This paper deals with structural issues concerning wavelet frames and their dual frames. It is known that there exist { a j / 2 ? ( ? ? k b ) } , ? Z in L R for which no frame has structure. We first generalize this result by proving approximately Motivated we show imposing very mild decay condition on the Fourier transform of generator certain oversampling N indeed an frame; most importantly, ...

A real valued function $f$ defined on a open interval $I$ is called $\Phi$-convex if, for all $x,y\in I$, $t\in[0,1]$ it satisfies $$ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+t\Phi\big((1-t)|x-y|\big)+(1-t)\Phi\big(t|x-y|\big), where $\Phi:\mathbb{R}_+\to\mathbb{R}_+$ nonnegative error function. If and $-f$ are simultaneously $\Phi$-convex, then said to be $\Phi$-affine In the main results of paper, ...

we show that  higher derivations on a hilbert$c^{*}-$module associated with the cauchy functional equation satisfying generalized hyers--ulam stability.