Summary Five interesting theorems of Konyushkov giving estimations for the best approximation in terms of the coefficients of a Fourier series are generalized or extended to the cases when the monotone or quasi-monotone coefficients are replaced by sequences of rest bounded variation of coefficients.
Recently we extended some interesting theorems of Konyushkov giving estimations for the best approximation by the coefficients
of the Fourier series of the function in question. We replaced the monotone or quasi-monotone coefficient sequences by coefficient
sequences of rest bounded variation. In this note both notions are generalized for such coefficient sequences where certain
restriction is given only in terms of the "rest variation" of the sequence.
Among others a general equivalence theorem on Fourier cosine series with monotone coefficients are generalized to coefficients
of rest bounded variation. Similarly some theorems of Aljančić are also extended, namely one of them in this generalized form
is required to the proof of the equivalence theorem.
Our aim is to find the source why the logarithm sequences play the crucial role in the L1-convergence of sine series. We define three new classes of sequences; one of them has the character of the logarithm sequences,
the other two are the extensions of the class defined by Zhou and named Logarithm Rest Bounded Variation Sequences. In terms
of these classes, extended analogues of Zhou’s theorems are proved.
Summary Utilizing the good properties of the sequences of rest bounded variation, the usual monotonicity hypothesis on the coefficients of Fourier cosine series given in previous theorems will be weakened in the sense that the sequence of coefficients is of rest bounded variation. The theorems in question reformulate the conditions in some theorems on embedding relations of Besov classes.