Title: Theory / Important / Math / Analysis / Zorich{ / 18 Fourier=square integrable
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On every note of this deck, unless a note says otherwise:
Chapter: 18

Name: Orthogonal vectors
Definition: \(x,y\in X\) such that \(\langle x,y\rangle=0\)
Page: 494
Section: 1a
Variables: \(X\) a vector space with inner product
Index: 1
Kind: Def

Name: An orthogonal system
Definition: A set \(K\) of vectors such that for each \(x,y\in K\) distinct, \(x\) is orthogonal to \(y\)
Page: 494-496
Section: 1a
Variables: \(X\) a vector space with inner product
Index: 2, 3
Kind: Def, ex
Example: the trigonometric system (in complex notation or not) for \(\mathcal ([-\pi,\pi],\mathbb C)\)
Example2: \(\sin{\left(\frac{\xi_n x}{l}\right)}\) for \(\xi_n\) a family of roots of \(\tan {\xi l}=c\xi\), for \(c\) fixed. For \(\mathcal ([0,l],\mathbb C)\)
Denoted by: \(K\), \(\{l_i\}\)
Example3: The \(i\)-th unit vector for \(\mathbb R^n\) or \(\mathbb R^{\mathbb N}\)

Name: An orthonormal system
Definition: An orthogonal system \(K\) such that \(\langle x,x\rangle=1\) for each \(x\in K\)
Page: 495
Section: 1a
Variables: \(X\) a vector space with inner product
Index: 3
Kind: Def
Example: \(\left\{\frac{1}{\sqrt{2\pi}},\frac{\cos nx}{\sqrt{\pi}},\frac{\sin nx}{\sqrt{\pi}}\mid n>0\right\}\) for \(\mathcal ([-\pi,\pi],\mathbb C)\)
Name2: An orthonormalized system
Denoted by: \(\{e_i\}\)

Name: A linearly independant system
Definition: A set \(K\) such that, for each \(\sum_{x\in K}\alpha_xx=0\) with \(\alpha_x\in\mathbb F\) finite support, \(\alpha_x=0\).
Page: 494
Section: 1a
Variables: \(\mathbb F\) a field,
\(X\) a \(\mathbb F\)-vector space
Index: 4
Kind: Def

Definition: The set of \(f:X\to Y\) that are locally \(p\)-integrable
Page: 494
Section: 1b
Variables: \(X, Y\) vector spaces with inner product,
\(p\ge1\)
Notation: \(\mathcal R_p(X,Y)\)
Kind: Def

Definition: [$](f,g)\mapsto\int_X (f\cdot\bar g)(x)\dd x[/$]
Page: 495
Section: 1b
Chapter: 18
Variables: \(f,g\in\mathcal R_2(\mathbb R^n,\mathbb C)\)
Notation: \(\langle f,g\rangle\)
Index: Def
Assuming: \(f,g\) locally square-integrable
Continuous: Continuous function, in \(\mathcal R_2(X,\mathbb C)\)

Question: [$]\int_{-\pi}^\pi e^{imx}\cdot e^{-inx}\dd x[/$]
Answer: equals \(0\) if \(m\ne n\)
equals \(2\pi\) if \(m=n\)
Page: 495
Chapter: 18
Section: 1b
Variables: \(m,n\in\mathbb N\)
Kind: Def

Question: [$]\int_{-\pi}^\pi \cos {mx}\cos {nx}\dd x[/$]
Answer: equals \(0\) if \(m\ne n\)
equals \(\pi\) if \(m=n\ne0\)
equals \(2\pi\) if \(m=n=0\)
Page: 495
Chapter: 18
Section: 1b
Variables: \(m,n\in\mathbb N\)
Kind: Def

Question: [$]\int_{-\pi}^\pi\cos {mx}\sin {nx}\dd x[/$]
Answer: equals \(0\)
Page: 495
Chapter: 18
Section: 1b
Variables: \(m,n\in\mathbb N\)
Kind: Def

Question: [$]\int_{-\pi}^\pi \sin {mx}\sin {nx}\dd x[/$]
Answer: equals \(0\) if \(m\ne n\)
equals \(\pi\) if \(m=n\ne 0\)
equals \(2\pi\) if \(m=n= 0\)
Page: 495
Chapter: 18
Section: 1b
Variables: \(m,n\in\mathbb N\)
Kind: Def

Name: The trigonometric system
Definition: \(\{\cos nx,\sin nx\mid n\in\mathbb N\}\)
Page: 495
Section: 1b
Index: 1
Kind: Ex
Typ: An orthogonal system for \(\mathcal R_2([-\pi,\pi],\mathbb C)\)

Name: The trigonometric system in complex notation
Definition: \(\{e^{inx}\mid n\in\mathbb Z\}\)
Page: 495
Section: 1b
Index: 1
Kind: Ex
Typ: An orthogonal system for \(\mathcal R_2([-\pi,\pi],\mathbb C)\)

Name: Set of orthonormal systems
Page: 496
Section: 1b
Chapter: 18
Index: 2
Kind: Ex
Under: cartesian products on domain and product on values.
Under2: subset

Page: 497
Implementation: Let \((\psi_i)_{i\in I}\) an enumeration of the elements of \(K\).
Let [$]\phi_i=\frac{\psi_i-\sum_{k=1}^{n-1}\langle\psi_i,\phi_k\rangle\phi_k}{\norm{\psi_i-\sum_{k=1}^{n-1}\langle\psi_i,\phi_k\rangle\phi_k}}[/$]. Take \(\left\{\phi_i\mid i\in I\right\}\)
Section: 1c
Chapter: 18
Variables: \(X\) a vector space with inner product
\(K\) a linear independent system of vectors
Name: Gram-Schmidt process
Returns: Obtaining an orthonormal system wich spans the same set than \(K\)
Assuming: \(K\) at most countable

Name: The system of Legendre polynomials
Definition: The result of applying Gramm-Schmidt procedure to \(\{x^i\mid i\in\mathbb N\}\) in \(\mathcal R_{2}([-1,1],\mathbb R)\)
Page: 497
Section: 1c
Index: 5
Kind: Ex

Name: The standard \(n\)-th legendre polynomial
Definition: [$]\frac{1}{n!2^n}\dv[n]{(x^2-1)^n}{x}[/$]
Page: 498
Section: 1c
Variables: \(n\in\mathbb N\)
Notation: \(P_n\)
Index: 5
Kind: Ex
Extra: Checked twice

Definition: Normalized version of \(P_n\). I.e. \(P_n\), divided by its leading coefficient
Page: 498
Section: 1c
Variables: \(n\in\mathbb N\)
Context: Fourier theory
Notation: \(\tilde P_n\)
Index: 5
Kind: Ex

Name: The orthonormalized Legendre polynomials
Definition: \(\sqrt{(n+.5)}P_n(x)\)
Page: 498
Section: 1c
Variables: \(n\in\mathbb N\)
Notation: \(\hat P_n\)
Index: 5
Kind: Ex

Question: \(P_n(1)\)
Answer: equals \(1\)
Page: 498
Chapter: 18
Section: 1c
Variables: \(n\in\mathbb N\)
\(P_n\) the standard Legendre polynomials
Index: 5
Kind: Ex

Smaller0: \(\langle\sum x_i,y\rangle\)
Greater0: \(\sum\langle x_i,y\rangle\)
Page: 499
Chapter: 18
Section: 1d
Variables: \(X\) a vector space with inner product \(\langle \cdot,\cdot\rangle\)
\(y,(x_i)\in X\)
Index: 1b
Kind: Lem
Smaller to greater: =
Assuming: The sum converge

Smaller0: \(\langle\sum x_ie_i,\sum y_ie_i\rangle\)
Greater0: \(\sum x_i\bar{y_i}\)
Page: 499
Chapter: 18
Section: 1d
Variables: \(X\) a complex vector space with inner product \(\langle \cdot,\cdot\rangle\),
\((e_i)\in X\) an orthonormal system
\((x_i,y_i)\in \mathbb C\)
Index: 1c
Kind: Lem
Smaller to greater: =
Assuming: The sums converge

Smaller0: \((x_i)\) is a system of mutually orthogonal vectors
Greater0: [$]\norm{\sum x_i}^2[/$]
Page: 499
Chapter: 18
Section: 1d
Variables: \(X\) a complex vector space with inner product \(\langle \cdot,\cdot\rangle\),
\((x_i)\in X\)
Index: 1a
Kind: Theo
Greater1: [$]\sum\norm{x_i^2}[/$]
Connect Greater0: =

Smaller0: \((e_i)\) is a system of mutually orthogonal vectors
Greater0: [$]\norm{\sum x_ie_i}^2[/$]
Page: 499
Chapter: 18
Section: 1d
Variables: \(X\) a complex vector space with inner product \(\langle \cdot,\cdot\rangle\),
\((e_i)\in X\)
\((x_i)\in\mathbb C\)
Index: 1a
Kind: Theo
Greater1: [$]\sum\abs{x_i}^2[/$]
Connect Greater0: =

Smaller0: \(x^i\)
Greater0: \(\frac{\langle x,l_i\rangle}{\langle l_i,l_i\rangle}\)
Page: 500
Chapter: 18
Section: 1.2
Variables: \(X\) a space
\(\langle,\rangle\) an inner product over \(X\),
\(\{l_i\}\) an orthogonal system
\(x=\sum_i x^i l_i\)
Smaller to greater: =

Name: The \(n\)-th Fourier coefficients of \(x\) in \(\{l_i\}_{i\in I}\)
Definition: \(\frac{\langle x,l_n\rangle}{\langle l_n,l_n\rangle}\)
Page: 500
Section: 1.2
Variables: \(X\) a space
\(\langle,\rangle\) an inner product over \(X\)
\(\{l_i\}_{i\in I}\) an orthogonal system
\(x=\sum_{i\in I} x^i l_i\)
\(n\in I\)
Index: 5
Kind: Def
Intuition: the projection of that vector in the direction of the unit vector \(\frac{l_i}{\langle l_i,l_i\rangle}\)

Name: The Fourier series of the vector \(x\) in the orthogonal system \(\{l_k\}\)
Definition: \(\sum_i \frac{\langle x, l_i\rangle}{\langle l_i,\rangle}\)
Page: 500
Section: 1.2
Variables: \(X\) a space
\(\langle,\rangle\) an inner product over \(X\)
\(\langle l_i\rangle_{i\in\mathbb N}\)
\(x\sim\sum_i \frac{\langle x, l_i\rangle}{\langle l_i,\rangle} l_i\)
Index: 6
Kind: Def
Assuming: \(\langle l_i\rangle\) an orthogonal system of nonzero vectors

Question: In the basis \(\{1,\cos kx,\sin kx\}\), the Fourier series of \(x\mapsto x\) is
Answer: \(\sum_{k=1}^\infty(-1)^{k+1}\frac 2k\sin kx\)
Page: 501
Chapter: 18
Section: 1.2
Variables: \(X=\mathcal R_2([-\pi,\pi],\mathbb R)\)

Question: \(x-x_l\) is
Answer: orthogonal to the closure of linear space generated by \(\{l_k\}\)
Page: 502
Chapter: 18
Section: 1.2b
Variables: \(X\) a space
\(\langle,\rangle\) an inner product over \(X\)
\(\{e_i\}\) an orthonormal system,
\(\{l_i\}\) an orthogonal system
\(x\in X\)
Assuming: \(x_l\) is defined

Smaller0: [$]\sum_{k=1}^\infty\frac{\abs{\langle x,l_k\rangle}^2}{\langle l_k,l_k\rangle}[/$]
Greater0: [$]\norm{x}^2[/$]
Page: 502
Chapter: 18
Section: 1.2b
Variables: \(X\) a space
\(\langle,\rangle\) an inner product over \(X\)
\(\{e_i\}\) an orthonormal system,
\(\{l_i\}\) an orthogonal system
\(x\in X\)
Smaller to greater: \(\le\)
Name: Bessel's inequality
Greater to smaller: \(\ge\)

Definition: [$]\frac1\pi\int_{-\pi}^\pi f(x)\cos kx\dd x[/$]
Page: 501
Section: 1.2b
Variables: \(X=\mathcal R_2([-\pi,\pi],\mathbb R)\)
\(f\in X\)
\(k\in\mathbb N\)
Notation: \(a_k(f)\)
Extra: checked
Intuition: The Fourier coefficient of \(f\) in the system \(\{1,\cos kx,\sin kx\}\) related to \(\cos kx\), or its double when \(k=0\)

Definition: [$]\frac1\pi\int_{-\pi}^\pi f(x)\sin kx\dd x[/$]
Page: 501
Section: 1.2b
Variables: \(X=\mathcal R_2([-\pi,\pi],\mathbb R)\)
\(f\in X\)
Notation: \(b_k(f)\)
Extra: Checked
Intuition: The Fourier coefficient of \(f\) in the system \(\{1,\cos kx,\sin kx\}\) related to \(\sin kx\)
Assuming: \(k\in\mathbb N^{>0}\)

Definition: [$]\frac1{2\pi}\int_{-\pi}^\pi f(x)e^{ikx}\dd x[/$]
Page: 501
Section: 1.2b
Variables: \(X=\mathcal R_2([-\pi,\pi],\mathbb C)\)
\(f\in X\)
Notation: \(c_k(f)\)
Definition2: \(\frac{a_k-ib_k}2\) if \(k\ge0\),
\(\frac{a_{-k}+ib_{-k}}2\) if \(k<0\),
Intuition: The \(k\)-th Fourier coefficient of \(f\) in the system \(\{e^{ikx}\mid k\in\mathbb Z\}\)
Assuming: \(k\in\mathbb Z\)

Name: The Fourier series of the vector \(x\) in the orthogonal system \(\{l_k\}\)
Definition: \(\sum_{k=1}^\infty\frac{\langle x,l_k\rangle}{\langle l_k,l_k\rangle}l_k\)
Page: 500
Section: 1.2b
Variables: \(X\) a space
\(\langle,\rangle\) an inner product over \(X\)
\(\langle l_i\rangle_{i\in\mathbb N}\)
\(x=\sum_i x^i l_i\)
Notation: \(x_l\)
Kind: Def
Assuming: This sum converges
\(\langle l_i\rangle\) an orthogonal system of nonzero vectors

Smaller0: [$]\norm{x-x_l}[/$]
Greater0: [$]\norm{x-y}[/$]
Page: 504
Chapter: 18
Section: 1.2b
Variables: \(X\) a space
\(\langle,\rangle\) an inner product over \(X\)
\(\{e_i\}\) an orthonormal system,
\(\{l_i\}\) an orthogonal system
\(x,y\in X\)
Kind: Lemma
Smaller to greater: \(\le\)
Assuming: \(y\) belongs to the vector space generated by the \(l_k\)
Name: Extremal property of the Fourier coefficient
Greater to smaller: \(\ge\)
Construction: [$]\norm{x-y}^2=\norm{(x-x_l)+(x_l-y)}^2[/$] by orthogonal complement lemma, both vectors are perpendicular. By Pythagorean theorem, this last value is equal to [$]\norm{x-x_l}^2+\norm{x_l-y}^2 \ge \norm{x-x_l}^2[/$]

Smaller0: \(\{l_i\}\) is orthonormal
Greater0: \(\langle \sum x^il_i,l_i\rangle\)
Page: 500
Chapter: 18
Section: 1.2
Variables: \(X\) a space
\(\langle,\rangle\) an inner product over \(X\)
\(\{l_i\}\) an orthogonal system
Greater1: \(x^i\)
Connect Greater0: \(=\)

Name: A complete system of vectors with respect to the set \(E\)
Definition: every vector \(x\in E\) can be approximated with arbitrary accuracy in the sens of the norm of \(X\) by finite linear combinations of vectors of \(\{x_\alpha\}\)
Page: 505-507
Section: 1c
Variables: \(X\) a normed space
\(E\subseteq X\)
Index: 7y,
Kind: def, Theo, Prop
Definition2: \(E\subseteq\overline L\{x_\alpha\}\)
Example: \(\{x^i\mid i\in\mathbb N\}\)
Conjdef: A set \(\{x_\alpha\mid \alpha\in A\}\) such that, equivalently
Definition3: for each \(x\in E\), \(x=\sum_{\alpha\in A}\frac{\langle x,x_\alpha\rangle}{\langle x_\alpha,x_\alpha\rangle}\alpha\) assuming \(A\) at most countable
Name2: A system of vectors complete in \(E\)
Example2: X
Definition4: for each \(x\in E\), Parseval's equality holds [$]\norm{x}^2=\sum_{\alpha\in A}\frac{\abs{\langle x,x_\alpha\rangle}^2}{\langle x_\alpha,x_\alpha\rangle}\alpha[/$] assuming \(A\) at most countable
Definition5: there are no \(x\in E\setminus\{0\}\) such that \(x\bot x_\alpha\) for each \(\alpha\) (assuming \(X\) complete (Hilbert) space)
Applied to: \(C[a,b]\), the set of continuous function
Counterexample: \(\{\cos kx,\sin kx\}\) is not complete in \(\mathcal R_2([-\pi,\pi],\mathbb C)\). \(x\mapsto 1\) can not be approximated
Counterexample2: Let \(e_i\) the \(i\)-th unit vector of \(\mathbb R^n\).
The set \(\{e_i\mid i>1\}\) is not complete in \(l_2\)

Definition: The set of linear combinations of a finite numbers of \(x_\alpha\)
Page: 505
Section: 1c
Variables: \(X\) a normed space
\(\{x_\alpha\}\) a system of vectors of \(X\)
Notation: \(L\{x_\alpha\}\)

Smaller0: [$]\norm{x}^2[/$]
Greater0: [$]\sum_{\alpha}\frac{\abs{\langle x,\alpha\rangle}^2}{\langle \alpha,\alpha\rangle}\alpha[/$]
Page: 505
Chapter: 18
Section: 1c
Variables: \(X\) a normed space
\(\{x_\alpha\}\) a system of vectors of \(X\)
\(x\in X\)
Smaller to greater: =
Assuming: \(\{x_\alpha\}\) is complete in \(x\)
Name: Parseval's equality

Smaller0: \(\{x_\alpha\}\) is complete in \(X\)
Greater0: \(x=0\)
Page: 507
Chapter: 18
Section: 1c
Variables: \(X\) a vector space with inner product
\(\{x_\alpha\}\) a system of vectors of \(X\)
\(x\in X\)
Kind: Prop
Smaller1: \(x\bot x_\alpha\) for each \(\alpha\)
Extra: \(\{x_\alpha\}\) linearly independant is added as hypothesis, but not used

Name: A basis of \(X\)
Definition: each finite subsystem are linearly independent vectors
Page: 508
Section: 1c
Variables: [$](X,\norm{}_X)[/$] a normed vector space
Definition2: where each \(x\in X\) is of the form \(\sum_\alpha c_\alpha x_\alpha\), with convergence in the sens of [$]\norm{}_X[/$]. i.e. is complete
Conjdef: A set \(\{x_\alpha\}\) such that

Smaller0: \(\{x_\alpha\}\) is complete in \(X\)
Greater0: \(\{x_\alpha\}\) spans \(X\)
Page: 508
Chapter: 18
Section: 1c
Variables: [$](X,\norm{}_X)[/$] a normed vector space
\(\{x_\alpha\}\subseteq X\)
Index: , 14
Kind: , Ex
Smaller1: \(X\) is finite dimensional
Counterexample: \(\{x^i\mid i\in\mathbb N\}\) is complete in \(C_2([-1,1],\mathbb R)\) but is not a basis.
Finite sums are infinitely derivable. However continuous functions are not all infinitely derivable.

Question: Difference between Bessel and Parseval ?
Answer: Parseval assume that the system is complete, and the inequality becomes equality
Page: 505
Chapter: 18
Section: 1.

Question: Difference between a complete set of vector for \(X\) and a basis for \(X\)?
Answer: A complete set is not necesseraly linearly independant
Variables: [$](X,\norm{}_X)[/$] a normed vector space