�9 116 �9 C.C. Lira et at: Comparisons o f Absolutely Conv..ergent

COMPARISONS OF ABSOLUTELY CONVERGENT

TRIGONOMETRIC SERIES

Chincheng Lin

(Central University, T aiwan )

Weichi Yang (Rad ford University, U. S. A )

Received Dec. I, 1994

Theorem.

Then

In this paper, we first discuss the methods of comparing two special absolutely convergent

sine series, ~ a. sinnx and ~ b.sinnx. We state the theorem in one dimensional ease as fol-

lows:

Theorem. Let ~-~a., ~-~b. be convergent series with nonnegative terms. Suppose

~ E z n ( g , nt-b.)<--~-(at--lh ). Then ~'~,7_la.sinnx>/ ~'~Elb, sinnx for all x E [O,r:].

l f , in addition, ~ - ~ nb-1ff~---2 b then

~_j a.~nnz • b.sinnx >/O, for all x C= [O,~r']. -.--1 ,,--1

By setting b. = 0 for all n E N, we can view Theorem 1 as providing us information on a

lower bound of a x , so that ~-'~7_ a,,sinnx/> 0 for all x E ['O,rc'].

Second, using the first and second derivatives on the function ~ a. cosnx, we could also

compare two cosine series in the inteval [ '-- a ' /2,a'/23. We state this theorem in one

dimensional ease as follows.

Let a.~O, h e N , satisfy ~-~7_,nZg,< ~/? at and { h a . } be decreasing.

a.cosnx >I a . c o s - - , for all x E" ~r ~r - = , . - 1 2 ' 2 "

Finally, we discuss the extensions of Theorems 1 and 2 to two dimensions. The inequali-

Approx. Theory ~ i t s Appl . , 12:1, Mar. , 1996 " 117 �9

ties achieved in this paper will not only enable us to compare two oscillating series, but also the

inequalities in two dimensions can give us information on the behavior of the solutions to cer-

tain heat equations without applying the maximal principle. For instance, let U , ( z , t ) =

kU,:z(x,t) for z E (0,rr).t > 0,k a positive constant, and

U (O,t) = U Or,t) = O.

Also let

U(x,O) = f ( x ) or U(x,O) = g ( x ) .

Then the solutions U I and U t to this heat equation with respect of f and g are c>~ I~Q

U l ( x , t ) = ~ ls innx .e -"z'' and U , ( x , t ) = ~ 3 �9 ~ - s l n r l x * e -nzkt ,

h E 1 n = l

and hence we have U1(x , t ) ~ U, (x,t) for all (x,t) 6 E0,~r] X [-0,c~).

Chin-cheng Lin

Department of Mathematics

Ce, tral University

Chung-Li, Taiwan 32054

China

Wei-Chi Yang

Department of Mathematics and St~histics

Radfor University

Radford, Virginia 24142

U.S.A.