CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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L-p convergence of a lacunary series is equivalent to ell-two coefficients

Statement

For a $q$ -Hadamard-lacunary sequence and coefficients $(a_{j})$ , the partial sums of $\sum_{j \geq 1} a_{j} e_{λ_{j}}$ converge in $L^{p} (T)$ if and only if $(a_{j}) \in ℓ^{2}$ , for every $1 \leq p < \infty$ . For $0 < p < 1$ , they converge if and only if $(a_{j}) \in ℓ^{2}$ in the complete metric $d_{p}$ of The $L^{p}$ distance for $0 < p < 1$ is a complete translation-invariant metric. Completeness above one is supplied by Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , and the finite estimate is L-p norm equivalence for finite Hadamard-lacunary sums.

Facts & Assumptions

Given: $p, q, (λ_{j})$ and $(a_{j})$ as in the Statement.

Proof

technique · apply the finite estimate to tails and use completeness

1.1

If $(a_{j}) \in ℓ^{2}$ , its coefficient tails tend to zero. The finite [given, algebra] estimate applied to differences of partial sums therefore makes them Cauchy in $L^{p}$ for $p \geq 1$ , and Cauchy in $d_{p}$ for $0 < p < 1$ (raise the displayed finite estimate to the power $p$ ).

givenalgebra

2.1

The relevant cited completeness theorem gives a limit in the respective [step 1.1] space, so the partial sums converge.

step 1.1

3.1

Conversely, convergence makes the partial sums Cauchy. The lower finite [step 1.1, step 2.1, algebra] estimate applied to every difference of two partial sums forces the corresponding coefficient tail to tend to zero in $ℓ^{2}$ . Thus $(a_{j}) \in ℓ^{2}$ , proving both implications.

step 1.1step 2.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Loukas Grafakos, Classical Fourier Analysis, 3rd ed., Theorem 3.6.4 (standard reference, not scraped)