referral request – higher order domestic products orthonormal base

Let $$pi$$ to be a measure of probability on a space $$mathcal {X}$$and leave $$Phi = { phi_k } _ {k geqslant 0}$$ be an orthonormal base (possibly with complex value) for $$L ^ 2 ( pi)$$, with $$phi_0 equiv 1$$. Let $$f in L ^ 2 ( pi)$$ be expressible on this basis as $$f = sum_ {k geqslant 0} f_k phi_k$$.

In some calculations, it became relevant for me to process the quantities (and linked) of the form

begin {align} Q ^ k = int pi (dx) | f (x) | ^ 2 | phi_k (x) | ^ 2 quad text {for} k geqslant 1, end {align}

and ideally I would like a form limit $$Q ^ k leqslant c ^ k sum_ {k geqslant 0} | f_k | ^ 2$$ for an explicit sequence $${c ^ k } _ {k geqslant 0}$$.

In principle, $$Q ^ k$$ is a quadratic form in the $${f_k } _ {k geqslant 0}$$, so it should be useful. However, the nature of this quadratic form is generally somewhat mysterious; it should end up involving quantities like

begin {align} Q ^ k_ {ij} & = int pi (dx) phi_i (x) overline { phi} _j (x) | phi_k (x) | ^ 2 \ & = int pi (dx) left ( phi_i (x) overline { phi} _k (x) right) cdot overline { left ( phi_j (x) overline { phi} _k (x) right)} quad text {for} i, j geqslant 0 end {align}

and these should depend quite strongly on the properties of the base $$Phi$$.

In the end, I think that a solution to this problem will only be possible once a base has been set, and that it will be something relatively treatable. In the case where $$Phi$$ is a Fourier basis, for example, things are pretty good, and we can take $$c ^ k equiv 1$$. I think it could also be possible in other cases where $$phi_a overline { phi_b}$$ can be written as linear combinations of others $$phi_c$$; beyond that, it could be quite tricky.

My question is: are there other orthonormal bases for which the limits of this form should be traceable? I would be particularly happy if there were families of classical orthogonal polynomials for which this is possible, but I don't know where to look for such results. All relevant references will also be received with pleasure.

orthogonality – Show that a finite set \$ B = left {1, x, x ^ 2 right } \$ is an orthonormal system with respect to the interior product

Here's my problem:

Show that a finite set $$B = left {1, x, x ^ 2 right }$$ is a system orthonormal to the inner product $$left langle f, g right rangle = int _ {- 1} ^ 1 : f left (t right) cdot g left (t right) dt$$ for all $$f, g in L ^ 2 left (-1, 1 right)$$.

And then there is a clue: assess $$left langle 1, x right rangle, left langle 1, x ^ 2 right rangle and left langle x, x ^ 2 right rangle$$. As far as I'm concerned, these are the indicators of orthogonality (if they are all zero then the whole is orthogonal). My results are consecutive $$2x$$, $$2x ^ 2$$, and $$2x ^ 3$$.

I don't understand – they are only worth zero if $$x = 0$$. Does this mean that the whole is not orthogonal – and therefore cannot be orthonormal? How to proceed with this problem?

Math – Find a quaternion of rotation from an orthonormal basis?

Given three unitary 3D vectors $$a$$, $$b$$, $$c$$ such as:

$$a times b = c$$

$$b times c = a$$

$$c times a = b$$

(that is to say $$a, b, c$$ for men orthonormal basis)

How do you calculate a unit quartile $$q$$ so that the product of the sand width (i.e. rotation) of $$q$$ the (1,0,0) is $$a$$, (0,1,0) is $$b$$ and (0,0,1) is $$c$$ ?

functional analysis – Orthonormal basis in the spectral theorem of compact operators in Hilbert spaces

I started to study compact operators on Hilbert spaces. I read the proof of the spectral theorem (taken from page 12 of https://www.iith.ac.in/~rameshg/NITKworkshop.pdf) and wondered if it was possible to complete by proving that the system of orthonormal vectors which appear is, in fact, an orthonormal basis.

Could someone help me? I do not see how.

linear algebra – Composition of the simetric matrix, an orthonormal basis of V is \$ {XD ^ {^ { frac {-1} {2}}} e_ {1}, …, XD ^ {^ { frac {-1} {2}}} e_ {n} } \$?

I saw this question

Let $$V = mathbb {R} ^ {n}$$ vector space, $$Q$$ is a definitive symmetric and positive matrix, it breaks down $$Q = XDX ^ {T}$$ gives an orthonormal basis for $$v$$ given by the columns put on the scale of $$X$$to know $${XD ^ {^ { frac {-1} {2}}} e_ {1}, …, XD ^ {^ { frac {-1} {2}}} e_ {n} }$$.

My attempt:
I know if $$Q$$ is symmetrical then diagonalisable, there is a base consisting of eigenvectors of $$Q$$, but I do not know what it looks like$$big ( {XD ^ {^ { frac {-1} {2}}} e_ {1}, …, XD ^ {^ { frac {-1} {2}}} e_ {n } } big)$$it is very general. Any help with the proof, thanks in advance

fa.functional analysis – Reproduction of the kernel and orthonormal basis of a multidimensional Sobolev space of different orders

Let $$Omega$$ to be an open subset of $$mathbb {R} ^ d$$. In regular conditions, we know that the $$s$$order of Sobolev $$H ^ s ( Omega)$$ with $$s geq d / 2$$ is a Hilbert space with reproductive core. On the other hand, $$H ^ s ( Omega)$$ with $$s is only a Hilbert space without the reproduction property.

My question concerns the construction of the orthonormal basis of $$H ^ s ( Omega)$$.

For $$s geq d / 2$$the proper function of the reproductive nucleus gives us an orthonormal basis which, up to a resizing of the quantities, is also an orthonormal basis of $$L ^ 2 ( Omega)$$ (which can be written in an equivalent way as $$H ^ 0 ( Omega)$$).

For $$s how can we build an orthonormal basis in the same way, since there is no longer a reproducible core? In addition, for $$0 leq s_1, s_2 , is it possible to align the orthonormal bases of $$H ^ {s_1} ( Omega)$$ and $$H ^ {s_2} ( Omega)$$ so that they differ only until scaling up the magnitudes?