Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in L^{1}(\mathbb R): \hat{f}\in L^{1}(\mathbb R) \}$$ $A(\mathbb R)$ is normed by: $$\left\|f\right\|:= \left\|\hat{f}\right\|_{L^{1}(\mathbb R)}=\int_{\mathbb R}|\hat{f}(\xi)| d\xi; \ (f\in A(\mathbb R)).$$ We note that $A(\mathbb R)$ is a Banach algebra under point wise addition and multiplication.

Let $M>0$ and fix it; and consider $B_{M}= \{f\in A(\mathbb R): f|f|\in A(\mathbb R) \ \text{and} \ ||f||\leq M \}.$

My Question: Can we expect, $\left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$; for $f,g \in B_{M}$ , where $C$ is some Constant ? If yes, what can we say about $C$ ? If not, can we produce counter example ?

Thanks,

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.