Wyniki wyszukiwania

1

Orthogonality and the maximum of Littlewood cosine polynomials

100%

Erdélyi T.

Acta Arithmetica

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2011

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tom 146

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nr 3

215-231

2

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

100%

Erdélyi T.

Studia Mathematica

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2003

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tom 155

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nr 2

145-152

EN

Denote by span{f₁,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 $span{1,x^{λ₁},x^{λ₂},...}$ 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 $span{1,x^{λ₁},x^{λ₂},...}$ can be represented as an analytic function on {z ∈ ℂ ∖ (-∞, 0]: |z| < 1} restricted to (0,1). This result improves an earlier result by P. Borwein and Erdélyi stating that if $∑_{j=1}^{∞} (λ_{j})/(λ_{j}²+1) < ∞$, then every function from the C[0,1] closure of $span{1,x^{λ₁},x^{λ₂},...}$ is in $C^{∞}(0,1)$. Our result may also be viewed as an improvement, extension, or completion of earlier results by Müntz, Szász, Clarkson, Erdős, L. Schwartz, P. Borwein, Erdélyi, W. B. Johnson, and Operstein.

3

The norm of the polynomial truncation operator on the unit disk and on [-1,1]

100%

Erdélyi T.

Colloquium Mathematicum

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2001

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tom 90

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nr 2

287-293

EN

Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by 𝒫ₙ (resp. $𝒫ₙ^{c}$) the set of all polynomials of degree at most~n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $Pₙ ∈ 𝒫ₙ^{c}$ of the form $Pₙ(z) := ∑_{j=0}^{n} a_{j}z_{j}$, $a_{j} ∈ C$, by $Sₙ(Pₙ)(z):= ∑_{j=0}^{n} ã_{j}z_{j}$, $ã_{j}:= a_{j}|a_{j}| min{|a_{j}|,1}$ (here 0/0 is interpreted as 1). We define the norms of the truncation operators by $∥Sₙ∥^{real}_{∞,∂D}:= sup_{Pₙ∈𝒫ₙ} (max_{z∈∂D} |Sₙ(Pₙ)(z)|)/(max_{z∈∂D}|Pₙ(z)|)$, $∥Sₙ∥^{comp}_{∞,∂D}:= sup_{Pₙ∈𝒫ₙ^{c}} (max_{z∈∂D} |Sₙ(Pₙ)(z)|)/(max_{z∈∂D} |Pₙ(z)|$. Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁ > 0 such that $c₁√(2n+1) ≤ ∥Sₙ∥^{real}_{∞,∂D} ≤ ∥Sₙ∥^{comp}_{∞,∂D} ≤ √(2n+1)$ This settles a question asked by S. Kwapień. Moreover, an analogous result in $L_{p}(∂D)$ for p ∈ [2,∞] is established and the case when the unit circle ∂D is replaced by the interval [−1,1] is studied.

4

Lower bounds for the number of zeros of cosine polynomials in the period: a problem of Littlewood

64%

Borwein P., Erdélyi T.

Acta Arithmetica

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2007

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tom 128

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nr 4

377-384

5

The multiplicity of the zero at 1 of polynomials with constrained coefficients

51%

Borwein P., Erdélyi T., Kós G.

Acta Arithmetica

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2013

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tom 159

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nr 4

387-395

EN

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 $P(x) = ∑_{j=0}^n{a_jx^j}$, $|a_0| ≥ L(∑_{j=1}^n{|a_j|^p} }^{1/p}$, $a_j ∈ ℂ$, such that $(x-1)^k$ divides P(x). For n ∈ ℕ and L > 0 let $κ_∞(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P(x) = ∑_{j=0}^n{a_jx^j}$, $|a_0| ≥ Lmax_{1 ≤ j ≤ n}{|a_j|}$, $a_j ∈ ℂ$, such that $(x-1)^k$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that $c_1 √(n/L) -1 ≤ κ_{∞}(n,L) ≤ c_2 √(n/L)$ for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈ (0,1]. Essentially sharp results on the size of κ₂(n,L) are also proved.

Ograniczanie wyników

3 Acta Arithmetica

1 Colloquium Mathematicum

1 Studia Mathematica

5 Erdélyi T.

2 Borwein P.

1 Kós G.

1 2013

1 2011

1 2007

1 2003

1 2001

Orthogonality and the maximum of Littlewood cosine polynomials

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

The norm of the polynomial truncation operator on the unit disk and on [-1,1]

Lower bounds for the number of zeros of cosine polynomials in the period: a problem of Littlewood

The multiplicity of the zero at 1 of polynomials with constrained coefficients