×

The Wiener algebra of absolutely convergent Fourier integrals: an overview. (English) Zbl 1258.42008

In this nice survey of 68 pages written by three experts in the area, the main results in the topic of representation of non-periodic functions by absolutely convergent Fourier integral are collected, classified, discussed and applied.
The history of the topic has its origin in the classical results of S. N. Bernstein and of A. Zygmund, who proved that if a periodic function \(f\) is from the Lip\(\alpha\) class with \(\alpha >1/2\), then for the absolutely convergence of its Fourier series, a sufficient condition is \(\int_{0}^{1}\frac{\omega(f, t)}{t^{3/2}}dt <+\infty\) (S. N. Bernstein’s result), and if in addition \(f\) is of bounded variation then a sufficient condition is \(\int_{0}^{1}\frac{\sqrt{\omega(f, t)}}{t}d t < +\infty\) (A. Zygmund’s result). Here \(\omega(f, t)\) denotes the modulus of continuity of \(f\). Extensions to the non-periodic case were started in the \(20-40\)s of the twentieth century, with E. C. Titchmarsh’s and N. Wiener’s results, then becoming a well-studied area of research until our days.
This survey consists of 13 sections and an updated list of references.
Sections 1, 2 and 3 contains the Introduction, Notation and Definitions and Preliminaries, respectively.
Section 4 presents general properties of the Wiener algebras. Sufficient conditions for a function \(f:\mathbb{R}\to \mathbb{C}\) to belong to the Wiener algebra \(W_{0}(\mathbb{R})=\{f ; f(x)=\int_{\mathbb{R}}\mathrm e^{-\mathrm i x y}g(y)d y , g\in L_{1}(\mathbb{R}^{n})\}\) are presented in the next sections 6-12. From a very long list, from each section we selected the following results.
Theorem 1. (Tests in one-dimensional case) (i) Let \(f\in L_{p}(\mathbb{R})\bigcap C_{0}(\mathbb{R})\), \(1<p\leq 2\). If \(\int_{\mathbb{R}}|f(x+h)-f(x)|^{p}dx\leq Ch^{\alpha p}\), for all \(0<h\leq 1\), where \(1/p < \alpha \leq 1\), then \(f\in W_{0}(\mathbb{R})\);
(ii) If \(f\) satisfies the condition \(|f(x)-f(y)|\leq C\frac{|x-y|^{\lambda}}{(1+|x|)^{\lambda}(1+|y|)^{\lambda}}\), for all \(x, y\in \overline{\mathbb{R}}\), with \(\lambda > 1/2\) and \(f(\infty)=0\), then \(f\in W_{0}(\mathbb{R})\);
(iii) Let \(f\in C_{0}(\mathbb{R})\) be twice continuously differentiable on \(\mathbb{R}\) such that \(\int_{0}^{\infty}x|f^{\prime \prime}(x)+f^{\prime \prime}(-x)|dx <\infty\) and there exists \(\sigma >0\) with \(\int_{0}^{\infty}x^{1+\sigma}\left |\left (\frac{f(x)-f(-x)}{x^{\sigma}}\right )^{\prime \prime }\right | dx <\infty\). Then \(f\in W_{0}(\mathbb{R})\).
Theorem 2. (Tests in terms of integrability of derivatives) (i) Let \(f\in L_{2}(\mathbb{R}^{n})\) and \((-\Delta)^{\alpha/2}(f)\in L_{e}(\mathbb{R}^{n})\) with \(\alpha \in (n/2, +\infty)\) not necessarily integer. Then \(f\in W_{0}(\mathbb{R}^{n})\) (here \(\Delta\) denotes the Laplacian);
(ii)If \(f\in L_{1}(\mathbb{R}^{n})\) and \(f\) has mixed distributional derivatives \(D^{j}(f)\in L_{p}(\mathbb{R}^{n})\) for all \(j\in [0, 1]^{n}\), \(j\not=0\), where \(1<p\leq 2\), then \(f\in W_{0}(\mathbb{R}^{n})\).
Theorem 3. (Tests in terms of finite difference) (i) (Bernstein-type result) Let \(f\in C(\mathbb{R}^{n})\) satisfying the condition \(\int_{0}^{\infty}\sup_{0<|h|<t}\|\Delta^{m}_{h}(f)\|_{L_{2}}\frac{dt}{t^{1+n/2}}<\infty\), with \(m>n/2\). Then \(f\) is representable as the sum of the Fourier transform of a function in \(L_{1}(\mathbb{R}^{n})\) and a polynomial of degree \(\leq m-1\);
(ii) (Zygmund-type result) Let \(f\in C^{q}(\mathbb{R}^{n})\) be compactly supported, where \(q=[(n-1)/2]\). If the partial derivative \(D_{q, j}(f):=\frac{\partial^{q} f}{\partial x_{j}^{q}}\) has , as a function of \(x_{j}\), a bounded, with respect to the other variables, number of points separating the intervals of convexity for each \(1\leq j\leq n\), then the condition \(\int_{0}^{1}\omega(t)t^{q-(n+1)/2} dt <\infty\), where \(\omega(t)=\max_{j=1,..., n}\{\omega(D_{q, j}(f) ;t)\}\), yields \(f\in W_{0}(\mathbb{R}^{n})\).
Theorem 4. (Tests for radial and quasi-radial functions) (i) If \(g\in BV_{k+1}\) (a subspace of functions with derivative of order \(k+1\) with bounded variation, see Definition 3.1 in the paper), where \(k=[(k+1)/2]\), then \(g(|x|^{\lambda})\in W_{0}(\mathbb{R}^{n})\);
(ii) Let \(f(x)=f_{0}(x)\), where \(N>(n-1)/2\) and \(f\) is differentiable up the order \(N\), with \(g\) and \({\mathcal{D}}^{N}(g)\) nonincreasing and tending to zero as \(r\to \infty\) (here \({\mathcal{D}}^{1}g(r)=-(1/r)g^{\prime}(r)\), \({\mathcal{D}}^{N}(g)={\mathcal{D}}^{1}({\mathcal{D}}^{N-1}(g))\)). Denote \(A_{N}:=\int_{0}^{\infty}r^{2N-1}|{\mathcal{D}}^{N}f_{0}(r)|dr<\infty\). Then \(f\in W_{0}(\mathbb{R}^{n})\) and \(\|f\|_{W_{0}}\leq C A_{n}\) with \(C\) independent of \(f\).
Theorem 5. (Tests with fractional derivatives) (i) Let \(f, \mathbb{D}^{\alpha}(f)\in L_{2}(\mathbb{R})\), where \(\mathbb{D}^{\alpha}(f)\) denotes the Riesz fractional derivative and \(\alpha > 1/2\). Then \(\|f\|_{W_{0}}\leq C \|f\|_{L_{2}}^{1-1/(2\alpha)}\cdot \|\mathbb{D}^{\alpha}(f)\|_{L_{2}}^{1/(2\alpha)}\).
(ii) Let \(f_{0}\in BV_{k+1}(\mathbb{R}_{+})\), where \(\alpha > (n-1)/2\). Then \(f(x)=f_{0}(|x|)\in W_{0}(\mathbb{R}^{n})\).
Theorem 6. (Test for functions with singular derivatives) Let \(f\in C^{N}(\mathbb{R}^{n}\setminus \{0\})\), \(N=1+[n/2]\) have compact support and let there exist constants \(C>0\) and \(\delta >0\) such that \(|D^{j}(f)(x)|\leq C|x|^{\delta -|j|}, x\in \mathbb{R}^{n}\setminus \{0\}\), for all \(j\) with \(|j|=0, 1, ..., N\). Then \(f\in W_{0}(\mathbb{R}^{n})\).
Theorem 7. (Tests for positive definite functions) (i) \(f:\mathbb{R}\to \mathbb{C}\) satisfying \(f(t)=0\) for \(|t|\geq \sigma >0\) belongs to \(W_{0}^{+}(\mathbb{R})=W_{0}(\mathbb{R})\bigcap \{\text{ the set of positive definite functions } \}\), if and only if there exists \(g\in L_{2}(\mathbb{R})\), \(g(t)=0\) for \(|t|\geq \sigma/2\), such that for any \(t\in \mathbb{R}\), \(f(t)=\int_{-\infty}^{+\infty}g(y)\overline{g(t+y)}dy\);
(ii) Each even, convex and monotone decreasing to zero function on \([0, \infty)\) belongs to \(W_{0}^{+}(\mathbb{R})\).
Theorem 8. (Test for convex and concave functions) (Zygmund) If an odd function \(f\) is concave on \(\mathbb{R}_{+}\), then \(f\in W_{0}(\mathbb{R})\) locally (that is can be extended from any interval in such a way that the extension will belong to \(W_{0}(\mathbb{R})\)) if and only if the integral \(\int_{0}^{1}\frac{f(t)}{t}dt\) converges.
Also, for functions in the intermediate classes \(V^{*}\) between that of functions with bounded variation and the class of functions that are linear combination of convex functions each, sufficient conditions to belong to \(W_{0}(\mathbb{R})\) are presented. Sections 12 ends with some necessary and sufficient conditions for the summability of Fourier series.
In the last section 13, two very useful tables, one containing Fourier transforms for various integrable functions and the other one containing positive definite functions depending on parameters, are presented.

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B15 Multipliers for harmonic analysis in several variables
42A82 Positive definite functions in one variable harmonic analysis
42B08 Summability in several variables
42B35 Function spaces arising in harmonic analysis
26A45 Functions of bounded variation, generalizations
26A33 Fractional derivatives and integrals
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Fizmatgiz, Moscow (1961, in Russian). English transl.: Oliver &amp; Boyd, Edinburgh and London (1965) · Zbl 0124.06202
[2] Askey, R.: Radial characteristic functions. Tech. Report No. 1262, Math. Research Center, University of Wisconsin-Madison (1973) · Zbl 0253.33009
[3] Balan R., Krishtal I.: An almost periodic noncommutative Wiener’s lemma. J. Math. Anal. Appl. 370, 339–349 (2010) · Zbl 1211.46043 · doi:10.1016/j.jmaa.2010.04.053
[4] Bary, N.K.: A Treatise on Trigonometric Series. Fizmatgiz, Moscow (1961, in Russian). English transl.: Pergamon Press, MacMillan, New York (1964)
[5] Beckenbach, E.F., Bellman, R.: Inequalities. Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 30. Springer, Berlin (1961)
[6] Belinsky, E.S., Trigub, R.M.: Summability on a Lebesgue set and a Banach algebra. Theory Func. Approx., Saratov 1982 Winter School Proc. Part II, pp. 29–34 (1983, in Russian)
[7] Belinsky E.S., Dvejrin M.Z., Malamud M.M.: Multipliers in L 1 and estimates for systems of differential operators. Russ. J. Math. Phys. 12, 6–16 (2005) · Zbl 1189.35052
[8] Belinsky E.S., Liflyand E., Trigub R.M.: The Banach algebra A* and its properties. J. Fourier Anal. Appl. 3, 103–129 (1997) · Zbl 0882.42002 · doi:10.1007/BF02649131
[9] Berens H., Görlich E.: Über einen Darstellungssatz für Funktionen als Fourierintegrale und Anwendungen in der Fourieranalysis. Tôhoku Math. J. 18(2), 429–453 (1966) · Zbl 0163.35503 · doi:10.2748/tmj/1178243385
[10] Berens H., Xu Y.: l-1 summability of multiple Fourier integrals and positivity. Math. Proc. Cambridge Philos. Soc. 122, 149–172 (1997) · Zbl 0881.42007 · doi:10.1017/S0305004196001521
[11] Bernstein, S.N.: On majorants of finite or quasi-finite growth. Dokl. Akad. Nauk SSSR. 65, 117–120 (1949, in Russian)
[12] Bernstein, S.N.: Complete Works, vol. II. Constructive Function Theory [1931–1953]. Izdat. Akad. Nauk SSSR Moscow (1954, in Russian)
[13] Berry A.C.: Necessary and sufficient conditions in the theory of Fourier transforms. Ann. Math. 32, 830–838 (1931) · JFM 57.0482.02 · doi:10.2307/1968324
[14] Besov, O.V.: Hörmander’s theorem on Fourier multipliers. Trudy Mat. Inst. Steklov 173, 164–180 (1986, in Russian). English transl.: Proc. Steklov Inst. Math. 4, 4–14 (1987) · Zbl 0653.42014
[15] Besov, O.V., Il’in, V.P., Nikol’skii, S.M.: Integral Representations of Functions and Embedding of Functions, 2nd edn. Nauka, Fizmatlit, Moscow (1996, in Russian)
[16] Besov, O.V., Lizorkin, P.I.: Singular integral operators and sequences of convolutions in L p -spaces. Mat. Sb. (N.S.) 73(115), 65–88 (1967, Russian). English transl.: Math. USSR-Sbornik 2(1), 57–76 (1967)
[17] Beurling, A.: Sur les integrales de Fourier absolument convergentes et leur application à une transformation fonctionelle. Proc. IX Congrès de Math. Scand. Helsingfors, pp. 345–366 (1938) · JFM 65.0483.02
[18] Beurling A.: On the spectral synthesis of bounded functions. Acta Math. 81, 225–238 (1949) · Zbl 0034.21301 · doi:10.1007/BF02395018
[19] Beurling A.: Construction and analysis of some convolution algebras. Ann. Inst. Fourier 14, 1–32 (1964) · Zbl 0133.07501 · doi:10.5802/aif.172
[20] Bochkarev, S.V.: On a problem of Zygmund. Izv. AN SSSR 37, 630–638 (1973, in Russian). English transl. in Math. USSR-Izvestiya 7, 629–637 (1973) · Zbl 0273.42004
[21] Bochner S.: Summation of multiple Fourier series by spherical means. Trans. Am. Math. Soc. 40, 175–207 (1936) · JFM 62.0293.03 · doi:10.1090/S0002-9947-1936-1501870-1
[22] Bochner S.: Lectures on Fourier Integrals. Princeton University Press, Princeton (1959) · Zbl 0085.31802
[23] Boas R.P.: Absolute convergence and integrability of trigonometric series. J. Ration. Mech. Anal. 5, 621–632 (1956) · Zbl 0071.28404
[24] Borwein D.: Linear functionals connected with strong Cesáro summability. J. Lond. Math. Soc. 40, 628–634 (1965) · Zbl 0143.36303 · doi:10.1112/jlms/s1-40.1.628
[25] Brenner Ph., Thomée V., Wahlbin L.B.: Besov Spaces and Applications to Difference Methods for Initial Value Problems. Lecture Notes in Mathematics, vol. 434. Springer, Berlin (1975) · Zbl 0294.35002
[26] Bugrov, Ya.S.: Summability of Fourier transforms and absolute convergence of multiple Fourier series. Trudy Mat. Inst. Steklov. 187, 22–30 (1989, in Russian). English transl.: Proc. Steklov Inst. Math. Studies in the theory of differentiable functions of several variables and its applications 13(3), 25–34 (1990) · Zbl 0683.42011
[27] Buhmann M.D.: A new class of radial basis functions with compact support. Math. Comp. 70(23), 307–318 (2000) · Zbl 0956.41002 · doi:10.1090/S0025-5718-00-01251-5
[28] Buhmann, M.D.: Radial Basis Functions: Theory and Implementation. Cambridge University Press (2004)
[29] Butzer, P.L., Nessel, R.J.: Fourier analysis and approximation. One-Dimensional Theory, vol. 1. Pure and Applied Mathematics, vol. 40. Academic Press, New York (1971) · Zbl 0217.42603
[30] Butzer P.L., Nessel R.J., Trebels W.: On summation processes of Fourier expansions in Banach spaces, II. Saturation theorems. Tohoku Math. J. 24(2), 551–569 (1972) · Zbl 0247.42013 · doi:10.2748/tmj/1178241446
[31] Carleman, T.: L’intégrale de Fourier et Questions qui s’y rattachent. Almquist and Wiksells Boktyckere (1944) · Zbl 0060.25504
[32] Carleson, L.: Appendix to the paper of J.-P. Kahane and Y. Katznelson. Stud. Pure Math. Mem. P. Turan, Budapest, pp. 411–413 (1983)
[33] Chelidze V.G.: On the absolute convergence of double Fourier series. Doklady AN SSSR 54(2), 117–120 (1946) · Zbl 0060.19005
[34] Cossar J.: A theorem on Cesàro summability. J. Lond. Math. Soc. 16, 56–68 (1941) · JFM 67.0997.02 · doi:10.1112/jlms/s1-16.1.56
[35] Dappa H., Trebels W.: On L 1-criteria for quasiradial Fourier multipliers with applications to some anisotropic function spaces. Anal. Math. 9, 275–289 (1983) · Zbl 0544.42011 · doi:10.1007/BF01910307
[36] Dappa H., Trebels W.: On hypersingular integrals and anisotropic Bessel potential spaces. Trans. Am. Math. Soc. 286, 419–429 (1984) · Zbl 0558.42009 · doi:10.1090/S0002-9947-1984-0756046-2
[37] Duoandikoetxea, J.: Fourier Analysis. Grad. Studies Math., vol. 29. AMS, Providence (2001) · Zbl 0969.42001
[38] Edwards R.E.: Criteria for Fourier transforms. J. Australian Math. Soc. 7, 239–246 (1967) · Zbl 0148.38001 · doi:10.1017/S1446788700005589
[39] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G.: Tables of Integral Transforms, vols. 1, 2. McGraw-Hill, New York (1954) · Zbl 0055.36401
[40] Fefferman Ch.: Inequalities for strongly singular convolution operators. Acta Math. 124, 9–36 (1970) · Zbl 0188.42601 · doi:10.1007/BF02394567
[41] Feichtinger H.G.: A characterization of Wiener’s algebra on locally compact groups. Arch. Math. (Basel) 29, 136–140 (1977) · Zbl 0363.43003 · doi:10.1007/BF01220386
[42] Flett T.M.: Temperatures, Bessel potentials and Lipschitz spaces. Proc. Lond. Math. Soc. 22, 385–451 (1971) · Zbl 0234.35032 · doi:10.1112/plms/s3-22.3.385
[43] Fomin, G.A.: A class of trigonometric series. Mat. Zametki 23, 213–222 (1978, in Russian). English transl.: Math. Notes 23, 117–123 (1978) · Zbl 0379.42004
[44] Fridli S.: Hardy spaces generated by an integrability condition. J. Approx. Theory 113, 91–109 (2001) · Zbl 0996.42003 · doi:10.1006/jath.2001.3614
[45] Fridli S.: An inverse Sidon type inequality. Stud. Math. 105, 283–308 (1993) · Zbl 0811.42001
[46] Gabisoniya O.D.: On absolute convergence of double Fourier series and integrals. Soobshch. AN GSSR 42, 3–9 (1966) (in Russian)
[47] Geĭsberg S.P.: Fractional derivatives of functions bounded on the axis. Izv. Vysš. Učebn. Zaved. Matematika 11(78), 51–69 (1968) (in Russian)
[48] Gel’fand I.M., Raikov A.A., Shilov G.E.: Commutative normed rings. Uspekhi Matem. Nauk. I, 48–146 (1946) (Russian)
[49] Gel’fand I.M., Raikov, A.A., Shilov, G.E.: Commutative Normed Rings. Moscow, GIFML (1960, in Russian). English transl.: AMS, Chelsea, Bronx (1964)
[50] Giang D.V., Móricz F.: Lebesgue integrability of double Fourier transforms. Acta Sci. Math. (Szeged) 58, 299–328 (1993) · Zbl 0788.42004
[51] Gneiting T.: Criteria of Pólya type for radial positive definite functions. Proc. Am. Math. Soc. 129, 2309–2318 (2001) · Zbl 1008.42012 · doi:10.1090/S0002-9939-01-05839-7
[52] Gohberg, I., Fel’dman, I.A.: Convolution type equations and projection methods for their solution. Nauka, Moscow (1971, in Russian). English transl.: Amer. Math. Soc. Transl. of Math. Monographs, Providence, R.I., vol. 41 (1974)
[53] Gohberg, I., Krein, M.: The fundamentals on defect numbers, root numbers, and indices of linear operators. Uspekhi Matem. Nauk 12, 44–118 (1957, in Russian). English transl.: Amer. Math. Soc. Transl. 13, 185–264 (1960)
[54] Gol’dman, M.L.: An isomorphism of generalized H ölder classes. Diff. Uravnenija 7, 1449–1458, 1541 (1971, in Russian). English transl.: Differ. Equ. 7, 1100–1107 (1973)
[55] Gol’dman, M.L.: Generalized kernels of fractional order. Diff. Uravnenija 7, 2199–2210 (1971, in Russian). English transl.: Differ. Equ. 7, 1655–1664 (1974)
[56] Goldman, M.L.: Estimates for Multiple Fourier Transforms of Radially Symmetric Monotone Functions. Sibirsk. Mat. Ž. 18, 549–569 (1977, in Russian). English transl.: Sib. Math. J. 18, 391–406 (1978)
[57] Golovkin, K.K., Solonnikov, V.A.: Estimates of convolution operators. Zap. Nauch. Semin. LOMI. 7, 6–86 (1968, in Russian). English transl.: Semin. Math., Steklov Math. Inst., Leningrad 7, 1–36 (1968) · Zbl 0205.14602
[58] Golovkin, K.K.: Uniform equivalence of parametric norms in ergodic and approximation theories. Izvestiya AN SSSR 35(4), 900–921 (1971, in Russian). English transl.: Math. USSR-Izvestiya 5(4), 915–934 (1972) · Zbl 0226.46039
[59] Golubov, B.I.: Multiple series and Fourier integrals. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow. Mathematical analysis, vol. 19, 3–54, 232 (1982, in Russian). English transl.: J. Soviet Math. 24, 639–673 (1984) · Zbl 0511.42021
[60] Gradshtein, I.S., Ryzhik, I.M.: Tables of Integrals, Sums, Series and Products, 5th edn. Academic Press (1994)
[61] Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, Upper Saddle River (2004) · Zbl 1148.42001
[62] Grishin, A.F., Skoryk, M.V.: Some properties of Fourier integrals. ArXiv: http://arxiv.org/abs/1108.2890v (2011, in Russian)
[63] Hardy G.H., Littlewood J.E., Pólya G.: Inequalities. Cambridge University Press, Cambridge (1934)
[64] Herz, C.S.: Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms. J. Math. Mech. 18, 283–323 · Zbl 0177.15701
[65] Izumi S.I., Tsuchikura T.: Absolute convergence of trigonometric expansions. Tôhoku Math. J. 7, 243–251 (1955) · Zbl 0066.31504 · doi:10.2748/tmj/1178245063
[66] Kahane J.-P.: Séries de Fourier absolument convergentes. Springer, Berlin (1970) · Zbl 0195.07602
[67] Karapetyants, A.: On a generalization of the Beurling inequality. Unpublished manuscript, Rostov State University (1992, in Russian) · Zbl 0772.26007
[68] Koldobsky, A.: Schoenberg’s problem on positive definite functions. Algebra i Analiz 3, 78–85 (1991, in Russian). English transl.: St.-Petersburg Math. J. 3, 563–570 (1991)
[69] Koldobsky, A.: Fourier Analysis in Convex Geometry. (Math. Surveys and Monographs.) AMS (2005) · Zbl 1082.52002
[70] Kolmogorov A.N.: Une série de Fourier-Lebesgue divergente partout. C.R. Acad. Sci. Paris 183, 1327–1328 (1926) · JFM 52.0269.02
[71] Kolomoitsev, Yu., Liflyand, E.: Sufficient conditions for absolute convergence of multiple Fourier integrals. ArXiv: http://arxiv.org/abs/1108.5470v (2010) · Zbl 1388.42019
[72] Kostetskaya, G.S., Samko, S.G.: A sufficient condition for a function to belong to the Wiener ring $${\(\backslash\)mathcal{R}\(\backslash\)it ({R}\^n )}$$ of Fourier integrals of absolutely integrable functions I. Deposited in VINITI, no. 3483-86 (1986, in Russian)
[73] Kostetskaya, G.S., Samko, S.G.: A sufficient condition for a function to belong to the Wiener ring $${\(\backslash\)mathcal{R}\(\backslash\)it ({R}\^n )}$$ of Fourier integrals of absolutely integrable functions, II. Deposited in VINITI, no. 5801-86 (1986, in Russian)
[74] Kostetskaya, G.S., Samko, S.G.: Inversion of potential type operators with a difference characteristic. Deposited in VINITI, no. 5800-86 (1986, in Russian)
[75] Kostetskaya, G.S., Samko, S.G.: A criterion for absolute integrability of Fourier integrals. Izv. Vuzov. Matematika. 3, 72–75 (1988, in Russian). English transl.: Soviet Math. (Izv. VUZ) 32(3), 103–108 (1988) · Zbl 0652.42003
[76] Krein M.G.: On the problem of extension of Hermite-positive continuous functions. Doklady AN SSSR 26(1), 17–20 (1940) (in Russian)
[77] Krein M.G.: On the representation of functions by Fourier-Stieltjes integrals. Uchenye Zapiski Kuibyshev Gos. Pedagog. Inst. 7, 123–148 (1943) (in Russian)
[78] Kuznetsova, O.I.: On the integrability of a class of N-dimensional trigonometric series. Ukr. Mat. Zh. 52, 837–840 (2000, in Russian). English transl.: Ukr. Math. J. 52, 960–963 (2000) · Zbl 0984.42005
[79] Larsson, L., Maligranda, L., Pecaric, J., Persson, L.E.: Multiplicative Inequalities of Carlson Type and Interpolation. World Scientific, Singapore (2006) · Zbl 1102.41001
[80] Lebedev, V.V., Olevskii, A.M.: L p -Fourier multipliers with bounded powers. Izv. RAN. Ser. Mat. 70(3), 129–166 (2006, in Russian). English transl.: Izvestiya: Mathematics 70(3), 549–585 (2006) · Zbl 1187.42004
[81] Liflyand, E.R.: Some questions of absolute convergence of multiple Fourier integrals. Theory of functions and mappings. Naukova Dumka, Kiev, pp. 110–132, 179 (1979, in Russian) · Zbl 0506.42018
[82] Liflyand E.: Fourier transform of functions from certain classes. Anal. Math. 19, 151–168 (1993) · Zbl 0794.42006 · doi:10.1007/BF02905077
[83] Liflyand E.: Fourier Transforms of Radial Functions. Integr. Transf. Special Funct. 4, 279–300 (1996) · Zbl 0856.42010 · doi:10.1080/10652469608819115
[84] Liflyand E.: On quasi-monotone functions and sequences. Comput. Methods Funct. Theory 1, 345–352 (2002) · Zbl 1018.46013 · doi:10.1007/BF03320995
[85] Liflyand, E.: Lebesgue constants of multidimensional Fourier series. Online J. Anal. Combin. 1, Art. 5, 112 pp (2006) · Zbl 1129.42344
[86] Liflyand E.: Necessary conditions for integrability of the Fourier transform. Georgian Math. J. 16, 553–559 (2009) · Zbl 1181.42010
[87] Liflyand, E.: On absolute convergence of Fourier integrals. Real Anal. Exchange (accepted)
[88] Liflyand E., Ournycheva E.: Two spaces conditions for integrability of the Fourier transform. Analysis 19, 331–366 (2001) · Zbl 1162.42004
[89] Liflyand E., Trebels W.: On asymptotics for a class of radial Fourier transforms. ZAA 17, 103–114 (1998) · Zbl 0911.42006
[90] Liflyand, E., Trigub, R.: Known and new results on absolute integrability of Fourier integrals. Preprint CRM 859, 29 pp (2009)
[91] Liflyand, E., Trigub, R.: On the representation of a function as an absolutely convergent Fourier integral. Trudy Mat. Inst. Steklov 269, 153–166 (2010, in Russian). English transl.: Proc. Steklov Inst. Math. 269, 146–159 (2010) · Zbl 1203.42011
[92] Liflyand E., Trigub R.: Conditions for the absolute convergence of Fourier integrals. J. Approx. Theory 163, 438–459 (2011) · Zbl 1217.42025 · doi:10.1016/j.jat.2010.11.001
[93] Lizorkin, P.I.: On the theory of Fourier multipliers. Studies in the theory of differentiable functions of several variables and its applications, 11. Trudy Mat. Inst. Steklov. 173, 149–163 (1986, in Russian). English transl.: Proc. Steklov Inst. Math. 4, 161–176 (1987) · Zbl 0642.42010
[94] Lizorkin, P.I.: Limit cases of theorems on $${\(\backslash\)mathcal{F} L_p}$$ -multipliers. Trudy Mat. Inst. Steklov 173, 164–180 (1986, in Russian). English transl.: Proc. Steklov Inst. Math. 4, 177–194 (1987) · Zbl 0638.42015
[95] Löfström J.: Some theorems on interpolation spaces with applications to approximations in L p . Math. Ann. 172, 176–196 (1967) · Zbl 0186.45701 · doi:10.1007/BF01351183
[96] Löfström J.: Besov spaces in theory of approximation. Ann. Mat. Pura Appl. 85(4), 93–184 (1970) · Zbl 0193.41401 · doi:10.1007/BF02413532
[97] Long F.-L.: Sommes partielles de Fourier des fonctions born’ees. C.R. Acad. Sc. Paris Ser. A 288, 1009–1011 (1979) · Zbl 0424.42009
[98] Lucacs, E.: Characteristic Functions, 2nd edn. Charles Griffin &amp; Co. Ltd, London (1970)
[99] Madych W.R.: On Littlewood-Paley functions. Stud. Math. 50, 43–63 (1974) · Zbl 0247.44005
[100] Madych W.R.: Absolute summability of Fourier transforms on R n . Indiana Univ. Math. J. 25, 467–479 (1976) · Zbl 0327.42008 · doi:10.1512/iumj.1976.25.25037
[101] Mateu, J., Porcu, E.: Positive Definite Functions: From Schoenberg to Space-Time Challenges. Universitat Jaume I (2008)
[102] Miyachi A.: On some Fourier multipliers for $${H\^p(\(\backslash\)mathbb R\^n)}$$ . J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 157–179 (1980) · Zbl 0433.42019
[103] Miyachi A.: On some singular Fourier multipliers. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28, 267–315 (1981) · Zbl 0469.42003
[104] Mukherjee R.N.: A note on the criteria for absolute integrability of Fourier transforms. Aligarh Bull. Math. 9(10), 57–59 (1980) · Zbl 0518.42018
[105] Müller, C.: Spherical Harmonics. Lect. Notes in Math., vol. 17. Springer, Berlin (1966) · Zbl 0138.05101
[106] Musielak J.: On the absolute convergence of multiple Fourier series. Ann. Polon. Math. 5, 107–120 (1958) · Zbl 0092.06305
[107] Nessel R.J., Trebels W.: Multipliers with respect to spectral measures in Banach spaces and approximation. II. One-dimensional Fourier multipliers. J. Approx. Theory 14, 23–29 (1975) · Zbl 0304.42004 · doi:10.1016/0021-9045(75)90064-7
[108] Nikol’skii, S.M.: The Approximation of Functions of Several Variables and the Imbedding Theorems, 2nd edn. Nauka, Moscow (1977, in Russian). English transl. of 1st edn.: Wiley, New York (1978)
[109] Peetre J.: Applications de la théorie des espaces d’interpolation dans l’analyse harmonique. Ricerche Mat. 15, 3–36 (1966) · Zbl 0154.15302
[110] Peetre, J.: New thoughts on Besov spaces. Duke Univ. Math. Series, No. 1. Math. Dept. Duke Univ. Durham, NC (1976) · Zbl 0356.46038
[111] Pitt H.R.: A note on the representation of functions by absolutely convergent Fourier integrals. Proc. Cambridge Phil. Soc. 44, 8–12 (1940) · Zbl 0029.30202 · doi:10.1017/S0305004100023926
[112] Reiter H., Stegeman J.D.: Classical Harmonic Analysis and Locally Compact Groups. Clarendon Press, Oxford (2000) · Zbl 0965.43001
[113] Ryan R.: Fourier transforms of certain classes of integrable functions. Trans. Am. Math. Soc. 105, 102–111 (1962) · Zbl 0107.09701 · doi:10.1090/S0002-9947-1962-0187020-0
[114] Salem, R.: Essais sur les séries trigonometriques. Actual. Sci. et Industr., vol. 862. Paris (1940) · JFM 66.0280.01
[115] Samko, S.G.: The spaces $${L_{p, r}\^{\(\backslash\)alpha}({R}\^n)}$$ and hypersingular integrals. Studia Math. 61, 193–230 (1977, in Russian) · Zbl 0367.47029
[116] Samko, S.G.: Hypersingular integrals and their applications. Rostov-on-Don, Izdat. Rostov Univ. (1984, in Russian) · Zbl 0577.42016
[117] Samko, S.G.: Hypersingular integrals and differences of fractional order. Trudy Mat. Inst. Steklov. 192, 164–182 (1990, in Russian). English transl.: Proc. Steklov Inst. Math. 192, 175–194 (1992) · Zbl 0747.42010
[118] Samko, S.G.: On absolute integrability of Fouriers transforms in terms of fractional derivatives. Unpublished manuscript, Rostov State University (1992, in Russian)
[119] Samko, S.G.: Hypersingular Integrals and their Applications. Analytical Methods and Special Functions, vol. 5. Taylor &amp; Francis, London-New-York (2002) · Zbl 0998.42010
[120] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, London-New-York (1993) · Zbl 0818.26003
[121] Samko, S.G., Umarkhadzhiev, S.M.: Applications of hypersingular integrals to multidimensional integral equations of the first kind. Trudy Mat. Inst. Steklov. 172, 299–312 (1985, in Russian). English transl.: Proc. Steklov Inst. Math. 3, 325–339 (1987) · Zbl 0578.45020
[122] Samko, S.G., Kostetskaya, G.S.: Absolute integrability of Fourier integrals. Vestnik RUDN (Russian Peoples Friendship Univ.), Math. 1, 138–168 (1994) · Zbl 0826.42011
[123] Schaback R.: Multivariate interpolation by polynomials and radial basis functions. Constr. Approx. 21(3), 293–317 (2005) · Zbl 1076.41003 · doi:10.1007/s00365-004-0585-2
[124] Schoenberg I.J.: A remark on the preceding note by Bochner. Bull. Am. Math. Soc. 40, 277–278 (1934) · Zbl 0009.24704 · doi:10.1090/S0002-9904-1934-05845-2
[125] Schoenberg I.J.: Metric spaces and completely monotone functions. Ann. Math. 39, 811–841 (1938) · JFM 64.0617.03 · doi:10.2307/1968466
[126] Seeger A.: Necessary conditions for quasiradial Fourier multipliers. Tohoku Math. J. 39(2), 249–257 (1987) · Zbl 0607.42015 · doi:10.2748/tmj/1178228328
[127] Shapiro, H.S.: Topics in approximation theory. With appendices by Jan Boman and Torbjörn Hedberg. Lecture Notes in Math., vol. 187. Springer, Berlin (1971)
[128] Stein, E.M.: Singular integrals, harmonic functions, and differentiability properties of functions of several variables. Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pp. 316–335. Amer. Math. Soc., Providence (1967)
[129] Stein E.M.: Note on the class L log L. Stud. Math. XXXII, 305–310 (1969) · Zbl 0182.47803
[130] Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press (1970) · Zbl 0207.13501
[131] Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Space. Princeton University Press (1971) · Zbl 0232.42007
[132] Stein E.M.: Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton (1993) · Zbl 0821.42001
[133] Stepanov, V.D.: Integral representation of the resolvent of a self-adjoint elliptic operator. Diff. Uravnenija. 12, 129–136 (1976, in Russian). English transl.: Differ. Equ. 12, 89–94 (1977) · Zbl 0328.35030
[134] Stepanov, V.D.: Absolute summability of the Fsourier transform of functions of several variables. Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, Trudy Sem. S.L. Soboleva, No. 1, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat. Novosibirsk, 116–121 (1978, in Russian)
[135] Sz-Nagy B.: Sur une classe générale de procédés de sommation pour les séries de Fourier. Hungarica Acta Math. 1, 14–52 (1948) · Zbl 0034.04401
[136] Telyakovskii, S.A.: Integrability conditions for trigonometric series and their applications to the study of linear summation methods of Fourier series. Izv. Akad. Nauk SSSR, Ser. Matem. 28, 1209–1236 (1964, in Russian)
[137] Telyakovskii, S.A.: On a sufficient condition of Sidon for the integrability of trigonometric series. Mat. Zametki 14, 317–328 (1973, in Russian). English transl.: Math. Notes 14, 742–748 (1973)
[138] Timan, M.F.: Absolute convergence of multiple Fourier series. Dokl. Akad. Nauk SSSR 137, 1074–1077 (1961, in Russian). English translation in Soviet Math. Dokl. 2 , 430–433 (1961) · Zbl 0102.28703
[139] Timan, M.F.: Approximation and Properties of Periodic Functions. Naukova dumka, Kiev (2009, in Russian) · Zbl 1224.42021
[140] Titchmarsh E.C.: A note on Fourier transforms. J. Lond. Math. Soc. 2, 148–150 (1927) · JFM 53.0274.02 · doi:10.1112/jlms/s1-2.3.148
[141] Titchmarsh E.C.: Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford (1937) · Zbl 0017.40404
[142] Trebels W.: On a Fourier L 1(E n ) multiplier criterion. Acta Sci. Math. (Szeged) 35, 21–26 (1973) · Zbl 0255.42007
[143] Trebels, W.: Fourier multipliers on L p (R n ) in connection with bounded Riesz means. In: Approximation Theory (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1973), pp. 505–510. Academic Press, New York (1973) · Zbl 0318.42010
[144] Trebels, W.: Multipliers for (C, {\(\alpha\)})-bounded Fourier expansions in Banach spaces and approximation theory. In: Lecture Notes in Mathematics, vol. 329. Springer, Berlin (1973) · Zbl 0293.41035
[145] Trebels W.: Some Fourier multiplier criteria and the spherical Bochner-Riesz kernel. Rev. Roumaine Math. Pures Appl. 20, 1173–1185 (1975) · Zbl 0328.42003
[146] Trebels, W.: Estimates for moduli of continuity of functions given by their Fourier transform. In: Constructive Theory of Functions of Several Variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976). Lecture Notes in Math., vol. 571, pp. 277–288. Springer, Berlin (1977)
[147] Trigub, R.M.: Integrability and asymptotic behavior of the Fourier transform of a radial function. In: Metric Questions of the Theory of Functions and Mappings, pp. 142–163. Naukova Dumka, Kiev (1977, in Russian) · Zbl 0485.42013
[148] Trigub, R.M.: Absolute convergence of Fourier integrals, summability of Fourier series, and polynomial approximation of functions on the torus. Izv. Akad. Nauk SSSR Ser. Mat. 44, 1378–1408 (1980, in Russian). English translation in Math. USSR Izv. 17, 567–593 (1981) · Zbl 0459.42019
[149] Trigub, R.M.: Certain properties of the Fourier transform of a measure and their application. In: Proc. Intern. Conf. Approx. Theory (Kiev, 1983), 439–443. Nauka, Moscow (1987, in Russian)
[150] Trigub, R.M.: Positive definite radial functions and splines. Constructive Function Theory ’87 (Varna, 25-31.05.1987), 123. Publ. House Bulgar. Acad. Sci., Sofia (1987)
[151] Trigub, R.M.: A criterion for a characteristic function and a polya-type criterion for radial functions of several variables. Teor. Ver. Pril. 34, 805–810 (1989, in Russian). English transl.: Theory Probab. Appl. 34, 738–742 (1989) · Zbl 0696.60023
[152] Trigub, R.M.: Multipliers of Fourier series and approximation of functions by polynomials in spaces C and L. Doklady Akad. Nauk SSSR, 306, 292–296 (1989, in Russian). English transl.: Soviet Math. Dokl. 39, 494–498 (1989)
[153] Trigub, R.M.: Multipliers of Fourier series. Ukr. Mat. Zh. 43, 1686–1693 (1991, in Russian). English transl.: Ukr. Math. J. 43, 1572–1578 (1991) · Zbl 0752.42007
[154] Trigub, R.M.: Positive definite functions and splines. Theory Funct. Approx. Proc. V Saratov Winter School (1990). Part I, Saratov, pp. 68–75 (1992, in Russian)
[155] Trigub, R.M.: Some Topics in Fourier Analysis and Approximation Theory. arXiv:funct-an/9612008v1 (1996)
[156] Trigub, R.M.: Multipliers in the Hardy spaces H p (D m ) with $${p\(\backslash\)in(0,1]}$$ and approximation properties of summability methods for power series. Mat. Sbornik 188, 145–160 (1997, in Russian). English transl.: Sbornik Math. 188, 621–638 (1997) · Zbl 0896.42003
[157] Trigub R.M.: Fourier multipliers and K-functionals in spaces of smooth functions. Ukr. Math. Bull. 2, 239–284 (2005) · Zbl 1148.42300
[158] Trigub, R.M.: On comparison of linear differential operators. Matem. Zametki 82, 426–440 (2007, in Russian). English transl.: Math. Notes 82, 380–394 (2007) · Zbl 1151.26012
[159] Trigub R.M.: On Fourier multipliers and absolute convergence of the Fourier series of radial functions. Ukr. Math. J. 62, 1280–1293 (2010) (Russian) · Zbl 1240.42030
[160] Trigub, R.M., Belinsky, E.S.: Fourier Analysis and Approximation of Functions. Kluwer/Springer (2004)
[161] Watson, G.N.: A treatise on the theory of Bessel functions. Cambridge Mathematical Library, Reprint of the second (1944) edition. Cambridge University Press, Cambridge (1995)
[162] Wendland H.: Piecewise positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396 (1995) · Zbl 0838.41014 · doi:10.1007/BF02123482
[163] Widder, D.V.: The Laplace Transform. Princeton University Press (1946) · Zbl 0060.24801
[164] Wiener N.: On the representation of functions by trigonometrical integrals. Math. Zeitschr. 24, 575–616 (1926) · JFM 51.0228.06 · doi:10.1007/BF01216799
[165] Wiener N.: Generalized harmonic analysis. Acta Math. 55, 117–258 (1930) · JFM 56.0954.02 · doi:10.1007/BF02546511
[166] Wiener N.: Tauberian theorems. Ann. Math. 33, 1–100 (1932) · JFM 58.0226.02 · doi:10.2307/1968102
[167] Wiener N.: The Fourier Integral and Certain of its Applications. Cambridge University Press, Cambridge (1935) · JFM 61.0408.03
[168] Wiener N., Pitt H.R.: On absolutely convergent Fourier-Stieltjes transforms. Duke Math. J. 4, 420–436 (1938) · Zbl 0019.16803 · doi:10.1215/S0012-7094-38-00435-1
[169] Yosida K.: Proc. Imp. Acad. Tokyo 20, 656–660 (1944)
[170] Zastavnyi, V.P.: Positive definite functions depending on the norm. Solution of a problem of Shoenberg. Preprint N1-35, Inst. Applied Math. Mech. Acad. Sci., Ukraine, Donetsk (1991, in Russian)
[171] Zastavnyi V.P.: Positive definite functions depending on the norm. Russ. J. Math. Phys. 1, 511–522 (1993) · Zbl 0906.43003
[172] Zastavnyi V.P.: On positive definiteness of some functions. J. Multivar. Anal. 73, 55–81 (2000) · Zbl 0956.42006 · doi:10.1006/jmva.1999.1864
[173] Zastavnyi, V.P.: Positive definite radial functions and splines. Doklady Ross. Akad. Nauk, 386, 446–449 (2002, in Russian). English transl.: Russ. Acad. Sci., Dokl. Math. 66, 1–4 (2002) · Zbl 1259.41017
[174] Zastavnyi, V.P., Trigub, R.M.: Positive-definite splines of special form. Mat. Sbornik 193(12), 41–68 (2002, in Russian). English transl.: Sb. Math. 193, 1771–1800 (2002)
[175] Zygmund A.: Trigonometric series, vols. I, II, 2nd edn. Reprinted with corrections and some additions. Cambridge University Press, London (1968)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.