Walsh series and transforms. Theory and applications.
(Ryady i preobrazovaniya Uolsha. Teoriya i primeneniya.)

*(Russian)*Zbl 0692.42009
Moskva: Nauka. 344 p. R. 3.30 (1987).

Considerable part of the book is devoted to systematical study of Walsh system. Here is the list of main topics: Lebesgue constants of Walsh system; absolutely and uniformly convergent Walsh-Fourier series; uniqueness theorems; summability of Walsh-Fourier series by Cesàro and other regular methods of summation; Walsh-Fourier series in \(L_ p\) \((1\leq p<\infty)\); Walsh-Fourier series with coefficients tending monotonically to zero; Rademacher and other lacunary subsystems of Walsh system; central limit theorem for lacunary subsystems; existence of \(L^ 1\)-functions with everywhere divergent Walsh-Fourier series; almost everywhere convergence of Walsh-Fourier series of \(L^ 2\)-functions; best approximation of functions by Walsh and Haar polynomials.

The rest of the book is devoted to multiplicative systems, which are generalizations of Walsh system, and to multiplicative transforms. Here are their definitions. Let \(P=(p_ 1,p_ 2,...,p_ j,...)\); \(p_ j\) are integers, \(p_ j\geq 2\). Let \(m_ 0=1\); \(m_ j=\prod^{j}_{s=1}p_ s\). For every natural number n we have a unique representation \(n=\sum^{k}_{j=1}a_ jm_{j-1}\), where \(a_ j\) are integers, \(0\leq a_ j\leq p_ j-1\). For every \(x\in [0,1)\) we have a representation \(x=\sum^{\infty}_{j=1}(x_ j/m_ j),\) where \(x_ j\) are integers, \(0\leq x_ j\leq p_ j-1\). Such representation is unique if we exclude representations in which all \(x_ j\) except a finite number of them are equal \(p_ j-1\). Multiplicative system corresponding to P is the system \(\chi_ n(x)=\exp (2\pi i\sum^{k}_{j=1}(a_ jx_ j/p_ j)).\) Let \({\mathcal P}=(...,p_{-j},...,p_{-1},p_ 1,...,p_ j,...),\) where \(p_ j\) are integers, \(p_ j\geq 2\). For every \(x,y\in [0,\infty)\) we can find representations \(x=\sum^{k}_{j=1}x_{-j}m_{- j+1}+\sum^{\infty}_{j=1}(x_ j/m_ j);\) \(y=\sum^{m}_{j=1}y_{- j}m_{j-1}+\sum^{\infty}_{j=1}(y_ j/m_{-j}),\) where \(x_ j\), \(y_ j\) are integers; \(0\leq x_ j\leq p_ j-1\); \(0\leq y_ j\leq p_{- j}-1\). These representations are unique if we make above mentioned exclusion. The multiplicative transform corresponding to \({\mathcal P}\) is defined in the following way. Let \(f\in L^ 1[0,\infty)\). Then \[ (1)\quad \hat f(y)=\int^{\infty}_{0}f(x)\exp (2\pi i(\sum^{m}_{j=1}(x_ jy_{-j}/p_ j)+\sum^{k}_{j=1}(x_{- j}y_ j/p_{-j})))dx. \] The book contains results on representation of functions using their multiplicative transforms. An interest to multiplicative transforms and their active use in applications are based upon the properties of their discretizations, i.e. transforms which is obtained if the integral in (1) is replaced by certain integral sums.

The book contains descriptions of the discretizations of the multiplicative transforms and investigation of the obtained discrete transforms. Here we find also descriptions of several applications of multiplicative systems and transforms.

The rest of the book is devoted to multiplicative systems, which are generalizations of Walsh system, and to multiplicative transforms. Here are their definitions. Let \(P=(p_ 1,p_ 2,...,p_ j,...)\); \(p_ j\) are integers, \(p_ j\geq 2\). Let \(m_ 0=1\); \(m_ j=\prod^{j}_{s=1}p_ s\). For every natural number n we have a unique representation \(n=\sum^{k}_{j=1}a_ jm_{j-1}\), where \(a_ j\) are integers, \(0\leq a_ j\leq p_ j-1\). For every \(x\in [0,1)\) we have a representation \(x=\sum^{\infty}_{j=1}(x_ j/m_ j),\) where \(x_ j\) are integers, \(0\leq x_ j\leq p_ j-1\). Such representation is unique if we exclude representations in which all \(x_ j\) except a finite number of them are equal \(p_ j-1\). Multiplicative system corresponding to P is the system \(\chi_ n(x)=\exp (2\pi i\sum^{k}_{j=1}(a_ jx_ j/p_ j)).\) Let \({\mathcal P}=(...,p_{-j},...,p_{-1},p_ 1,...,p_ j,...),\) where \(p_ j\) are integers, \(p_ j\geq 2\). For every \(x,y\in [0,\infty)\) we can find representations \(x=\sum^{k}_{j=1}x_{-j}m_{- j+1}+\sum^{\infty}_{j=1}(x_ j/m_ j);\) \(y=\sum^{m}_{j=1}y_{- j}m_{j-1}+\sum^{\infty}_{j=1}(y_ j/m_{-j}),\) where \(x_ j\), \(y_ j\) are integers; \(0\leq x_ j\leq p_ j-1\); \(0\leq y_ j\leq p_{- j}-1\). These representations are unique if we make above mentioned exclusion. The multiplicative transform corresponding to \({\mathcal P}\) is defined in the following way. Let \(f\in L^ 1[0,\infty)\). Then \[ (1)\quad \hat f(y)=\int^{\infty}_{0}f(x)\exp (2\pi i(\sum^{m}_{j=1}(x_ jy_{-j}/p_ j)+\sum^{k}_{j=1}(x_{- j}y_ j/p_{-j})))dx. \] The book contains results on representation of functions using their multiplicative transforms. An interest to multiplicative transforms and their active use in applications are based upon the properties of their discretizations, i.e. transforms which is obtained if the integral in (1) is replaced by certain integral sums.

The book contains descriptions of the discretizations of the multiplicative transforms and investigation of the obtained discrete transforms. Here we find also descriptions of several applications of multiplicative systems and transforms.

Reviewer: M.Ostrovskij

##### MSC:

42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |

42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |

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

40G05 | Cesàro, Euler, Nörlund and Hausdorff methods |