Answer to Question #97505 in Calculus for Rachel

Question #97505
Let f(x) = x^a, xER^n, where a is a multi-index. Compute f-hat.
1
Expert's answer
2019-10-28T18:40:18-0400

f^(ω)=1(2π)n2Rnf(x)ei(x,ω)dx\hat{f}(\omega)=\frac{1}{(2\pi)^{\frac{n}{2}}}\int\limits_{\mathbb R^n}f(x)e^{-i(x,\omega)}dx

f^(ω)=1(2π)n2Rnxαei(x,ω)dx=1(2π)n2Rnk=1nxkαkeixkωkdx=\hat{f}(\omega)=\frac{1}{(2\pi)^{\frac{n}{2}}}\int\limits_{\mathbb R^n}x^{\alpha}e^{-i(x,\omega)}dx=\frac{1}{(2\pi)^{\frac{n}{2}}}\int\limits_{\mathbb R^n}\prod\limits_{k=1}^n x_k^{\alpha_k}e^{-ix_k\omega_k}dx=

=k=1n(12πRxkαkeixkωkdxk)=\prod\limits_{k=1}^n\bigl(\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb R}x_k^{\alpha_k}e^{-ix_k\omega_k}dx_k\bigr)

Consider 12πRxαeixωdx\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb R}x^{\alpha}e^{-ix\omega}dx. Prove by induction that 12πRxαeixωdx=iα2πδ(α)(ω)\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb R}x^{\alpha}e^{-ix\omega}dx=i^{\alpha}\sqrt{2\pi}\delta^{(\alpha)}(\omega).

Since δ(ω)=δ(ω)\delta(\omega)=\overline{\delta(\omega)} , for α=0\alpha=0 we have 12πReixωdx=12πReixωdx=\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb R}e^{-ix\omega}dx=\overline{\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb R}e^{ix\omega}dx}=

=2πδ(ω)=2πδ(ω)=\sqrt{2\pi}\cdot\overline{\delta(\omega)}=\sqrt{2\pi}\delta(\omega).

Suppose that 12πRxkeixωdx=ik2πδ(k)(ω)\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb R}x^ke^{-ix\omega}dx=i^k\sqrt{2\pi}\delta^{(k)}(\omega), where k0k\ge 0.

Then ik+12πδ(k+1)(ω)=iddω(ik2πδ(k)(ω))i^{k+1}\sqrt{2\pi}\delta^{(k+1)}(\omega)=i\frac{d}{d\omega}\bigl(i^k\sqrt{2\pi}\delta^{(k)}(\omega)\bigr). By the induction hypothesis we obtain ik+12πδ(k+1)(ω)=iddω(12πRxkeixωdx))=i^{k+1}\sqrt{2\pi}\delta^{(k+1)}(\omega)=i\frac{d}{d\omega}\bigl(\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb R}x^ke^{-ix\omega}dx)\bigr)=

=12πRixkdeixωdωdx=12πRxk+1eixωdx=\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb R}ix^k\frac{de^{-ix\omega}}{d\omega}dx=\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb R}x^{k+1}e^{-ix\omega}dx

So we prove this statement for α=k+1\alpha=k+1.

By the principle of mathematical induction we obtain that 12πRxαeixωdx=iα2πδ(α)(ω)\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb R}x^{\alpha}e^{-ix\omega}dx=i^{\alpha}\sqrt{2\pi}\delta^{(\alpha)}(\omega) for all α0\alpha\ge 0.

So f^(ω)=k=1n(12πRxkαkeixkωkdxk)=\hat{f}(\omega)=\prod\limits_{k=1}^n\bigl(\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb R}x_k^{\alpha_k}e^{-ix_k\omega_k}dx_k\bigr)=

=k=1n(iαk2πδ(αk)(ωk))=iα(2π)n2k=1nδ(αk)(ωk)=\prod\limits_{k=1}^n\bigl(i^{\alpha_k}\sqrt{2\pi}\delta^{(\alpha_k)}(\omega_k)\bigr)=i^{|\alpha|}(2\pi)^\frac{n}{2}\prod\limits_{k=1}^n\delta^{(\alpha_k)}(\omega_k)

Answer: iα(2π)n2k=1nδ(αk)(ωk)i^{|\alpha|}(2\pi)^\frac{n}{2}\prod\limits_{k=1}^n\delta^{(\alpha_k)}(\omega_k)



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