...all odd functions are clearly subspaces of T. We claim that Т = ,Feven 0 Fodd. In fact, an arbitrary function / can be written as a sum of an even and an odd function; namely, / = /t+/2 where /,(z) := \(f(x)+f(-x)) and/2(x) := |(/(x)-/(-x)). On the other hand, if / is...

...function. Proof. at Remark. The same holds for odd functions. Theorem. An arbitrary function u can always be written as a sum of an even and an odd function. Proof. Using the identity _ 1 1 we find that u(q) + u(-q) is an even function and u(q) - u(-q) is an...

...Fourier transform of f(x) with even and odd parts given by E and O, respectively, then Note that e~tk*x can be written as a sum of an even and an odd function in x. 1.4 Write the convolution theorem for ^ /j^ F(kx)G(kx)H(kx)eik'xdkx. 1.5 The zero-order Hankel...

...involved powers NI,..., Nm. Using V.3.1, g(z) is a polynomial in p(z), and we are done. D Any elliptic function can be written as a sum of an even and an odd function, both elliptic: because together with z i—> f(z) also z \^ f(-z) is an elliptic function. Let's look...

...interval. Most functions are neither even nor odd, but any function that is defined on a symmetric interval can be written as a sum of an even and an odd function: f(x) = It is easy to show that the first term is an even function and the second is odd. Even and odd functions...