Long-term trends and dependences on solar activity were derived for wave disturbances at mesopause altitudes (80−100 km) based on temperature variations obtained by spectral observations of the hydroxyl airglow at the Zvenigorod Science Station, Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, in 2000−2024, as well as by methods of statistical analysis. Their activities were determined by digital frequency filtering in three regions of wave periods, such as 0.7−2.0, 1.4−4.1, and 2.7−8.2 h with maxima at 1, 2, and 4 h. The root-mean-square values of temperature half-differences served as an indicator of wave activity. Both year-round and average seasonal (winter and summer) values were analyzed. According to the study results, the wave activity is characterized by positive trends with their dependence on the frequency band of disturbances (in winter, the trend is greater in the high-frequency band, and in summer, the trend is greater in the low-frequency band). The dependence on solar activity is positive. Its values are greater for the high-frequency band of disturbances, as well as in winter.

The study analyzes the long-term average annual OH* temperature trend, the values of which were obtained from nighttime spectral measurements of hydroxyl airglow bands at Zvenigorod research station (56° N, 37° E) from 1957 to 2022. At present, this OH* temperature series, which reflects the thermal state of the mesopause region, is the longest in the world. On its basis, the linear trend and response of temperature to changes in solar activity are estimated both in general for the entire data set and for individual time intervals. In the first case, the trend was –0.23 ± 0.04 K/year. In the second case, the analysis showed a strong cooling in the mesopause region (–0.53 ± 0.34 K/yr) until the 1970s, which subsequently slowed to –0.14 ± 0.03 K/yr. Comparison of the results with other measurements and model calculations shows that the latter have lower trend values. It is suggested that the causes of the temperature trend, in addition to the increase in greenhouse gases, the main one being CO2, can be due to long-term changes in the dynamics of the upper atmosphere.

The influence of the 27-day solar oscillation on the temperature and intensity of the airglow of the mesopause region based on the hourly midnight observations of the О 2 А(0-1) and ОН(6-2) bands at the Zvenigorod Scientific Station in 2000−2021 has been studied. It is found that the response to solar activity changes is observed in all characteristics of the atmospheric emissions both by the yearly and seasonal data. They are statistically significant for all characteristics of the atmospheric emissions in winter and only for the О 2 А(0-1) intensity in summer. The positive response of the molecular oxygen emission is lower in the summer than in the winter by approximately a factor of 2. It is found that the variations in the atmospheric emission characteristics are close to antiphase with a 27-day solar oscillation. At the same time, the temperature of the emitting hydroxyl in winter undergoes oscillations with a shift by a third of the period relative to the solar variation maximum.

Spectral observations of the mesopause airglow at the Zvenigorod Scientific Station have been used to obtain the midnight emission intensities of molecular oxygen (О2А(0-1) band) and hydroxyl (OH (6-2) band) for 2000–2019. Spectral analysis of the variations has made it possible to determine the annual variability for each emission, which is described by the sum of four harmonics. The time lag in seasonal variations of the hydroxyl emission relative to variations in the emission of molecular oxygen is 5–18 days. Long-term changes in the average annual emission intensities have been studied. The linear trend (–3.3 ± 0.3% per year for О2А(0-1) and –2.6 ± 0.2% per year for ОН(6-2)), the dependences on the 11-year solar cycle (response to changes in the Lyman-alpha solar radiation (18.5 ± 3.3% per 1011 photons cm–2 s–1 for О2А(0-1) and 10.5 ± 2.5% per 1011 photons cm–2 s–1 for OH (6-2)) and the 22-year solar cycle (response to changes in the solar magnetic field strength (23.2 ± 4.5% per 100 μТ for О2А(0-1) and 12.1 ± 3.5% per 100 μТ for ОН (6-2)), as well as quasi-eight-year oscillations have been found.

Long-term series of midnight temperature in the mesopause region have been obtained from spectral observations of hydroxyl airglow emission (OH(6-2) λ840 nm band) at the Tory station (52° N, 103° E) in 2008–2016 and Zvenigorod (56° N, 37° E) station in 2000–2016. On their basis, the Lomb-Scargle spectra of the variations in the period range from ~12 days to ~11 years have been determined. Estimates of the amplitudes of statistically significant temperature fluctuations are made. The dominant oscillations are the first and second harmonics of the annual variation, the amplitudes of which are 23–24 K and 4–7 K, respectively. The remaining variations, the number of which was 16 for the Tory and 22 for Zvenigorod stations, have small amplitudes (0.5–3 K). Oscillations with combinational frequencies, which arise from modulation of the annual variation harmonics, are observed in a structure of the variation spectra in addition to interannual oscillations (periods from ~2 to ~11 years) and harmonics of the annual variation (up to its tenth harmonic).

Using spectral measurements of the hydroxyl airglow at the Zvenigorod station (56°N, 37°E), Moscow region, over 2000–2016, we obtained the long-term set of data comprising 1822 midnight values of the OH∗ temperature in the mesopause region. These data revealed a 17-year series of its mean annual values, as well as amplitudes and phases of the first two harmonics of its annual variation. The obtained parameters were analyzed to determine statistically relevant characteristics of their long-term variations. As a result, we found that the long-term behaviour of the mean annual OH∗ temperature features a small negative linear trend (−0.07±0.03K/year) over the addressed period. Besides, its dependence on solar activity is shown to be 4.1±0.5K/100 SFU. Regarding the long-term behaviour of the mean annual OH∗ temperature, we revealed the existence of two oscillations with 3-year (the amplitude being 1.3±0.2K) and 4.1-year (the amplitude being 0.6±0.2K) periods. We obtained empirical relations describing year-to-year variations in the amplitudes and phases of the annual and semi-annual harmonics.

The variations in different (7–21 years) activity periods (F10.7) in the atmospheres of stars of the late spectral types (G2–K5) and the Sun have been considered. F10.7 quasiperiodic oscillations with periods varying from 1.8 to 12.5 years have been detected. The oscillation amplitudes account for ~20% of the amplitude of the long-term cycle maximum. It has been found that the logarithm of quasiperiodic oscillations correlates well (r = 0.900 ± 0.051) with the period of the long-term activity cycle.

A brief overview of applications of Schoenberg’s polynomial B-splines of odd degrees in mathematical statistics, computational mathematics, and statistical radio engineering is provided. Exact formulas for the found Schoenberg B-spline of 15th degree are presented. High-quality approximations of smooth functions with an infinite Fourier transform by functions with a finite Fourier transform are found.

A Welch-type estimator (an estimator obtained by averaging over the time shift) for the spectral density function of a stationary continuous-time random process is considered. The bias and variance of the statistical estimator are studied, and some recommendations concerning the choice of its parameters are formulated.

The Welch-type spectral density estimator is considered once more. Some practical recommendations that make it possible to facilitate and speed up considerably the calculation of the estimator are formulated.

One of the most efficient tools of studying meteorological and climatic time series is wavelet analysis, which allows local properties of a nonstationary time series to be examined. New sets of continuous and discrete wavelet-forming functions are proposed. All proposed wavelets are finite (have a bounded carrier), even, and well localized in the spectral space.

Temperature distributions with height (70–130 km) are constructed from EISCAT radar measurements at high latitudes and spectrophotometric measurements of hydroxyl emission (87 km), sodium (92 km) and of atomic oxygen emission of 557.7 nm (97–102 km) at mid latitudes for the period 1984–1998. It is revealed that the temperature dependence on solar activity at heights of 87 and 108 km is seasonal with negative correlation in winter and positive correlation in summer. Estimates of the long-term change of temperature for the specified period of observation are made.

Some regularities of the quasibiennial variations (QBV) in atmospheric temperature at the altitudes of the mesopause and lower thermosphere have been studied based on the data of radiophysical and emission measurements. It has been shown that these variations are closely related to the quasibiennial variations in solar activity and form a decaying oscillation train within the solar activity cycle, whose time behavior is described by the Airy function. The train appears in the period of minimum solar activity. In the period of maximum solar activity, the oscillation period is similar to38 months and linearly (r = -0.98) decreases to similar to21 months by the end of the 11-year cycle.