{VERSION 4 0 "IBM INTEL LINUX" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "# CESARO SUMMATION O F A TRIANGLE WAVE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "aodd := k -> 1/(2*k + 1)^2; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "partialsum := K -> (Pi/ 2) - (4/Pi)*sum(aodd(k)*cos((2*k+1)*x), k=0..K);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "mean := M -> (1/(M+1))*sum(partialsum(K), K=0 ..M);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot([partialsum(1 ), mean(1)], x=-1..7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "p lot([partialsum(5), mean(5)], x=-1..7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([partialsum(20), mean(20)], x=-1..7);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "# Close inspection shows tha t the Cesaro mean curve is BLUNTER than the partial sum curve." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "# When the partial sums are already well convergent, Cesaro summation just slows down the converg ence." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# This is shown cl early by examining the error in the approximations:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot([abs(x) - partialsum(5), abs(x) - mean(5)], x=0..1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot([abs(x) - partialsum(10 ), abs(x) - mean(10)], x=0..1);" }}}}{MARK "14" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }