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Present work contains a method to obtain Jackson and Stechkin typeinequalities of approximation by integral functions of finite degree (IFFD) insome variable exponent Lebesgue space of real functions defined on$\boldsymbol{R}:=\left( -\infty ,+\infty \right) $. To do this we employ atransference theorem which produce norm inequalities starting from norminequalities in $\mathcal{C}(\boldsymbol{R})$, the class of bounded uniformlycontinuous functions defined on $\boldsymbol{R}$. Let $B\subseteq\boldsymbol{R}$ be a measurable set, $p\left( x\right) :B\rightarrow \lbrack1,\infty )$ be a measurable function. For the class of functions $f$ belongingto variable exponent Lebesgue spaces $L_{p\left( x\right) }\left(B\right) $ weconsider difference operator $\left( I-T_{\delta }\right) ^{r}f\left( \cdot\right) $ under the condition that $p(x)$ satisfies the Log H\"{o}ldercontinuity condition and $1\leq \mathop{\rm ess \; inf}\limits\nolimits_{x\inB}p(x)$, $\mathop{\rm ess \; sup}\limits\nolimits_{x\in B}p(x)<\infty $ where$I$ is the identity operator, $r\in \mathrm{N}:=\left\{ 1,2,3,\cdots \right\}$, $\delta \geq 0$ and \begin{equation*} T_{\delta }f\left( x\right)=\frac{1}{\delta }\int\nolimits_{0}^{\delta }f\left( x+t\right) dt )\end{equation*} is the forward Steklov operator. We obtain main properties ofdifference operator $\left\Vert \left( I-T_{\delta }\right) ^{r}f\right\Vert_{p\left( \cdot \right) }$ in $L_{p\left( x\right) }\left( B\right) .$ We giveproof of direct and inverse theorems of approximation by IFFD in $L_{p\left(x\right) }\left( \boldsymbol{R}\right). $

Exponential approximation in variable exponent Lebesgue spaces on the real line