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A characterization of Fourier transforms

Philippe Jaming (2010)

Colloquium Mathematicae

The aim of this paper is to show that, in various situations, the only continuous linear (or not) map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups ℤ/nℤ, the integers ℤ, the torus 𝕋 and the real line. We also ask a related question for the twisted convolution.

A convolution property of the Cantor-Lebesgue measure, II

Daniel M. Oberlin (2003)

Colloquium Mathematicae

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 ( ) to L q ( ) . We also give a condition on p which is necessary if this operator maps L p ( ) into L²().

A formula for the number of solutions of a restricted linear congruence

K. Vishnu Namboothiri (2021)

Mathematica Bohemica

Consider the linear congruence equation x 1 + ... + x k ≡ b ( mod n s ) for b ∈ ℤ , n , s ∈ ℕ . Let ( a , b ) s denote the generalized gcd of a and b which is the largest l s with l ∈ ℕ dividing a and b simultaneously. Let d 1 , ... , d τ ( n ) be all positive divisors of n . For each d j ∣ n , define 𝒞 j , s ( n ) = { 1 ≤ x ≤ 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...

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