For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from $L^p\left(\right)$ to $L^q\left(\right)$. We also give a condition on p which is necessary if this operator maps $L^p\left(\right)$ into L²().

An improvement of a lemma of Calderón and Zygmund involving singular spherical harmonic kernels is obtained and a counter-example is given to show that this result is best possible. In a particular case when the singularity is O(|log r|), let $f\in C¹\left({ℝ}^N\setminus 0\right)$ and suppose f vanishes outside of a compact subset of ${ℝ}^N$, N ≥ 2. Also, let k(x) be a Calderón-Zygmund kernel of spherical harmonic type. Suppose f(x) = O(|log r|) as r → 0 in the $L^p$-sense. Set $F\left(x\right)={\int }_{{ℝ}^N}k(x-y)f\left(y\right)dy\forall x\in {ℝ}^N\setminus 0$. Then F(x) = O(log²r) as r → 0 in the $L^p$-sense, 1 < p < ∞....

We propose a cryptographic algorithm utilizing integer wavelet transform via a lifting scheme. In this research, we construct some predict and update operators within the lifting scheme of wavelet transforms employing operations in min-max-plus algebra, termed as lifting scheme integer wavelet transform over min-max-plus algebra (MMPLS-IWavelet). The analysis and synthesis process on MMPLS-IWavelet is implemented for both encryption and decryption processes. The encryption key comprises a sequence...

The aim of this paper is to prove a quantitative version of Shapiro's uncertainty principle for orthonormal sequences in the setting of Gabor-Hankel theory.

Consider the linear congruence equation $x_1+...+x_k\equiv b(mod{n}^s)$ for $b\in \mathbb{Z}$, $n,s\in \mathbb{N}$. Let ${(a,b)}_s$ denote the generalized gcd of $a$ and $b$ which is the largest $l^s$ with $l\in \mathbb{N}$ dividing $a$ and $b$ simultaneously. Let $d_1,...,d_{\tau \left(n\right)}$ be all positive divisors of $n$. For each $d_j\mid n$, define ${𝒞}_{j,s}\left(n\right)=\{1\le x\le n^s:{(x,n^s)}_s=d_j^s\}$. K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on $x_i$. We generalize their result with generalized gcd restrictions on $x_i$ and prove that for the above linear congruence, the number of solutions...

The object of this note is to generalize some Fourier inequalities.