Vector space V and Z are Ndimensional with e0en1 as an ortho

Vector space V and Z are N-dimensional with {e_?0,...e(n-1)} as an orthogonal basis.

For two subsets A, B Z(N) let A + B = {a + b; a epsilon A, b epsilon B}. (This is the addition modulo N.) Show that for any two functions f, g epsilon V(N) supp(f * g) supp(f) + supp(g).

Solution

Let V be an inner product space. Two vectors u, v V are said to be orthogonal if hu, vi = 0. Example 6.1. For inner product space C[, ], the functions sin t and cost are orthogonal as hsin t, costi = Z sin t cost dt = 1 2 sin2 t ¯ ¯ ¯ = 0 0 = 0. Example 6.2. Let u = [a1, a2, . . . , an] T R n. The set of all vector of the Euclidean n-space R n that are orthogonal to u is a subspace of R n. In fact, it is the solution space of the single linear equation hu, xi = a1x1 + a2x2 + · · · + anxn = 0. Example 6.3. Let u = [1, 2, 3, 4, 5]T , v = [2, 3, 4, 5, 6]T , and w = [1, 2, 3, 3, 2]T R 5 . The set of all vectors of R 5 that are orthogonal to u, v, w is a subspace of R 5 . In fact, it is the solution space of the linear system x1 + 2x2 + 3x3 + 4x4 + 5x5 = 0 2x1 + 3x2 + 4x3 + 5x4 + 6x5 = 0 x1 + 2x2 + 3x3 + 3x4 + 2x5 = 0 Let S be a nonempty subset of an inner product space V . We denote by S the set of all vectors of V that are orthogonal to every vector of S, called the orthogonal complement of S in V . In notation, S := n v V ¯ ¯ hv,ui = 0 for all u S o . If S contains only one vector u, we write u = n v V ¯ ¯ hv,ui = 0o .