@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Mathematics, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Cross, George Elliot"@en ; dcterms:issued "2012-02-05T21:14:17Z"@en, "1958"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms: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.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,… ."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/40492?expand=metadata"@en ; skos:note "Wat P t t f o e r s t i g of B r i t i s h Oloiuxnbia Faculty of Graduate Studies PROGRAMME OF T H E F I N A L O R A L E X A M I N A T I O N F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y of G. E. CROSS B.A., Dalhousie University, 1952 M . A . , Dalhousie University, 1954 FRIDAY, SEPTEMBER 26, 1958, at 3:30 p.m. IN ROOM 315, BUCHANAN BUILDING C O M M I T T E E IN C H A R G E D E A N G . M . S H R U M , Chairman R. D. J A M E S B A R N E T T S A V E R Y F. M . C. G O O D S P E E D A V R U M S T R O L L W. H . SIMONS - B. C. B I N N I N G P. S. B U L L E N R. E . BURGESS External Examiner: R. L. J E F F E R Y Queen's University O N T H E UNIQUENESS O F M U L T I P L E T R I G O N O M E T I C SERIES ABSTRACT The multiple trigonometric series 2 cM exv(Hm,x)), where m = (mv ... , mn), x = (xv . .. , xn), (m,x) = mxx^ + ... + mnxn, and mi > 0 is said to be summable (T, k) if the series 2 Cp(x) is summable (C, k), where Cp(x) denotes the tri-angular sum 2 cmexp(*(m>x))> /»i + . . . + mn = p. The series is said to be bounded (T, k) if the series 2 Cp(x) is bounded (C, k). Using the fact that the triangular summation of multiple trigonometric series considered is equivalent to the Cesaro sum-mation 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 2 cnexp(int) is summable (C, k) then the coefficients c„are given in terms of the p k + 2 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 £+i-P-integral [Proc. London Math. Soc. (2), vol. 39, 1935, pp. 541-552]. It is shown that if f(t) is periodic and CrP-integrabIe, then the definite CrP-integral is a constant multiple of the definite flr+l -integral of f(x). This gives a Fourier representation of the coefficients in terms of the jo k+2 -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+i 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 «-tuples of the form ( * , . . . , xfll fl , xn . ) for fixed x.q,..., xn_x 0 and i ^rrl. G R A D U A T E S T U D I E S F I E L D O F S T U D Y : M A T H E M A T I C S Modern Algebra Rimhak Ree Integral Equations T . E . Hull Theory of Functions W. H . Simons Functional Analysis R. R. Christian O T H E R STUDIES Symbolic Logic Avrum Stroll Contemporary Philosophy Peter Remnant ON THE UNIQUENESS OF MULTIPLE TRIGONOMETRIC SERIES • by GEORGE ELLIOT CROSS M. A., Dalhousie University, 195*+ A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of MATHEMATICS We accept this thesis as conforming to the standard required from candidates for the degree of Doctor of Philosophy:: THE UNIVERSITY OF BRITISH COLUMBIA October, 1958 i i Abstract The multiple trigonometric series E c exp(i(m,x)), where m = (m-^ , ... ,mn), x = (x^, x n ) , (m,x) = mq^ x1+- ... +• m x , and m. 0, is said to be summable (T.k) i f the n n' j 7 7 series z~ C (x) is summable (C.k) where C (x) denotes the ° P P triangular sum ZI c exp(i(m,x)), m, + ... 4-m = p. The m 1 n oo series is said to be bounded (T,k) i f the series c p ^ x ^ is bounded (C,k). Using the fact that the triangular summation of the multiple trigonometric series considered is equivalent to the Cesaro summation of a single series of a particular form, this thesis obtains uniqueness theorems for multiple trigonometric series by f i r s t proving the required theorems for the single series. It is shown that i f the series c exp(int) o n is summable (C,k) then the coefficients c^ are given in terms of the ? ° k + 2 - integral defined by James [Trans. Amer. Math. Soc. vol. 76 (195^) pp. l1+9-176, section 8] . When the series is bounded (C,k) a Fourier representation is obtained in terms of Burkill's Ck+-^P - integral [Proc. London Math. Soc. (2) vol. 39 (1935) pp. 5^1-552]. It is shown that i f f ( t ) is periodic and C rP -integrable, then the C rP - (definite) integral is a constant multiple of t h e ^ r + 1 - (definite) integral of f ( x ) . This gives a Fourier representation of the coefficients in terms of t h e ^ k + 2 - integral when the series is bounded (C,k). i i l These results are then extended to multiple trigonometric series. A representation for the coefficients in terms of the C^^P - Integral is demonstrated i f 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 (x^o ' x 20 ' **• , x n - 1 0 , x n i ^ ' f o r fixed X ^ Q , X 2 Q , . . . »Xfl_]_ 0 ' ^ \" >^ 3) ••• • In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e . I t i s under-stood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without rny w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia, Vancouver #, Canada. i v Table of Contents Chapter 1 Introduction 1 Chapter 2 Notation and Definitions 3 Chapter 3 The Integrated Series . 9 Chapter k The Expression of Coefficients in Terms of the £° k + 2 - integral . 15 Chapter 5 The Expression of Coefficients in Terms of the Cfc + -^P - integral . . 21 Chapter 6 The C^ + .]P.- integral and the , ?° k + 2 - integral 23 Chapter 7 Uniqueness Theorems for Multiple Trigonometric Series . . . . . . . 25 Bibliography . * . 30 V Acknowledgment The author wishes to acknowledge with gratitude the generous assistance given by Dr. R. D. James in the preparation of this thesis. 1 1. Introduction Using a process of spherical summation introduced by Bochner [l] , Cheng has proved that i f a multiple trigonometric series (1.1) cmexp(i(m,x)) is everywhere (C,l) summable to zero by spherical means and satisfies a certain condition, then i t vanishes identically. Shapiro [l3, l1*] has extended these results to allow ex-ceptional sets of capacity zero. More generally Cheng has shown that i f the series is everywhere (C,l) summable by spherical means to a function f ( x 1 , x 2 , ... ,xn) then under certain conditions the series is the Fourier series of f. This thesis is concerned with uniqueness theorems for multiple trigonometric series of a particular form summed by a \"triangular\" method. This method reduces the summation of the multiple series to that of a single series of the form (1.2) ^ c nexp(inx) o where c„ = a - i b . n n n James [?] has shown that i f the real part of series (1.2) is summable (C,k) and an additional condition involving the imaginary part is satisfied, then the coefficients are given in modified Fourier form, where the integral involved is his P - integral. It is shown here that i f series (1.2) 2 is summable (C,k) then the coefficients have a similar 70 it + 2 Fourier representation in terms of James10 - integral. Burkill [3] has shown that, i f (1.2) is bounded (C,0) except on a countable set and i f the series obtained by integrating series (1.2) once converges everywhere, then the coefficients can be written in Fourier form using the C]_P - integral. An analogous result i s shown to be true In this thesis when (1.2) is bounded (C,k). The proof of this depends on a powerful result by Marcinkiewicz and Zygmund [8J , and on generalizations of theorems by Verblunsky and Zygmund. James has proved that i f f(x) is CrP - integrable i t is also f°T integrable and has given a representation of the P r + - 1 -(indefinite) Integral in terms of the C rP -(indefinite) integral. The two preceding results suggest the relationship between the ? ° r ^ and C rP -(definite) integrals given in the theorem of Chapter 6. Series (1.1) is said to be bounded (T,k) i f the corresponding series (1.2) Is bounded (C,k). It is shown that i f (1.1) is bounded (T,k) for a l l values (xj + t, xn+- t) then the coefficients are given by repeated integrals of dimension n+-l, the inner integral being a C^ + ^P - integral. It is then shown that i f there is a countable set of n - tuples such that series (1.1) is summable (T,k) to 0 for a l l values ( X 1 0 + t , ••• » x n-l 0 *\" * ^ n i 4 \" ^ ' 1 = 1 » 2 >3 ? . . . > then the series vanishes identically. 3 2. Notation and Definitions Consider the multiple trigonometric series (2.1) exp(i(m,x)), where c m may be complex, m - (m^, ... ,mn), (m,x) = m i x i + ••• +- mnxn, and the summation is over a l l non-negative integers m-j.. DEFINITION 2.1. The series is said to be summable (T.,k) to sum f(x) i f the series (2.2) £ l C (x) P s 0 V is summable (C,k) to f(x), where C p(x) denotes the triangular sum (2.3) Z I c mexp(i(m,x)), where the summation is over a l l non-negative integers m^ such that m^ + ... +• mQ•••= p and (x) =• (x^, ... >xn). DEFINITION 2.2 The series (2.1) is said to be bounded (T,k) at (x) i f the series (2.2) is bounded (C,k) at (x). DEFINITION 2.3. Let g(t) be a function defined in the interval [a,b] . If. for a given t Q in [a,b] , g ( t 0 + h) = c Q + c xh + c 2h 2 /2 I + ... + c ^ / k l 0 ( h k ) , where the numbers c^ = c^(t Q) are independent of h, then c k Is called the k - th de la Vallee Poussin derivative of g at the point t Q and is denoted by ^ d ^ C ^ ) . DEFINITION 2.If. If g ( k ) ( t Q ) exists for 0 < k i n - 1, define V n ( t Q , h ) by. n -1 h n/nl r G ( t 0 , h ) = g(t 0+ h) - g ( o ) ( t 0 ) - ^ [ h V k l ] g ( k ) ( t D ) , and let ^ S r n ^ t J = l i m sup V (t h) 6Cn) o h-> 0 n 0 DEFINITION 2.5. Let CP(t) = * ( x 1 0 + > x n 0 «\" t) = f ( x 0 + t) = u(x 0 + t) + i v(x 0 -t- t ) . If for a given t O J CP(t 0 - f t ) « • * 0 . * + ° C 2 t 2 / 2 1 * ••• + C^p £ r , t ) t r / r l for 1 < r < k, where f p^t\"* 0 with t T and the < ^ are complex constants Independent of t, then °f k w i l l be called the k - th generalized derivative of U> at the point ( X q -*• t Q ) and w i l l be denoted bv ^ ( k ) C x 0 + t0)« It w i l l be convenient to write C0(k)(t o) for 6 0 ( k ) ( x o t Q ) wherever i t w i l l cause no misunderstanding. Thus i f U(t) s u(x 0 ••- t) and V(t) = v(x 0 t) then = U(k)^o> + 1 v ( k ) ^ o ) w h e r e u(k><*o> a n d v(k) ( to> denote the k - th de la Vallee Poussin derivatives of U(t) and V(t) at (x Q + t Q ) . For reference purposes some definitions and theorems required for the development of the ^° n - integral [6] and the C nP - integral [2] are given below. Let u(t) be a real valued function of the single variable t defined in [a,b] and l e t a^, 1 - 1 , 2 , ... ,n be fixed points such that a - a, < a~ < ... < a_= b. 5 DEFINITION 2.6. The functions Q(t) and q(t) are called major and minor functions, respectively, of u(t) over (a^) = (2.^.1) Q(t) and q(t) are continuous in [a.b] , and, for 1 < k 5 n - 1, each Q(j c)(t), ^ ( k ) ^ exists and Is f i n i t e in (a,b)j (2.^.2) QCa^ = qCa^ =\"0, 1 = 1 , ... ,n, where a = a-^ < ... < a n = b ; (2A.3) fQ(n)(t) 2 u(t) > A q ( n ) ( t ) , In (a,b); (2.^.lf) ^ Q ( n ) ( t ) ^ - oo, A q ( n ) ( t ) 1= + co in (a,b). DEFINITION 2.7. For each major and minor function of u(t) over (a^), the functions defined by Q*(t) = ( - l ) r + - n Q ( t ) , q f t(t) = C - l ) r * n q ( t ) , a p < t < a r „ l f are called associated ma.lor and minor functions, respectively, of u(t) over (a^). DEFINITION 2.8. Let c be a point of (a^a ) such that c * a^,I = 1,2, ... ,n. Then u(t) is said to be fr°n -lntegrable over (a^;c) i f for every € > 0 there is a pair Q(t ) ,q(t) satisfying conditions (2.1+.1) - (2 A A ) such that |Q(c) - q(c)| V 6 • THEOREM 2.1. If u(t) is Tn - lntegrable over (a^c) there is a function U*(t) which is the inf of a l l associated ma.jor functions of u(t) over (a^) and the sup of a l l assoc- iated minor functions. 6 DEFINITION 2.9. I£ u(t) I4 f ° n - lntegrable over (ajL;c) and If U*(t) is the function of Theorem 2.1. define U(t) by ( - l ) r * nU(t) = U A ( t ) , when a p < t < a r ^ If a s < c < a s + 1 , the ? ° n - integral of u(t) over (a^jc) is defined to be ( -1) s * nU(c). Since ( - l ) s * ^ ( a ^ - 11(8^ = 0, the integral is defined to be zero i f c = 8 ^ 1 =1,2, ... , n, The notation is ( - l ) s + n U ( c ) = j u C t ) d„t. THEOREM 2.2. If u(t) is £ ° n - lntegrable over ( a ^ c ) , i t is also ? ° n - lntegrable over (a^;t) for every t in [a,b] . If U(t) is the function of Definition 2.9. then, for a s * t < as+l» ( - l ) s + nU(t) = ft u(t) d t. The C nP - integral is defined by induction. Suppose that for n > 1 the C n _ ^ P - integral has been defined taking as the C 0P - integral the Perron integral [9, p. 20l] . Assuming that u(t) is c n _ i p - lntegrable, l e t A * h C n(u,t,t *- h) = (n/h n)C n _ XP / (t + h - J ) n \" \\ ( f )d f DEFINITION 2.10. The function u(t) is said to be C n - continuous at t 0 i f C n ( u , t 0 , t Q + h) u(t Q) as h 0. DEFINITION 2.11. The upper and lower C n - derlvates of u(t) denoted by C nD 4u(t) and C n D A u ( t ) , respectively, are 7 defined to be the lim SUP and the lim inf. respectively, as h -> 0 of the expression (n + l/h)(C (u,t,t + h) - u(t)). n 4 DEFINITION 2.12. If CQD u(t) = C nD 4u(t), their common value is defined to be the C n - derivative of u(t) and is denoted by CnD u(t). DEFINITION 2.13. The functbn M(t) is said to be a C Q - ma.ior function of u(t) over fa.b] i f (2.5.1) M(t) is C n - continuous: (2.5.2) M(a) = 0; (2.5.3) C nD AM(t) > u(t), p.p in \\a,b] ; (2 .5A) C nD 4M(t) > -oo in [a,b] . A C n - minor function m(t) is defined in a similar way. DEFINITION 2.1k, If, for every € > 0, there Is a pair M(t), m(t) satisfying the conditions of Definition 2.13 and such that )M(b) - m(b)| < e , then u(t) is said to be CnP -Integrable over [a,b] . DEFINITION 2.15. £et 1(b) = lower bound of a l l M(b) and J(b) = upper bound of a l l m(b). For a CnP - integrable function u(t) the bounds have a common limit [2] which is called the definite CnP - integral of u(t) over farb1 . Suppose that ^ ( t ) = u(t) + 1 v(t) where u(t) and v(t) a r e ^ n - integrable and in the notation of Theorem 2.2 8 (- l ) s * \"iKc)= J u(t) d t (- l ) s + nV(c)= / v(t) d t . If a g < c < a s + 1 ? the T ° n - integral of ^ ( t ) over (a i;c) is defined to be (- 1) S * ^ ( c ) + i ( - 1 ) S *\" 'VCc), i.e. / ^ ( t ) d n t - / ° u(t)d_t + i / ° v(t)d nt = - / ( a 1 ) n - / ( a 1 ) n • / ( a 1 ) n (- l ) s * n [u(c) +-1 V(c)] = (- l ) s * n $ ( c ) . The C nP - integral of CP (t) is defined in an analogous way. When a series of the form oo 2> c exp(int) n T T ( in) j is summable (C, r - j - 1) to a function, say, F r \" J ( t ) where r and j are positive integers, i t w i l l be convenient to write simply, (2.6) y ^ c n e x p d n t ) = F r \" Ht) - i G p \" 3 C t ) , (C, r - J -n = 1 (In)1 without stating e x p l i c i t l y that F r \" ^(t) - 1 G r \" J ( t ) is defined by (2.6). 9 3. The Integrated Series A solution of the representation problem for the series (3.1) S _ c nexp(int) n =1 in terms of the £° k 2 - integral, when the series is summable (C,k), involves the following theorem stated by Zygmund [l6, p. 226, problem 12J to be proved as an exercise: THEOREM 3.1. Let the series (3.1) be summable (C,* ), for a fixed * = 0, 1, 2, ... and t = t Q , to sum s, |s/ < co . Let r be an integer > ' « + 1. Then the series integrated term by term r times converges uniformly for a l l t to a continuous function f (t ) , and *P( r)(t 0) exists and is equal to s. When the series (3.1) is bounded (C,k) a Fourier representation for the coefficients in terms of the C k + ^ P - integral can be obtained using a generalization of Theorem 3.1. THEOREM 3.2. Let the series (3.1) be bounded (C, « ), for a fixed « = 0, 1, 2, ... and t 6 E, /E/ > 0. If r = <* +• 2, then for each t e E, \\ (3.2.j) c nexp(int) n = 1 (in)3 F r \" Ht) - i G r ~ J ( t ) , . (C,r - j - l ) , 10 where j = 1, 2, ... , r - 1, and (3.2.r) y ~ c nexp(int) = F(t) - i G(t) - CP(t), n = 1 ( i n ) r where the last series is absolutely and^uniformly convergent for a l l t. In addition. ^ ( s ) ( t ) exists and is f i n i t e for 0 < s < r - 1, t e E, and (3.3) ^ ( t - h) = L£(t) + h ^ ( 1 ) ( t ) + ... -[h r ' V(r - l)l\\&{t_ 1 } ( t ) + [ ^ ( t , h ) / r l ] h r where u> (t,h) - 0(1) as h -» 0. Furthermore. ( 3-^) (P^ + 2 - j ) ( t ) \" Y ~ c n e * P ( l n t > (t,h) = 0(1) as h-» 0. It follows from a theorem due to Marcinkiewicz and Zygmund [8, Lemma 7, p. 15] that ^( r)(' t) exists p.p. in E. Equation (3-5) gives -0, 4 , (0) = A = 51 s* A ^ C l n ) \" K vr- j ; r - j n and since the (C,1* ) sum of the series c /(n)^ equals the (0,0) sum of the series (3.*+) is established. THEOREM 3.3. If under the hypotheses of Theorem 3.2, the set E is an open interval, then (3.6) C DF a(t) - F. . ( t ) , C DG a(t) - G. n x ( t ) , a? (a + 1) a (a + 1) 0 < a < «< , 11 E, C „I>4F, ,>(t), «* +• 1 ( OC +• 1) C DP. , x ( t ) , C nD*G, ,x(t), C D G, , x ( t ) are f i n i t e for te.E, and ^ + 1 A (°< +- 1) 12 (3.7) C IJF * + l ( t ) - F. ( t ) , C ^ _DG * + 1 ( t ) -G ( R ) ( T ) > P«P* in E. For the proof a lemma is required. LEMMA. If y ° a is summable (C, r -t- 1 ) , where r > -n = 1 n then a necessary and sufficient condition that i t should be bounded (C,r) is that B £ - 0(n r + 1) where bfl = na n and B°, B 1, B 2, ... are formed from the b as A 0, A 1, A 2, ... n* n' n' n n' n' n* are from the a f l (cf. 5, p. 96). Proof. It is easy to verify that (ri +- r +• l ) A r = B r +• (r + l ) A r 1, n n n 1 and hence that B A ^ (r - l / n + r + D C p f - y + A £ + h, and the lemma follows. The relations (3.6) w i l l be proved by induction and in virtue of the symmetry of the enunciation only the f i r s t of the relations w i l l be considered. The result is well known for a = 0 [ l5 , Lemma 5, p. 206] . By the lemma, series (3.2.1) is bounded (C,c< - 1 ) , series (3.2.2) is bounded (C,°<- 2) , ... , series (3.2.r - 1) is bounded (C,0). Assume that the relation holds for a l l s, » s < °< and hence that F (t) is the C SP - integral of F ( s + 1 ) ^ f o r a 1 1 s < ^ • T n e t l (°< - 1) integrations by parts gives 13 t + h lira h-> o ( */h )C « - 1 p j (t + h - / )e< \" cr )df - F * (t) \"t h/( * + 1) lim [( « +• l) ! h * x F(t f h) - F(t) - XI (h k/ki)F k(t) k - 1 Hence C DF (t) = lim [( <* + l)J/h * 1 h-*0 1 1 [F(t +-h) - F(t) - ( hk / k i ) F ( k ) ( t ) F(<* +• l ) ( t ) > b v T h e o rem 3.2. It follows from (3.3) that (3-8) [( « + 2)!/h °\" *\" 2 ] [ F(t i- h) - F(t) -: = 1 as h -> 0 and so C JLLl , -I J E Z ( h k / k ! ) F ( k ) ( t ) J = 0(1), 1 D ^ ( ~ !)<*> a n d C « l V ( * + l ) ( t ) are f i n i t e . Finally, C c* 1 lim + 2)!/h IwO DF ^ + 1 ( t ) oC + 1 F(t h) - F(t) - ^ T\" ( h k / k i F ( k ) ( t ) k = 1 i f the limit on the right hand side exists. By Theorem 3.2 C DF «r 4- 1 exists p.p. in E and is equal to F, x(t) ( <* +• 2) THEOREM 3.1+. Suppose that the real part of series (3.1) is bounded (C,*< ), (x « E, )E| > 0, °< > - 1 ) . Then the 11+ series (3.1) Is summable (C, «* ^ ), f > 0, almost everywhere In E to sum ^ ( r ) ( t ) , where <^(t) - F(t) - 1 G(t) is the function of Theorem 3.2 and r = +- 2. Proof. Marcinkiewicz and Zygmund have proved |^ 8, Theorems 2 and 3J that i f the real part of series (3.1) is bounded (C,« ), (x £ E, |E| > 0, °< > - 1), then the series (3.1) is bounded (C,«0 and summable (C, * *• f ) , ^ > 0, p.p. in E. Now, to f i x ideas take <=< = 1. In view of the proof of Theorem 3.2, the series oo oo N> - a nsin(nt) +- b ncos(nt) and -^T~ a ncos(nt) +• b nsin(nt) n = 1 n^ n = 1 n^ are uniformly convergent everywhere and hence are the Fourier series of F(t) and G(t) respectively. It follows also from Theorem 3*2 that F ( 3 ) ( ' t ) a n d G ( 3 ) ^ exist almost everywhere in E. By a well known theorem (Cf. [l6] , p. 257) the Fourier series of F(t) and G(t) differentiated term by term three times are, almost everywhere In E, summable (C,k), k > 3 to the values F ( 3 ) ( t ) a n d G ( 3 ) ^ ^ respectively. Hence series (3.1) is summable (C, 1 + f ), 0, p.p. in E and also summable (0,*+) to ^ ( 3 ) ( t ) p.p. in E . It follows that the series (3.1) is almost everywhere summable (C, 1 + in (a,b). Thus Q(t) is both a major and a minor function of F ( k + 2 ) ^ o v e r ^ a i ^ # J t f o l l o w s t h a t F ( k + 2 ) ^ i s - integrable over ( a ^ t ) , and (h.k) follows from the definition of t h e ^ ° k + 2 - integral. THEOREM *t.3. If the series (3.1) is summable (C,k) for a l l t in [a,b] to a function £P(t) - u(t) + i v(t), where [a,b] Is a f i n i t e , closed interval and , | ^ ( t ) l < oo , then (Pit) i s r k + 2 - Integrable over (a^;t) for every t In [a»bj where a a^ t ... < a n = b. Proof. Series (3.1) integrated term by term k •+• 2 times 17 converges uniformly to a continuous function H(t) + i K(t). By Theorem 3.1 H ( r ) ( t ) a n d K ( r ) ^ t ) exist and are f inite for a l l t in [a,b] , 1 < r < k + 2, and H ( k + 2 ) ( t ) = u ( t ) ' K ( k •*- 2 ) ( t ) = v ( t ) * B y t h e P r e v i o u s theorem H ^ ^ 2 ) ^ ~ u ^ a n d K ( k + 2 ) ^ = v ^ a r e * 2 - integrable. Hence u(t) + i v ( t ) - C^(t) i s >^k -f 2 _ integrable over (a^;t). LEMMA. Under the hypotheses of Theorem C l f t l e x p f - l p t l , p > 1, is. + 2 - Integrable over (a^jt). Proof. By hypothesis c nexp(int) - ^ ( t ) , (C,k), n= o for a l l t in [a,b] . It follows that c n exp(i(n - p)t) - ^ ( t ) e x p ( - i p t ) , (C,k), n - o for a l l t in [a,b] , and therefore that (**#5) 2 c nexp(i(n - p)t) - - c Dexp(- ipt) -n= p ciexp(- i ( l - p)t) - Cp _ ^expC- i t ) + CP(t)exp(- i p t ) ' (9(t), (C,k). By Theorem ^ . 3 , 0(t) is ^ k + 2 - integrable over ( a ^ t ) for every t in [a,b] . It follows that (^(t)exp(- ipt) is ?° k + 2 - integrable since each of the other functions on the right hand side of (*t.5) is ^ ° k + 2 - integrable. 18 Theorem k.l can now be proved. Proof. Consider f i r s t the expression for c Q when k is even. It follows from Theorem k,2 and the proof of Theorem k.3 that, th.6) <- 1) [ ( k + 2 ) / 2 i + k + 2 f * CP(t)d k + 2 t <- i)Ck + 2)/2 / * # ( t ) d pt=H(t) - ky\" 2X(t;^,)H(^ - 4 ^ ) k + 2 1 ^ 1 1 i ^K(t) - ^ \" ^ X c t ; ^j . ) K ( « < I ) J = Q(t) i R ( t ) , (k + 2)/2 5 * [(k\" + 2)/2] + 1» where C ^ ) is \"the set (^.1). The function H(t) + iK(t)may be written in the form ^ P ( t ) + B^Ct) + i H 2 ( t ) , where SP(t) = c Q t k * 2/(k + 2 ) ' and B^Ct) and H 2(t) are periodic with period 2 T T . If t = 0, the right hand side of (*+.6) becomes 0*.7) ^ p (0) - ^ ^ X ( o ; ^ ^ P c ^ ) + [%((» i i H 2 ( O ) ] k +• 2 - A ( 0 ; ^ [ ^(o) t i H 2 ( o ) ] . This reduces to k + 2 C Q T T (- oq ) (, 1) Oc +2)/2 C o ( 2 r r ) k ^ 2 + 2TT I \" \" n (k )1 • • i = 1 'k (cf. James, £7] » p. 106). 19 Hence < 2 * r ) k + 2 y ( * ) The expression for c Q when k is odd can be obtained in a similar manner since (if.8) ( - 1 ) + k + 2 ^ ( t ) d ' ?t (- 1) I* * ^ + 1 / * ^ ( t ) d k + Y (t) + H^t) + i H 2(t), • (k + l)/2 s r \" £k + l)/2] + 1» 2* = •where (* ^) is the set (*f.2). To obtain the formulae for c p,p > 1 , rewrite equation C+.5) in the form A . 9 ) a_exp(Irt)= ©(t), (C,k) r = 0 r where r - n - p, n = p, p +' 1 , ... , and a„ = a 7 7 7 r n - p The constant term in (*+.9) is a Q = Cp, and for k even, k + 2 cP T P Tk~r 2TT i - 1 (- 1) L(k + 2 ) / 2 ] - k + 2 A g 20 (- i ) ( k + 2 ^ 2 ^ ( t ) e x p ( - i p t ) d k + 2 t , since s-0 exp(- imt)d k ± 2 t = 0. (^ ^) It follows that c = ^ 6?(t)exp(- ipt)d t. P k T ~ 2 y k •»•2 For k odd the argument is similar. Remark. The proof of Theorem M-.l.is simpler than the proof of the analogous result by James JV» pp. 105 -106] since the results on the formal multiplication of trigonometric series are not needed. 21 5» The Expression of Coefficients in Terms of the C <* +- l p ~ Integral THEOREM. Suppose that the series (1.2) is bounded (C,<* ), for a fixed * - 0 , 1, 2, ... jLn (- TT, 7r ) and let r = «* +- 2. If F(t) and G(t) are the functions of Theorem 3.2, the coefficients of series (1.2) are given by J (r) TT (5.1) a = _L_ f F , v ( t ) co s (n t )d t = n TT /-rr G. N ( t ) s i n ( n t ) d t , (r) (5.2) b - _L / F , . ( t ) s i n (n t )d t = n ^ J (r) - n TT ^ (r) (t)cos(nt)dt, where the integrals are C ^ ^ - integrals. Proof. To f i x ideas take <* - 2. It follows from Theorem 3«2, Theorem 3.3, and the definition of the C N P -integral that F S ( t ) - F S(--r) = F ( s ) ( t ) - F ( s ) ( - - ) = v / / ( s + 1 ) ( x ) d x , 0 £ s < 3. Hence, using the property of Integration by parts for the C n P - integral £2] , C ^ P F ( 1 + ) ( t ) c o s ( n t ) d t ^ C 2 P / ^ F ( 3 ) ( t ) n s i n ( n t ) d t = C 1 P f F ( 2 ) ( t ) n 2 c o s ( n t ) d t \" \" c0^/^F<1>Ct>n3slnCJa-t)« = 22 77 C 0P / F(t)n l fcos(nt)dt - a^tr , since [a ncos(nt) •*- b Qsin(nt ) J /n is the uniformly n - 1 convergent Fourier series of F(t)., The other results follow similarly. 23 6. The C k + ± ? - integral and the ^° k + 2 - integral It has been shown by James [6] that C k + -jP -integrability implies ¥° k + 2 - i i n t e g r a b i l i t y and that the indefinite V k * 2 - integral is equal to an (r + 1) - fold integral in which the innermost integral is an indefinite C rP - integral, the next one an indefinite C- -jP -integral, and so on, the outermost integral being an indefinite C DP - Integral. A relationship between the definite ( 2 T ) r - 1 / ! 7 - 1 « * . P- 18). fc- 1 , « t ) a t - _ i _ c p / f ( t )at , the required result. 25 7« Uniqueness Theorems for Multiple Trigonometric Series Since the triangular method of summation defined in Chapter 2 reduces the summation of a multiple trigonometric series to the Cesaro summation of a single series, i t is possible to use the results of the preceding chapters to obtain uniqueness theorems for multiple trigonometric series. THEOREM 7.1. If the series (1.1) is summable (T,k) to a function CPjx^ x g, ... , x n, t) = #?(x,t), |6?(x,t)) < co , for a l l values (x^ -*• t, x 2 +• t, ... , x Q •+• t) where 0 5 x^, x 2, ... , x Q, t < 2 \"f then the coefficients of the series are given by ^ ( x , t ) e \" i p t d k + 2 t dx x ... dx n, where mn + ... + m = p, Ik and ( «• .) are defined as in 1 n 1 — : Theorem ^.1, and the n outer integrals are Riemann integrals. Proof. For simplicity in writing, the proof w i l l be given for n = 2. By hypothesis, ^ + p]>Z C p(x 1,x 2)e iP t = ^ ( X ; L , x 2 , t ) , (C ,k) , for a l l t and 0 < x 1 ? x 2 < 2-rr, It follows from Theorem *Kl 26 C (x 1,x 2) = P \\ i(m,x, +• m 0x 0) (7.2) > c m m e 1 1 2 2 k / 6^(x 1,x 2,t)e- i P * ^ , 2 t , 0 < x^,x 2 < 2TT . NOW let m and n be any two positive integers such that m + n = p. Multiplying both sides of (7.2) by e\" i m x l and integrating with respect to x^ over (0,2'\"') yields (2T)(c )e i^P ' m ) x 2 m,p - m . • — (7.3) 2 t c e l n x2 -mn -2 - V i » 'o ( 2 7 r Then, multiplying both sides of (7.3) by e\" l n x 2 and integrating with respect to x~ over (0,2 t t) gives (7.W . ( 2 ^ ) 2 = r a n 2TT 2-\"-- i(mx e P 0 1 + n X 2 } / k C ^ ) ( x 1 , x 2 , t ) e - i P t d k v 2 t d x ^ , (2 7)\\ 0 0 ' ( ^ ) the required result. 27 A more general result is obtainable in a similar way in virtue of the theorem of Chapter 5. THEOREM 7.2. If series X c e i ( m > x \\ m 7 is bounded (T,k) for a l l values (x.^ + t, ... ,x n + t ) , 0 < x 1,x 2, ... »xn,t i 2tr, and i f r - k + 2, then the series C (x ,x , ... ,x ) ipt p l 7 2 7 7 n e ( i p ) r converges uniformly for a l l t and each (x) to a continuous function ^ ( t ) , and the coefficients of the series are given by 1' ' n 2 TT 2 2 r71\" n integrals e\" 1 ( m> x^ /^/ ( r )(t)e\" : LP tdtdx 1 ... ( S O \" ' ' 2 T 0 '0 '0 where the inner integral is a C^ + - Integral. Proof, (for n = 2) The convergence property of the integrated series is obvious. The theorem of Chapter 5 gives C (x,.x) -p 1 2 \\ i(m,x, * m x„) (7.6) /_ c e l l 2 2 m ,——— _ m-i .jij + m0 s p J-' c-1 ' m2 2 T ° k . l P f > ( r ) ( t ) j \" 1 P t d t » ^0 28 and (7.5) follows from (7.6) in the same way that (7.1) follows from (7.2). A uniqueness theorem which is stronger than Theorem 7.1 may be obtained in a very interesting manner. It is stronger in the sense that the set of n - tuples (x^,x 2, ... , X q ) in the hypothesis is reduced to a countable set. THEOREM 7.3. If for a countable set A of n - tuples ( x 10 ' X 20> , x n - 1 0 ' x n i * ' f o r f l x e d x i o ' x 20> , x n - 1 and 1 = 1, 2, 3, ... , series (1.1) is summable (T,k) to 0 for a l l values ( x 1 Q + t ,x 2 0 «• t, ... ,x n i + t ) , 1 = 1 , 2 , 3 , * . . , 0 < t < 2 t t, then the series vanishes ident- i c a l l y . Proof. Only the case n = 3 w i l l be considered. In view of the proof of Theorem 7.1, i t is clear that (7.7) Y L °»»-e 1(-\"1*10* \" 2 X 2 ° 4 \" 3 X 3 l ) = m * m + m = p 1 2 3 ' 1 2 3 for fixed x 1 0 , x 2 0 a n d i = 1 » 2 > 3 , ... . This gives y fcm ,m ,1 m. + m. =• o L x * m e 1 ( m i X 1 0 + m 2 X 2 0 ) m^ + m^ * m^ =• p «- •*• *~ J im.x_ e 3 3 i 29 5 m-^ + m2 + m^ = P m l , m 2 , D 1 3 e i m 3 x 3 i = r / c m l , m 2 * P ~ m l ~ m 2 e i p x 3 i e- i ( m i , + m2) m l m 2 = ^ 'm^ ,m2,p - m^ - nig 1 e- i(m1 + m2) i = 1, 2, 3» ... • It follows that the equation (7.8) I O 1 ' 2 , P 1 2 e - iu m l * ra2 has a countable number of roots, x21,x32iX339 ... . But the l e f t hand side of (7.8) is a polynomial of degree p and cannot have a countable number of zeros. The only possibility is that c' _ _ _ _ r c l m m = 0, r J m2,m2,p- m^ - m2 m^ ,m2,m3 ' and hence that c 1' 2»^3 1 2 3 such that m l * m 2 * m 3 = p < 30 BIBLIOGRAPHY 1. S. Bochner, \"Summation of multiple Fourier series by spherical means\", Trans. Amer. Math. Soc. vol. *t0 (1936) pp. 175 - 207. s 2. J. C. Burki l l , \"The Cesaro-Perron scale of integration\", Proc. London Math. Soc. (2) vol. 39 (1935) pp. 5^1 -552. 3. , \"The expression of trigonometrical series in Fourier form\", J. London Math. Soc. vol. 11 (1936) pp. hZ - **8. k, Min-Teh Cheng, \"Uniqueness of multiple trigonometric series\", Ann. of Math. vol. 52 (1950) pp. h03 - *4-l6. 5. G. H. Hardy, \"Divergent series\", Oxford, 19*+9. 6. R. D. James, \"Generalized n - th primitives\", Trans. Amer. Math. Soc. vol. 76 (1951*) pp. 1**9 - 176. 7. , \"Summable trigonometric series\", Pacific J. Math. vol. 6 (1956) pp. 99 - 110. 8. J. Marcinkiewicz and A. Zygmund, \"On the different-i a b i l i t y of functions and summability of trigonometrical series\", Fund. Math. vol. 26 (1936) pp. 1 - 1*3. 9. S. Saks, \"Theory of the integral\", Warsaw, 1937. 10. W. L. C. Sargent, \"On the Cesaro derivates of a function\", Proc. London Math. Soc. (2) vol. *+0 (1936) pp. 235 - 25^. 31 11. W. L. C. Sargent (cont.), \"A descriptive definition of Cesaro-Perron integrals\", Proc. London Math. Soc. (2) vol. k? (19^2) pp. 212 - 2*+7. 12. , \"On generalized derivatives and Cesaro-Denjoy integrals\", Proc. London Math. Soc. (2) vol. 52 (1951) pp. 365 - 376. 13. V. L. Shapiro, \"An extension of results in the uniqueness theory of double trigonometric series\", Duke Math. J. vol. 20 (1935) pp. 359 - 365. l h . , \"A note on the uniqueness of double trigonometric series\", Proc. Amer. Math. Soc. vol. (1953) pp. 692 - 695. 15. - S. Verblunsky, \"On the theory of trigonometric series ¥11\", Fund. Math. vol. 23 (193*0 pp. 193 -236. 16. A. Zygmund, \"Trigonometrical series\", Warsaw, 1935. "@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0080652"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Mathematics"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "On the uniqueness of multiple trigonometric series"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/40492"@en .