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Research in the theory of trigonometric series has been carried out
for over two centuries. The results obtained have greatly
influenced various fields of mathematics, mechanics, and physics.
Nowadays, the theory of simple trigonometric series has been
developed fully enough (we will only mention the monographs by
Zygmund [15, 16] and Bari [2]). The achievements in the theory of
multiple trigonometric series look rather modest as compared to
those in the one-dimensional case though multiple trigonometric
series seem to be a natural, interesting and promising object of
investigation. We should say, however, that the past few decades
have seen a more intensive development of the theory in this field.
To form an idea about the theory of multiple trigonometric series,
the reader can refer to the surveys by Shapiro [1], Zhizhiashvili
[16], [46], Golubov [1], D'yachenko [3]. As to monographs on this
topic, only that ofYanushauskas [1] is known to me. This book
covers several aspects of the theory of multiple trigonometric
Fourier series: the existence and properties of the conjugates and
Hilbert transforms of integrable functions; convergence (pointwise
and in the LP-norm, p > 0) of Fourier series and their
conjugates, as well as their summability by the Cesaro (C,a), a>
-1, and Abel-Poisson methods; approximating properties of Cesaro
means of Fourier series and their conjugates.
Research in the theory of trigonometric series has been carried out
for over two centuries. The results obtained have greatly
influenced various fields of mathematics, mechanics, and physics.
Nowadays, the theory of simple trigonometric series has been
developed fully enough (we will only mention the monographs by
Zygmund [15, 16] and Bari [2]). The achievements in the theory of
multiple trigonometric series look rather modest as compared to
those in the one-dimensional case though multiple trigonometric
series seem to be a natural, interesting and promising object of
investigation. We should say, however, that the past few decades
have seen a more intensive development of the theory in this field.
To form an idea about the theory of multiple trigonometric series,
the reader can refer to the surveys by Shapiro [1], Zhizhiashvili
[16], [46], Golubov [1], D'yachenko [3]. As to monographs on this
topic, only that ofYanushauskas [1] is known to me. This book
covers several aspects of the theory of multiple trigonometric
Fourier series: the existence and properties of the conjugates and
Hilbert transforms of integrable functions; convergence (pointwise
and in the LP-norm, p > 0) of Fourier series and their
conjugates, as well as their summability by the Cesaro (C,a), a>
-1, and Abel-Poisson methods; approximating properties of Cesaro
means of Fourier series and their conjugates.
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