end{aligned} (32) Now, suppose that (f(x)) is an arbitrary square-integrable function on the intervals (Omega_{i}) ((i=1,2,ldots,n+1)).
The Shannon sampling theorem asserts that a continuous square-integrable function on the real line which has a compactly supported Fourier transform is uniquely determined by its restriction to a uniform lattice of points whose density is determined by the support of the Fourier transform.
The second derivative (Omega t)) is a square-integrable function on ([0,1]), i.e., (Omega t)in L_{2}[0,1]); The second derivative (Omega t)) is bounded on ([0,1]), i.e., (vert Omega^{primeprime}(t) vert leqell) for some constant ℓ.
When the Hilbert space is the space L2(D) of square-integrable functions on a domain D, the quantity: :\varphi_n \varphi_n^\dagger, is an integral operator, and the expression for f can be rewritten as: :f(x) = \sum_{n=1}^\infty \int_D\, \left( \varphi_n (x) \varphi_n^*(\xi \right) f(\xi) \, d \xi.
At this stage, Theorem 1 can be invoked to bound the inner integrals, after noting that u 2, l, u 3, l, u 4, l, v l are all square-integrable functions on ℝ. Apropos of the r.v.'s X, Y, Z and W, one can start by noting that the bounds | Y | ≤ M 3 n, | W | ≤ M 4 n are in force directly from the definition of these random quantities.
Let L2 be the Lebesgue space of square-integrable functions on the unit circle.
He began with the space of square-integrable functions on the real line.
Here the one particle space is (L^{2}(mathbb{R})), the space of complex-valued square-integrable functions on (mathbb{R}).
Let (V=L^{2}[0,1]) be the vector space of real square-integrable functions on ([0,1]) with inner product langle f,grangle= int_{0}^{1}f(x g(x),dx. (4.12).
This last element is interpreted in terms of the irreducible factors in the Plancherel decomposition of the Hilbert space of square-integrable functions on the Euclidean plane, the sphere, and the hyperbolic plane, respectively.
The Hilbertian tensor product of two copies of L2 [0, 1]) is isometrically and linearly isomorphic to the space L2 [0, 1]2) of square-integrable functions on the square [0, 1]2.
