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zoom RSS テイラー展開の研究 その1  多項式や級数を無限次元のベクトルと見なす。

<<   作成日時 : 2014/04/17 15:38   >>

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多項式を各係数を成分とするベクトルと同一視してみる。

例えば、2次式f(x)=3−2x+7x^2 を (3,−2,7)と言う具合。

こうすると多項式のスカラー倍や、多項式同士の和がベクトルのスカラー倍や和と対応することが分かる。
しかし、2次式は3次元ベクトル空間の元としておくと、2次式同士の積が4次式となり、空間からはみ出してしまうので都合が悪い。

 だから、最初から無限次元ベクトル空間Vを考えて、成分が0でないものが有限個しかない場合を多項式としておく方が都合がいい。これだと級数もベクトルの元と見なせる。

3−2x+7x^2に対応するのは、(3,−2,7,0,0,0,・・・)という具合。
級数1+x+x^2+x^3+・・・には、(1,1,1,1,・・・・)が対応する。

スカラー倍や微分、級数同士の和や積、合成などを、ベクトルで定義し直してみる。

スカラー倍と和に関しては通常のベクトルと同様だから割愛する。

【微分】 VからVへの線型写像Dを

u=(a__0,a__1,a__2,a__3,・・・)に対して、Du=((a__1,2a__2,3a__3,・・・)などと定義する。


【積】 Vの元の2項演算を以下のように定める。

u=(a__0,a__1,a__2,a__3,・・・)  v=(b__0,b__1,b__2,b__3,・・・) とするとき

u*v=(a__0*b__0,a__1*b__0+a__0*b__1,a__2*b__0+a__1*b__1+a__0*b__2,a__3*b__0+a__2*b__1+a__1*b__2+a__0*b__3,・・・・)


【交換法則 結合法則 分配法則】 積と和に関して、交換法則 結合法則 分配法則 などが成り立つことが確認できる。


【累積】 積を繰り返すことでu^nが定義できる。uがなんであれ、u^0=(1,0,0,0,・・・)と定義する。

きちんと書くと、u^0=(1,0,0,0,・・・) u^(n+1)=(u^n)*u という具合に帰納的に定義する。


【合成積】 級数を関数と見なしたときの関数の合成に対応する操作は以下のようになる。

u=(a__0,a__1,a__2,a__3,・・・) とするとき

u○v=a__0*v^0+a__1*v^1+a__2*v^2+a__3*v^3+・・・ と定義する。

f(x)に対応するベクトルがu
g(x)に対応するベクトルがv

とすると、u○vは、f(g(x))に対応するベクトルとなる。

合成積に関しては、無限和が出てくるので必ずしも値が確定するとは限らない。
また一般には可換ではない。

左側の元に関しては線型性が成り立つ。
つまり、(u+v)○w=u○w+v○w   (ku)○w=k(u○w) k:スカラー


注意 「合成積」「○」という言葉・記号は、ここで便宜的に使っているだけで、一般的にそのような言葉が使われているわけではない。


【微分と積・合成積の関係】

よく知られた微分の性質 (f*g)'=f'*g+f*g'  (f(g))'=f'(g)*g' などに対応するのは

D(u*v)=Du*v+u*Dv  D(u○v)=(Du)○(Dv) などだが、これは容易に証明できる。

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