ is bounded; since each $j^{- a} > 0$ for $j \leq N$ , its range is exactly $span {e_{1}, \dots, e_{N}}$ , so [F4] makes it compact. For every $x$ , $∥ (K_{a} - K_{a, N}) x ∥_{2}^{2} = \sum_{j > N} j^{- 2 a} ∣ x_{j} ∣^{2} \leq (N + 1)^{- 2 a} ∥ x ∥_{2}^{2}$ , with equality at $x = e_{N + 1}$ ; therefore $∥ K_{a} - K_{a, N} ∥ = (N + 1)^{- a}$ . Given $ε > 0$ , [F8] supplies a threshold $N$ with $1 / j < ε^{1 / a}$ for all $j \geq N + 1$ ; monotonicity of $t \mapsto t^{a}$ gives $(N + 1)^{- a} < ε$ . Thus $K_{a, N} \to K_{a}$ in norm, and [F4] makes $K_{a}$ compact. Each diagonal coefficient is positive, so $K_{a} x = 0$ forces every $x_{j} = 0$ and $K_{a}$ is injective.

3.1givenA1step 1.2step 2.1F1F5F6algebra

Since $K_{a}$ is compact, self-adjoint and positive by steps 1.2–2.1, $K_{a}$ is the unique positive square root of $K_{a}^{*} K_{a} = K_{a}^{2}$ ; [F5] therefore gives $∣ K_{a} ∣ = K_{a}$ . Each $e_{j}$ is an eigenvector of $∣ K_{a} ∣$ with eigenvalue $j^{- a}$ . Applying the compact spectral multiplier [F5] coordinatewise yields $(∣ K_{a} ∣^{ν} w)_{j} = j^{- a ν} w_{j}$ and $(ψ_{q} (∣ K_{a} ∣) w)_{j} = ψ_{q} (j^{- a}) w_{j} = (1 + a \log j)^{- q} w_{j}$ , proving the stated power and logarithmic coordinates.

4.1givenstep 3.1F1F7constructalgebra∎

For each $j \geq 1$ take $w^{(j)} = ρ e_{j}$ and $x^{(j)} = ∣ K_{a} ∣^{ν} w^{(j)} = ρ j^{- a ν} e_{j}$ . Then $∥ K_{a} x^{(j)} ∥_{2} = ρ j^{- a (ν + 1)} = δ_{j}$ and $∥ x^{(j)} ∥_{2} = ρ j^{- a ν} = δ_{j}^{ν / (ν + 1)} ρ^{1 / (ν + 1)}$ . This feasible source-ball element attains the universal upper bound from [F7], so equality holds at $δ_{j}$ .

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Sources

Christian Clason, Regularization of Inverse Problems, lecture notes — Section 4.2, printed pp. 36–43 (standard reference, not scraped)
Bernd Hofmann and Peter Mathé, A Note on the Modulus of Continuity for Ill-Posed Problems in Hilbert Space (standard reference, not scraped)