Parseval's Identity
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematical analysis, Parseval''s identity is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is the Pythagorean theorem for inner-product spaces. Informally, the identity asserts that the sum of the squares of the Fourier coefficients of a function is equal to the square integral of the function. This is directly analogous to the Pythagorean theorem, which asserts that the sum of the squares of the components of a…mehr

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematical analysis, Parseval''s identity is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is the Pythagorean theorem for inner-product spaces. Informally, the identity asserts that the sum of the squares of the Fourier coefficients of a function is equal to the square integral of the function. This is directly analogous to the Pythagorean theorem, which asserts that the sum of the squares of the components of a vector in an orthonormal basis is equal to the squared length of the vector. One can recover the Fourier series version of Parseval''s identity by letting H be the Hilbert space L2[ , ], and setting en = e inx for n Z. More generally, Parseval''s identity holds in any inner-product space, not just separable Hilbert spaces. Thus suppose that H is an inner-product space. Let B be an orthonormal basis of H; i.e., an orthonormal set which is total in the sense that the linear span of B is dense in H.