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On the uniqueness of multiple trigonometric series Cross, George Elliot

Abstract

The multiple trigonometric series ∑ c(m) exp(i(m,x)), where m = (m₁, ... ,m(n)), x = (x₁, …x(n)), (m,x) = m₁^x₁+ ...+ m(n) x(n), and m(j) ≥ 0, is said to be summable (T,k) if the series [symbol omitted] C(p) (x) is summable (C,k) where C(p)(x) denotes the triangular sum ∑c(m)exp(i(m,x)), m₁ + ... + m(n) = p. The series is said to be bounded (T,k) if the series [symbol omitted] C(p)(x) is bounded (C,k). Using the fact that the triangular summation of the multiple trigonometric series considered is equivalent to the Cesàro summation of a single series of a particular form, this thesis obtains uniqueness theorems for multiple trigonometric series by first proving the required theorems for the single series. It is shown that if the series [symbol omitted] c(n) exp(int) is summable (C,k) then the coefficients c(n) are given in terms of the [formula omitted] - integral defined by James [Trans. Amer. Math. Soc. vol. 76 (1954) pp.149-176, section 8]. When the series is bounded (C,k) a Fourier representation is obtained in terms of Burkill's C(k+1)P - integral [Proc. London Math. Soc. (2) vol. 39 (1935) pp. 541-552]. It is shown that if f(t) is periodic and C(r)P - integrable, then the C(r)P - (definite) integral is a constant multiple of the [formula omitted] - (definite) integral of f(x). This gives a Fourier representation of the coefficients in terms of the [formula omitted] - integral when the series is bounded (C,k). These results are then extended to multiple trigonometric series. A representation for the coefficients in terms of the C(k+1)P - integral is demonstrated if the series is bounded (T,k). Finally a uniqueness theorem is proved where the summability set is a countable set of n - tuples of the form (x₁₀,x₂₀, …,x(n)-₁₀,x(ni), for fixed x₁₀x₂₀,…,x(n)₋₁₀, and I = 1,2,3,… .