By G. Hardy

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25) If moreover, f, 9 E L2 the symmetric equation is also valid: F(fg) = * F(g). 26) These equations hold for the conjugated transform F* as well. 25) to tions F(f)r, F(g)r. These yields 1 exp( -j f(x - y)g(y)dydx 1 exp( -j (~, y) )f(z)g(y)dydz 1 exp( -j (~, y) )g(y)dy. = x - y is eligible, since the integrand belongs to x V). The right side is equal to F(f)F(g). 7. 16. Similarly F*(F(g)r) -+ 9 in mean. 27) converges to the function f 9 in the space L1 by the triangle inequality. Apply the Fourier transform to both sides and get uniform convergence of the sequence FF*(F(f)r * F(g)r) -+ f g.

A Euclidean space has zero curvature. Any straight line is a geodesic and vice versa. An elliptic space of dimension n is the real projective space JP>n = sn /Z2 with the metric inherited from the unit sphere sn c E where E is a Euclidean space of dimension n+ 1. The sectional curvature of the elliptic space is equal everywhere to 1. For a subspace FeE of dimension two, the intersection F n sn is a big circle; its image in the elliptic space JP>n is a closed geodesic curve 'Y. For an arbitrary subspace F the manifold Y ~ F n sn /Z2 is a projective subspace.

IR for simplicity. Take a finite interval, say I = [a, b] c IR and denote by hI the indicator of this interval. 21) for f = hI,g = hJ for arbitrary intervals I, J. We have , Ib hI = a 1 exp( ~j~x)dx = - j~ [exp (-jb~) - exp (-ja~)] = _ exp (_j~a + b) sin(7r~(b 2 7r~ a)) . 22 Chapter 1. 8). Similarly for J = [c, d] we have There is no pole at the point ~ = 0 and we can integrate over the line ~ = 'f/ - z. The first term in the bracket gives zero after integration, if d - b < 0, since the function exp (j (d - b)~) decreases rapidly in the bottom half-plane.