- #1

SVD

- 6

- 0

Let ∫ |f(t)|^2dt for 0≤t<∞ equals F.

Show that lim(σ→∞) ∫(1-x/σ)|f(x)|^2 dx for0≤x≤σ converges to F.

I know that it needs to prove that lim(σ→∞) ∫(x/σ)|f(x)|^2 dx for0≤x≤σ converges to 0. Can anyone give me a hint??

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- Thread starter SVD
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- #1

SVD

- 6

- 0

Let ∫ |f(t)|^2dt for 0≤t<∞ equals F.

Show that lim(σ→∞) ∫(1-x/σ)|f(x)|^2 dx for0≤x≤σ converges to F.

I know that it needs to prove that lim(σ→∞) ∫(x/σ)|f(x)|^2 dx for0≤x≤σ converges to 0. Can anyone give me a hint??

- #2

mathman

Science Advisor

- 8,084

- 550

For σ > T, split the integral of (x/σ)|f|² into 2 parts at T.

The integral from 0 to σ of (x/σ)|f|² < integral of (x/σ)|f|² from 0 to T + the integral of |f|² from T to ∞.

The first term -> 0 as σ -> ∞, while the second term < ε. Since ε is arbitrarily small, the integral -> 0.

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