# What is the exact difference between Parseval's theorem and Plancherel Theorem?

This question also discusses what I am asking but the answer states that Parseval's theorem is for periodic functions as per wikipedia's notation here.

However, Parseval's relationship also holds for continuous-time aperiodic signals as follows:

$$\int_{-\infty}^{\infty}|x(t)|^2 dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}|\mathrm{X}(j\omega)|^2 d\omega$$

so then what difference remains between the two other than the $$2\pi$$ correction and how should they be interpreted differently?

## 1 Answer

In most (engineering) literature, the two names are used interchangeably, with Parseval's Theorem being used more often. WolframMathWorld agrees with the view that Plancherel's Theorem is used for non-periodic continuous functions (i.e., using the Fourier transform), whereas Parseval's Theorem is used for Fourier series (i.e., for periodic functions). However, many authors don't seem to adhere to this convention.

Another distinction I've seen used is that the preservation of energy is called Plancherel's Theorem, whereas the preservation of inner products

$$\int_{-\infty}^{\infty}x(t)y^*(t)dt=\int_{-\infty}^{\infty}X(f)Y^*(f)df$$

is called Parseval's Theorem (cf. [1]). However, the author of [1] then suggests

We shall use "Parseval's Theorem" for both.

Sometimes you also see the term Parseval-Plancherel identity (e.g., here), which doesn't help in clearing up the confusion.

In sum, more often than not Plancherel's Theorem and Parseval's Theorem are used to mean the same thing, namely preservation of energy and preservation of inner products for Fourier series as well as Fourier transforms (in the $$L_2$$ sense).

Note that this answer represents an engineering point of view, as seems appropriate for this site. But even in some of the more mathematical literature I couldn't find a consequent distinction between the two names.

[1] A.Lapidoth, A Foundation in Digital Communication, p. 72

• It's funny, I have known of Parsval's theorem for at least 45 years. And I have seen variations of it, essentially $$\int\limits_{-\infty}^{\infty} x(t) \overline{y(t)} \, \mathrm{d}t = \int\limits_{-\infty}^{\infty} X(f) \overline{Y(f)} \, \mathrm{d}f$$ the edit you just made, Matt. And discrete-time and discrete-frequency versions of it. But I have never heard of it as "Plancherel". Commented May 25, 2023 at 17:52
• $$\int\limits_{-\infty}^{\infty} \big|x(t) \big|^2 \, \mathrm{d}t = \int\limits_{-\infty}^{\infty} \big| X(f) \big|^2 \, \mathrm{d}f$$ is, of course, a special case of the more general form above. Commented May 25, 2023 at 18:00
• @robertbristow-johnson: Yes, in signal processing we see "Parseval" much more often than "Plancherel". You can find "Plancherel" in the more mathematically inclined literature. Commented May 25, 2023 at 18:04
• I mean, it's not even in my Papoulis text. Not that I can see. Commented May 25, 2023 at 18:07
• @robertbristow-johnson: As for engineering books, I've seen it in some Wavelet texts and in some Communication books (Gallager, Lapidoth). Commented May 25, 2023 at 18:11