Quadratic spectral concentration of characteristic functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913399899684864 |
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| author | Oganesyan, Kristina |
| author_facet | Oganesyan, Kristina |
| contents | It is known that the inequality \begin{align*}\int_{-W/2}^{W/2}|\widehat{f}(ξ)|^2dξ\leq \int_{-W/2}^{W/2}|\widehat{|f|^*}(ξ)|^2dξ\end{align*} between the quadratic spectral concentration of a function and that of its decreasing rearrangement holds for any function $f\in L^2,\;|\text{supp} f|=T,$ if and only if the product $WT$ does not exceed the critical value $\approx 0.884$. We show that by restricting ourselves to characteristic functions we can enlarge this range up to $WT\leq 4/3$. Besides, we establish various properties of minimizers of the difference $\int_{-W/2}^{W/2}|\widehat{χ_A^*}(ξ)|^2dξ-\int_{-W/2}^{W/2}|\widehat{χ_A}(ξ)|^2dξ$ over sets $A$ of finite measure and prove that this difference is non-negative for all $W,T>0$ if $A$ is the union of two intervals. As a corollary, we obtain a sharp (up to a constant) estimate for the $L_2$-norms of non-harmonic trigonometric polynomials with alternating coefficients $\pm 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_14921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quadratic spectral concentration of characteristic functions Oganesyan, Kristina Classical Analysis and ODEs 42A38, 42A05 It is known that the inequality \begin{align*}\int_{-W/2}^{W/2}|\widehat{f}(ξ)|^2dξ\leq \int_{-W/2}^{W/2}|\widehat{|f|^*}(ξ)|^2dξ\end{align*} between the quadratic spectral concentration of a function and that of its decreasing rearrangement holds for any function $f\in L^2,\;|\text{supp} f|=T,$ if and only if the product $WT$ does not exceed the critical value $\approx 0.884$. We show that by restricting ourselves to characteristic functions we can enlarge this range up to $WT\leq 4/3$. Besides, we establish various properties of minimizers of the difference $\int_{-W/2}^{W/2}|\widehat{χ_A^*}(ξ)|^2dξ-\int_{-W/2}^{W/2}|\widehat{χ_A}(ξ)|^2dξ$ over sets $A$ of finite measure and prove that this difference is non-negative for all $W,T>0$ if $A$ is the union of two intervals. As a corollary, we obtain a sharp (up to a constant) estimate for the $L_2$-norms of non-harmonic trigonometric polynomials with alternating coefficients $\pm 1$. |
| title | Quadratic spectral concentration of characteristic functions |
| topic | Classical Analysis and ODEs 42A38, 42A05 |
| url | https://arxiv.org/abs/2406.14921 |