DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Hadamard-lacunary sequences and lacunary trigonometric series

Definition

Use the circle, normalized measure, and characters $e_{k} (x) = e^{2 π i k x}$ from Period-one Fourier coefficients, partial sums, and convolution on the torus. A strictly increasing sequence $(λ_{j})_{j \geq 1}$ of positive integers is Hadamard-lacunary with ratio $q$ if one fixed $q > 1$ satisfies

$λ_{j + 1} \geq q λ_{j} (j \geq 1) .$

A finite lacunary trigonometric sum is $\sum_{j \in J} a_{j} e_{λ_{j}}$ for a finite $J \subset N_{> 0}$ ; a lacunary trigonometric series is the corresponding formal series over $j \geq 1$ . The ratio condition, rather than merely $λ_{j + 1} - λ_{j} \to \infty$ , is part of this definition.

Depends on

Period-one Fourier coefficients, partial sums, and convolution on the torus

Used by

Dependency tree · one level

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Sources

Loukas Grafakos, Classical Fourier Analysis, 3rd ed., Definition 3.6.1 (standard reference, not scraped)