Best Approximation

Definition

1. With $W$ as a Subspace of a Inner Product Space $V$
2. With Vector $\beta \in V$
3. The best approximation to $\beta$ by vectors $V$ is the vector $\alpha \in W$ such that $||\beta -\alpha ||<+||\beta -\gamma ||,\forall \gamma \in W$

Alternate Definitions

• $||\beta -\alpha ||=mi{n}_{\forall \gamma \in W}||\beta -\gamma ||$
• $d(\beta ,\alpha )=mi{n}_{\forall \gamma \in W}d(\beta ,\gamma )$
• The Orthogonal Projection of vector $\beta$ onto the Subspace $W$, is the same as the best approximation

Formula

If $W$ is in finite dimension, and $\{{\alpha }_1,\dots ,{\alpha }_n\}$ is any Orthogonal Basis for $W$. Then, $\alpha ={\sum }_{i=1}^n\frac{⟨\alpha |{\alpha }_k⟩}{||{\alpha }_k|{|}^2}\cdot {\alpha }_k$

Theorems

• Best Approximations are Characterized by Orthogonality
• Best Approximations Computed from Formula
• Best Approximations are Unique