We claim that we can express f t as the superposition

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: piecewise smooth functions which are periodic of period 2π . We say that V is a vector space because elements of V can be added and multiplied by scalars, these operations satisfying the same rules as those for addition of ordinary vectors and multiplication of ordinary vectors by scalars. We define an “inner product” between elements of V by means of the formula f, g = 1 π π f (t)g (t)dt. −π Thus for example, if f (t) = sin t and g (t) = 2 cos t, then f, g = 1 π π π 2 sin t cos tdt = −π −π sin(2t)dt = − cos(2t)|π π = 0. − The remarkable fact is that this inner product has properties quite similar to those of the standard dot product on R n : • f , g = g , f , whenever f and g are elements of V . • f + g, h = f , h + g , h . • cf, g = c f , g , when c is a real constant. • f , f ≥ 0, with equality holding only if f = 0 (at all points of continuity). This suggests that we might use geometric terminology for elements of V just as we did for vectors in R n . Thus, for example, we will say that an...
View Full Document

This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online