Mapping of $L^2 -$norm of Two Multiplied $2\pi-$Periodic Functions to Their Fourier Coefficients (Part II)

Abstract

This work derives an identity that maps between the $2$-norm of two multiplied $2\pi$-periodic functions in $L^2$ space (i.e., $||f.g||^2_{L^2 (-\pi,\pi)}$) to the individual Fourier coefficients of $f$ and $g$. Alternately, it maps between the $2$-norm of two multiplied discrete-time Fourier transforms (i.e., $||\mathscr{F}\{f\}.\mathscr{F}\{g\}||^2_{L^2 (-\pi,\pi)}$) to the discrete-time samples of $f$ and $g$. The results are equality to Cauchy–Schwarz inequality, and extend the results of our previous paper that map between $||f||^4_{L^4 (-\pi,\pi)}$ to the Fourier coefficients of $f$, alternately $||\mathscr{F}\{f\}||^4_{L^4 (-\pi,\pi)}$ to the discrete-time samples of $f$.

Keywords

• $L^2$-norm of two multiplied exponential Fourier series
• $L^4$-norm of exponential Fourier series
• $L^4$-norm of a DFT/IDFT sequence
• Equality of Cauchy–Schwarz inequality

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