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Let $\rho $ be a symmetric measure of Lebesgue type, i.e., \begin{equation*} \rho =\frac{1}{2}(\mu \times \delta _0+\delta _0\times \mu ), \end{equation*} where the component measure $\mu $ is the Lebesgue measure supported on $[t, t+1]$ for $t\in \mathbb{Q}\setminus \lbrace -\frac{1}{2}\rbrace $ and $\delta _{0}$ is the Dirac measure at $0$. We prove that $\rho $ is a spectral measure if and only if $ t \in \frac{1}{2}\mathbb{Z}$. In this case, $L^2(\rho )$ has a unique orthonormal basis of the form \[ \left\lbrace e^{2\pi i (\lambda x-\lambda y)}:\lambda \in \Lambda _0\right\rbrace , \] where $\Lambda _0$ is the spectrum of the Lebesgue measure supported on $[-t-1,-t]\cup [t,t+1]$. Our result answers some questions raised by Lai, Liu and Prince [JFA, 2021].

BACKGROUND: Plant diseases caused by phytopathogenic fungi and oomycetes pose a serious threat to ensuring crop yield and quality. Finding novel fungicidal candidates based on natural products is one of the critical methods for developing effective and environmentally friendly pesticides. In this study, a series of salicylaldehyde derivatives containing an α‐methylene‐γ‐butyrolactone moiety were designed, synthesized, and their fungicidal activities were evaluated. RESULTS: The bioassay studies indicated that compound C3 displayed an excellent in vitro activity against Rhizoctonia solani with a half‐maximal effective concentration (EC50) value of 0.65 μg/mL, higher than that of pyraclostrobin (EC50 = 1.44 μg/mL) and comparable to that of carbendazim (EC50 = 0.33 μg/mL). For Valsa mali and Phytophthora capsici, compound C3 also showed good fungicidal activities with EC50 values of 0.91 and 1.33 μg/mL, respectively. In addition, compound C3 exhibited promising protective in vivo activity against R. solani (84.1%) at 100 μg/mL, which was better than that of pyraclostrobin (78.4%). The pot experiment displayed that compound C3 had 74.8% protective efficacy against R. solani at 200 μg/mL, which was comparable to that of validamycin (78.2%). The antifungal mode of action research indicated that compound C3 could change the mycelial morphology and ultrastructure, increase cell membrane permeability, affect respiratory metabolism by binding to complex III, and inhibit the germination and formation of sclerotia, thereby effectively controlling the disease. CONCLUSION: The present study provides support for the application of these salicylaldehyde derivatives as promising potential pesticides with remarkable and broad‐spectrum fungicidal activities against phytopathogenic fungi and oomycetes in crop protection. © 2023 Society of Chemical Industry. [ABSTRACT FROM AUTHOR]