Displaying similar documents to “On the approximation of real continuous functions by series of solutions of a single system of partial differential equations”

Three-space problems for the approximation property

A. Szankowski (2009)

Journal of the European Mathematical Society

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It is shown that there is a subspace Z q of q for 1 < q < 2 which is isomorphic to q such that q / Z q does not have the approximation property. On the other hand, for 2 < p < there is a subspace Y p of p such that Y p does not have the approximation property (AP) but the quotient space p / Y p is isomorphic to p . The result is obtained by defining random “Enflo-Davie spaces” Y p which with full probability fail AP for all 2 < p and have AP for all 1 p 2 . For 1 < p 2 , Y p are isomorphic to p .

Approximation properties of β-expansions

Simon Baker (2015)

Acta Arithmetica

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Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ( ϵ i ) i = 1 0 , 1 a β-expansion for x if x = i = 1 ϵ i β - i . We call a finite sequence ( ϵ i ) i = 1 n 0 , 1 n an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given Ψ : 0 , we introduce the following subset of ℝ: W β ( Ψ ) : = m = 1 n = m ( ϵ i ) i = 1 n 0 , 1 n [ i = 1 n ( ϵ i ) / ( β i ) , i = 1 n ( ϵ i ) / ( β i ) + Ψ ( n ) ] In other words, W β ( Ψ ) is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities 0 x - i = 1 n ( ϵ i ) / ( β i ) Ψ ( n ) . When n = 1 2 n Ψ ( n ) < , the Borel-Cantelli lemma tells us that the Lebesgue measure...

On the multiples of a badly approximable vector

Yann Bugeaud (2015)

Acta Arithmetica

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Let d be a positive integer and α a real algebraic number of degree d + 1. Set α ̲ : = ( α , α ² , . . . , α d ) . It is well-known that c ( α ̲ ) : = l i m i n f q q 1 / d · | | q α ̲ | | > 0 , where ||·|| denotes the distance to the nearest integer. Furthermore, c ( α ̲ ) n - 1 / d c ( n α ̲ ) n c ( α ̲ ) for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that c ( n α ̲ ) C n - 1 / d for any integer n ≥ 1.

On a question of Schmidt and Summerer concerning 3 -systems

Johannes Schleischitz (2020)

Communications in Mathematics

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Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper 3 -system ( P 1 , P 2 , P 3 ) with the property ϕ ¯ 3 = 1 . In fact, our method generalizes to provide n -systems with ϕ ¯ n = 1 , for arbitrary n 3 . We visualize our constructions with graphics. We further present explicit examples of numbers ξ 1 , ... , ξ n - 1 that induce the n -systems in question.

A pure smoothness condition for Radó’s theorem for α -analytic functions

Abtin Daghighi, Frank Wikström (2016)

Czechoslovak Mathematical Journal

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Let Ω n be a bounded, simply connected -convex domain. Let α + n and let f be a function on Ω which is separately C 2 α j - 1 -smooth with respect to z j (by which we mean jointly C 2 α j - 1 -smooth with respect to Re z j , Im z j ). If f is α -analytic on Ω f - 1 ( 0 ) , then f is α -analytic on Ω . The result is well-known for the case α i = 1 , 1 i n , even when f a priori is only known to be continuous.

An approximation property of quadratic irrationals

Takao Komatsu (2002)

Bulletin de la Société Mathématique de France

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Let α &gt; 1 be irrational. Several authors studied the numbers m ( α ) = inf { | y | : y Λ m , y 0 } , where m is a positive integer and Λ m denotes the set of all real numbers of the form y = ϵ 0 α n + ϵ 1 α n - 1 + + ϵ n - 1 α + ϵ n with restricted integer coefficients | ϵ i | m . The value of 1 ( α ) was determined for many particular Pisot numbers and m ( α ) for the golden number. In this paper the value of  m ( α ) is determined for irrational numbers  α , satisfying α 2 = a α ± 1 with a positive integer a .

A note on the weighted Khintchine-Groshev Theorem

Mumtaz Hussain, Tatiana Yusupova (2014)

Journal de Théorie des Nombres de Bordeaux

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Let W ( m , n ; ψ ̲ ) denote the set of ψ 1 , ... , ψ n –approximable points in m n . The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions ψ ̲ . Removing monotonicity from the Khintchine–Groshev theorem is attributed to different authors for different cases of m and n . It can not be removed for m = n = 1 as Duffin–Schaeffer provided the counter example. We deal with the only remaining case m = 2 and thereby remove all unnecessary conditions from the Khintchine–Groshev theorem. ...

Some theorems of Korovkin type

Tomoko Hachiro, Takateru Okayasu (2003)

Studia Mathematica

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We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, C ( X ) ) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., C ( Y ) ), and ϕ a linear isometry from M into C(Y) (resp., C ( Y ) ). We show, under the assumption that Π N Π T , where Π N is...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Variations on a question concerning the degrees of divisors of x n - 1

Lola Thompson (2014)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we examine a natural question concerning the divisors of the polynomial x n - 1 : “How often does x n - 1 have a divisor of every degree between 1 and n ?” In a previous paper, we considered the situation when x n - 1 is factored in [ x ] . In this paper, we replace [ x ] with 𝔽 p [ x ] , where p is an arbitrary-but-fixed prime. We also consider those n where this condition holds for all p .

Theoretical analysis for 1 - 2 minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

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We investigate the recovery of k -sparse signals using the 1 - 2 minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume k -sparse signals 𝐱 with the prior support T which is composed of g true indices and b wrong...

The "Full Clarkson-Erdős-Schwartz Theorem" on the closure of non-dense Müntz spaces

Tamás Erdélyi (2003)

Studia Mathematica

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Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose ( λ j ) j = 1 is a sequence of distinct positive numbers. Then s p a n 1 , x λ , x λ , . . . is dense in C[0,1] if and only if j = 1 ( λ j ) / ( λ j ² + 1 ) = . Moreover, if j = 1 ( λ j ) / ( λ j ² + 1 ) < , then every function from the C[0,1] closure of s p a n 1 , x λ , x λ , . . . can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an...

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Fourier approximation and embeddings of Sobolev spaces

D. E. Edmunds, V. B. Moscatelli

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CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into W m , p ( Ω ) into L S ( Ω ) (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding W m , p ( Ω ) into L φ ( Ω ) ...............................................................

Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1

(2016)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≢ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p ) 1 / p , a j , such that ( x - 1 ) k divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let μ q ( n , L ) be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that | Q ( 0 ) | > 1 / L ( j = 1 n | Q ( j ) | q ) 1 / q . We find the size of κ p ( n , L ) and μ q ( n , L ) for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about μ ( n , L ) is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...