Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

We investigate the approximation properties of the trigonometric system in $ L_{2\pi }^{p\left( \cdot \right)} $
 . We consider the moduli of smoothness of fractional order and obtain direct and inverse approximation theorems together with a constructive characterization of a Lipschitz-type class.

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Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent

We investigate the approximation properties of the partial sums of the Fourier series and prove some direct and inverse theorems for approximation by polynomials in weighted Orlicz spaces. In particular we obtain a constructive characterization of the generalized Lipschitz classes in these spaces.

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Approximation by trigonometric polynomials in weighted Orlicz spaces

We consider the Lebesgue space L p(x) 2 π with variable exponent p(x). It consist of measurable functions f(x) for which the integral ∫ 0 2 π |f(x)| p(x) dx exist. We establish an analogue Jackson's second theorem in the case when the 2 π -periodic variable exponent p(x) ≥ 1 satisfies the condition |p(x) - p(y)| · ln(2 π )/(|x - y|) ≤ d, x,y ∈ [0,2 π ]. Results obtained in present paper radically differ from other authors results on this subject because we don't require from variable exponent p(x) the fulfillment of additional condition p(x) ≥ π which is closely related with boundedness of Hardy - Littlewood maximal function M(f) in L p(x) 2 π . In the definition of the modulus of continuity of a function f(x) ∈ L 2 π p(x) , we replace the ordinary shift f h (x)=f(x+h) by an averaged shift determined by Steklov's function s h (f)(x)=1/h ∫ 0 h f(x+t)dt.

On Direct And Inverse Theorems Of Approximation Theory In Variable Lebesgue And Sobolev Spaces

This work considers the Riemann boundary value problem with the piecewise continuous coefficient in Morrey– Hardy classes. Under some conditions on the coefficient, the Fredholmness of this problem is studied and the general solution of homogeneous and nonhomogeneous problems in Morrey–Hardy classes is constructed.

https://journals.tubitak.gov.tr/math/issues/mat-16-40-5/mat-40-5-14-1507-10.pdf

On the solvability of the Riemann boundary value problem in Morrey--Hardy classes

Let G\subset C be a finite domain with a regular Jordan boundary L . In this work, the approximation properties of a p -Faber polynomial series of functions in the weighted Smirnov class E

 p
 
(G,ω) are studied and the rate of polynomial approximation, for f∈ E

 p
 
( G,ω) by the weighted integral modulus of continuity, is estimated. Some application of this result to the uniform convergence of the Bieberbach polynomials π

 n
 in a closed domain \overline G with a smooth boundary L is given.

Approximation by p-Faber polynomials in the weighted Smirnov class Ep (G, ω) and the Bieberbach polynomials

The approximation properties of Norlund (Nn) and Riesz (Rn) means of trigono- metric Fourier series are investigated in generalized Lebesgue spaces L p(x) . The deviations � f −Nn(f)� p(x) andf −Rn(f)� p(x) are estimated by n −α for f ∈ Lip(α,p(x)) (0 < α 1).

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TRIGONOMETRIC APPROXIMATION IN GENERALIZED LEBESGUE SPACES L p(x)

In this work some direct and inverse theorems of approximation theory in the weighted Smirnov-Orlicz classes, defined in the domains with a Dini-smooth boundary, are proved. In particular, a constructive characterization of the generalized Lipschitz classes $Lip^{*}\alpha (M,\omega )$, $\alpha >0$, is obtained.

/pdf/approximation-in-weighted-smirnov-orlicz-classes-2oeteznr5f.pdf

Approximation in weighted Smirnov-Orlicz classes

Let L ⊂ C be a regular Jordan curve. In this work, the approximation properties of the p-Faber-Laurent rational series expansions in the ω weighted Lebesgue spaces L p(L, ω) are studied. Under some restrictive conditions upon the weight functions the degree of this approximation by a kth integral modulus of continuity in L p(L, ω) spaces is estimated.

/pdf/approximation-by-p-faber-laurent-rational-functions-in-the-1m724drd48.pdf

Daniyal M. Israfilov

Papers

Approximation by p-Faber polynomials in the weighted Smirnov class Ep (G, ω) and the Bieberbach polynomials

TRIGONOMETRIC APPROXIMATION IN GENERALIZED LEBESGUE SPACES L p(x)

Approximation by trigonometric polynomials in weighted Orlicz spaces

Approximation in weighted Smirnov-Orlicz classes

Approximation by p-Faber-Laurent Rational Functions in the Weighted Lebesgue Spaces