Correction: Young Sik, K. Partial Derivative Approach to the Integral Transform for the Function Space in the Banach Algebra. Entropy 2020, 22, 1047

1. Correction for Equations

In the original article [1], there were some mistakes in Equations as published.

(1)

We mistyped $\frac{1-z}{2}$ as $\frac{z-1}{2z}$, and $\frac{1-\lambda }{2}$ as $\frac{\lambda -1}{2\lambda }$, and $\frac{1-{\lambda }_n}{2}$ as $\frac{{\lambda }_n-1}{2{\lambda }_n}$ in Equation (23), Equation (25), Equations (27)–(29), Equation (41), Equation (43) and Equations (45)–(47). We correct them:

$$\underset{n\to \infty }{lim}{z}^{\frac{n}{2}}·E_x(exp\{\frac{1-z}{2}{\displaystyle \sum _{k=1}^n}{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\left\}[D,F,x+y,w]\right).$$

(23)

$$\begin{array}{c}\underset{n\to \infty }{lim}{z}^{\frac{n}{2}}·E_x(exp\{\frac{1-z}{2}{\displaystyle \sum _{k=1}^n}{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\left\}[D,F,x+y,w]\right)\hfill \\ =\underset{n\to \infty }{lim}{z}^{\frac{n}{2}}{\int }_{L_2[0,T]}{E}_x(exp\{\frac{1-z}{2}{\displaystyle \sum _{k=1}^n}{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2+i[I,v\left(t\right),x\left(t\right)]\left\}\right)\hfill \\ ·\left(i\right[I,v\left(t\right),w\left(t\right)\left]\right)·exp\left\{i\right[I,v\left(t\right),y\left(t\right)\left]\right\}df\left(v\right).\hfill \end{array}$$

(25)

$$\underset{n\to \infty }{lim}{\lambda }^{\frac{n}{2}}·E_x(exp\{\frac{1-\lambda }{2}{\displaystyle \sum _{k=1}^m}{\left[I{\varphi }_k\left(t\right)x\left(t\right)\right]}^2\left\}[D,F,x+y,w)\right]).$$

(27)

$$\underset{n\to \infty }{lim}{{\lambda }_n}^{\frac{n}{2}}·E_x(exp\{\frac{1-{\lambda }_n}{2}{\displaystyle \sum _{k=1}^m}{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\left\}[D,F,x+y,w)\right]).$$

(28)

$$\underset{n\to \infty }{lim}{{\lambda }_n}^{\frac{n}{2}}·E_x(exp\{\frac{1-{\lambda }_n}{2}{\displaystyle \sum _{k=1}^m}{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\left\}[D,F,x+y,w)\right]).$$

(29)

$$\underset{n\to \infty }{lim}{z}^{\frac{\nu n}{2}}·E_{\overrightarrow{x}}(exp\{\frac{1-z}{2}{\displaystyle \sum _{j=1}^{\nu }}{\displaystyle \sum _{k=1}^n}{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w})\left]\right).$$

(41)

$$\begin{array}{c}\underset{n\to \infty }{lim}{z}^{\frac{\nu n}{2}}·E_{\overrightarrow{x}}(exp\{\frac{1-z}{2}{\displaystyle \sum _{j=1}^{\nu }}{\displaystyle \sum _{k=1}^n}{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w})]\hfill \\ =\underset{n\to \infty }{lim}{z}^{\frac{\nu n}{2}}{\int }_{L_2^{\nu }[0,T]}{E}_{\overrightarrow{x}}(exp\{\frac{1-z}{2}{\displaystyle \sum _{j=1}^{\nu }}{\displaystyle \sum _{k=1}^n}{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\}\hfill \\ ·exp\left\{i{\displaystyle \sum _{j=1}^{\nu }}[I,v_j\left(t\right),x_j\left(t\right)]\right\})\hfill \\ ·\left(i{\displaystyle \sum _{j=1}^{\nu }}[I,v_j\left(t\right),w_j\left(t\right)]\right)·exp\left\{i{\displaystyle \sum _{j=1}^{\nu }}[I,v_j\left(t\right),y_j\left(t\right)]\right\}df\left(\overrightarrow{v}\right).\hfill \end{array}$$

(43)

$$=\underset{n\to \infty }{lim}{\lambda }^{\frac{\nu n}{2}}·E_{\overrightarrow{x}}(exp\{\frac{1-\lambda }{2}{\displaystyle \sum _{j=1}^{\nu }}{\displaystyle \sum _{k=1}^m}{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w})\left]\right).$$

(45)

$$=\underset{n\to \infty }{lim}{{\lambda }_n}^{\frac{\nu n}{2}}·E_{\overrightarrow{x}}(exp\{\frac{1-{\lambda }_n}{2}{\displaystyle \sum _{j=1}^{\nu }}{\displaystyle \sum _{k=1}^m}{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w})\left]\right).$$

(46)

$$=\underset{n\to \infty }{lim}{{\lambda }_n}^{\frac{\nu n}{2}}·E_{\overrightarrow{x}}(exp\{\frac{1-{\lambda }_n}{2}{\displaystyle \sum _{j=1}^{\nu }}{\displaystyle \sum _{k=1}^m}{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w})\left]\right).$$

(47)

(2)

In Equations (26)–(29):

(a). We mistyped $[D,F,x+y,w]$ as $D[F,x+y,w]$ in Equations (26)–(29).

(b). We mistyped $[D,F,\rho x+y,w]$ as $D[F,\rho x+y,w]$ in Equation (26).

(3)

In Equations (31)–(47):

(a). We mistyped $[D,F,\overrightarrow{x},\overrightarrow{w}]$ as $D[F,\overrightarrow{x},\overrightarrow{w}]$ in Equations (31) and (32).

(b). We mistyped $[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]$ as $D[F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]$ in Equations (34)–(41) and Equations (43)–(47).

(c). We mistyped $[D,F,z^{-\frac{1}{2}}\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]$ as $D[F,z^{-\frac{1}{2}}\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]$ in Equations (42) and (43).

(d). We mistyped $[D,F,\rho \overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]$ as $D[F,\rho \overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]$ in Equation (44).

(e). We mistyped $[D,F,{\lambda }^{-\frac{1}{2}}\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]$ as $D[F,{\lambda }^{-\frac{1}{2}}\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]$ in Equation (45).

The authors apologize for any inconvenience caused and state that the scientific conclusions are unaffected. The original article has been updated.