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Finite lacunary Fourier sums have their coefficient ell-two norm

Statement

If $J \subset N_{> 0}$ is finite and $f = \sum_{j \in J} a_{j} e_{λ_{j}}$ is a lacunary sum in the sense of Hadamard-lacunary sequences and lacunary trigonometric series, then

$∥ f ∥_{L^{2} (T)}^{2} = \sum_{j \in J} ∣ a_{j} ∣^{2} .$

Facts & Assumptions

Given: A finite set $J$ and the displayed lacunary sum $f$ .

Proof

technique · direct character integration

1.1givenalgebra

Expanding $∣ f ∣^{2}$ gives [given, algebra] $∣ f (x) ∣^{2} = \sum_{j, k \in J} a_{j} \overline{a_{k}} e_{λ_{j} - λ_{k}} (x) .$ The frequencies are distinct because the defining sequence is strictly increasing.

2.1step 1.1algebra∎

For an integer $n$ , direct integration gives [step 1.1, algebra] $\int_{0}^{1} e_{n} (x) d x = 1$ when $n = 0$ and $0$ otherwise. Thus integration of step 1.1 retains precisely the $j = k$ terms and gives $\int_{0}^{1} ∣ f (x) ∣^{2} d x = \sum_{j \in J} ∣ a_{j} ∣^{2} .$

Depends on

Hadamard-lacunary sequences and lacunary trigonometric series

Used by

L-p norm equivalence for finite Hadamard-lacunary sums Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Loukas Grafakos, Classical Fourier Analysis, 3rd ed., §3.6.2 (standard reference, not scraped)