Non-differentiable-function Smoothing-curve. Abstract: The interpolation by polynomial of degree n, , may smooth any curve

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1 Issue, Volume, Smoothig curve b orthogoal olomials Sua Pasomboo ad Poga Rattaathaawa Kig Mogut s Istitute of Techolog Ladrabag Ladrabag Bago Thailad Maitree Podisu Kasem Budit Uiversit Sualuag Bago Thailad Abstract: The iterolatio b olomial of degree, ), ma smooth a curve f ) with the coditio that this olomial must asses through oits of the curve f ) This coditio ma mae the shae of the curve of ) differet from the eact shae of curve of f ) eseciall i the case that the oits of f ) are obtaied from a eerimet which ma ot be the eact oit of the curve f ) But the least square aroimatio b orthogoal olomial, q ), ma give the better shae tha the iterolatio b olomial, ) However the curve of q ) ma ot ass through a eact oit of the curve f ) MPodisu, PRattaathaawa ad PPhataraavi i [] itroduced the sequeces of orthogoal olomials with ste fuctios as their weight fuctios but the did ot use them for least square aroimatio I this aer, the sequeces of orthogoal olomials with ste fuctios as their weight fuctios will be used for least square aroimatio Kewords: Iterolatio Orthogoalolomial Least-squarearoimatio, Stefuctio No-differetiable-fuctio Smoothig-curve Itroductio The iterolatio of the fuctio, f ), b olomial, ), is of the form ) a a a a where or L )f ) () L ) j j j j ) j ) ) a a a a f ) f[, ] ) f[,, ] ) ) () The equatio () is called Lagrage method ad the equatio () is called Newto divided differece method The least square aroimatio method of the fuctio f ) b olomial q ) is of the form q ) d ) d ) () where ),, ) are the orthogoal olomials with resect to the weight fuctio w ) i the iterval [ a,b] ad j

2 Issue, Volume, a b a b w) w) )f )d d () ) )d Orthogoal Polomials I this aer, sequeces of orthogoal olomials are costructed The orthogoal olomials u to degree are obtaied for each sequece of orthogoal olomials The weight fuctios are ste fuctios ad the are, resectivel First weight fuctio, < w ), Secod weight fuctio, < w ), <, Third weight fuctio, < w ), <, Fourth weight fuctio, < w ), <, Fifth weight fuctio, < w ), Sith weight fuctio, < w ), <, The sequeces of orthogoal olomials of the above weight fuctios are as follow (u to orthogoal olomial of degree for each sequece) First Sequece ) ) 8 ) 98 ( 9) 88) ) ) 8) 898) ) ) 8) 8) 9) ) ) 9) 9) 989) 9) ) ) ) 8) 9) 8) 99)

3 ) 9) 88) ) 8) ) ) 8) Secod Sequece ) ) ) 89) 98 ) 9 8 ) 88) ) 8 ( ) 9) 89) 8) 9) 8 88 ) 9) 89) ) ) 988) 89 8 ) 99) 8) 8) 98) 8) ) ) 9) 89) 8) ) 88) 9) 9) Third Sequece ) ) 88) 9) 9 ) ) 898) ) 8) ( ) 9) 9) 8) 9) 98 8 ) 9) 8) ) 9) ) 99) 89) 9) 899) 9) ) ) 9) 89) 8) ) ) ) 9) Fourth Sequece ) ) 88) 89) ) 8 9 ) 89) ) 9) ) Issue, Volume,

4 99) ) ) 9) 9 8 ) 99) 8) 8) ) 8) ) 99) 89) ) 8) 888) 99) ) 9) 89) 8) 898) ) 9) 89) Fifth Sequece ) ) ) 9) 9 ) 89 9 ) 88) ) 8) ) 9) ) 99) ) ) 9) 9) 9) ) 99) ) 99) 89) 99) ) 9) ) 98 ) 98) 88) 988) ) 88) ) 9) Sith Sequece ) 8 ) ) 8) 8 9 ) 9 9 ) 89) 8) 9) ) 9) ) 9) 8) 9 8 ) 989) 99) ) ) ) ) Issue, Volume,

5 Issue, Volume, 8 ) 8) ) 89) 88) 9) ) ) ) 8) 8) ) 8) 9) Eamle Eamle Use the first sequece ad the fifth to smooth the curve, [,) f ) ( ), [,] For the first weight fuctio, we obtai ad their grahs are i the followig figure For the fifth weight, we obtai ad their grahs are i the followig figure Eamle Use the secod sequece to smooth the curve

6 Issue, Volume, 9 9 f ), [,] We obtai the followig olomials ad their grahs are i the followig figure ad their grahs are i the followig figure Eamle Use the third sequece to smooth the curve 9 f ), [,] We obtai the followig olomials Eamle Use the fourth sequece ad the sith sequece to smooth the curve of, [, ) f ), [, ), [,] For the fourth, we obtai -9

7 Issue, Volume, ad grahs are i the followig figure ad their grahs are i the followig figure Coclusio All results are as good as the are eected We recommed all si sequeces of orthogoal olomials For the sith sequece, we obtai Refereces [] Maitree Podisu, Poga Rattaathaawa ad Pima Phataraavi, Sequeces of Orthogoal Polomials with Ste Fuctios as Their Weight Fuctios, Proc of the 8 th WSEAS It Coferece o Mathematical Method ad Comutatioal Techiques i Electrical Egieerig, Bucharest, Romaia, October -8,