Ascoldcaves

Euclidian metric spaces

Euclidian metric spaces are 1,2,3,…,n dimensional, where n is a natural number. In this article Euclidian metric spaces are extended to spaces with a complex number of dimensions or even non number at all. One can insert in place of n natural, complex or another type of quantity of dimensions and obtain the corresponding type of Euclidian metric space. For example 1.5 dimensional space. Something between line and plane.

Let there be an open, connected region $S$ in complex plane which includes points 1 and $c+1$. Let there be a set $R^{c}$ of functions $f(z)$ defined in the region $S$. Let there be the following norm, distance and scalar product defined for the functions $f(z)$ of the set $R^{c}$ :

$\|f\|={\sqrt {{\overline {\text{sgn}}}_{f}\sum _{x=1}^{c}f(x){\overline {f(x)}}}}.$

$\rho (f,g)={\sqrt {{\overline {\text{sgn}}}_{g-f}\sum _{x=1}^{c}{\big (}g(x)-f(x){\big )}{\overline {{\big (}g(x)-f(x){\big )}}}}}$

$(f,g)={\sqrt {{\overline {\text{sgn}}}_{f}{\overline {\text{sgn}}}_{g}}}\sum _{x=1}^{c}f(x){\overline {g(x)}},\,\$ where ${\text{sgn}}_{f}={\frac {\sum \limits _{x=1}^{c}f(x){\overline {f(x)}}}{\left|\sum \limits _{x=1}^{c}f(x){\overline {f(x)}}\right|}}$

More details one can see at www.oddmaths.info.

Summation in the case of analytical functions is taken with the Caves summation formula for indefinite sum:

Summation

Indefinite sum

$\sum _{k}^{x-1}\varphi (k+z)=\sum _{\nu =0}^{\infty }{\frac {B_{\nu }-A_{\nu }}{\nu !}}\varphi ^{(\nu -1)}(x+z)-$

$-\int \limits _{0}^{x}{\frac {A_{N}'(x-\xi )}{N!}}\varphi ^{(N-1)}(z+\xi )d\xi -\int \limits _{0}^{x}\sum _{m=0}^{\infty }\varphi ^{(N+m)}(z+\xi ){\begin{array}{l}k_{N+1+m}\\A_{N+m}(x-\xi )\\k_{N+m}+1\end{array}}d\xi +H(x,z)=$

$=F(x,z,N)-f(x,z,N)-f\varepsilon (x,z,N),\,\lim _{N\to \infty }f\varepsilon (x,z,N)=0\,$

where$A_{\nu }=0,\nu =0,1,2,...N-1\,$ and periodical function with the period one $\,\,\,\ \ H(x,z)=$

$=\int \limits ^{\zeta =0}\int \limits _{0}^{x}\left({\frac {A''_{N}(\xi +\zeta )}{N!}}\varphi ^{(N-1)}(z-\zeta )\sum _{m=0}^{\infty }\varphi ^{(N+m)}(z-\zeta ){\begin{array}{l}k_{N+1+m}\\A_{N+m}'(\xi +\zeta )\\k_{N+m}+1\end{array}}\right)d\xi d\zeta =$

$=h_{N}(x,z)+h\varepsilon _{N}(x,z),{\text{ and }}\lim _{N\to \infty }h\varepsilon _{N}(x,z)=0,$

where $z\,$ is a parameter, $B_{\nu }\,$ are Bernoulli numbers and

$A_{N}'(\alpha )={\begin{cases}\displaystyle {2(-1)^{\lfloor {\frac {N}{2}}\rfloor +1}N!\sum _{k=1}^{k_{N}}{\frac {-\sin 2\pi k\alpha }{(2\pi k)^{N-1}}}},&{\text{when N is even}}\\.\\\displaystyle {2(-1)^{\lfloor {\frac {N}{2}}\rfloor +1}N!\sum _{k=1}^{k_{N}}{\frac {\cos 2\pi k\alpha }{(2\pi k)^{N-1}}}},&{\text{when N is odd}}\end{cases}}$

$A_{\nu }(0)=A_{\nu }\ \ {\text{and}}$ $\left.{\begin{array}{l}k_{N+1+m}\\A_{N+m}(x)\\k_{N+m}+1\end{array}}\right.={\begin{cases}\displaystyle {2(-1)^{\lfloor {\frac {N+m}{2}}\rfloor +1}}\sum _{k=k_{N+m}+1}^{k_{N+1+m}}{\frac {\cos 2\pi kx}{(2\pi k)^{N+m}}},&{\text{when N+m even}}\\.\\\displaystyle {2(-1)^{\lfloor {\frac {N+m}{2}}\rfloor +1}}\sum _{k=k_{N+m}+1}^{k_{N+1+m}}{\frac {\sin 2\pi kx}{(2\pi k)^{N+m}}},&{\text{when N+m odd}}\end{cases}}$

The Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {floor}}} of $x$ ($x$ is real) $\lfloor x\rfloor$ is the largest integer less then $x$. The boundaries of summation $k_{\nu }$ are determined for example from the folloving condition

$|B_{\nu }(x)-A_{\nu }(x)|\leq {\frac {1}{\nu !\nu ^{\nu }}},\ 0\leq x\leq 1\ \,\ \$ or $\quad \,\ |B_{\nu }(x)-A_{\nu }(x)|=r$ where $r$ is a constant.

$k_{\nu }\$ are chosen the least that satisfy the inequality.

Definite sum is defined as:

$\sum _{k=a}^{x-1}\varphi (k+z)=\sum _{k}^{x-1}\varphi (k+z)-\sum _{k}^{a}\varphi (k+z)$

More details one can see at www.oddmaths.info/indefinitesum.

Summation of non-analytical functions

Let there be a set $S$ of functions $\varphi (x)$ such that a $\varphi (x)\in S$ streams to zero when $|x|$ streams to infinity faster then any power of the inverse of $x$, i.e. $\displaystyle {\lim _{|x|\to \infty }\varphi (x)x^{n}}=0$ for any $n=1,2,\dots$ . The set $S$ is the space of basic functions. Let there on the space of basic functions be defined a functional

$(f,\varphi )=\sum _{k=-\infty }^{\infty }f(k)\varphi (k)$

The functional of finite difference of a function $f(x)$ is defined as follows:

$(\nabla f,\varphi )=-(f,\Delta \varphi )$ where $\nabla f(x)=\Delta f(x-1).$

Definition of the functional of the sum of a function $f(x)$.

A function $\Psi (x)=\Delta \varphi (x)\,$ belongs to the space of basic functions $S$. First I define the functional of sum on the functions $\Psi (x)\in S_{1}\subset S$. From the previous result $(f(x),\Delta \varphi (x))=-(\Delta f(x-1),\varphi (x))\,$ therefore

$\left(\sum _{k}^{x}f(k),\Psi \right)=-{\big (}f(x),$ $\sum$ $\Psi {\big )}$

where $\sum \Psi$ is an indefinite sum of $\Psi (x)\,$. For the rest functions $\varphi (x)\notin S_{1}$ I choose

$\left(\sum _{k}^{x}f(k),\varphi \right)=0.$

Therefore the functional is defined on the entire space $S.$

Examples

Heaviside function of the second type $U(x)=\Theta (x)\,$ and Dirac delta function of the second type $\delta (x)\,$

$(\Theta ,\varphi )=\sum _{k=-\infty }^{\infty }\Theta (k)\varphi (k)=\sum _{k=0}^{\infty }\varphi (k)\,,$ and $(\delta ,\varphi )=\sum _{k=-\infty }^{\infty }\delta (k)\varphi (k)=\varphi (0),$

or their shifted forms

$(\Theta ,\varphi )=\sum _{k=-\infty }^{\infty }\Theta (k-x_{0})\varphi (k)=\sum _{k=x_{0}}^{\infty }\varphi (k),$ and $(\delta ,\varphi )=\sum _{k=-\infty }^{\infty }\delta (k-x_{0})\varphi (k)=\varphi (x_{0}).$

Summation with non-number boundaries

Let $A,\ B,\$ zero matrix Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle \ \,{\emph {0}}\ ,} and identity matrix $I\ \$ are {$n\times n$} matrices. $U\ \$ is {$n\times n$} orthonormal matrix with $n\,$ orthonormal vectors and $|U|=1\,$. Let $A=UD_{a}U^{*},\ B=UD_{b}U^{*}$, where $U^{*}\,$ is Hermitian conjugate of matrix $U\,$ and

$D_{a}={\begin{pmatrix}d_{a,1}&0&\cdots &0\\0&d_{a,2}&\cdots &0\\...&...&...&...\\0&0&\cdots &d_{a,n}\end{pmatrix}},\ \ D_{b}={\begin{pmatrix}d_{b,1}&0&\cdots &0\\0&d_{b,2}&\cdots &0\\...&...&...&...\\0&0&\cdots &d_{b,n}\\\end{pmatrix}},$

then by definition

$\sum _{k=A}^{B}\varphi (k)=\sum _{k=UD_{a}U^{*}}^{UD_{b}U^{*}}\varphi (k)=U\left(\sum _{k=D_{a}}^{D_{b}}\varphi (k)\right)U^{*}=$

$=U{\begin{pmatrix}\displaystyle {\sum _{k=d_{a,1}}^{d_{b,1}}\varphi (k)}&0&0&\cdots &0\\0&\displaystyle {\sum _{k=d_{a,2}}^{d_{b,2}}\varphi (k)}&0&\cdots &0\\0&0&\displaystyle {\sum _{k=d_{a,3}}^{d_{b,3}}\varphi (k)}&\cdots &0\\...&...&...&\cdots &...\\0&0&0&\cdots &\displaystyle {\sum _{k=d_{a,n}}^{d_{b,n}}\varphi (k)}\end{pmatrix}}U^{*}$

If $A=U_{a}D_{a}U_{a}^{*},\ B=U_{b}D_{b}U_{b}^{*},$ then

$\sum _{k=A}^{B}\varphi (k)=\sum _{k=A=U_{a}D_{a}U_{a}^{*}}^{U_{a}0U_{a}^{*}}\varphi (k)+\sum _{k=U_{b}IU_{b}^{*}}^{U_{b}D_{b}U_{b}^{*}}\varphi (k)$

--Ascoldcaves (talk) 00:37, 3 November 2011 (UTC)