Yes, the sum of these potentials is zero everywhere with the true ground-state density. But the true ground-state density is generally only expressible when the basis set is complete (if you know the true ground-state density in advance, you can of course construct a finite basis set that reprodu...
I know that a set of functions are said to form a complete basis on an inteval if any function on that interval can be expressed as a linear combination of the functions in the set. I also know that every function in the set are orthogonal. Now what is what is the condition(s) that a set of fun...
I don't know if it's true but I think in a finite dimensional veccter space, an orthonormal set which is complete becomes a basis. But in a Hilbert space, the books say that the set of finite linear combinations of the elements in a complete orthonormal set is only dense in the Hilbert space. So ...
I am reading the technical report by Leo Breimann entitled "Some Infinity Theory for Predictor Ensembles". On page 4 he has the following: Definition 1 Let $L_2(P)$ be the space of functions on $\mathbb{R}^D$ that are square - integrable wrt $P(dx)$. A set of functions $F$ in $L_2(P)$ will ...
If I start with the set of functions $e^{-nx}$ for all integers $n>1$, can I use them as basis to create a complete set of orthogonal functions on the interval $(0,+\infty)$? By complete I mean that they can be used in an expansion that will converge in the mean to any square integrable function...
Today I was working on solving a partial differential equation in the unit square, $[0,1] \times [0,1].$ Using the method of separation of variables on the spatial problem gave solutions of the form $f_i(x) g_j(y)$, where $\{f_i\}$ and $\{g_j\}$ each form complete orthonormal families of function...
The following excerpt is from the book Operators, Functions, and Systems, Vol. 1 by Nikolai K. Nikolski, pages 216-217 What is a "complete set of vectors" in the proof of Lemma 2.4.4? Im putting my money on "A subset of the basis which is linearly independent and whose span is dense is ca...
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