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# Marcinkiewicz- $$\theta$$ -Summability of Fourier Transforms

Acta Mathematica Hungarica
Author: Ferenc Weisz

## Abstract

A general summability method of two-dimensional Fourier transforms is given with the help of an integrable function

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. Under some conditions on
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\theta$$ \end{document}
we show that the maximal operator of the Marcinkiewicz-
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\theta$$ \end{document}
-means of a tempered distribution is bounded from
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$H_p \left( {R^2 } \right)$$ \end{document}
to
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$L_p \left( {R^2 } \right)$$ \end{document}
for all
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$p_0 < p \leqq \infty$$ \end{document}
and, consequently, is of weak type
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left( {1,1} \right)$$ \end{document}
, where
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depends only on
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\theta$$ \end{document}
. As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkievicz and Zhizhiashvili, more exactly, the Marcinkiewicz-
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\theta$$ \end{document}
-means of a function
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$f \in L_1 \left( {R^2 } \right)$$ \end{document}
converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz-
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\theta$$ \end{document}
-means are uniformly bounded on the spaces
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$H_p \left( {R^2 } \right)$$ \end{document}
and so they converge in norm
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left( {p_0 < p < \infty } \right)$$ \end{document}
. Some special cases of the Marcinkievicz-
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-summation are considered, such as the Weierstrass, Picar, Bessel, Fejr, de la Valle-Poussin, Rogosinski and Riesz summations.

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# Random noise reduction capability of the hermit polynomial based Least Squares Fourier Transform method

Acta Geodaetica et Geophysica Hungarica
Author: P. Vass

Brigham E O 1974: The Fast Fourier Transform. Prentice-Hall Inc., Englewood Cliffs, New Jersey Brigham E. O. The

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# New Technique for Fourier Transform Thermal Analysis

Journal of Thermal Analysis and Calorimetry
Authors: J. Morikawa and T. Hashimoto

## Abstract

A new technique for the simultaneous measurement of higher order harmonic of temperature wave under temperature scan was proposed. The mathematical rule for the propagation of harmonies in the film shaped specimen was examined and the principle of Fourier transform thermal analysis was theoretically and experimentally justified. This principle applied to a technique called ‘Fourier transform thermal analysis’, makes it possible to determine simultaneously thermal diffusivity, heat capacity per unit volume and thermal conductivity as a function of frequency and temperature. The results on thermoplastics were shown and the glass transition and the crystallization were discussed.

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# Thermogravimetric Fourier Transform Infrared Spectroscopy of Hydrocarbon Fuel Residues

Journal of Thermal Analysis and Calorimetry
Author: A. Zanier

## Abstract

Although thermogravimetric analysis (TG) has become an indispensable tool for the analysis and characterization of materials, its scope is limited as no information is obtained about the qualitative aspects of the evolved gases during the thermal decomposition. For processes involving mass loss, a powerful technique to provide this missing information is Fourier transform infrared spectroscopy (FT-IR) in combination with TG. It supplies a comprehensive understanding of thermal events in a reliable and meaningful way as data are obtained from a single sample under the same conditions. The coupling TG/FT-IR is used in fuel analysis for the identification of residual volatiles, to determine their sequence of release and to resolve thermogravimetric curves. In this work, the usefulness of TG/FT-IR for characterizing middle distillate fuel residues is illustrated with some typical examples of recent application. A Bio-Rad FTS 25 FT-IR spectrometer coupled with a TA Instruments TGA 2950 thermogravimetric analyzer was used for data aquisition. The results obtained demonstrate the utility of this combined technique in determining the decomposition pathway of tarry materials at various stages of pyrolysis, thereby allowing new insights into the complex thermal behaviour of hydrocarbon residual systems.

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# On the factorization and the zeros of fourier transforms of infinitely differentiable functions with compact support

Analysis Mathematica
Authors: А. Седлецкии and A. Sedletskiį

## Abstract

LetF denote the class of Fourier transforms of infinitely differentiable functions on the real line with compact support. We prove that if each zero of a functionF

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lies in the union of a horizontal strip with a finite number of semistrips, them a factorizationF=F 1 F 2 holds, where
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. We give estimates of |F 1(z)/F 2(z)| from above and from below. The zero sets of functions fromF are described in terms of integral sequences.

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# Thermal degradation of poly(lactic acid) measured by thermogravimetry coupled to Fourier transform infrared spectroscopy

Journal of Thermal Analysis and Calorimetry
Authors: Hantao Zou, Changhai Yi, Luoxin Wang, Hongtao Liu and Weilin Xu

## Abstract

Thermal decomposition of poly(lactic acid) (PLA) has been studied using thermogravimetry coupled to Fourier transform infrared spectroscopy (TGA-FTIR). FTIR analysis of the evolved decomposition products shows the release of lactide molecule, acetaldehyde, carbon monoxide and carbon dioxide. Acetaldehyde and carbon dioxide exist until the end of the experiments, whereas carbon monoxide gradually decreases above the peak temperature in that the higher temperature benefits from chain homolysis and the production of carbon dioxide. A kinetic study of thermal degradation of PLA in nitrogen has been studied by means of thermogravimetry. It is found that the thermal degradation kinetics of PLA can be interpreted in terms of multi-step degradation mechanisms. The activation energies obtained by Ozawa–Flynn–Wall method and Friedman’s method are in good agreement with that obtained by Kissinger’s method. The activation energies of PLA calculated by the three methods are 177.5 kJ mol−1, 183.6 kJ mol−1 and 181.1 kJ mol−1, respectively.

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# On the degree of continuity and smoothness of sine and cosine Fourier transforms of Lebesgue integrable functions

Acta Mathematica Hungarica
Author: Ferenc Móricz

## Abstract.

We consider complex-valued functions fL 1(ℝ+), where ℝ+:=[0,∞), and prove sufficient conditions under which the sine Fourier transform and the cosine Fourier transform belong to one of the Lipschitz classes Lip (α) and lip (α) for some 0<α≦1, or to one of the Zygmund classes Zyg (α) and zyg (α) for some 0<α≦2. These sufficient conditions are best possible in the sense that they are also necessary if f(x)≧0 almost everywhere.

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# Reactions of H atoms and OH radicals with ascorbic acid: A pulse radiolysis Fourier transform ESR study

Journal of Radioanalytical and Nuclear Chemistry
Authors: H. Sprinz, D. Beckert and O. Brede

## Abstract

Using pulse radiolysis, free radicals of ascorbic acid were generated by reactions of the primary radicals H and OH in acidic and basic aqueous solutions. The formation and the decay of several radicals of ascorbic acid were detected by time resolved Fourier transform electron spin resonance within a time interval of 100 ns to 1 ms. The rate constant of addition of H atoms to ascorbic acid (1.3·108 dm3· mol−1·s−1) was directly determined by the change of line width of the low field line of the H atom in the presence of ascorbic acid. The addition of OH radicals to ascorbic acid results in different radical structures, detected by highly resolved Fourier transform ESR spectra.

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# Determination of protein, lipid and sugar contents of bakery products by using Fourier-transform near infrared spectroscopy

Acta Alimentaria
Authors: T. Szigedi, M. Dernovics and M. Fodor

The use of Fourier-transform near infrared spectroscopy (FT-NIR) to measure the content of protein, lipid and sugar contents of bakery products was investigated. The samples were dried, homogenized, sieved and measured in the wavelength range of 780–2500 nm. The calibration was based on partial least squares (PLS) regression with cross-validation. The performance of the final model was evaluated according to root mean square of cross-validation (RMSECV), root mean square error of estimation (RMSEE) and the determination coefficient (R2).The developed models use the ranges of 1100–1245 nm and 1590–2600 nm for protein determination, 1330–1840 nm and 2170–2350 nm for lipid, 1400–1630 nm, 2000–2170 nm and 2230–2570 nm for sugar determination, respectively. Protein, lipid and sugar could be determined directly with R2 values of 98.93, 99.07 and 98.81, and RMSECV values of 0.16 m/m%, 0.79 m/m% and 0.28 m/m%, respectively. It can be concluded that FT-NIR spectroscopy can be used for the routine determination of protein, lipid and sugar content of bakery products and it can contribute to the estimation of calorie content in a fast and non-destructive way.

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# Using Lanthanum Fluoride Fine Particles as Stationary Phase for Thin-Layer Chromatography/Fourier Transform Infrared Spectroscopy Analysis

JPC - Journal of Planar Chromatography - Modern TLC
Authors: Qing Zhu, Haijun Wu, Fang Wang, Anqi He, Kun Huang, Yongju Wei, Cuige Liu, Yanjun Zhai, Shifu Weng, Zhanlan Yang, Yizhuang Xu, Isao Noda and Jinguang Wu

.G. Modern FTIR Spectroscopy, Technology and Application 1994 S.F. Weng , Fourier Transform Infrared Spectrometer

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