Parseval equality

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1: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►which is Parseval’s equality.

2: 2.5 Mellin Transform Methods

… ►When $x=1$, this identity is a Parseval-type formula; compare §1.14(iv). … ►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 …

3: 1.8 Fourier Series

… ►If $f(-x)=f(x)$, then $b_n=0$ for all $n$. ►If $f(-x)=-f(x)$, then $a_n=0$ for all $n$. … ►

Parseval’s Formula

… ►If $f(x)$ and $g(x)$ are continuous, have the same period and same Fourier coefficients, then $f(x)=g(x)$ for all $x$. … ►The convergence is non-uniform, however, at points where $f(x-)\ne f(x+)$; see §6.16(i). …

4: 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

… ►These bounds are sharp, and equality holds when $p=2$. …

5: 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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The constraint $a\ne 0$ was added.

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The titles have been changed to $\mu =0$ , $\nu =0,1$ , and Addendum to §14.5(ii): $\mu =0$, $\nu =2$ , respectively, in order to be more descriptive of their contents.

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Figure 20.3.1 ${\theta }_j(\pi x,0.15)$, $0\le x\le 2$, $j=1,2,3,4$.

The locations of the tick marks denoting $1.5$ and $2$ on the $x$-axis were corrected.

Reported 2017-05-22 by Paul Abbott.

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A new Subsection Addendum to §14.5(ii) $\mu =0$, $\nu =2$ , containing the values of Legendre and Ferrers functions for degree $\nu =2$ has been added.

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