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- W2028358313 abstract "shall have the smallest possible value, n being a given positive integer (or zero) and m a given number ? 1, not necessarily integral.t The purpose of the present note is to inquire what becomes of the problem for values of m less than 1. For m < 0, it is clear that the value of (1) can be brought arbitrarily near to zero by taking a sufficiently large constant for Tmn (x), but can not be made equal to zero, so that the problem has no significance. For mn 0, the integral (1) is always equal to 2 n, unless there is a trigonometric sum which is identically equal to f(x) throughout a point set of measure different from zero, in which case the integral is not defined. Attention may therefore be restricted to values of m between 0 and 1. 2. Existence of an approximating function. Let m be a number of the interval 0 cm < 1. When m and n are given, the value of the integral (1) has a lower limit rmn, which is positive or zero. It is to be inquired first whether there exists a trigonometric sum Tmn (x) for which the lower limit Ymtn is actually attained. The proof of existence given in the paper A for m ' 1, and in B for in = 1, does not apply when m < 1. It is possible nevertheless" @default.
- W2028358313 created "2016-06-24" @default.
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- W2028358313 date "1923-03-01" @default.
- W2028358313 modified "2023-09-26" @default.
- W2028358313 title "Note on an ambiguous case of approximation" @default.
- W2028358313 doi "https://doi.org/10.1090/s0002-9947-1923-1501249-3" @default.
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