Tables of Fourier transforms and Fourier transforms of distributions. Transl. from the German.
Rev. and enlarged ed.

*(English)*Zbl 0694.42017
Berlin etc.: Springer-Verlag. viii, 259 p. DM 64.00 (1990).

These tables represent a new, revised and enlarged version of the previously published book by this author, entitled “Tabellen zur Fourier-Transformation” (1957; Zbl 0077.120). Known errors have been corrected, apart from the addition of a considerable number of new results, which involve almost exclusively higher functions. Again, the following tables contain a collection of integrals of the form \((A)\quad g_ c(y)=\int^{\infty}_{0}f(x)\cos (xy)dx\) Fourier Cosine Transform, \((B)\quad g_ s(y)=\int^{\infty}_{0}f(x)\sin (xy)dx\) Fourier Sine Transform, \((C)\quad g_ e(y)=\int^{\infty}_{- \infty}f(x)e^{ixy}dx\) Exponential Fourier Transform.

Clearly, (A) and (B) are special cases of (C) if f(x) is respectively an even or an odd function. The transform parameter y in (A) and (B) is assumed to be positive, while in (C) negative values are also included. A possible analytic continuation to complex parameters \(y^*\) should present no difficulties. In some cases the result function g(y) is given over a partial range of y only. This means that g(y) for the remaining part of y cannot be given in a reasonably simple form. Under certain conditions the following inversion formulas for (A), (B), (C) hold: \((A')\quad f(x)=\frac{2}{\pi}\int^{\infty}_{0}g_ c(y)\cos (xy)dy,\) \((B')\quad f(x)=\frac{2}{\pi}\int^{\infty}_{0}g_ s(y)\sin (xy)dy,\) \((C')\quad f(x)=(2\pi)^{-1}\int^{\infty}_{-\infty}g_ e(y)e^{- ixy}dy.\)

In the following parts I, II, III tables for the transforms (A), (B) and (C) are given. The parts I and II are subdivided into 23 sections each involving the same class of functions. The first and the second column (in parenthesis) refers to the location of the correspondent page number for the cosine- and sine-transform respectively.

Compared with the before-mentioned previous edition, a new part IV titled “Fourier Transforms of Distributions” has been added. In this, those functions f(x) occurring in the parts I-III have been singled out which represent so-called probability density (or frequency distribution) functions. The corresponding normalization factors are likewise listed.

Clearly, (A) and (B) are special cases of (C) if f(x) is respectively an even or an odd function. The transform parameter y in (A) and (B) is assumed to be positive, while in (C) negative values are also included. A possible analytic continuation to complex parameters \(y^*\) should present no difficulties. In some cases the result function g(y) is given over a partial range of y only. This means that g(y) for the remaining part of y cannot be given in a reasonably simple form. Under certain conditions the following inversion formulas for (A), (B), (C) hold: \((A')\quad f(x)=\frac{2}{\pi}\int^{\infty}_{0}g_ c(y)\cos (xy)dy,\) \((B')\quad f(x)=\frac{2}{\pi}\int^{\infty}_{0}g_ s(y)\sin (xy)dy,\) \((C')\quad f(x)=(2\pi)^{-1}\int^{\infty}_{-\infty}g_ e(y)e^{- ixy}dy.\)

In the following parts I, II, III tables for the transforms (A), (B) and (C) are given. The parts I and II are subdivided into 23 sections each involving the same class of functions. The first and the second column (in parenthesis) refers to the location of the correspondent page number for the cosine- and sine-transform respectively.

Compared with the before-mentioned previous edition, a new part IV titled “Fourier Transforms of Distributions” has been added. In this, those functions f(x) occurring in the parts I-III have been singled out which represent so-called probability density (or frequency distribution) functions. The corresponding normalization factors are likewise listed.

##### MSC:

42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |

42-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces |