Parseval formula

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1: 1.14 Integral Transforms

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Parseval’s Formula

… ►(1.14.7_5) and (1.14.8) are Parseval’s formulas. … ►

Parseval’s Formula

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Parseval-type Formulas

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2: 1.8 Fourier Series

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Parseval’s Formula

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3: 2.5 Mellin Transform Methods

… ►When $x=1$, this identity is a Parseval-type formula; compare §1.14(iv). … ►Now suppose that there is a real number $p_{jk}$ in $D_{jk}$ such that the Parseval formula (2.5.5) applies and … ►Since $ℳ{\mathrm{e}}^{-t}\left(z\right)=\mathrm{\Gamma }\left(z\right)$, by the Parseval formula (2.5.5), there are real numbers $p_1$ and $p_2$ such that , , and …

4: Errata

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The following additions were made in Chapter 1:

• New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

• The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).

• In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).

• In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

• New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

• New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

• Another form of Parseval’s formula, (1.14.7_5).

• We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

• Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

• An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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5: 1.3 Determinants, Linear Operators, and Spectral Expansions

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Krattenthaler’s Formula

… ►which is Parseval’s equality.