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 On the uniqueness of multiple trigonometric series
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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.149176, 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. 541552]. 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,… .
Item Metadata
Title 
On the uniqueness of multiple trigonometric series

Creator  
Publisher 
University of British Columbia

Date Issued 
1958

Description 
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.149176, 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. 541552].
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,… .

Subject  
Genre  
Type  
Language 
eng

Date Available 
20120205

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080652

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.