Symbols:Real Analysis

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Symbols used in Real Analysis

Convolution Integral

$\map f t * \map g t$


Let $f$ and $g$ be real functions which are integrable.

The convolution integral of $f$ and $g$ is defined as:

$\displaystyle \map f t * \map g t := \int_{-\infty}^\infty \map f u \map g {t - u} \rd u$


The $\LaTeX$ code for \(\map f t * \map g t\) is \map f t * \map g t .


Convolution of Real Sequences

$\sequence {f_i} * \sequence {g_i}$


Let $\sequence f$ and $\sequence g$ be real sequences.

The convolution of $f$ and $g$ is defined as:

$\displaystyle \sequence {f_i} * \sequence {g_i} := \sum_{j \mathop \in \Z_{\ge 0} } f_i g_{i - j}$


The $\LaTeX$ code for \(\sequence {f_i} * \sequence {g_i}\) is \sequence {f_i} * \sequence {g_i} .


Cross-Correlation Integral

$\map f t \star \map g t$


The cross-correlation of $f$ and $g$ is defined as:

$\displaystyle \map f t \star \map g t := \int_{-\infty}^\infty \map f u \map g {t + u} \rd u$


The $\LaTeX$ code for \(\map f t \star \map g t\) is \map f t \star \map g t .


Symbols commonly used in both Real Analysis and Number Theory

Ceiling

$\ceiling x$

The ceiling function of $x$: the smallest integer greater than or equal to $x$.


The $\LaTeX$ code for \(\ceiling x\) is \ceiling x .


Floor

$\floor x$

The floor function of $x$: for $x \in \R$, the greatest integer less than or equal to $x$.


The $\LaTeX$ code for \(\floor x\) is \floor x .


Nearest Integer

$\nint x$


The nearest integer function is defined as:

$\forall x \in \R: \nint x = \begin {cases} \floor {x + \dfrac 1 2} & : x \notin 2 \Z + \dfrac 1 2 \\ x - \dfrac 1 2 & : x \in 2 \Z + \dfrac 1 2 \end{cases}$

where $\floor x$ is the floor function.


The $\LaTeX$ code for \(\nint x\) is \nint x .