The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Parseval Equality

an equation of the form

where a₀ a_n, and b_n are the Fourier coefficients for f(x). It was established in 1805 by the French mathematician M. Parseval by assuming that trigonometric series could be termwise integrated. In 1896, A. M. Liapunov showed that the equality is valid if the function is bounded on the interval (—π, π) and if the integral ∫^π_-πf(x) dx exists. It was later proved that the Parseval equality holds for any function whose square is integrable. V. A. Steklov demonstrated that the Parseval equality is valid for series in other orthogonal systems of functions.

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By the expansions (11) and (14), using Parseval equality, we obtain that

Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations with Superlinear Nonlinearities

and applying the Parseval equality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we obtain

Using the last theorem and Parseval equality, we obtain

High-order compact difference scheme for the numerical solution of time fractional heat equations

By the Parseval equality we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The [L.sup.p]-version of the generalized Bohl-Perron principle for vector equations with infinite delay

[K.sub.d] ([psi], [chi]) can be evaluated either by direct integration or using convolution formula and Parseval equality for harmonic analysis on [S.sub.d-1].

By Parseval equality for harmonic analysis on [S.sub.d-1] we obtain the following expression:

Variances of length and surface area estimates by spatial grids: preliminary study

Based on Parseval equality (4), instead of (3) it is possible to write the following:

Calculation of the braking force transitional processes for linear induction motor/Tiesiaeigio asinchroninio variklio stabdymo jegos pereinamuju procesu skaiciavimas

Since the spectrum is simple and discrete, then Parseval equality (6) yields for any sequence of values F([[lambda].sub.n]) such that

Blind sampling

Then the Fourier coefficients of the difference [f.sub.1] - f are zero and applying the Parseval equality (3.9) to the function [f.sub.1] - f we get that [f.sub.1] - f = 0, so that the sum of series (4.1) is equal to f(t).

Eigenfunction expansions for a Sturm-Liouville problem on time scales

and that f (b) = 0 in case b is left-scattered, we have the Parseval equality (see [6])

We will show that the Parseval equality for problem (1.1), (1.2) can be obtained from (3.4) by letting b [right arrow] [infinity].

Now we apply Parseval equality (3.4) to the function (assuming b > [t.sub.0])

An expansion theorem for a Sturm-Liouville operator on semi-unbounded time scales