第B湘 Dimensions of spline spaces over general T-meshes 579 vertices in parts labeled with N have to he 2(a+12(n+1). determinod.Following the analysis procedure, we know that the Bezier ordinates around interior T-vertices and inner b-vertices will be determined by the Bezier ordinates around interior crossing vertices and free b-vertices.So we only have to consider free h-vertices and interinr croxsing vertices For every non-singular free b-vertex or interior crossing vertex.we have (g+1)(2+1) free Bezier ordinates.For every singular b-vertex. 下ig8 A general T-mesh we have2(a+1)(9+1》-(a+1)-(3+1》free Example 3.4 Suppoxse we are given a general Bezier ordinates.Totally we have T-mesh as shown in Fig,9.In :.8, d=(-g++)(a+1)(8+1)+ E=8.E=8.V =0,and V=1,then the [2(a十1)3+1)-(a+1)-(3+1)]= dimension of the spline space &(m.n)is (9+8+)(a+1)(3+1)-(a++2) dims(mn:)= free Bezier ordinates. 8(m+1)(n+1)-(a+3+2)- Now the dimension of the spline spare is the 8(m十1)(3+1》-8(a+1)(n+1). sum of d,i=1,2,3,4): dims(m)=d.= =1 F(m+1)(m+1)一(2F--E.)(a+1)(m+1)- (2F--E)(m十1)(3+1)十 (4F-2E-2E-2E-2E+9++V)· (a+1)(g+1)-(a+3+2)= F(m+1)(n+1)-E(a+1)(#+1)- E(m+1)(0+1)+a+10+1)-(a+3+2). Fig.9 A general T-mesh since 4F-2-2E-2E-2E.+%++V- 4 Periodic spline spaces over regular (2F--E)+(2F-E-E)+ T-meshes (V+V-m-E)-E-E++ Es+E-E-E+V+=V+V+=V. In many applications.it is nerexsary to ▣ approximate a function that is known to he This completes the proof of the theoremn. periodic.As it is usually desirable to work with Now we illustrate some examples to show how periodic approximation functions in such cases,we to use Theorem 3.2 to calculate the dimensions of devote this section to the study of periodic spline spline spaces. spaces over regular T'-meshes. Example 3.3 Suppose we are given a general Given a regular T-mesh and-[e.]x T-mesh as shown in Fig.8.In 1.F=5.E= 2.E -2,and V-V-0,then the dimension of [yo,y ]we define the spline space s(m.n8.1)is ,(m,na,月，)= dim(m,,c3,1)= {s(r,y)s∈(mn,a,3,),sp(x,y》= 5(m+1)(n+1)-2(m+1)(0+1)一 (1y)为≤y≤y…i=01u小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’$K/>2J0I2,C0$,$O/>2CG%0,2CG?120C/>2 C+I$O4-$-;S#(#!#T%& J0I#$’#(#"###!%; % T -;S 4- ; G$’DS%$(DS%C$(GCE): 1 CE)1%$"DS%$(DS%C $(GCE): , CE),%$’DS%$#DS%D $TGC(E): , C(E): 1 C(E), C(E)1 DF: * DF: / DFD%’ $"DS%$#DS%CF: /$"D#D(%; G$’DS%$(DS%CE1$"DS%$(DS%C E,$’DS%$#DS%DF$"DS%$#DS%CF: /$"D#D(%# C0,12 TGC(E): , C(E): 1 C(E), C(E)1 DF: * DF: / DFD; $(GCE): , CE),%D$(GCE): 1 CE)1%D $F: * DF: / CE): , CE): 1%CE), CE)1 DFD; E, DE1 CE), CE)1 DFD;F*DFD;F< F>0C1$IG%2/2C/>2G.$$O$O/>2/>2$.2I& ’ ’$KK20%%+C/.?/2C$I22Z?IG%2C/$C>$K>$K /$+C2F>2$.2I!Y(/$1?%1+%?/2/>2J0I2,C0$,C$O CG%0,2CG?12C& OP+#)*$HJHW+GG$C2K2?.2@0H2,?@2,2.?% F6I2C>!S?CC>$K,0,\0@Y8&3,!S#G ;5#E, ; (#E1 ;(#?,JF ;F: / ;)#/>2,/>2J0I2,C0$,$O />2CG%0,2CG?12#$’#(#"###!S%0C J0I#$’#(#"###!S%; 5$’DS%$(DS%C($’DS%$#DS%C ($"DS%$(DS%< D"/GQ -@2,2.?%F6I2C>!S OP+#)*$HJI W+GG$C2K2?.2@0H2,?@2,2.?% F6I2C>!( ?CC>$K,0,\0@Y 9&3, !(#G ; 8# E, ;8#E1 ; 8#F ; )#?,JF: / ; S#/>2,/>2 J0I2,C0$,$O/>2CG%0,2CG?12#$’#(#"###!(%0C J0I#$’#(#"###!(%; 8$’DS%$(DS%C$"D#D(%C 8$’DS%$#DS%C8$"DS%$(DS%< D"/GR -@2,2.?%F6I2C>!( I L$."’9",&)*"%$&)+,$&’-$..$/<*+. 01#$&2$& 3, I?,U ?GG%01?/0$,C#0/ 0C ,212CC?.U /$ ?GG.$Z0I?/2 ? O+,1/0$, />?/0C [,$K, /$ M2 G2.0$J01&-C0/0C+C+?%%UJ2C0.?M%2/$K$.[K0/> G2.0$J01?GG.$Z0I?/0$,O+,1/0$,C0,C+1>1?C2C#K2 J2H$/2/>0CC21/0$,/$/>2C/+JU$OG2.0$J01CG%0,2 CG?12C$H2..2@+%?.F6I2C>2C& <0H2,?.2@+%?.F6I2C>!#?,J!;!>)#>S"? !2)#2S"#K2J2O0,2 # ) ,$’#(#"###!%=; (/$>#2%/# #$’#(#"###!%#/$@%$>)#2%; /$@%$>S#2%#2) +2 +2S#@;)#S#)#"*< K"2 40I2,C0$,C$OCG%0,2CG?12C$H2.@2,2.?%F6I2C>2C 579