 Finite-interval Parseval identities

ID: finite-interval-parseval-identities

F (t) = \sum_{k = 1}^{N} b_{k} e (k t)

\int_{0}^{1} ∣ F ∣^{2} = \sum_{k = 1}^{N} ∣ b_{k} ∣^{2}, \int_{0}^{1} ∣ F' ∣^{2} = 4 π^{2} \sum_{k = 1}^{N} k^{2} ∣ b_{k} ∣^{2} \leq 4 π^{2} N^{2} \sum_{k = 1}^{N} ∣ b_{k} ∣^{2} .

Indeed, expand each square and use

\int_{0}^{1} e (j t) d t = 0

for every nonzero integer

j

, and one for

j = 0

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