# Symbols:Real Analysis

## 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 .